{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2025-12-14T01:33:56.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2025-12-14T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":57565,"title":"Easy Sequences 92: Number of Roots of a System of Linear Diophantine Equations","description":"One of the tasks that Matlab is very good at, is in solving systems of linear equations. \r\nIn this problem we shall tackle a system of linear Diophantine Equations in which the roots are limited to certain range.\r\nGiven the number of variables , positive integers, ,  and a root limit, , create the function, numRoots(n,a,b,L), that outputs the number of posible integer root sets of the following system of equations:\r\n                \r\n                \r\n        with:\r\n                .\r\nFor example, if , the system of equations:  and , where , has only one root set, namely: . Therefore, numRoots(2,10,4,1) = 1.\r\nIf , two of the possible roots of:\r\n                ; and\r\n                \r\n                where:  \r\nare and . In fact, there are  possible root sets. Therefore, numRoots(4,20,6,3) = 16.\r\nThere are no possible roots for  and , therefore in these cases the function should return: numRoots = 0.\r\n-------------------------\r\nNOTE: As an added challenge, only those solutions with Cody program size of less than or equal to 200 will be accepted.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 607.5px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 303.75px; transform-origin: 407px 303.75px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 270px 8px; transform-origin: 270px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eOne of the tasks that Matlab is very good at, is in solving systems of linear equations. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 153px 8px; transform-origin: 153px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn this problem we shall tackle a system of linear \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Diophantine_equation\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eDiophantine Equations\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 145.5px 8px; transform-origin: 145.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in which the roots are limited to certain range.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.75px; text-align: left; transform-origin: 384px 31.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 109px 8px; transform-origin: 109px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eGiven the number of variables \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64px 8px; transform-origin: 64px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003epositive integers, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eb\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58px 8px; transform-origin: 58px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e and a root limit, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eL\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 75px 8px; transform-origin: 75px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, create the function, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 68px 8px; transform-origin: 68px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 68px 8.5px; transform-origin: 68px 8.5px; \"\u003enumRoots(n,a,b,L)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 281px 8px; transform-origin: 281px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, that outputs the number of posible integer root sets of the following system of equations:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32px 8px; transform-origin: 32px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg 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style=\"vertical-align:-6px\"\u003e\u003cimg 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style=\"width: 378.5px; height: 21px;\" width=\"378.5\" height=\"21\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.5px 8px; transform-origin: 32.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e        with:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32px 8px; transform-origin: 32px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 132.5px; height: 20px;\" width=\"132.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 48.5px 8px; transform-origin: 48.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor example, if \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 152.5px; height: 18.5px;\" width=\"152.5\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 83px 8px; transform-origin: 83px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, the system of equations: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 77px; height: 20px;\" width=\"77\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 69.5px; height: 20px;\" width=\"69.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 25px 8px; transform-origin: 25px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 58.5px; height: 20px;\" width=\"58.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.5px 8px; transform-origin: 32.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, has only one root set, namely: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 96px; height: 20px;\" width=\"96\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 39.5px 8px; transform-origin: 39.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Therefore, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 72px 8px; transform-origin: 72px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 72px 8.5px; transform-origin: 72px 8.5px; \"\u003enumRoots(2,10,4,1)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 4px 8.5px; transform-origin: 4px 8.5px; \"\u003e=\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 4px 8.5px; transform-origin: 4px 8.5px; \"\u003e1\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.5px 8px; transform-origin: 5.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 152.5px; height: 18.5px;\" width=\"152.5\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 90.5px 8px; transform-origin: 90.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, two of the possible roots of:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32px 8px; transform-origin: 32px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 134px; height: 20px;\" width=\"134\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e; and\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32px 8px; transform-origin: 32px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 126.5px; height: 20px;\" width=\"126.5\" height=\"20\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 57px 8px; transform-origin: 57px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                where:  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 94.5px; height: 20px;\" width=\"94.5\" height=\"20\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 43.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.75px; text-align: left; transform-origin: 384px 21.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.5px 8px; transform-origin: 12.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eare \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg 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style=\"width: 166.5px; height: 20px;\" width=\"166.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58px 8px; transform-origin: 58px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. In fact, there are \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 18px; height: 18px;\" width=\"18\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 97px 8px; transform-origin: 97px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e possible root sets. Therefore, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 72px 8px; transform-origin: 72px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 72px 8.5px; transform-origin: 72px 8.5px; \"\u003enumRoots(4,20,6,3)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 4px 8.5px; transform-origin: 4px 8.5px; \"\u003e=\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6px 8px; transform-origin: 6px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e 1\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 4px 8.5px; transform-origin: 4px 8.5px; \"\u003e6\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.25px; text-align: left; transform-origin: 384px 21.25px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 100px 8px; transform-origin: 100px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThere are no possible roots for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 185px; height: 18.5px;\" width=\"185\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 175px; height: 18.5px;\" width=\"175\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 82.5px 8px; transform-origin: 82.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, therefore in these cases the function should return: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 48px 8px; transform-origin: 48px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 48px 8.5px; transform-origin: 48px 8.5px; \"\u003enumRoots = 0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 62.5px 8px; transform-origin: 62.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e-------------------------\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 23px 8px; transform-origin: 23px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eNOTE: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 76.5px 8px; transform-origin: 76.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAs an added challenge, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 89.5px 8px; transform-origin: 89.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eonly those solutions with \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/content/cody/about.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-weight: 700; text-decoration-line: underline; \"\u003eCody program size\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 85px 8px; transform-origin: 85px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e of less than or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12px 8px; transform-origin: 12px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; text-decoration-line: underline; \"\u003e200\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.5px 8px; transform-origin: 26.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e will be accepted.