HMMT 二月 2005 · GEN2 赛 · 第 5 题

英文解析

5. You and I play the following game on an 8 × 8 square grid of boxes: Initially, everybox is empty. On your turn, you choose an empty box and draw an X in it; if anyof the four adjacent boxes are empty, you mark them with an X as well. (Two boxesare adjacent if they share an edge.) We alternate turns, with you moving first, andwhoever draws the last X wins. How many choices do you have for a first move thatwill enable you to guarantee a win no matter how I play?
   Solution: 0
   I can follow a symmetry strategy: whenever you play in the box S , I play in the image
   °
   of S under the 180 rotation about the center of the board. This ensures that the boardwill always be centrally symmetric at the beginning of your turn. Thus, if you play in
   ′
   an empty box S , its symmetric image S is also empty at the beginning of your turn,
   and it remains so after your turn, since the even size of the board ensures that S can
   ′
   be neither equal to nor adjacent to S . In particular, I always have a move available.
   Since the first person without an available move loses, you are guaranteed to lose. Sothe answer is that you have 0 choices for a first move that will guarantee your win.