NonCoplanar Points
Description  Submit An Entry  Standings  Final Report 
The ProblemOn March 20, 2014 Ed Pegg reported on mathpuzzle.com, "There are 38 ways to pick 13 points from a 5 Ã— 5 Ã— 5 grid so that no four points are in a plane. There is a unique solution for 8 points from 3 Ã— 3 Ã— 3, and 232 solutions for 10 points from 4 Ã— 4 Ã— 4." For example, no 4 of these points from a 4 Ã— 4 Ã— 4 grid are coplanar: (1,3,1), (3,2,2), (3,3,3), (1,0,3), (0,3,2), (0,1,3), (0,0,1), (2,0,1), (1,1,0), (3,2,0) What is the largest number of points you can choose from an n Ã— n Ã— n grid so that no 4 are coplanar? The ContestFor each value of n, where n is one of the first 25 primes, submit the largest set of points you can find in an n Ã— n Ã— n grid so that no 4 points are in a plane. The first 25 primes are { 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 }. See How to Enter, below, for instructions on how to submit a set of points. For each value of n you can submit more than one set of points if you wish, but only your largest will count. See The Scoring System, below, to learn how we determine the winner. The PrizesFirst prize is a $500 credit at Bathsheba Sculpture. Second prize is a $100 credit. How to EnterJust paste your points into the large box on the Submit page and click the Submit Entry button. Format your sets of points as follows:
We will infer the size of your grid from the largest coordinate. Specifically, if the largest coordinate is x, we take n to be x + 1.
For example, if you were permitted to submit points for a
4 Ã— 4 Ã— 4 grid, you could submit the
example
from above by entering:
Do not create more than one account for submitting solutions. This is important. Do not create more than one account for submitting solutions. The Scoring SystemThe entrant with the highest "contest score" wins. Here's how we calculate your contest score:
If two entrants have the same contest score, we break the tie by giving preference to the entrant whose last improvement was submitted least recently. Let's walk through a simplified example. Suppose that the contest asks you to submit solutions only for n = 4, 6 and 8. Further suppose that we have 3 entrants (Oscar, Edgar and Hugo) and that the sizes of their best solutions are as follows:
Their raw scores are therefore:
We note the largest raw score for each value of n, as follows:
Finally, we compute the subscores and contest score for each entrant:
Edgar has the highest contest score and therefore wins. Getting Your Questions AnsweredFirst, check the FAQ section below. If you can't find the information you need there, send your question to the discussion group. If your question is of a personal nature, and not of general interest, send an email directly to Al Zimmermann. The Discussion GroupIf you think you might enter the contest, you should join the contest discussion group. You can join either by sending a blank email here or by visiting the group on Yahoo!. The discussion group serves two purposes. First, it allows contestants to ask for clarifications to the rules. Be aware that sometimes these requests result in changes to the rules, and the first place those changes are announced is in the discussion group. Second, the discussion group allows contestants to interact with each other regarding programming techniques, results and anything else relevant to the contest. You do not need to have a Yahoo account to join the group. My Lawyer Would Want Me To Say ThisI reserve the right to discontinue the contest at any time. I reserve the right to disqualify any entry or entrant for any reason that suits me. I reserve the right to interpret the rules as I see fit. I reserve the right to change the contest rules in midcontest. In all matters contestrelated, my word is final. Frequently Asked Questions