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function c = numRoots(n,a,b,L)\r\n    n = 'number of unknown variables';\r\n    a = 'sum of the unknown variables';\r\n    b = 'alternating sum and difference of the unknown variables';\r\n    L = 'limit of the roots (i.e. all roots should be \u003e L');\r\n    c = 'the number of root sets';\r\nend","test_suite":"%%\r\n[n,a,b,L] = deal(4,20,6,3);\r\nc_correct = 16;\r\nassert(isequal(str2num(numRoots(n,a,b,L)),c_correct))\r\n[n,a,b,L] = deal(4,20,6,-3);\r\nc_correct = 280;\r\nassert(isequal(str2num(numRoots(n,a,b,L)),c_correct))\r\n[n,a,b,L] = deal(3,20,6,4);\r\nc_correct = 6;\r\nassert(isequal(str2num(numRoots(n,a,b,L)),c_correct))\r\n%%\r\n[n,a,b,L] = deal(2,10,4,1);\r\nc_correct = 1;\r\nassert(isequal(str2num(numRoots(n,a,b,L)),c_correct))\r\n[n,a,b,L] = deal(10,55,10,-10);\r\nc_correct = 0;\r\nassert(isequal(str2num(numRoots(n,a,b,L)),c_correct))\r\n[n,a,b,L] = deal(42,10,2,100);\r\nc_correct = 0;\r\nassert(isequal(str2num(numRoots(n,a,b,L)),c_correct))\r\n%%\r\n[n,a,b,L] = deal(10,25,15,-5);\r\nc_correct = '9825961376';\r\nassert(isequal(numRoots(n,a,b,L),c_correct))\r\n[n,a,b,L] = deal(25,50,25,5);\r\nc_correct = '0';\r\nassert(isequal(numRoots(n,a,b,L),c_correct))\r\n%%\r\n[n,a,b,L] = deal(8,88,8,8);\r\nc_correct = '159885';\r\nassert(isequal(numRoots(n,a,b,L),c_correct))\r\n[n,a,b,L] = deal(88,88,8,8);\r\nc_correct = '0';\r\nassert(isequal(numRoots(n,a,b,L),c_correct))\r\n[n,a,b,L] = deal(8,888,88,-8);\r\nc_correct =  '322979008681245';\r\nassert(isequal(numRoots(n,a,b,L),c_correct))\r\n%%\r\n[n,a,b,L] = deal(11,111,11,-1);\r\nc_correct =  '6367915454544';\r\nassert(isequal(numRoots(n,a,b,L),c_correct))\r\n%%\r\n[n,a,b,L] = deal(111,11111,111,11);\r\nc_correct = '2909165067998259047855867968538898760414304923603195356244648530663146995238036844997557527483252003388541616691119031429859046897494224647322949977583143171656492664325583077702371945252381217441222592230772490362933347417600378555600078275316289441596896000';\r\nassert(isequal(numRoots(n,a,b,L),c_correct))\r\n%%\r\n[n,a,b,L] = deal(1234,56789,1011,12);\r\nc = numRoots(n,a,b,L);\r\ns = [sum(c) histc(c,50:55) sum(c\u003c58) sum(c\u003e47)]\r\ns_correct = [127146 260 249 257 231 226 247 2428 2428];\r\nassert(isequal(s,s_correct))\r\n%%\r\nfiletext = fileread('numRoots.m');\r\nnot_allowed = contains(filetext, 'persistent') || contains(filetext, 'global') || contains(filetext, 'regexp') || contains(filetext, 'eval') || contains(filetext, 'assignin');\r\nassert(~not_allowed)\r\nassert(mtree(filetext).count\u003c=150)","published":true,"deleted":false,"likes_count":0,"comments_count":1,"created_by":255988,"edited_by":255988,"edited_at":"2023-02-07T05:06:35.000Z","deleted_by":null,"deleted_at":null,"solvers_count":1,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-01-18T14:33:46.000Z","updated_at":"2023-02-07T05:06:35.000Z","published_at":"2023-02-02T11:44:46.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOne of the tasks that Matlab is very good at, is in solving systems of linear equations. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn this problem we shall tackle a system of linear \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Diophantine_equation\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eDiophantine Equations\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e in which the roots are limited to certain range.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eGiven the number of variables \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003epositive integers, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e and a root limit, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eL\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, create the function, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enumRoots(n,a,b,L)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, that outputs the number of posible integer root sets of the following system of equations:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1+x_2+x_3+x_4...+x_{n-2}+x_{n-1}+x_n=a\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1-x_2+x_3-x_4...+(-1)^{n-1}x_{n-2}+(-1)^{n}x_{n-1}+(-1)^{n+1}x_n=b\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e        with:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1,x_2,x_3...x_{n-1},x_n \\\\ge L\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, if \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(n,a,b,L) = (2,10,4,1)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, the system of equations: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1+x_2=10\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1-x_2 = 4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1,x_2\\\\ge 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, has only one root set, namely: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(x_1,x_2)=(7,3)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Therefore, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enumRoots(2,10,4,1)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e=\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(n,a,b,L) = (4,20,6,3)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, two of the possible roots of:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1+x_2+x_3+x_4=20\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e; and\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1-x_2+x_3-x_4=6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                where:  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1,x_2,x_3,x_4 \\\\ge 3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eare \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(x_1,x_2,x_3,x_4)=(5,4,8,3)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eand \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(x_1,x_2,x_3,x_4)=(10,3,3,4)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. In fact, there are \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e16\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e possible root sets. Therefore, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enumRoots(4,20,6,3)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e=\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e 1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e6\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThere are no possible roots for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(n,a,b,L) = (10,55,10,-10)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(n,a,b,L) = (42,10,2,100)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, therefore in these cases the function should return: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enumRoots = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e-------------------------\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNOTE: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eAs an added challenge, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eonly those solutions with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/content/cody/about.html\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody program size\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e of less than or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e200\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e will be accepted.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":58847,"title":"Possible American Football Scoring Combinations","description":"With inspiration from Problem 56195. Possible Ruby Scores, given the final score for two teams in an American football game (score1, score2), return two matrices (team1_combs, team2_combs) representing all possible combinations of scores by each team (Official NFL Scoring Plays). \r\nIn American football, the 7 possible ways of scoring are:\r\nTouchdown - 6 points\r\nTry after touchdown - 1 point\r\nTry after touchdown - 2 points\r\nSafety - 2 points\r\nField Goal - 3 points\r\nDefensive safety on try after touchdown (i.e. conversion safety) - 1 points\r\nDefensive \"touchdown\" on try after touchdown - 2 points\r\nValues for score1 and score2 include all natural numbers (realistically \u003c100), and the resulting matrices will have dimensions n1 x 7 and n2 x 7, where n1, n2 are the total number of possible scoring combinations for each team, respectively. The order of the elements in the rows for team1_combs and team2_combs should follow the above list.\r\n*** Attention - There is added complexity in the scoring possibilities in that the number of tries after touchdowns (also known as extra points or point after attempts) must be less than or equal to number of touchdowns. This constraint is further complicated by the possibility of defensive scores on tries after touchdowns, in which the possible number of scores is constrained by the touchdowns scored by the opposing team.\r\nExample\r\n[t1_combs, t2_combs] = amerfootballscores(5, 8)\r\n\r\nt1_combs =\r\n\r\n     0     0     0     0     1     0     1\r\n     0     0     0     1     1     0     0\r\n     0     0     0     2     0     1     0\r\n\r\n\r\nt2_combs =\r\n\r\n     0     0     0     1     2     0     0\r\n     0     0     0     4     0     0     0\r\n     1     0     0     1     0     0     0\r\n     1     0     1     0     0     0     0\r\nThis system is known as a linear Diophantine equation, an equation where only the integer--and in this case only nonnegative--solutions are of interest.\r\nEach team's score can be modeled by the following equation:\r\n6(tds) + 1(off. 1 pt. pats) + 2(off. 2 pt. pats) + 2(safs) + 3(fgs) + 1(def. 1 pt. pats) + 2(def. 2 pt. pats) = score\r\n    \u003e\u003e\u003e with the added constraints:\r\n                off. 1 pt. pats + off. 2 pt. pats \u003c= tds\r\n                def. 1pt. pats + def. 2pt. pats \u003c= tds for opponent","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 970.625px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 485.312px; transform-origin: 407px 485.312px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eWith inspiration from \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/56195-possible-rugby-scores\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eProblem 56195. Possible Ruby Scores\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, given the final score for two teams in an American football game (score1, score2), return two matrices (team1_combs, team2_combs) representing all possible combinations of scores by each team (\u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://operations.nfl.com/the-rules/nfl-video-rulebook/scoring-plays/\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eOfficial NFL Scoring Plays\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e). \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eIn American football, the 7 possible ways of scoring are:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003col style=\"block-size: 143.062px; font-family: Helvetica, Arial, sans-serif; list-style-type: decimal; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 391px 71.525px; transform-origin: 391px 71.5312px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2125px; text-align: left; transform-origin: 363px 10.2188px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eTouchdown - 6 points\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2125px; text-align: left; transform-origin: 363px 10.2188px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eTry after touchdown - 1 point\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2125px; text-align: left; transform-origin: 363px 10.2188px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eTry after touchdown - 2 points\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2125px; text-align: left; transform-origin: 363px 10.2188px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eSafety - 2 points\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2125px; text-align: left; transform-origin: 363px 10.2188px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eField Goal - 3 points\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2125px; text-align: left; transform-origin: 363px 10.2188px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eDefensive safety on try after touchdown (i.e. conversion safety) - 1 points\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2125px; text-align: left; transform-origin: 363px 10.2188px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eDefensive \"touchdown\" on try after touchdown - 2 points\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eValues for score1 and score2 include all natural numbers (realistically \u0026lt;100), and the resulting matrices will have dimensions n1 x 7 and n2 x 7, where n1, n2 are the total number of possible scoring combinations for each team, respectively. The order of the elements in the rows for team1_combs and team2_combs should follow the above list.