Problem 42765. Maximize Non-Co-Planar Points in an N-Cube

Created by Richard Zapor

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{"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":[],"contentType":"application/vnd.mathworks.matlab.code.document+xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\"?>\n<w:document xmlns:w=\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\"><w:body><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>This Challenge is to find a set with the maximum number of integer points that create planar surfaces with a maximum of three points from the set. No four points may be co-planar. Given the size N and the number of expected points Q find a set of Q points. Only N=2/Q=5 and N=3/Q=8 will be tested. N=4/Q=10 or N=5/Q=13 are too large to process.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"code\"/></w:pPr><w:r><w:t><![CDATA[N=2 contains 8 points [0,0,0;0,1,0;1,0,0;1,1,0;0,0,1;0,1,1;1,0,1;1,1,1]\nN=3 contains 27 points [0,0,0;0,0,1;0,0,2;...2,2,2]]]></w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Output is a Qx3 matrix of the non-co-planar points.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Reference: The</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"http://68.173.157.131/Contest/Tetrahedra\"><w:r><w:t>March 2016 Al Zimmermann Non-Coplanar contest</w:t></w:r></w:hyperlink><w:r><w:t> is N=primes less than 100. Maximize the number of points in an NxNxN cube with no 4 points in a common plane.</w:t></w:r></w:p><w:p><w:pPr><w:pStyle w:val=\"text\"/></w:pPr><w:r><w:t>Theory: The N=2 and N=3 cases can be processed by brute force if care is taken. Assumption of [0,0,0] greatly reduces number of cases. Solving</w:t></w:r><w:r><w:t> </w:t></w:r><w:hyperlink w:docLocation=\"http://www.mathworks.com/matlabcentral/cody/problems/42762-is-3d-point-set-co-planar\"><w:r><w:t>Cody Co-Planar Check</w:t></w:r></w:hyperlink><w:r><w:t> may improve speed.</w:t></w:r></w:p></w:body></w:document>"},{"partUri":"/matlab/output.xml","contentType":"text/xml","content":"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\" ?><embeddedOutputs><metaData><evaluationState>manual</evaluationState><layoutState>code</layoutState><outputStatus>ready</outputStatus></metaData><outputArray type=\"array\"/><regionArray type=\"array\"/></embeddedOutputs>"}]}

This Challenge is to find a set with the maximum number of integer points that create planar surfaces with a maximum of three points from the set. No four points may be co-planar.
Given the size N and the number of expected points Q find a set of Q points. Only N=2/Q=5 and N=3/Q=8 will be tested. N=4/Q=10 or N=5/Q=13 are too large to process.<pre class="language-matlab">N=2 contains 8 points [0,0,0;0,1,0;1,0,0;1,1,0;0,0,1;0,1,1;1,0,1;1,1,1]
N=3 contains 27 points [0,0,0;0,0,1;0,0,2;...2,2,2]
</pre>Output is a Qx3 matrix of the non-co-planar points.Reference: The <a href = "http://68.173.157.131/Contest/Tetrahedra">March 2016 Al Zimmermann Non-Coplanar contest</a> is N=primes less than 100. Maximize the number of points in an NxNxN cube with no 4 points in a common plane.Theory: The N=2 and N=3 cases can be processed by brute force if care is taken. Assumption of [0,0,0] greatly reduces number of cases. Solving <a href = "http://www.mathworks.com/matlabcentral/cody/problems/42762-is-3d-point-set-co-planar">Cody Co-Planar Check</a> may improve speed.

This Challenge is to find a set with the maximum number of integer points that create planar surfaces with a maximum of three points from the set. No four points may be co-planar.
Given the size N and the number of expected points Q find a set of Q points. Only N=2/Q=5 and N=3/Q=8 will be tested. N=4/Q=10 or N=5/Q=13 are too large to process.

N=2 contains 8 points [0,0,0;0,1,0;1,0,0;1,1,0;0,0,1;0,1,1;1,0,1;1,1,1]
  N=3 contains 27 points [0,0,0;0,0,1;0,0,2;...2,2,2]

Output is a Qx3 matrix of the non-co-planar points.

Reference: The <http://68.173.157.131/Contest/Tetrahedra March 2016 Al Zimmermann Non-Coplanar contest> is N=primes less than 100. Maximize the number of points in an NxNxN cube with no 4 points in a common plane.

Theory: The N=2 and N=3 cases can be processed by brute force if care is taken. Assumption of [0,0,0] greatly reduces number of cases. Solving <http://www.mathworks.com/matlabcentral/cody/problems/42762-is-3d-point-set-co-planar Cody Co-Planar Check> may improve speed.

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Last Solution submitted on Apr 20, 2020

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Problem 42765. Maximize Non-Co-Planar Points in an N-Cube

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Problem 42765. Maximize Non-Co-Planar Points in an N-Cube

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