tailieunhanh - Báo cáo toán học: "Dynamic One-Pile Blocking Nim"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Dynamic One-Pile Blocking Nim | Dynamic One-Pile Blocking Nim Achim Flammenkamp Mathematisierung Universităt Bielefeld Federal Republic of Germany POB 100131 achim@ Arthur Holshouser 3600 Bullard St. Charlotte NC USA Harold Reiter Department of Mathematics University of North Carolina Charlotte Charlotte NC 28223 USA hbreiter@ Submitted Jun 3 2002 Accepted Apr 18 2003 Published May 20 2003 MR Subject Classifications 11B37 11B39 05A10 Abstract The purpose of this paper is to solve a class of combinatorial games consisting of one-pile counter pickup games for which the number of counters that can be removed on each successive turn changes during the play of the game. Both the minimum and the maximum number of counters that can be removed is dependent upon the move number. Also on each move the opposing player can block some of the moving player s options. This number of blocks also depends upon the move number. There is great interest in generalizations and modifications of simple deterministic two-player take-away-games for a nice survey see chapter 4 of 1 . We discuss here a modification where the player-not-to-move may effect the options of the other player. Modifications of this type have been called Muller twists in the literature. See 4 . In 3 we discuss games in which the number of counters that can be removed depends on the number removed in the previous move. THE ELECTRONIC JOURNAL OF COMBINATORICS 10 2003 N4 1 We begin with some notation. The set of integers is denoted by Z the positive integers by N and the nonnegative integers by No. If a b E Z with a b then a b denotes x E Z a x b . Rules of the Game We are given three sequences ci E N0 ieN mi E N ieN and Mi E N ieN which satisfy the following conditions di E N Ci Ci 1 1 and ui Mi mi ci for each i E N and di E N 0 ui ui 1. 2 These two conditions imply that Mi mi E N0 ieN is a nondecreasing sequence. There are two players and a pile of counters. These two players alternate removing counters from the single