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e*** Attention - There is added complexity in the scoring possibilities in that the number of tries after touchdowns (also known as extra points or point after attempts) must be less than or equal to number of touchdowns. This constraint is further complicated by the possibility of defensive scores on tries after touchdowns, in which the possible number of scores is constrained by the touchdowns scored by the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; text-decoration: underline; text-decoration-line: underline; \"\u003eopposing team\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExample\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 306.562px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 153.275px; transform-origin: 404px 153.281px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e[t1_combs, t2_combs] = amerfootballscores(5, 8)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003et1_combs =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     0     0     0     0     1     0     1\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     0     0     0     1     1     0     0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     0     0     0     2     0     1     0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003et2_combs =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     0     0     0     1     2     0     0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     0     0     0     4     0     0     0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     1     0     0     1     0     0     0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     1     0     1     0     0     0     0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThis system is known as a linear \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Diophantine_equation\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eDiophantine equation\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, an equation where only the integer--and in this case only nonnegative--solutions are of interest.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eEach team's score can be modeled by the following equation:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e6(tds) + 1(off. 1 pt. pats) + 2\u003c/span\u003e\u003cspan style=\"border-block-end-style: solid; border-block-end-width: 0.8px; border-bottom-style: solid; border-bottom-width: 0.8px; \"\u003e(\u003c/span\u003e\u003cspan style=\"\"\u003eoff. 2 pt. pats\u003c/span\u003e\u003cspan style=\"border-block-end-style: solid; border-block-end-width: 0.8px; border-bottom-style: solid; border-bottom-width: 0.8px; \"\u003e)\u003c/span\u003e\u003cspan style=\"\"\u003e + 2(safs) + 3(fgs) + 1(def. 1 pt. pats) + 2(def. 2 pt. pats) = score\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e    \u0026gt;\u0026gt;\u0026gt; with the added constraints:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                off. 1 pt. pats + off. 2 pt. pats \u0026lt;= tds\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                def. 1pt. pats + def. 2pt. pats \u0026lt;= tds for opponent\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [t1_combs,t2_combs] = amerfootballscores(s1,s2)  \r\nt1_combs = [];  \r\nt2_combs = [];\r\n\r\nend","test_suite":"%%\r\ns1 = 5;\r\ns2 = 8;\r\nteam1 = [0     0     0     0     1     0     1;\r\n         0     0     0     1     1     0     0;\r\n         0     0     0     2     0     1     0];\r\nteam2 = [0     0     0     1     2     0     0;\r\n         0     0     0     4     0     0     0;\r\n         1     0     0     1     0     0     0;\r\n         1     0     1     0     0     0     0];\r\n[t1_combs,t2_combs] = amerfootballscores(s1,s2);\r\nassert(isequal([t1_combs;t2_combs],[team1;team2]))\r\n\r\n%%\r\ns1 = 0;\r\ns2 = 28;\r\nteam1 = [0     0     0     0     0     0     0];\r\nteam2 = [0     0     0     2     8     0     0;\r\n         0     0     0     5     6     0     0;\r\n         0     0     0     8     4     0     0;\r\n         0     0     0    11     2     0     0;\r\n         0     0     0    14     0     0     0;\r\n         1     0     0     2     6     0     0;\r\n         1     0     0     5     4     0     0;\r\n         1     0     0     8     2     0     0;\r\n         1     0     0    11     0     0     0;\r\n         1     0     1     1     6     0     0;\r\n         1     0     1     4     4     0     0;\r\n         1     0     1     7     2     0     0;\r\n         1     0     1    10     0     0     0;\r\n         1     1     0     0     7     0     0;\r\n         1     1     0     3     5     0     0;\r\n         1     1     0     6     3     0     0;\r\n         1     1     0     9     1     0     0;\r\n         2     0     0     2     4     0     0;\r\n         2     0     0     5     2     0     0;\r\n         2     0     0     8     0     0     0;\r\n         2     0     1     1     4     0     0;\r\n         2     0     1     4     2     0     0;\r\n         2     0     1     7     0     0     0;\r\n         2     0     2     0     4     0     0;\r\n         2     0     2     3     2     0     0;\r\n         2     0     2     6     0     0     0;\r\n         2     1     0     0     5     0     0;\r\n         2     1     0     3     3     0     0;\r\n         2     1     0     6     1     0     0;\r\n         2     1     1     2     3     0     0;\r\n         2     1     1     5     1     0     0;\r\n         2     2     0     1     4     0     0;\r\n         2     2     0     4     2     0     0;\r\n         2     2     0     7     0     0     0;\r\n         3     0     0     2     2     0     0;\r\n         3     0     0     5     0     0     0;\r\n         3     0     1     1     2     0     0;\r\n         3     0     1     4     0     0     0;\r\n         3     0     2     0     2     0     0;\r\n         3     0     2     3     0     0     0;\r\n         3     0     3     2     0     0     0;\r\n         3     1     0     0     3     0     0;\r\n         3     1     0     3     1     0     0;\r\n         3     1     1     2     1     0     0;\r\n         3     1     2     1     1     0     0;\r\n         3     2     0     1     2     0     0;\r\n         3     2     0     4     0     0     0;\r\n         3     2     1     0     2     0     0;\r\n         3     2     1     3     0     0     0;\r\n         3     3     0     2     1     0     0;\r\n         4     0     0     2     0     0     0;\r\n         4     0     1     1     0     0     0;\r\n         4     0     2     0     0     0     0;\r\n         4     1     0     0     1     0     0;\r\n         4     2     0     1     0     0     0;\r\n         4     2     1     0     0     0     0;\r\n         4     4     0     0     0     0     0];\r\n[t1_combs,t2_combs] = amerfootballscores(s1,s2);\r\nassert(isequal([t1_combs;t2_combs],[team1;team2]))\r\n\r\n%%\r\ns1 = 37;\r\ns2 = 49;\r\n[t1_combs,t2_combs] = amerfootballscores(s1,s2);\r\n% Matrices are rather large, so test case is dependent on the size of each output matrix, \r\n% where the number of rows will equate to the total number of scoring combinations\r\n[t1_rows,t1_cols] = size(t1_combs);\r\n[t2_rows,t2_cols] = size(t2_combs);\r\nassert(isequal([t1_rows,t2_rows],[2894,5625]))\r\n\r\n%%\r\ns1 = 37;\r\ns2 = 49;\r\n[t1_combs,t2_combs] = amerfootballscores(s1,s2);\r\n% Matrices are rather large, so test case is dependent on the sums of each column. These\r\n% values don't have any practical application.\r\nt1_sums = [6724, 2210, 1774, 10752, 6776, 7566, 5789];\r\nt2_sums = [19476, 6398, 5272, 28518, 18222, 11185, 9470];\r\nassert(isequal([t1_sums;t2_sums],[sum(t1_combs);sum(t2_combs)]))","published":true,"deleted":false,"likes_count":0,"comments_count":8,"created_by":3499438,"edited_by":3499438,"edited_at":"2023-08-17T23:02:26.000Z","deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-08-09T17:04:20.000Z","updated_at":"2026-02-23T23:47:16.000Z","published_at":"2023-08-10T05:17:56.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWith inspiration from \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/56195-possible-rugby-scores\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 56195. Possible Ruby Scores\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, given the final score for two teams in an American football game (score1, score2), return two matrices (team1_combs, team2_combs) representing all possible combinations of scores by each team (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://operations.nfl.com/the-rules/nfl-video-rulebook/scoring-plays/\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eOfficial NFL Scoring Plays\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e). \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn American football, the 7 possible ways of scoring are:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTouchdown - 6 points\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTry after touchdown - 1 point\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTry after touchdown - 2 points\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSafety - 2 points\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eField Goal - 3 points\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDefensive safety on try after touchdown (i.e. conversion safety) - 1 points\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDefensive \\\"touchdown\\\" on try after touchdown - 2 points\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eValues for score1 and score2 include all natural numbers (realistically \u0026lt;100), and the resulting matrices will have dimensions n1 x 7 and n2 x 7, where n1, n2 are the total number of possible scoring combinations for each team, respectively. The order of the elements in the rows for team1_combs and team2_combs should follow the above list.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e*** Attention - There is added complexity in the scoring possibilities in that the number of tries after touchdowns (also known as extra points or point after attempts) must be less than or equal to number of touchdowns. This constraint is further complicated by the possibility of defensive scores on tries after touchdowns, in which the possible number of scores is constrained by the touchdowns scored by the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eopposing team\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExample\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[[t1_combs, t2_combs] = amerfootballscores(5, 8)\\n\\nt1_combs =\\n\\n     0     0     0     0     1     0     1\\n     0     0     0     1     1     0     0\\n     0     0     0     2     0     1     0\\n\\n\\nt2_combs =\\n\\n     0     0     0     1     2     0     0\\n     0     0     0     4     0     0     0\\n     1     0     0     1     0     0     0\\n     1     0     1     0     0     0     0]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis system is known as a linear \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Diophantine_equation\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eDiophantine equation\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, an equation where only the integer--and in this case only nonnegative--solutions are of interest.