From a 3x3x3 grid, select 8 points such that no 4 lie in a plane. The solution is unique. (answer below)
The no-3-in-line, in 2-space, is a well known problem (by Dudeney, 1917). http://www.research.att.com/~njas/sequences/A000769 I did find "No-three-in-line-in-3D" by Attila Por, David R. Wood http://citeseer.ist.psu.edu/por04nothreelined.html I haven't found any references to the no-4-in-plane problem, though. I was looking for small grid embeddings of complete graphs with non-crossing edges, and the no-4-in-plane seemed like the simplest criteria for selecting the points. The unique solution for 8 points in the 3x3x3 grid is 000 011 012 101 110 120 201 222. For 2x2x2, 5 points. I haven't yet solved the 4x4x4 case. --Ed Pegg Jr