OFFSET
1,1
COMMENTS
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.
First step: generate all the P-positions, where, for example, n = 15000. Second step: run the algorithm.
From Petros Hadjicostas, Apr 11 2020: (Start)
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."
"In 3-by-n Chomp, a game position is (p,q,r) with p not less than q and q not less than r.
If r = 0 then this is really 2-by-n Chomp, and the P-positions (previous player wins) are those with p = q+1.
If r = 1, the only P-positions are (3,1,1) and (2,2,1).
If r = 2, the P-positions are those with p = q+2.
If r = 3, the P-positions are (6,3,3), (7,4,3) and (5,5,3).
If r = 4, the P-positions are (8,4,4), (9,5,4), (10,6,4) and (7,7,4).
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)
LINKS
Emily Bergman, Game theory and an explanation of a 3 x n Chomp! Boards, Senior Mathematics Project, Lynchburg College, 2014. (See also here.)
Andries E. Brouwer, The game of Chomp.
Eva Elduque, The game of CHOMP, Madison Math Circle, Department of Mathematics, UW-Madison.
E. J. Friedman and A. S. Landsberg, Scaling, renormalization, and universality in combinatorial games: The geometry of Chomp, 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).
David Gale, A curious Nim-type game, American Mathematical Monthly 81(8) (1974), 876-879.
T. Khandhawit and L. Ye, Chomp on graphs and subsets, arXiv:1101.2718 [math.CO], 2011.
Wikipedia, Chomp.
Doron Zeilberger, Three-rowed CHOMP, Advances in Applied Mathematics 26 (2001), 168-179.
Doron Zeilberger, Chomp, recurrences and Chaos(?), Journal of Difference Equations and Applications 10(13-15) (2004), 1281-1293.
Doron Zeilberger, All the winning bites for a by b Chomp for a and b up to 14 and two computational challenges, Personal Journal of Shalosh B. Ekhad and Doron Zeilberger, 2018; Local copy, pdf file only, no active links
CROSSREFS
KEYWORD
nonn
AUTHOR
Csaba Beretka (bcs183(AT)freemail.hu), Nov 18 2008
EXTENSIONS
Name edited by Petros Hadjicostas, Apr 11 2020
STATUS
approved