tailieunhanh - Báo cáo toán học: "Dynamic Single-Pile Nim Using Multiple Bases"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Dynamic Single-Pile Nim Using Multiple Bases. | Dynamic Single-Pile Nim Using Multiple Bases Arthur Holshouser 3600 Bullard St Charlotte NC 28208 USA Harold Reiter Department of Mathematics University of North Carolina Charlotte Charlotte NC 28223 USA hbreiter@ Submitted Jun 25 2004 Accepted Mar 23 2006 Published Mar 30 2006 Subject classifications 91A46 Abstract In the game G0 two players alternate removing positive numbers of counters from a single pile and the winner is the player who removes the last counter. On the first move of the game the player moving first can remove a maximum of k counters k being specified in advance. On each subsequent move a player can remove a maximum of f n t counters where t was the number of counters removed by his opponent on the preceding move and n is the preceding pile size where f N X N N is an arbitrary function satisfying the condition 1 9t 2 N such that for all n x 2 N f n x f n t x . This note extends our paper 5 that appeared in E-JC. We first solve the game for functions f N X N N that also satisfy the condition 2 8 11 x 2 N f n x 1 f n x 1. Then we state the solution when f N X N N is restricted only by condition 1 and point out that the more general proof is almost the same as the simpler proof. The solutions when t 2 use multiple bases. Introduction Notation 1 N is the set of positive integers and N0 0 u N. Let f N X N N be a function satisfying the condition 1 9t 2 N such that for all n x 2 N f n x f n fix . If x y 2 N0 then is defined by x y x y mod t and x y 2 0 1 2 . t 1 . Thus x y is uniquely specified. Note that 0 1 2 . t 1 is a cyclic group. THE ELECTRONIC JOURNAL OF COMBINATORICS 13 2006 N7 1 The Games. Suppose game Go with move function f is given. Then we dehne the games Gi i 1 2 t 1 where Gi is the same as game Go except the move function f i n x is used in the place of f n x . Example 1 In game Go suppose the moving player is facing a pile size of 10 counters and the preceding pile size was 15 counters. This means his opponent removed 5 .