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A sequence from a 3 X n Chomp game, see the first comment.
1

%I #48 Jan 08 2025 10:29:10

%S 120,400,402,422,424,513,576,583,585,593,605,610,861,888,890,892,904,

%T 1013,1015,1059,1129,1141,1143,1163,1216,1281,1291,1293,1295,1419,

%U 1448,1508,15231525,1537,1561,1723,1747,1824,1868,1870,1875,1889,2003,2010

%N A sequence from a 3 X n Chomp game, see the first comment.

%C Values of r where a P-position (p,q,r) has a pattern. This algorithm is very efficient. When the length of a period and the start matrix are not of interest (only whether r has or does not have a pattern), the algorithm generates the sequence: 120, 400, 402, ..., 10501, 10583, 10585 very quickly.

%C First step: generate all the P-positions, where, for example, n = 15000. Second step: run the algorithm.

%C From _Petros Hadjicostas_, Apr 11 2020: (Start)

%C From A. E. Brouwer's website (see below): "Chomp is a game played on a partially ordered set P with smallest element 0. A move consists of picking an element x of P and removing x and all larger elements from P. Whoever picks 0 loses."

%C "In 3-by-n Chomp, a game position is (p,q,r) with p not less than q and q not less than r.

%C If r = 0 then this is really 2-by-n Chomp, and the P-positions (previous player wins) are those with p = q+1.

%C If r = 1, the only P-positions are (3,1,1) and (2,2,1).

%C If r = 2, the P-positions are those with p = q+2.

%C If r = 3, the P-positions are (6,3,3), (7,4,3) and (5,5,3).

%C If r = 4, the P-positions are (8,4,4), (9,5,4), (10,6,4) and (7,7,4).

%C If r = 5, the P-positions are (10,5,5), (9,6,5) and (a+11,a+7,5) for nonnegative a." (For more details, see his web page.) (End)

%H Emily Bergman, <a href="https://cpb-us-w2.wpmucdn.com/sites.udel.edu/dist/e/3522/files/2015/08/Game-Theory-and-an-Exploration-of-3-x-n-Chomp-190t6o8.pdf">Game theory and an explanation of a 3 x n Chomp! Boards</a>, Senior Mathematics Project, Lynchburg College, 2014. (See also <a href="https://sites.udel.edu/ebergman/">here</a>.)

%H Andries E. Brouwer, <a href="https://www.win.tue.nl/~aeb/games/chomp.html">The game of Chomp</a>.

%H Eva Elduque, <a href="https://www.math.wisc.edu/wiki/images/Chomp_Sol.pdf">The game of CHOMP</a>, Madison Math Circle, Department of Mathematics, UW-Madison.

%H E. J. Friedman and A. S. Landsberg, <a href="https://doi.org/10.1007/978-3-540-73556-4_23">Scaling, renormalization, and universality in combinatorial games: The geometry of Chomp</a>, in: A. Dress, Y. Xu, and B. Zhu (eds), Combinatorial Optimization and Applications, COCOA 2007 (Lecture Notes in Computer Science, vol 4616. Springer, Berlin, Heidelberg).

%H David Gale, <a href="https://www.jstor.org/stable/2319446">A curious Nim-type game</a>, American Mathematical Monthly 81(8) (1974), 876-879.

%H T. Khandhawit and L. Ye, <a href="https://arxiv.org/abs/1101.2718">Chomp on graphs and subsets</a>, arXiv:1101.2718 [math.CO], 2011.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chomp">Chomp</a>.

%H Doron Zeilberger, <a href="https://doi.org/10.1006/aama.2000.0714">Three-rowed CHOMP</a>, Advances in Applied Mathematics 26 (2001), 168-179.

%H Doron Zeilberger, <a href="https://doi.org/10.1080/10236190410001652720">Chomp, recurrences and Chaos(?)</a>, Journal of Difference Equations and Applications 10(13-15) (2004), 1281-1293.

%H Doron Zeilberger, <a href="https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/chompc.html">All the winning bites for a by b Chomp for a and b up to 14 and two computational challenges</a>, Personal Journal of Shalosh B. Ekhad and Doron Zeilberger, 2018; <a href="/A147983/a147983.pdf">Local copy, pdf file only, no active links</a>

%Y Cf. A029899, A029900, A029901, A029902, A029903, A029904, A029905.

%K nonn

%O 1,1

%A Csaba Beretka (bcs183(AT)freemail.hu), Nov 18 2008

%E Name edited by _Petros Hadjicostas_, Apr 11 2020