HMMT 二月 2011 · ALGCALC 赛 · 第 5 题

HMMT February 2011 — ALGCALC Round — Problem 5

Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins, or else indefinitely. If Nathaniel goes first, determine the probability that he ends up winning.

解析

Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins, or else indefinitely. If Nathaniel goes first, determine the probability that he ends up winning. 5 Answer: For 1 ≤ k ≤ 6, let x be the probability that the current player, say A , will win when k 11 the number on the tally at the beginning of his turn is k modulo 7. The probability that the total 1 1 is l modulo 7 after his roll is for each l 6 ≡ k (mod 7); in particular, there is a chance he wins 6 6 immediately. The chance that A will win if he leaves l on the board after his turn is 1 − x . Hence for l 1 ≤ k ≤ 6, ∑ 1 1 x = (1 − x ) + . k l 6 6 1 ≤ l ≤ 6 , l 6 = k ∑ 6 x − s 5 x s k k Letting s = x , this becomes x = + 1 or = − + 1. Hence x = · · · = x , and 6 x = s l k 1 6 k l =1 6 6 6 11 x 6 k for every k . Plugging this in gives = 1, or x = . k 6 11 Since Nathaniel cannot win on his first turn, he leaves Obediah with a number not divisible by 7. 6 5 Hence Obediah’s chance of winning is and Nathaniel’s chance of winning is . 11 11