PUMaC 2016 · 组合（B 组） · 第 5 题

PUMaC 2016 — Combinatorics (Division B) — Problem 5

Alice, Bob, Charlie, Diana, Emma, and Fred sit in a circle, in that order, and each roll a six-sided die. Each person looks at his or her own roll, and also looks at the roll of either the person to the right or to the left, deciding at random. Then, at the same time, Alice, Bob, Charlie, Diana, Emma and Fred each state the expected sum of the dice rolls based on the information they have. All six people say different numbers; in particular, Alice, Bob, Charlie, and Diana say 19, 22, 21, and 23, respectively. Compute the product of the dice rolls.

解析

The sum of the two rolls each person sees is what they say minus 14 (the expected sum of the rolls they don’t see). Since the stated numbers are all different, the sum of the two rolls each person sees is a different number, which means that no two people look at each other’s dice, so everyone looks in the same direction. Assume that Alice looks at Bob, Bob looks at Charlie, and so on. (The other case is identical.) The sum of the two rolls that Alice, Bob, Charlie, and Diana see, respectively, is 5, 8, 7, and 9. If we let a through f be the rolls of Alice through Fred, this means that a + b = 5, b + c = 8, c + d = 7, and d + e = 9. Adding the second and fourth equations and subtracting the first and third gives e − a = 5, meaning that e = 6 and a = 1. Thus, b = 4, c = 4, and d = 3. It remains to determine f , and note that e + f and f + a do not belong to the set { 5 , 7 , 8 , 9 } , for this would violate the condition that everyone says a different number. This rules out everything but f = 5. Thus, we have abcdef = 1 · 4 · 4 · 3 · 6 · 5 = 1440 . Problem written by Eric Neyman.