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEach team's score can be modeled by the following equation:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e6(tds) + 1(off. 1 pt. pats) + 2(off. 2 pt. pats) + 2(safs) + 3(fgs) + 1(def. 1 pt. pats) + 2(def. 2 pt. pats) = score\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e    \u0026gt;\u0026gt;\u0026gt; with the added constraints:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                off. 1 pt. pats + off. 2 pt. pats \u0026lt;= tds\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                def. 1pt. pats + def. 2pt. pats \u0026lt;= tds for opponent\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":2737,"title":"Prouhet–Tarry–Escott (basic)","description":"Inspired by \u003chttp://www.mathworks.com/matlabcentral/cody/problems/660-find-a-subset-that-divides-the-vector-into-equal-halves problem 660.\u003e\r\n\r\nGiven n return two disjoint sets of integers _A_ and _B_ with same cardinality having following property:\r\n\r\n\u003c\u003chttps://i.imgur.com/gSW7nWy.png\u003e\u003e\r\n\r\nfor i = 1:n\r\n\r\nTry to minimize sets cardinality. ","description_html":"\u003cp\u003eInspired by \u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/660-find-a-subset-that-divides-the-vector-into-equal-halves\"\u003eproblem 660.\u003c/a\u003e\u003c/p\u003e\u003cp\u003eGiven n return two disjoint sets of integers \u003ci\u003eA\u003c/i\u003e and \u003ci\u003eB\u003c/i\u003e with same cardinality having following property:\u003c/p\u003e\u003cimg src = \"https://i.imgur.com/gSW7nWy.png\"\u003e\u003cp\u003efor i = 1:n\u003c/p\u003e\u003cp\u003eTry to minimize sets cardinality.\u003c/p\u003e","function_template":"function [A, B] = prouhet(n)\r\n  A = 1:n;\r\n  B = -A;\r\nend","test_suite":"%%\r\nn = 1;\r\n[A, B] = prouhet(n);\r\nassert(isequal(A, round(A)) \u0026\u0026 isequal(B, round(B)))\r\nassert(isempty(intersect(A, B)));\r\nassert(isequal(numel(A), numel(B), numel(unique(A)), numel(unique(B))));\r\nassert(isequal(sum(A(:).^(1:n),1), sum(B(:).^(1:n),1)));\r\ndisp(sprintf('Each set has %i elements.', numel(A)))\r\n%%\r\nn = 2;\r\n[A, B] = prouhet(n);\r\nassert(isequal(A, round(A)) \u0026\u0026 isequal(B, round(B)))\r\nassert(isempty(intersect(A, B)));\r\nassert(isequal(numel(A), numel(B), numel(unique(A)), numel(unique(B))));\r\nassert(isequal(sum(A(:).^(1:n),1), sum(B(:).^(1:n),1)));\r\ndisp(sprintf('Each set has %i elements.', numel(A)))\r\n%%\r\nn = 5;\r\n[A, B] = prouhet(n);\r\nassert(isequal(A, round(A)) \u0026\u0026 isequal(B, round(B)))\r\nassert(isempty(intersect(A, B)));\r\nassert(isequal(numel(A), numel(B), numel(unique(A)), numel(unique(B))));\r\nassert(isequal(sum(A(:).^(1:n),1), sum(B(:).^(1:n),1)));\r\ndisp(sprintf('Each set has %i elements.', numel(A)))\r\n%%\r\nn = 7;\r\n[A, B] = prouhet(n);\r\nassert(isequal(A, round(A)) \u0026\u0026 isequal(B, round(B)))\r\nassert(isempty(intersect(A, B)));\r\nassert(isequal(numel(A), numel(B), numel(unique(A)), numel(unique(B))));\r\nassert(isequal(sum(A(:).^(1:n),1), sum(B(:).^(1:n),1)));\r\ndisp(sprintf('Each set has %i elements.', numel(A)))\r\n%%\r\n%n = 9;\r\n%[A, B] = prouhet(n);\r\n%assert(isequal(A, round(A)) \u0026\u0026 isequal(B, round(B)))\r\n%assert(isempty(intersect(A, B)));\r\n%assert(isequal(numel(A), numel(B), numel(unique(A)), numel(unique(B))));\r\n%assert(isequal(sum(uint64(A(:)).^uint64(1:n)), sum(uint64(A(:)).^uint64(1:n))));\r\n%disp(sprintf('Each set has %i elements.', numel(A)))\r\n%if numel(A) \u003c=20\r\n%    disp('A:')\r\n%    disp(A)\r\n%    disp('B:')\r\n%    disp(B)\r\n%end\r\n%%\r\n% test info\r\n%\r\n% larger n will be added later\r\n%\r\n% scoring function will be added later as well\r\n% scoring will be entirely based on size of output: smaller output == better score\r\n% something like this:\r\n%\r\n% score = 0;\r\n% for n = 1:25\r\n%   [A, B] = prouhet(n)\r\n%   assert(...);\r\n%   score = score + numel(A);\r\n% end\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":2,"created_by":14358,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":7,"test_suite_updated_at":"2016-10-08T00:11:00.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2014-12-08T09:56:20.000Z","updated_at":"2020-07-22T23:41:27.000Z","published_at":"2016-10-07T08:07:12.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/media/image1.JPEG\"}],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eInspired by\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/660-find-a-subset-that-divides-the-vector-into-equal-halves\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eproblem 660.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven n return two disjoint sets of integers\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eA\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eB\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e with same cardinality having following property:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003efor i = 1:n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTry to minimize sets cardinality.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"},{\"partUri\":\"/media/image1.JPEG\",\"contentType\":\"image/JPEG\",\"content\":\"data:image/JPEG;base64,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\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":57565,"title":"Easy Sequences 92: Number of Roots of a System of Linear Diophantine Equations","description":"One of the tasks that Matlab is very good at, is in solving systems of linear equations. \r\nIn this problem we shall tackle a system of linear Diophantine Equations in which the roots are limited to certain range.\r\nGiven the number of variables , positive integers, ,  and a root limit, , create the function, numRoots(n,a,b,L), that outputs the number of posible integer root sets of the following system of equations:\r\n                \r\n                \r\n        with:\r\n                .\r\nFor example, if , the system of equations:  and , where , has only one root set, namely: . Therefore, numRoots(2,10,4,1) = 1.\r\nIf , two of the possible roots of:\r\n                ; and\r\n                \r\n                where:  \r\nare and . In fact, there are  possible root sets. Therefore, numRoots(4,20,6,3) = 16.\r\nThere are no possible roots for  and , therefore in these cases the function should return: numRoots = 0.\r\n-------------------------\r\nNOTE: As an added challenge, only those solutions with Cody program size of less than or equal to 200 will be accepted.","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 607.5px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 303.75px; transform-origin: 407px 303.75px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 270px 8px; transform-origin: 270px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eOne of the tasks that Matlab is very good at, is in solving systems of linear equations. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 153px 8px; transform-origin: 153px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIn this problem we shall tackle a system of linear \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Diophantine_equation\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eDiophantine Equations\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 145.5px 8px; transform-origin: 145.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e in which the roots are limited to certain range.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 63.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.75px; text-align: left; transform-origin: 384px 31.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 109px 8px; transform-origin: 109px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eGiven the number of variables \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003en\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 64px 8px; transform-origin: 64px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003epositive integers, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ea\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eb\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58px 8px; transform-origin: 58px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e and a root limit, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003eL\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 75px 8px; transform-origin: 75px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, create the function, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 68px 8px; transform-origin: 68px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 68px 8.5px; transform-origin: 68px 8.5px; \"\u003enumRoots(n,a,b,L)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 281px 8px; transform-origin: 281px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e, that outputs the number of posible integer root sets of the following system of equations:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32px 8px; transform-origin: 32px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg 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style=\"width: 378.5px; height: 21px;\" width=\"378.5\" height=\"21\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.5px 8px; transform-origin: 32.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e        with:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32px 8px; transform-origin: 32px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 132.5px; height: 20px;\" width=\"132.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 44px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 22px; text-align: left; transform-origin: 384px 22px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 48.5px 8px; transform-origin: 48.