Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Another Form of Matrix Nim. | Another Form of Matrix Nim Thomas S. Ferguson Mathematics Department UCLA Los Angeles CA 90095 USA tom@ Submitted February 28 2000 Accepted February 6 2001. MR Subject Classifications 91A46 Abstract A new form of 2-dimensional nim is investigated. Positions are rectangular matrices of non-negative integers. Moves consist of chosing a positive integer and a row or column and subtracting the integer from every element of the chosen row or column. Last to move wins. The 2 X 1 case is just Wythoff s Game. The outcomes of all 2 X 2 positions are found in both the impartial and partizan cases. Some hope is given of being able to solve sums of 2 X 2 games in the partizan case. 1. Introduction. There exist in the literature on combinatorial games several generalizations to two dimensions of the game of nim. The earliest is the game of matrix nim found in Holladay 1958 . In this game call it Nn there is an m X n rectangular board of piles of counters. A move consists of either 1 removing any number of counters in the piles in one row or 2 removing any number of counters from any of the piles provided one column is left untouched. Last to move wins. The game N1 is the game nim. In the Two-dimensional Nim found on page 313 of Winning Ways Berlekamp et al. 1982 there are a finite number of counters in the non-negative integer lattice of the plane. A move consists of either 1 moving any counter to the left or 2 moving any counter to any position in a lower row. When all counters are on the lowest row the game is nim. This game is used to illustrate transfinite nim values. In the Two-Dimensional Nim due to Eggleston Fraenkel and Rothschild found in Fraenkel 1994 a player may remove any number of counters from any row or any column. This is related to the game of Nimby usually played on a triangular board see for example Fraenkel and Herda 1980 in which a move consists of removing any number of contiguous counters in an arbitrary line. In the game of Shrage 1985 a .
