Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: Winning Positions in Simplicial Nim. | Winning Positions in Simplicial Nim David Horrocks Department of Mathematics and Statistics University of Prince Edward Island Charlottetown Prince Edward Island Canada C1A 4P3 dhorrocks@ Submitted Jun 9 2009 Accepted May 27 2010 Published Jun 7 2010 Mathematics Subject Classifications 91A05 91A43 91A44 91A46 Abstract Simplicial Nim introduced by Ehrenborg and Steingrimsson is a generalization of the classical two-player game of Nim. The heaps are placed on the vertices of a simplicial complex and a player s move may affect any number of piles provided that the corresponding vertices form a face of the complex. In this paper we present properties of a complex that are equivalent to the P-positions winning positions for the second player being closed under addition. We provide examples of such complexes and answer a number of open questions posed by Ehrenborg and Steingrimsson. 1 Introduction Simplicial Nim as defined by Ehrenborg and Steingrlmsson in 2 is a generalization of the classical game of Nim. It is a combinatorial game for two players who move alternately and as usual the last player able to make a move is the winner. Moreover like Nim Simplicial Nim is played with a number of piles of chips and a legal move consists of removing a positive number of chips. A simplicial complex A on a finite set V is defined to be a collection of subsets of V such that v G A for every v G V and B G A whenever A G A and B c A. The elements of V and A are called vertices or points and faces respectively. A face that is maximal with respect to inclusion is called a facet. To play Simplicial Nim begin with a simplicial complex A and place a pile of chips on each vertex of V. On his turn a player may remove chips from any nonempty set of piles provided that the vertices corresponding to the affected piles form a face of A. Note that a player may remove any number of chips from each of the piles on which he chooses to play and that at least one chip must be removed. .
