Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: The game of End-Nim. | The game of End-Nim Michael H. Albert Dept. of Computer Science Otago University Dunedin New Zealand malbert@ Richard J. Nowakowski Dept. of Mathematics Statistics Dalhousie University Halifax . Canada B3J 3H5 rjn@ Abstract In the game of End-Nim two players take turns in removing one or more boxes from a string of non-empty stacks. At each move boxes may only be taken from the two stacks which form the ends of the string unless only one stack remains . We give a solution for both impartial and partizan versions of the game and explain the significance of the mystic hieroglyphs AMS subject classifications 91A46 primary 05B99 secondary . Submitted Aug 20 2000 Accepted Feb 6 2001. 1 Introduction Lorraine and Roger are fork-lift operators with a penchant for combinatorial games. Many of the warehouses from which they need to remove boxes have the boxes in stacks with the stacks arranged in a row. Only boxes belonging to the stacks at the end of a row are accessible but the fork-lifts are sufficiently powerful that they can move an entire stack of boxes if necessary. The game which Lorraine and Roger play most often is won by the player who removes the last box from a row of stacks. Usually they play fair and allow each other to remove boxes from either end. Sometimes in particularly narrow warehouses each of them is assigned a specific end to work from. We have dubbed the game which they play End-Nim and the two versions are of course the impartial and partizan versions of the game. Formally an End-Nim position is a sequence of positive integers and the legal moves in the impartial version are to reduce Partially supported by a grant from NSERC and the Beverley Trust of the Department of Mathematics and Statistics at Otago University. THE ELECTRONIC JOURNAL OF COMBINATORICS 8 no. 2 2001 R1 1 the first or last element of the sequence by at least one or to delete it entirely. In the partizan version one player is restricted to work at
