HMMT 二月 2002 · 团队赛 · 第 9 题

HMMT February 2002 — Team Round — Problem 9

[30] In this problem suppose that s = s . Prove that for each board configuration, the first 1 2 1 player wins with probability strictly greater than . 2

解析

[30] In this problem suppose that s = s . Prove that for each board configuration, the first 1 2 1 player wins with probability strictly greater than . 2 Solution. Let σ and σ denote the sequence of the next twelve die rolls that players 1 and 2 1 2 respectively will make. The outcome of the game is completely determined by the σ . Now player i 1 wins in all cases in which σ = σ , for then each of player 2’s moves bring her piece to a square 1 2 already occupied by player 1’s piece. It is sufficient, therefore, to show that player 1 wins at least half the cases in which σ 6 = σ . But all these cases can be partitioned into disjoint pairs 1 2 { ( σ , σ ) , ( σ , σ ) } , 1 2 2 1 and player 1 wins in at least one case in each pair. For if player 2 wins in the case ( σ , σ ), say 1 2 th on her n turn, the first n elements of σ do not take player 1 beyond space 12, while the first n 1 th elements of σ must take player 2 beyond space 12. Clearly, then, player 1 wins ( σ , σ ) on his n 2 2 1 turn.