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eFor example, if \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAATEAAAAlCAYAAAAqVd5RAAALtUlEQVR4Xu2d28t30xbHX3+AHK9s9OZQZNcWGyUU5VwiCnGh9m5vJBeSc0nOp5Sco94LNkKk2HtTexdSjlHKhcOFC1cOW/sPYHz0G4xma8455lxzrd/6Pe9cNS6e55lrzXH8zjHHHGs9u2zrV9dA10DXwAZrYJcN5r2z3jXQNdA1sK2DWHeCroGugY3WQAexjTZfZ75roGugFMT2EpWdJ/R4V93iNXCccPh/oU8Xz2kdg3+S2/4g9Hrd7f2uBWrgAOHpkFKbloAYAPaM0LVbODAWaNdqlrDXw0KvCD1X/ZRl3giA3S10kdD3y2Sxc1WpgWvkvu1Cl3vv94IYAfEvob/u5ADGSrGP0LFCLwp97VV0xTh0fmjmvs8zQax2u38LARkA9qTQaQ4AU3uhxpyuKkwUvYV5f3LwZx9geX2nJTOOZ6FTrqmzdq9eHlnx4wIyL4gBYP8RusehkK045O8iFAD+ZyOcV3e1+lAQu0IecH7wkDvk538KeZyd57wndO4MTlorq/c+ZPlC6KyM7GfI3x8UOjB48L/lZwJjqsWHIL1a6DKhw536vkDG3Sb0odDHQtuF+N31QiyUU2ea6Oo1oedX83ptUTKuRi9gzg6h7C7CE4ikd38TOqiE6y069ueVXAQDmcBcl87LfADYjYUTKwgfVXjf0obj2ABQaoW+Xf5+Q4LxH+VvLEYtgQxwvVUI8NLLA2JkHNwT2hR7PSaEn025ZdZFYQ+ZZwoQq9ULOiQ7/K/HVjkQUyFxmiwiLs3jG/ODLr5bPZO64FxZKcb8xMhCdlETgF/KfU/MyHdj9W/joOJtoZT8OuZRGXffSk/Y7WQhsh3NzMh6WgE6mcwfhT4TumUVdMieAzEFqq9k7DFCYcYFYJ8qVLNoeXWvczC+NYjV6sXyDsiTxSUThhyI9Szsd5WS4j+7+jHnoF4n8oyz8+LwtRkxtrxOaE/PpAsc48nCPhC+oaFMzWYdHpCpUQE65sAh93zLSwykpvY3jW3NTFuDmNWfVy+hznVRSsZbDsR+kKeyqpVuX2ocYOn3aOqP0ecEAjJgrYlhC1exc0CZrGiA4IVCm5ZVazZ6pvAea6lgzEtCQ1mNqsNuNS+VX7ZuFfIGq2Zh8BWTSe3FmNbZmOrzeHn2TUJkfEsEMWRnB/FGyu9TIKYFv5TjLB14WvIHoE9VO0jxqfOmHN4rJw7BVopVfpMuBZ+9helYoRt/3S8DTLqyI/s6QYxsUQ+JUttjtf2YDDy0sx70aGlBt5RLBTGSB/w1mjikQMzjOEOBwArCZes2IP+uQlMec+sR9bdm7lZHx7YuZTOZqY/Fw3pYKog9oEQGdkrKITwPWcMYgp5rbB3LghhZiOd0t0RcbyamBzW5rN7WrGproSH/YZ1p6SCmWWt0S5kCMRUut+Vkkv2FjhQiLeXSCXGaHUK2oOrp7ylxHD2ixiFeEDpCiJWLUx8ypxYrrk3/kQVApvah8sIvc17SODDsvC1ORDXIagJCF6IS2wyNrQEOgr6F/FpnihXTx8rmATG7MJWAWIsdkbae2HabpYOYLjzRMkgKoEhl3xfKtRLo6x/0mmgwU3zGYUB9ajhnC2ldp4UxmEffIABIQqCy6XqLIrwamq0Y/WIc/d4l9K4Qja9azG0dHHYlbnEiqkFWk4VYXmqDPRe0Q8/VoG8hv9Y1WzxriFcPiNlsMKcPq/OxizG7Bvw37DTYFBCL2iwFYiWrX9h+wHEzzYZaZLXGrckCQofRTnTqCkPGVcO0qCVY2SisA84nCtnuZg0O+KwBiBgo2P6wFs/NrmoJdKK8QLY95vrfSn8lz1CexwKPngiyOKeK/yW8hWNLQSxXh7LPGwtisdPdpYMYOiYOoodarUBMDwGYkEzraSHbUKgnbC36cyyAxQTTTGzMaZ46qJWNlfNiofCEzDpbq0zTzgsvuW29J/haAYJnrlZjWvE8Jgv1ylIKYrktcqvTVG2nSPWj5QDVq4OhcR69pJ6fTKhagZgqmyAn+wnf1eNUjAxs7GqKoNawQ4Vumzm1ABSbZcVA0fLUYvsaytnKwVoBwhiHLr23Bc+6JZ26vcQTrHNvJ1X2mF9uSiYWBfxWIKYghYOGgW4LmWO3RJ7eGdskOPY0D3msbLHnaeaXq3GUBLCt643dSoRZZQtwL5FlzNixIKYtBSwEU/c7ekAMXdScTtYsjio7fVb/iBjhAfk9uyZAgrcauFp/wsmrlyEWNSmJ9sqlQIzghXKFfQssMBEGup6wtQhwmxXFamsKOi22rhaAY1mYlb9FpokObTbJzzUOPOQQY7ZU6zqdVP3W6FZLDwTx1ACGvr3Bqj6aiwnNkmrf97RZX8lC0irz1zm9ehniMbuItWqx4GVVrqGMQethLRSjzX8xgLJZWI3Th0q0LQ4xILGd/AfLA1p8daDmVSOMfXomWNWZajLUdZ1OYpOSQya14dwAVgJidjFOLVCasdUuyCw8enIeA7Gj5Q+0IgGUdCNwtf5iTQsQi+4eUiDm7Smyr8UMBYcaYmw9wmZFQ6ll+G5ciy2TyhY75ZyqedJThwudEpB5WSj1Kg1jaH+pef9yXaeTyFnDN9vxXAbG4Ul4SBMLds/vvcFq/SYWF9bfW5UThmRYek0si0MpEFMl5sBAQWoo07InbGz/+FAcSrOd12QdNMvSlpFyKGv4EMR01QVstB8NQN1XiGxK3zdka8LnteHjTaHc1yBUtqGtpAXNmJN5ZQudy9bhPOCvGWOufYVMFmCuff/SE8hTjFH5vBkki0Du6wfY5gSjC+yJb+wmRH2o5gOBXhBDRwoesYK1HhYN9R4SC/QnfrPy4zHZfw7E5tZL6D/4KzW76MKbO7bPvXyZW1E0oyAdpraGwm42YGW3TTCfWnFsnch+ZwlnBUCfFAJ46dS3Tam2p8uCQ64x1QJwCJo6J8qNgUyJbNZw4atGuUXEftIllWGprcYerkwBUrlnlry8rj6HzVishq7tKz+xurDZL/fU1CFLQEybT9nKhXPZBTK0V1iDHlumyYHY3HoJ7UUikSwN5UAMo1wnFKv15NoddDvGfpss4F4hu92xNSeYzzWnWoUyFlCiY1+BxNZtmDNsSrUvUzNfKsuxIKaZmK5Kd654JeuLZXOlsqnx7Pac3+GkDw1E4mHyO/u12VwNEN3x3mTNVnIQCWb+JbblSh00hQGXYjGsM4V6L+0xBFzYvuordjl7wBsLFm9/cKmv2gVyaAELFznuzWXgKT3kQGwdelF+NRFIypcDsdxHEbVO8pHMOnT6o4VFOrUJxPC9OZ7PFzExnAZXLsgAh3NWUlIDsp/wJdu4SQh+nhIKAQZgunJ1rwW/mJHhiw/q6XyMo+jp2W6UyqaF+dqu+NRnl9WOQ426KQdf0t80k4w5tLWth+8dMsh+kghf5ZPQuwtR7PZuu/GRvwgN2Q3/Y1uaqlPqwgiIMTexArClyh0kFycJcS+7gTEgpjEcK+avSy/YEICNYctvNs6BGANxDjrwW528xRwMw2wXmqteQ5o6xvieQNExc8sW8qaOmmuXKZFpHWPnkoPg2SFkQW4d8ubmBNjhM7fw557j/fucenHjjgfEEJA0nWsqgNGmvHNljpqCqtcIOo5sjpWPdHXqa27ZQnlw9FeFpl6EptYjz9cDnCn/exOZR+7jinPI6pkDkCVra/1xx6G559SL7hxy/xDmVz69IMbYqVBYHdMW/D0GrB0DcF0iNOU/YFDe5pYt1Mm6AbTWRqn7ppSJQOWAaBP+NSGJBXXfOZp459ZLEdaUgBiOheLeEmqZZrPV4pM2Nd+ZKg0SjEEtjHrZmGNp77xzyjYEYFv1nx0DZK1l45lXrXwj13rjtf9U43QH0TIOY7zOrZdijCkFMQRt3SA4laF39ueyjZzyS7rr1i/BxT8XnmPxW7esO8v8HJLwwdGiklINiO0sCu1ydg10DWyABjqIbYCROotdA10DcQ10EOve0TXQNbDRGuggttHm68x3DXQNdBDrPtA10DWw0Rr4BaLeVFMZ7zjkAAAAAElFTkSuQmCC\" style=\"width: 152.5px; height: 18.5px;\" width=\"152.5\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 83px 8px; transform-origin: 83px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, the system of equations: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJoAAAAoCAYAAAAG/eNpAAAFtElEQVR4Xu2bScsdRRSGkx/gFF25EHFYiAuFOIDoRnBAQQgRNAoSUNAouHMWXDhnIwSiiWtDdCtxiC4EJ3CCCIILB0TElSP+AH0f7SPnq1vVXXVvd3/d362Gd3G7azqn3jp1zqm627fVp2pgBA1sH6GP2kXVwLZKtEqCUTRQiTaKmmsnlWiVA6NooBJtFDXXTirRKgfQwEXCl4WqoM5Jwl85dSvRCrW7xYpfKXkeFy4TdmTIdrrK3Ck8LLwk/ClcLZwn3C+8mWqjEi1Du1uwCNboeeG6RrbfM4gGyd4WLhFuDEj1qn7fIjwjPBbTVyXaFmRRh0h36/spwlfCK8JpQg7RjEyvqfytQR+Q8JumrZCE/xadE9FYTazAe4TD68ePQSQ2nXYRjS32g2YEUSLp24vCPuE74XLhVz/iSrRB5m82jeYSzawZhGTr/D4iIVbuaIqMlWiz4cQgA80hGtviL03vWCsc/9iD33ei+UCgcG+pRcNsxkLYc/T+TOHr0EwOopL/HNGhtk4L1T8Mxo6SLxB+TqzigUQdrdkcovltM+af+cH+nSJkyqLhMO5qJpa6ocl8Wu8ebRpNRho9q6tvot2g8d0hXCvgEPN4/4OtAL+Db8eF23tcUEbsVVUULozS9nKIBhcONQ3nEo3iZ3h9pYhmCTy/75oTDsmYnM8EnL+xnPO+iWYyYpnZEnhMkZCQiIwtABnfEcJIq3RSfXmTZZU2uhz4nLZziPagGiIV4vWTatssGt8vFv5PAuf4aFYZpR8TDgipfTpHuGXL9E202MR/rpd3Ce8JKad32fH7eizWnSs29EcP5C8lWpdR8QuomGhWmVVP9vh8YUPouqLCrLr5fKnmXmgmn636rZY+l/EZ/aqFbE8IySx3T/JOoZlSoj2kQe9vGTi7HAuUp5hofhK6GL2K8nw/q7SzQcDMhkoc3swmZ1GslGglPlox0XKSdV6rdrzxhV5GjyMSU4APtLdlejiPwzHHsn7bUo6wOpbn6Zp5cxHaghui0CcFfNRzBfyk5zpWeVe/m/k9h2gli9B0uOA/5vhoPj/SZjrZ+m4WOHCFEH1Ho0P6aBDok4Y8RJjXR2bfzvoIDNi6LxSebWRN1UmRaE5RZyxYSsllRMP9uNQX6iKanwDqLTTgGmNAnOZzuk+UMieiWebbxNkQmjcvCfPPEryV9ouwZMueU9SJ+OwgZsFTtzy85VuY+y6iMQG/CT805KHT2CR48pqvNReiQaAHhL1C23keuiB3GG7L5gCnzgBjq39OUSfj9/4zhIu5Jj7ftrDo2ohGRUJ9TGDMT+PdFUIYhcyJaGaRbPGEfhoW/aDQlkMz61Ri0WLk24x3OT4a4/K3M1Luky24aMDgiYbSOeLh+sjJAllxn8rw+TRuT5BriqU6pkw0u7j3o8b+k/C6cJNgGXZTPC4C96tQWleqwwKThRsLm8Gcwj5ziUazJLHfEGK3M8wQ8Y1AacHieaKFfkO4QkM/JrWCp0y0MIUSpmu8+Ue5Xekcc5T3qCz6mdPjdynGfZVbcCk57KSI4Meie9phweJi7RaiV8I90WAs13HJOBNNhRWweI80I4h9t8FNmWgmw6kaLCccYVLW0hcQKPY9nACsPs7xfcIQSewhiAsxbhOQMXxISX0U0YsvZ9kFrnDzYL3eF95t00FXMLCMoEMRjdVE1PexsOph8jJyhXWYME4r2GKXydv1MYbZtDEnok1Jqaxq8mlRf2RKA53KWCrRymeC7fWI8NRELGu5BJtQoxKtTOl2OhCLRHEZcJJL/x9ZNoKZlh6CaHYpsu+E7RRUTK6I47WXg8Gcrd9sp31ejpyCvL2NoU+ioehrBDv/42CVEPjTLeIs2798UsrvSoX0NmlzbKhPos1R/jrmkTRQiTaSote9m0q0dWfASPJXoo2k6HXvphJt3RkwkvyVaCMpet27qURbdwaMJP8/YupiOGKJgUwAAAAASUVORK5CYII=\" style=\"width: 77px; height: 20px;\" width=\"77\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 69.5px; height: 20px;\" width=\"69.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 25px 8px; transform-origin: 25px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, where \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 58.5px; height: 20px;\" width=\"58.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32.5px 8px; transform-origin: 32.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, has only one root set, namely: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 96px; height: 20px;\" width=\"96\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 39.5px 8px; transform-origin: 39.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. Therefore, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 72px 8px; transform-origin: 72px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 72px 8.5px; transform-origin: 72px 8.5px; \"\u003enumRoots(2,10,4,1)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 4px 8.5px; transform-origin: 4px 8.5px; \"\u003e=\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 4px 8.5px; transform-origin: 4px 8.5px; \"\u003e1\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 5.5px 8px; transform-origin: 5.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eIf \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 152.5px; height: 18.5px;\" width=\"152.5\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 90.5px 8px; transform-origin: 90.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, two of the possible roots of:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32px 8px; transform-origin: 32px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 134px; height: 20px;\" width=\"134\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e; and\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 32px 8px; transform-origin: 32px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 126.5px; height: 20px;\" width=\"126.5\" height=\"20\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 22px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 11px; text-align: left; transform-origin: 384px 11px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 57px 8px; transform-origin: 57px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e                where:  \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 94.5px; height: 20px;\" width=\"94.5\" height=\"20\"\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 43.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.75px; text-align: left; transform-origin: 384px 21.75px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12.5px 8px; transform-origin: 12.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eare \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-6px\"\u003e\u003cimg 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style=\"width: 166.5px; height: 20px;\" width=\"166.5\" height=\"20\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 58px 8px; transform-origin: 58px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. In fact, there are \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 18px; height: 18px;\" width=\"18\" height=\"18\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 97px 8px; transform-origin: 97px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e possible root sets. Therefore, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 72px 8px; transform-origin: 72px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 72px 8.5px; transform-origin: 72px 8.5px; \"\u003enumRoots(4,20,6,3)\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 4px 8.5px; transform-origin: 4px 8.5px; \"\u003e=\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6px 8px; transform-origin: 6px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e 1\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 4px 8px; transform-origin: 4px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 4px 8.5px; transform-origin: 4px 8.5px; \"\u003e6\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42.5px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21.25px; text-align: left; transform-origin: 384px 21.25px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 100px 8px; transform-origin: 100px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eThere are no possible roots for \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 185px; height: 18.5px;\" width=\"185\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16px 8px; transform-origin: 16px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" style=\"width: 175px; height: 18.5px;\" width=\"175\" height=\"18.5\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 82.5px 8px; transform-origin: 82.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, therefore in these cases the function should return: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 48px 8px; transform-origin: 48px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-family: Menlo, Monaco, Consolas, \u0026quot;Courier New\u0026quot;, monospace; perspective-origin: 48px 8.5px; transform-origin: 48px 8.5px; \"\u003enumRoots = 0\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 2px 8px; transform-origin: 2px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 62.5px 8px; transform-origin: 62.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e-------------------------\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space: pre-wrap; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 23px 8px; transform-origin: 23px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eNOTE: \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 76.5px 8px; transform-origin: 76.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eAs an added challenge, \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 89.5px 8px; transform-origin: 89.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eonly those solutions with \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/content/cody/about.html\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"font-weight: 700; text-decoration-line: underline; \"\u003eCody program size\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 85px 8px; transform-origin: 85px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e of less than or equal to \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 12px 8px; transform-origin: 12px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; text-decoration-line: underline; \"\u003e200\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 26.5px 8px; transform-origin: 26.5px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003e will be accepted.