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兰多·雷克斯教授

Professor Rando Redux

专题
Probability / 概率
难度
L7

题目详情

您可能已经从 December 的谜题中注意到，如果 Rando 教授选择了他的 统一从 1 到 N 范围内的两个整数，其中 N > 5，有一定的机会 游戏永远不会结束。

但是，教授最近迎来了一位非常聪明的新学生蒂姆，他想要 包含在游戏中。兰多决定游戏的运作方式与 之前：

对于某个正整数N，他独立地生成两个随机整数并且 统一从 1 到 N（含），然后告诉每个学生一个 关于这两个数字的不同事实。他告诉：

  • Daphne 整数的绝对差，

  • Max 的最大值 整数，

  • Mindy 是整数中的最小值，

  • Sam 整数之和，并且

  • Tim 整数的乘积。

然后，每天直到比赛结束，他都会聚集他的学生并问他们， 按字母顺序排列，表示两个整数的同一性。每个学生都有 每天只有一次回答的机会，即当她或他被要求时。每个学生 仅当他或她明确知道答案时才回答，否则 那天没有给出答复。所有的学生都知道这一点，并且没有 勾结。一旦学生给出答案（这将是正确的），该学生 获胜，游戏结束。

这个游戏最终总会结束吗？如果是，请提交“是”。如果没有，请提交 最小的 N，使得游戏不一定会结束。

You may have noticed from December’s puzzle that if Professor Rando selects his two integers uniformly from the range 1 to N, where N > 5, there is some chance that the game never ends.

But, the professor has recently gotten a very bright new pupil, Tim, who wants to be included in the game. Rando decides the game will work similarly as before:

For some positive integer N, he generates two random integers independently and uniformly from 1 to N (inclusive), and then tells each of his students a different fact about the two numbers. He tells:

  • Daphne the absolute difference of the integers,
  • Max the maximum of the integers,
  • Mindy the minimum of the integers,
  • Sam the sum of the integers, and
  • Tim the product of the integers.

Then, each day until the game ends, he congregates his students and asks them, in alphabetical order, for the identity of the two integers. Each student has only one chance to answer each day, when she or he is called upon. Each student answers ONLY when the answer is definitively known to him or her, and otherwise gives no answer that day. All of the students know this, and there is no collusion. Once a student gives an answer (which will be correct), that student wins and the game ends.

Will this game always eventually end? If so, submit “yes”. If not, submit the smallest N such that the game might not necessarily ever end.

解析

游戏并不总是结束！当N = 15（或更高）时，有数字 兰多教授可以选择这样的游戏永远不会结束。

恭喜本月所有正确的解算者，尤其是 Mdavis，这 本月随机选出的 Jane Street T 恤优胜者！

Original Explanation

The game does not always end! When N = 15 (or higher), there are numbers Professor Rando could pick such that the game would never end.

Congratulations to all of this month’s correct solvers, especially Mdavis, this month’s randomly-selected winner of a Jane Street t-shirt!