PUMaC 2019 · 团队赛 · 第 6 题
PUMaC 2019 — Team Round — Problem 6
题目详情
- Pavel and Sara roll two, fair six-sided dice (with faces labeled from 1 to 6) but do not look at the result. A third-party observer whispers the product of the face-up numbers to Pavel and the sum of the face-up numbers to Sara. Pavel and Sara are perfectly rational and truth-telling, and they both know this. Pavel says, “With the information I have, I am unable to deduce the sum of the two numbers rolled.” Sara responds, “Interesting! With the information I have, I am unable to deduce the product of the two numbers rolled.” Pavel responds, “Wow! I still cannot deduce the sum. But I’m sure you know the product by now!” 1 What is the product?
解析
- Pavel and Sara roll two, fair six-sided dice (with faces labeled from 1 to 6) but do not look at the result. A third-party observer whispers the product of the face-up numbers to Pavel and the sum of the face-up numbers to Sara. Pavel and Sara are perfectly rational and truth-telling, and they both know this. Pavel says, “With the information I have, I am unable to deduce the sum of the two numbers rolled.” Sara responds, “Interesting! With the information I have, I am unable to deduce the product of the two numbers rolled.” Pavel responds, “Wow! I still cannot deduce the sum. But I’m sure you know the product by now!” What is the product? Proposed by: Jacob Wachspress Answer: 6 The only products that may arise in multiple ways are 12 = 4 · 3 = 6 · 2, 6 = 3 · 2 = 6 · 1, and 4 = 2 · 2 = 4 · 1. Thus, Pavel must have received one of { 4 , 6 , 12 } , or else he would have been able to deduce the two numbers and their sum. The possible sums for numbers with a product of 12 are 7 and 8, the possible sums for numbers with a product of 6 are 5 and 7, and possible sums for numbers with a product of 4 are 4 and 5. Only 5 and 7 appear multiple times, so Sara must have received one of these numbers; otherwise knowing that Pavel received one of { 4 , 6 , 12 } (which she learns when he says he does not know the sum) would allow her to deduce the numbers and their product. 2 If Pavel does not know the sum of the numbers despite knowing the product and that the sum is 5 or 7, then the product must be 6, since of the remaining options only 6 can be written as the product of two numbers that sum to 5 and as the product of two numbers that sum to 7.