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function c = numRoots(n,a,b,L)\r\n    n = 'number of unknown variables';\r\n    a = 'sum of the unknown variables';\r\n    b = 'alternating sum and difference of the unknown variables';\r\n    L = 'limit of the roots (i.e. all roots should be \u003e L');\r\n    c = 'the number of root sets';\r\nend","test_suite":"%%\r\n[n,a,b,L] = deal(4,20,6,3);\r\nc_correct = 16;\r\nassert(isequal(str2num(numRoots(n,a,b,L)),c_correct))\r\n[n,a,b,L] = deal(4,20,6,-3);\r\nc_correct = 280;\r\nassert(isequal(str2num(numRoots(n,a,b,L)),c_correct))\r\n[n,a,b,L] = deal(3,20,6,4);\r\nc_correct = 6;\r\nassert(isequal(str2num(numRoots(n,a,b,L)),c_correct))\r\n%%\r\n[n,a,b,L] = deal(2,10,4,1);\r\nc_correct = 1;\r\nassert(isequal(str2num(numRoots(n,a,b,L)),c_correct))\r\n[n,a,b,L] = deal(10,55,10,-10);\r\nc_correct = 0;\r\nassert(isequal(str2num(numRoots(n,a,b,L)),c_correct))\r\n[n,a,b,L] = deal(42,10,2,100);\r\nc_correct = 0;\r\nassert(isequal(str2num(numRoots(n,a,b,L)),c_correct))\r\n%%\r\n[n,a,b,L] = deal(10,25,15,-5);\r\nc_correct = '9825961376';\r\nassert(isequal(numRoots(n,a,b,L),c_correct))\r\n[n,a,b,L] = deal(25,50,25,5);\r\nc_correct = '0';\r\nassert(isequal(numRoots(n,a,b,L),c_correct))\r\n%%\r\n[n,a,b,L] = deal(8,88,8,8);\r\nc_correct = '159885';\r\nassert(isequal(numRoots(n,a,b,L),c_correct))\r\n[n,a,b,L] = deal(88,88,8,8);\r\nc_correct = '0';\r\nassert(isequal(numRoots(n,a,b,L),c_correct))\r\n[n,a,b,L] = deal(8,888,88,-8);\r\nc_correct =  '322979008681245';\r\nassert(isequal(numRoots(n,a,b,L),c_correct))\r\n%%\r\n[n,a,b,L] = deal(11,111,11,-1);\r\nc_correct =  '6367915454544';\r\nassert(isequal(numRoots(n,a,b,L),c_correct))\r\n%%\r\n[n,a,b,L] = deal(111,11111,111,11);\r\nc_correct = '2909165067998259047855867968538898760414304923603195356244648530663146995238036844997557527483252003388541616691119031429859046897494224647322949977583143171656492664325583077702371945252381217441222592230772490362933347417600378555600078275316289441596896000';\r\nassert(isequal(numRoots(n,a,b,L),c_correct))\r\n%%\r\n[n,a,b,L] = deal(1234,56789,1011,12);\r\nc = numRoots(n,a,b,L);\r\ns = [sum(c) histc(c,50:55) sum(c\u003c58) sum(c\u003e47)]\r\ns_correct = [127146 260 249 257 231 226 247 2428 2428];\r\nassert(isequal(s,s_correct))\r\n%%\r\nfiletext = fileread('numRoots.m');\r\nnot_allowed = contains(filetext, 'persistent') || contains(filetext, 'global') || contains(filetext, 'regexp') || contains(filetext, 'eval') || contains(filetext, 'assignin');\r\nassert(~not_allowed)\r\nassert(mtree(filetext).count\u003c=150)","published":true,"deleted":false,"likes_count":0,"comments_count":1,"created_by":255988,"edited_by":255988,"edited_at":"2023-02-07T05:06:35.000Z","deleted_by":null,"deleted_at":null,"solvers_count":1,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-01-18T14:33:46.000Z","updated_at":"2023-02-07T05:06:35.000Z","published_at":"2023-02-02T11:44:46.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eOne of the tasks that Matlab is very good at, is in solving systems of linear equations. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn this problem we shall tackle a system of linear \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Diophantine_equation\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eDiophantine Equations\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e in which the roots are limited to certain range.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eGiven the number of variables \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003en\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003epositive integers, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ea\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eb\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e and a root limit, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003eL\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, create the function, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enumRoots(n,a,b,L)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e, that outputs the number of posible integer root sets of the following system of equations:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1+x_2+x_3+x_4...+x_{n-2}+x_{n-1}+x_n=a\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1-x_2+x_3-x_4...+(-1)^{n-1}x_{n-2}+(-1)^{n}x_{n-1}+(-1)^{n+1}x_n=b\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e        with:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1,x_2,x_3...x_{n-1},x_n \\\\ge L\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eFor example, if \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(n,a,b,L) = (2,10,4,1)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, the system of equations: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1+x_2=10\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1-x_2 = 4\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, where \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1,x_2\\\\ge 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, has only one root set, namely: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(x_1,x_2)=(7,3)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. Therefore, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enumRoots(2,10,4,1)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e=\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIf \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(n,a,b,L) = (4,20,6,3)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, two of the possible roots of:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1+x_2+x_3+x_4=20\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e; and\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"true\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1-x_2+x_3-x_4=6\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                where:  \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ex_1,x_2,x_3,x_4 \\\\ge 3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eare \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(x_1,x_2,x_3,x_4)=(5,4,8,3)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003eand \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(x_1,x_2,x_3,x_4)=(10,3,3,4)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. In fact, there are \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e16\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e possible root sets. Therefore, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enumRoots(4,20,6,3)\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e=\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e 1\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e6\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThere are no possible roots for \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(n,a,b,L) = (10,55,10,-10)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e(n,a,b,L) = (42,10,2,100)\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, therefore in these cases the function should return: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:rFonts w:cs=\\\"monospace\\\"/\u003e\u003c/w:rPr\u003e\u003cw:t\u003enumRoots = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e-------------------------\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eNOTE: \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003eAs an added challenge, \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eonly those solutions with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/content/cody/about.html\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eCody program size\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e of less than or equal to \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e200\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e will be accepted.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":58847,"title":"Possible American Football Scoring Combinations","description":"With inspiration from Problem 56195. Possible Ruby Scores, given the final score for two teams in an American football game (score1, score2), return two matrices (team1_combs, team2_combs) representing all possible combinations of scores by each team (Official NFL Scoring Plays). \r\nIn American football, the 7 possible ways of scoring are:\r\nTouchdown - 6 points\r\nTry after touchdown - 1 point\r\nTry after touchdown - 2 points\r\nSafety - 2 points\r\nField Goal - 3 points\r\nDefensive safety on try after touchdown (i.e. conversion safety) - 1 points\r\nDefensive \"touchdown\" on try after touchdown - 2 points\r\nValues for score1 and score2 include all natural numbers (realistically \u003c100), and the resulting matrices will have dimensions n1 x 7 and n2 x 7, where n1, n2 are the total number of possible scoring combinations for each team, respectively. The order of the elements in the rows for team1_combs and team2_combs should follow the above list.\r\n*** Attention - There is added complexity in the scoring possibilities in that the number of tries after touchdowns (also known as extra points or point after attempts) must be less than or equal to number of touchdowns. This constraint is further complicated by the possibility of defensive scores on tries after touchdowns, in which the possible number of scores is constrained by the touchdowns scored by the opposing team.\r\nExample\r\n[t1_combs, t2_combs] = amerfootballscores(5, 8)\r\n\r\nt1_combs =\r\n\r\n     0     0     0     0     1     0     1\r\n     0     0     0     1     1     0     0\r\n     0     0     0     2     0     1     0\r\n\r\n\r\nt2_combs =\r\n\r\n     0     0     0     1     2     0     0\r\n     0     0     0     4     0     0     0\r\n     1     0     0     1     0     0     0\r\n     1     0     1     0     0     0     0\r\nThis system is known as a linear Diophantine equation, an equation where only the integer--and in this case only nonnegative--solutions are of interest.