My observations from the results in (1) for n1=2, n2=n3 are the numbers not divisible by four and game length = 2*n2 - 1 - is_even(n2) you could try whether this patterns continues for n2=n3>16... Christoph ________________________________________ From: math-fun [math-fun-bounces@mailman.xmission.com] on behalf of Mike Beeler [mikebeeler@verizon.net] Sent: Wednesday, March 16, 2016 2:42 AM To: math-fun Subject: Re: [math-fun] Grabbing Cubes Some further Grabbing Cubes results. The strangeness continues. -- Mike Consider distinct odd prime boxes with (p1=3) < p2 < p3 < 100. There are 23 odd primes between 3 and 100; C(23,2) = 253. For each of the 253 box sizes, 1000 games were played, with random choice to break ties among maximal-yield cubes. In all games for all 253 boxes, the game length = p1(=3) * p2, as in the original problem. None were seen to have varying game length. It seems a box must have minimum dimension > 3 to have the "shortcuts" phenomenon that causes varying game lengths on these box sizes. Then 1000 games, with random tie breaking, were played on boxes with not necessarily distinct and not necessarily prime dimensions. Box dimensions 2 <= n1 <= n2 <= n3 <= 16 were run. This is 680 cases. Results: (1) In 14 box sizes, the 1000 games had constant length, but that length was not n1*n2. The box sizes and the game lengths are: 2 2 2, game length = 2 2 3 3, game length = 5 2 5 5, game length = 9 2 6 6, game length = 10 2 7 7, game length = 13 2 9 9, game length = 17 2 10 10, game length = 18 2 11 11, game length = 21 2 13 13, game length = 25 2 14 14, game length = 26 2 15 15, game length = 29 3 5 5, game length = 11 3 6 7, game length = 16 4 5 5, game length = 13 (2) Over all 680 cases, only the 14 noted above had a constant game length not equal to n1*n2. All others had length=n1*n2, or had varying length. (3) For n1 = 3, most (82) box sizes had constant game length. The 23 others with varying game length were: 3 4 4 3 4 5 3 5 6 3 6 6 3 7 7 3 7 8 3 8 8 3 8 9 3 9 9 3 9 10 3 10 10 3 10 11 3 11 11 3 11 12 3 12 12 3 12 13 3 13 13 3 13 14 3 14 14 3 14 15 3 15 15 3 15 16 3 16 16 (4) For various n1, the following table shows n1, the total number of boxes analyzed, the number with varying game lentgh, the number with constant n1*n2 game length, and the number with constant < n1*n2 game length. This covers all 680 cases, including those in (1)-(3) above. n1 cases varying const=n1*n2 const<n1*n2 2 120 0 109 11 3 105 23 80 2 4 91 35 55 1 5 78 42 36 0 6 66 45 21 0 7 55 45 10 0 8 45 42 3 0 9 36 36 0 0 10 28 28 0 0 11 21 21 0 0 12 15 15 0 0 13 10 10 0 0 14 6 6 0 0 15 3 3 0 0 16 1 1 0 0 The table seems to suggest that larger boxes are more likely to have varying length games, but that may be an artifact of the cutoff (the search limited all dimensions to at most 16). For example, consider: 4 10 10, varying length 4 10 11, varying length 4 10 12, varying length 4 10 13, constant length = 40 4 10 14, constant length = 40 4 10 15, constant length = 40 4 10 16, constant length = 40 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun