By Shalosh B. Ekhad and Doron Zeilberger
Exclusively published in the Personal Journal of Shalosh B. Ekhad and Doron Zeilberger
First Written: Aug. 11, 2018
In the classic book, Winning Ways, by Berlekamp, Conway, and Guy they briefly (volume 2, Chapter 18, pp. 598-599) mention the game of Chomp, and present a table of the winning bites for small boards. Here we extend their table to boards with length and width up to 14, and present two computational challenges, that would make the OEIS 500 and 100 dollars richer, respectively.
Acknowledgment: This short article was inspired by Purui Zhang and Lu Yan who just found an intriguing extension to Multiplayer CHOMP . Purui Zhang and Lu Yan are entering freshmen(!) from Fudan and Tsinguha Universities, respectively.
Here is what he says: This was found by an exhaustive retrograde solve of the full ideal lattice below the bar, and verified twice over: re-derived from a second, different box, and counter-signed by an independent implementation with a different iteration order. All validated against brute force and against your published table (Thompson's 8 times 10, your 6 times 13 and 10 times 14). The same computation extended your table considerably. There are exactly eighteen bars with exactly two winning moves in the searched region, and a small theorem organizes them all. The winning bites of any bar form a strict staircase (deeper row, strictly smaller column), which the triple exemplifies. A short note is attached and submitted to the arXiv. It includes the complete bite table for all nineteen multi-move bars, and some curious quasi-periodic structure in where multiple winning moves occur. This includes Beatty-type laws with rotation numbers in Q(sqrt(2)) for the diagonal families of three-row Chomp, which may amuse you given the title of your 2004 paper. Code, both verification logs, and all data is in this GitHub repository I have pushed the project to: my github site
the input file generates the output file.
the input file generates the output file.