\r\nEach team's score can be modeled by the following equation:\r\n6(tds) + 1(off. 1 pt. pats) + 2(off. 2 pt. pats) + 2(safs) + 3(fgs) + 1(def. 1 pt. pats) + 2(def. 2 pt. pats) = score\r\n    \u003e\u003e\u003e with the added constraints:\r\n                off. 1 pt. pats + off. 2 pt. pats \u003c= tds\r\n                def. 1pt. pats + def. 2pt. pats \u003c= tds for opponent","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.44px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: none solid rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 970.625px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 485.312px; transform-origin: 407px 485.312px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eWith inspiration from \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/56195-possible-rugby-scores\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eProblem 56195. Possible Ruby Scores\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, given the final score for two teams in an American football game (score1, score2), return two matrices (team1_combs, team2_combs) representing all possible combinations of scores by each team (\u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://operations.nfl.com/the-rules/nfl-video-rulebook/scoring-plays/\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eOfficial NFL Scoring Plays\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e). \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eIn American football, the 7 possible ways of scoring are:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003col style=\"block-size: 143.062px; font-family: Helvetica, Arial, sans-serif; list-style-type: decimal; margin-block-end: 20px; margin-block-start: 10px; margin-bottom: 20px; margin-top: 10px; perspective-origin: 391px 71.525px; transform-origin: 391px 71.5312px; margin-top: 10px; margin-bottom: 20px; \"\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2125px; text-align: left; transform-origin: 363px 10.2188px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eTouchdown - 6 points\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2125px; text-align: left; transform-origin: 363px 10.2188px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eTry after touchdown - 1 point\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2125px; text-align: left; transform-origin: 363px 10.2188px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eTry after touchdown - 2 points\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2125px; text-align: left; transform-origin: 363px 10.2188px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eSafety - 2 points\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2125px; text-align: left; transform-origin: 363px 10.2188px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eField Goal - 3 points\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2125px; text-align: left; transform-origin: 363px 10.2188px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eDefensive safety on try after touchdown (i.e. conversion safety) - 1 points\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003cli style=\"block-size: 20.4375px; display: list-item; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-start: 56px; margin-left: 56px; margin-top: 0px; perspective-origin: 363px 10.2125px; text-align: left; transform-origin: 363px 10.2188px; white-space-collapse: preserve; margin-left: 56px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-inline-start: 0px; margin-left: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eDefensive \"touchdown\" on try after touchdown - 2 points\u003c/span\u003e\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e\u003cdiv style=\"block-size: 63px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 31.5px; text-align: left; transform-origin: 384px 31.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eValues for score1 and score2 include all natural numbers (realistically \u0026lt;100), and the resulting matrices will have dimensions n1 x 7 and n2 x 7, where n1, n2 are the total number of possible scoring combinations for each team, respectively. The order of the elements in the rows for team1_combs and team2_combs should follow the above list.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 84px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 42px; text-align: left; transform-origin: 384px 42px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e*** Attention - There is added complexity in the scoring possibilities in that the number of tries after touchdowns (also known as extra points or point after attempts) must be less than or equal to number of touchdowns. This constraint is further complicated by the possibility of defensive scores on tries after touchdowns, in which the possible number of scores is constrained by the touchdowns scored by the \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; text-decoration: underline; text-decoration-line: underline; \"\u003eopposing team\u003c/span\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-style: italic; \"\u003e.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"font-weight: 700; \"\u003eExample\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgb(247, 247, 247); block-size: 306.562px; border-bottom-left-radius: 4px; border-bottom-right-radius: 4px; border-end-end-radius: 4px; border-end-start-radius: 4px; border-start-end-radius: 4px; border-start-start-radius: 4px; border-top-left-radius: 4px; border-top-right-radius: 4px; margin-block-end: 10px; margin-block-start: 10px; margin-bottom: 10px; margin-inline-end: 3px; margin-inline-start: 3px; margin-left: 3px; margin-right: 3px; margin-top: 10px; perspective-origin: 404px 153.275px; transform-origin: 404px 153.281px; margin-left: 3px; margin-top: 10px; margin-bottom: 10px; margin-right: 3px; \"\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e[t1_combs, t2_combs] = amerfootballscores(5, 8)\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003et1_combs =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     0     0     0     0     1     0     1\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     0     0     0     1     1     0     0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     0     0     0     2     0     1     0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003et2_combs =\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     0     0     0     1     2     0     0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     0     0     0     4     0     0     0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     1     0     0     1     0     0     0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"background-color: rgba(0, 0, 0, 0); block-size: 20.4375px; border-bottom-left-radius: 0px; border-bottom-right-radius: 0px; border-end-end-radius: 0px; border-end-start-radius: 0px; border-inline-end-color: rgb(233, 233, 233); border-inline-end-style: solid; border-inline-end-width: 0.8px; border-inline-start-color: rgb(233, 233, 233); border-inline-start-style: solid; border-inline-start-width: 0.8px; border-left-color: rgb(233, 233, 233); border-left-style: solid; border-left-width: 0.8px; border-right-color: rgb(233, 233, 233); border-right-style: solid; border-right-width: 0.8px; border-start-end-radius: 0px; border-start-start-radius: 0px; border-top-left-radius: 0px; border-top-right-radius: 0px; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; min-block-size: 18px; min-height: 18px; padding-inline-start: 4px; padding-left: 4px; perspective-origin: 404px 10.2125px; text-wrap: nowrap; transform-origin: 404px 10.2188px; \"\u003e\u003cspan style=\"block-size: auto; border-inline-end-color: rgb(0, 0, 0); border-inline-end-style: none; border-inline-end-width: 0px; border-inline-start-color: rgb(0, 0, 0); border-inline-start-style: none; border-inline-start-width: 0px; border-left-color: rgb(0, 0, 0); border-left-style: none; border-left-width: 0px; border-right-color: rgb(0, 0, 0); border-right-style: none; border-right-width: 0px; display: inline; margin-inline-end: 45px; margin-right: 45px; min-block-size: 0px; min-height: 0px; padding-inline-start: 0px; padding-left: 0px; perspective-origin: 0px 0px; tab-size: 4; transform-origin: 0px 0px; white-space-collapse: preserve; margin-right: 45px; \"\u003e\u003cspan style=\"margin-inline-end: 0px; margin-right: 0px; \"\u003e     1     0     1     0     0     0     0\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 42px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 10px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 10px; perspective-origin: 384px 21px; text-align: left; transform-origin: 384px 21px; white-space-collapse: preserve; margin-left: 4px; margin-top: 10px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eThis system is known as a linear \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Diophantine_equation\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"\"\u003eDiophantine equation\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e, an equation where only the integer--and in this case only nonnegative--solutions are of interest.\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003eEach team's score can be modeled by the following equation:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e6(tds) + 1(off. 1 pt. pats) + 2\u003c/span\u003e\u003cspan style=\"border-block-end-style: solid; border-block-end-width: 0.8px; border-bottom-style: solid; border-bottom-width: 0.8px; \"\u003e(\u003c/span\u003e\u003cspan style=\"\"\u003eoff. 2 pt. pats\u003c/span\u003e\u003cspan style=\"border-block-end-style: solid; border-block-end-width: 0.8px; border-bottom-style: solid; border-bottom-width: 0.8px; \"\u003e)\u003c/span\u003e\u003cspan style=\"\"\u003e + 2(safs) + 3(fgs) + 1(def. 1 pt. pats) + 2(def. 2 pt. pats) = score\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e    \u0026gt;\u0026gt;\u0026gt; with the added constraints:\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                off. 1 pt. pats + off. 2 pt. pats \u0026lt;= tds\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 0px 0px; transform-origin: 0px 0px; \"\u003e\u003cspan style=\"\"\u003e                def. 1pt. pats + def. 2pt. pats \u0026lt;= tds for opponent\u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function [t1_combs,t2_combs] = amerfootballscores(s1,s2)  \r\nt1_combs = [];  \r\nt2_combs = [];\r\n\r\nend","test_suite":"%%\r\ns1 = 5;\r\ns2 = 8;\r\nteam1 = [0     0     0     0     1     0     1;\r\n         0     0     0     1     1     0     0;\r\n         0     0     0     2     0     1     0];\r\nteam2 = [0     0     0     1     2     0     0;\r\n         0     0     0     4     0     0     0;\r\n         1     0     0     1     0     0     0;\r\n         1     0     1     0     0     0     0];\r\n[t1_combs,t2_combs] = amerfootballscores(s1,s2);\r\nassert(isequal([t1_combs;t2_combs],[team1;team2]))\r\n\r\n%%\r\ns1 = 0;\r\ns2 = 28;\r\nteam1 = [0     0     0     0     0     0     0];\r\nteam2 = [0     0     0     2     8     0     0;\r\n         0     0     0     5     6     0     0;\r\n         0     0     0     8     4     0     0;\r\n         0     0     0    11     2     0     0;\r\n         0     0     0    14     0     0     0;\r\n         1     0     0     2     6     0     0;\r\n         1     0     0     5     4     0     0;\r\n         1     0     0     8     2     0     0;\r\n         1     0     0    11     0     0     0;\r\n         1     0     1     1     6     0     0;\r\n         1     0     1     4     4     0     0;\r\n         1     0     1     7     2     0     0;\r\n         1     0     1    10     0     0     0;\r\n         1     1     0     0     7     0     0;\r\n         1     1     0     3     5     0     0;\r\n         1     1     0     6     3     0     0;\r\n         1     1     0     9     1     0     0;\r\n         2     0     0     2     4     0     0;\r\n         2     0     0     5     2     0     0;\r\n         2     0     0     8     0     0     0;\r\n         2     0     1     1     4     0     0;\r\n         2     0     1     4     2     0     0;\r\n         2     0     1     7     0     0     0;\r\n         2     0     2     0     4     0     0;\r\n         2     0     2     3     2     0     0;\r\n         2     0     2     6     0     0     0;\r\n         2     1     0     0     5     0     0;\r\n         2     1     0     3     3     0     0;\r\n         2     1     0     6     1     0     0;\r\n         2     1     1     2     3     0     0;\r\n         2     1     1     5     1     0     0;\r\n         2     2     0     1     4     0     0;\r\n         2     2     0     4     2     0     0;\r\n         2     2     0     7     0     0     0;\r\n         3     0     0     2     2     0     0;\r\n         3     0     0     5     0     0     0;\r\n         3     0     1     1     2     0     0;\r\n         3     0     1     4     0     0     0;\r\n         3     0     2     0     2     0     0;\r\n         3     0     2     3     0     0     0;\r\n         3     0     3     2     0     0     0;\r\n         3     1     0     0     3     0     0;\r\n         3     1     0     3     1     0     0;\r\n         3     1     1     2     1     0     0;\r\n         3     1     2     1     1     0     0;\r\n         3     2     0     1     2     0     0;\r\n         3     2     0     4     0     0     0;\r\n         3     2     1     0     2     0     0;\r\n         3     2     1     3     0     0     0;\r\n         3     3     0     2     1     0     0;\r\n         4     0     0     2     0     0     0;\r\n         4     0     1     1     0     0     0;\r\n         4     0     2     0     0     0     0;\r\n         4     1     0     0     1     0     0;\r\n         4     2     0     1     0     0     0;\r\n         4     2     1     0     0     0     0;\r\n         4     4     0     0     0     0     0];\r\n[t1_combs,t2_combs] = amerfootballscores(s1,s2);\r\nassert(isequal([t1_combs;t2_combs],[team1;team2]))\r\n\r\n%%\r\ns1 = 37;\r\ns2 = 49;\r\n[t1_combs,t2_combs] = amerfootballscores(s1,s2);\r\n% Matrices are rather large, so test case is dependent on the size of each output matrix, \r\n% where the number of rows will equate to the total number of scoring combinations\r\n[t1_rows,t1_cols] = size(t1_combs);\r\n[t2_rows,t2_cols] = size(t2_combs);\r\nassert(isequal([t1_rows,t2_rows],[2894,5625]))\r\n\r\n%%\r\ns1 = 37;\r\ns2 = 49;\r\n[t1_combs,t2_combs] = amerfootballscores(s1,s2);\r\n% Matrices are rather large, so test case is dependent on the sums of each column. These\r\n% values don't have any practical application.