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Báo cáo toán học: "One Pile Nim with Arbitrary Move Function"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: One Pile Nim with Arbitrary Move Function | One Pile Nim with Arbitrary Move Function Arthur Holshouser 3600 Bullard St. Charlotte NC USA Harold Reiter Department of Mathematics University of North Carolina Charlotte Charlotte NC 28223 USA hbreiter@email.uncc.edu Submitted Feb 8 2002 Accepted Jun 11 2003 Published Jul 27 2003 MR Subject Classifications 91A46 11B37 Abstract This paper solves a class of combinatorial games consisting of one-pile counter pickup games for which the maximum number of counters that can be removed on each successive move equals f t where t is the previous move size and f is an arbitrary function. The purpose of this paper is to solve a class of combinatorial games consisting of one-pile counter pickup games for which the maximum number of counters that can be removed on each successive move changes during the play of the game. Two players alternate removing a positive number of counters from the pile. An ordered pair N x of positive integers is called a position. The number N represents the size of the pile of counters and x represents the greatest number of counters that can be removed on the next move. A function f Z - Z is given which determines the maximum size of the next move in terms of the current move size. Thus a move in a game is an ordered pair of positions N x N k f k where 1 k min N x . The game ends when there are no counters left and the winner is the last player to move in a game. In this paper we will consider f Z Z to be completely arbitrary. That is we place no restrictions on f. This paper extends a previous paper by the authors 2 which in turn extended two other papers 1 and 3 . The paper by Epp and Ferguson 1 assumed f is non-decreasing and the paper 3 assumed f is non-decreasing and f n n. Our previous paper 2 assumed more restrictive conditions on f including as a special case all f Z Z that satisfy f n 1 f n 1. THE ELECTRONIC JOURNAL OF COMBINATORICS 10 2003 N7 1 The main theorem of this paper will also allow the information concerning the strategy of a .