HMMT 二月 2002 · 团队赛 · 第 10 题

HMMT February 2002 — Team Round — Problem 10

[30] Exhibit a configuration of the board and a choice of s and s so that s > s , yet the 1 2 1 2 1 second player wins with probability strictly greater than . 2

解析

[30] Exhibit a configuration of the board and a choice of s and s so that s > s , yet the 1 2 1 2 1 second player wins with probability strictly greater than . 2 Solution. Let s = 3 and s = 2 and place an arrow on all the even-numbered squares. In this 1 2 configuration, player 1 can move at most six squares in a turn: up to three from his roll and an additional three if his roll landed him on an arrow. Hence player 1 cannot win on his first or second turn. Player 2, however, wins immediately if she ever lands on an arrow. Thus player 2 has probability 1 / 2 of winning on her first turn, and failing that, she has probability 1 / 2 of winning on her second turn. Hence player 2 wins with probability at least 1 / 2 + (1 / 2)(1 / 2) = 3 / 4.