\r\nt1_sums = [6724, 2210, 1774, 10752, 6776, 7566, 5789];\r\nt2_sums = [19476, 6398, 5272, 28518, 18222, 11185, 9470];\r\nassert(isequal([t1_sums;t2_sums],[sum(t1_combs);sum(t2_combs)]))","published":true,"deleted":false,"likes_count":0,"comments_count":8,"created_by":3499438,"edited_by":3499438,"edited_at":"2023-08-17T23:02:26.000Z","deleted_by":null,"deleted_at":null,"solvers_count":5,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2023-08-09T17:04:20.000Z","updated_at":"2026-02-23T23:47:16.000Z","published_at":"2023-08-10T05:17:56.000Z","restored_at":null,"restored_by":null,"spam":null,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWith inspiration from \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/56195-possible-rugby-scores\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eProblem 56195. Possible Ruby Scores\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, given the final score for two teams in an American football game (score1, score2), return two matrices (team1_combs, team2_combs) representing all possible combinations of scores by each team (\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://operations.nfl.com/the-rules/nfl-video-rulebook/scoring-plays/\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eOfficial NFL Scoring Plays\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e). \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eIn American football, the 7 possible ways of scoring are:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTouchdown - 6 points\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTry after touchdown - 1 point\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTry after touchdown - 2 points\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eSafety - 2 points\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eField Goal - 3 points\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDefensive safety on try after touchdown (i.e. conversion safety) - 1 points\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"ListParagraph\\\"/\u003e\u003cw:numPr\u003e\u003cw:numId w:val=\\\"2\\\"/\u003e\u003c/w:numPr\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eDefensive \\\"touchdown\\\" on try after touchdown - 2 points\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eValues for score1 and score2 include all natural numbers (realistically \u0026lt;100), and the resulting matrices will have dimensions n1 x 7 and n2 x 7, where n1, n2 are the total number of possible scoring combinations for each team, respectively. The order of the elements in the rows for team1_combs and team2_combs should follow the above list.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e*** Attention - There is added complexity in the scoring possibilities in that the number of tries after touchdowns (also known as extra points or point after attempts) must be less than or equal to number of touchdowns. This constraint is further complicated by the possibility of defensive scores on tries after touchdowns, in which the possible number of scores is constrained by the touchdowns scored by the \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eopposing team\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003e.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eExample\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[[t1_combs, t2_combs] = amerfootballscores(5, 8)\\n\\nt1_combs =\\n\\n     0     0     0     0     1     0     1\\n     0     0     0     1     1     0     0\\n     0     0     0     2     0     1     0\\n\\n\\nt2_combs =\\n\\n     0     0     0     1     2     0     0\\n     0     0     0     4     0     0     0\\n     1     0     0     1     0     0     0\\n     1     0     1     0     0     0     0]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eThis system is known as a linear \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Diophantine_equation\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eDiophantine equation\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e, an equation where only the integer--and in this case only nonnegative--solutions are of interest.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eEach team's score can be modeled by the following equation:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e6(tds) + 1(off. 1 pt. pats) + 2(off. 2 pt. pats) + 2(safs) + 3(fgs) + 1(def. 1 pt. pats) + 2(def. 2 pt. pats) = score\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e    \u0026gt;\u0026gt;\u0026gt; with the added constraints:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                off. 1 pt. pats + off. 2 pt. pats \u0026lt;= tds\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e                def. 1pt. pats + def. 2pt. pats \u0026lt;= tds for opponent\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"},{"id":2737,"title":"Prouhet–Tarry–Escott (basic)","description":"Inspired by \u003chttp://www.mathworks.com/matlabcentral/cody/problems/660-find-a-subset-that-divides-the-vector-into-equal-halves problem 660.\u003e\r\n\r\nGiven n return two disjoint sets of integers _A_ and _B_ with same cardinality having following property:\r\n\r\n\u003c\u003chttps://i.imgur.com/gSW7nWy.png\u003e\u003e\r\n\r\nfor i = 1:n\r\n\r\nTry to minimize sets cardinality. ","description_html":"\u003cp\u003eInspired by \u003ca href = \"http://www.mathworks.com/matlabcentral/cody/problems/660-find-a-subset-that-divides-the-vector-into-equal-halves\"\u003eproblem 660.\u003c/a\u003e\u003c/p\u003e\u003cp\u003eGiven n return two disjoint sets of integers \u003ci\u003eA\u003c/i\u003e and \u003ci\u003eB\u003c/i\u003e with same cardinality having following property:\u003c/p\u003e\u003cimg src = \"https://i.imgur.com/gSW7nWy.png\"\u003e\u003cp\u003efor i = 1:n\u003c/p\u003e\u003cp\u003eTry to minimize sets cardinality.\u003c/p\u003e","function_template":"function [A, B] = prouhet(n)\r\n  A = 1:n;\r\n  B = -A;\r\nend","test_suite":"%%\r\nn = 1;\r\n[A, B] = prouhet(n);\r\nassert(isequal(A, round(A)) \u0026\u0026 isequal(B, round(B)))\r\nassert(isempty(intersect(A, B)));\r\nassert(isequal(numel(A), numel(B), numel(unique(A)), numel(unique(B))));\r\nassert(isequal(sum(A(:).^(1:n),1), sum(B(:).^(1:n),1)));\r\ndisp(sprintf('Each set has %i elements.', numel(A)))\r\n%%\r\nn = 2;\r\n[A, B] = prouhet(n);\r\nassert(isequal(A, round(A)) \u0026\u0026 isequal(B, round(B)))\r\nassert(isempty(intersect(A, B)));\r\nassert(isequal(numel(A), numel(B), numel(unique(A)), numel(unique(B))));\r\nassert(isequal(sum(A(:).^(1:n),1), sum(B(:).^(1:n),1)));\r\ndisp(sprintf('Each set has %i elements.', numel(A)))\r\n%%\r\nn = 5;\r\n[A, B] = prouhet(n);\r\nassert(isequal(A, round(A)) \u0026\u0026 isequal(B, round(B)))\r\nassert(isempty(intersect(A, B)));\r\nassert(isequal(numel(A), numel(B), numel(unique(A)), numel(unique(B))));\r\nassert(isequal(sum(A(:).^(1:n),1), sum(B(:).^(1:n),1)));\r\ndisp(sprintf('Each set has %i elements.', numel(A)))\r\n%%\r\nn = 7;\r\n[A, B] = prouhet(n);\r\nassert(isequal(A, round(A)) \u0026\u0026 isequal(B, round(B)))\r\nassert(isempty(intersect(A, B)));\r\nassert(isequal(numel(A), numel(B), numel(unique(A)), numel(unique(B))));\r\nassert(isequal(sum(A(:).^(1:n),1), sum(B(:).^(1:n),1)));\r\ndisp(sprintf('Each set has %i elements.', numel(A)))\r\n%%\r\n%n = 9;\r\n%[A, B] = prouhet(n);\r\n%assert(isequal(A, round(A)) \u0026\u0026 isequal(B, round(B)))\r\n%assert(isempty(intersect(A, B)));\r\n%assert(isequal(numel(A), numel(B), numel(unique(A)), numel(unique(B))));\r\n%assert(isequal(sum(uint64(A(:)).^uint64(1:n)), sum(uint64(A(:)).^uint64(1:n))));\r\n%disp(sprintf('Each set has %i elements.', numel(A)))\r\n%if numel(A) \u003c=20\r\n%    disp('A:')\r\n%    disp(A)\r\n%    disp('B:')\r\n%    disp(B)\r\n%end\r\n%%\r\n% test info\r\n%\r\n% larger n will be added later\r\n%\r\n% scoring function will be added later as well\r\n% scoring will be entirely based on size of output: smaller output == better score\r\n% something like this:\r\n%\r\n% score = 0;\r\n% for n = 1:25\r\n%   [A, B] = prouhet(n)\r\n%   assert(...);\r\n%   score = score + numel(A);\r\n% end\r\n","published":true,"deleted":false,"likes_count":0,"comments_count":2,"created_by":14358,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":7,"test_suite_updated_at":"2016-10-08T00:11:00.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2014-12-08T09:56:20.000Z","updated_at":"2020-07-22T23:41:27.000Z","published_at":"2016-10-07T08:07:12.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/image\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/media/image1.JPEG\"}],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eInspired by\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://www.mathworks.com/matlabcentral/cody/problems/660-find-a-subset-that-divides-the-vector-into-equal-halves\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eproblem 660.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven n return two disjoint sets of integers\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eA\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e and\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:i/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eB\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e with same cardinality having following property:\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:customXml w:element=\\\"image\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"height\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"width\\\" w:val=\\\"-1\\\"/\u003e\u003cw:attr w:name=\\\"relationshipId\\\" w:val=\\\"rId1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003c/w:customXml\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003efor i = 1:n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTry to minimize sets cardinality.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray 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