返回题库

备用

Alter/Nate

专题
Optimization
难度
L6

题目详情

两个朋友 Alter 和 Nate 进行对话：

**改变：**内特，我们来玩个游戏吧。我会选择一个 1 到 10 之间的整数 （含），那么您将选择 1 到 10（含）之间的整数， 然后我再去，然后你再去，依此类推。 我们将继续将我们的数字加在一起以得出总计。并且 使总分大于或等于 100 的人就输了。 你先走。

**内特：**这不公平！每当我选择一个数字X时，你就会选择 11-X，然后我总是会被困在 99 中，我会得到总数 大于 100。

更改： 好吧。那么新规则，没有人可以选择一个数字 使该数字与前一个数字之和等于 11。您 还是先走吧。现在我们可以玩了吗？

**内特：**嗯……当然。

谁会获胜，他们的策略是什么？

Two friends, Alter and Nate, have a conversation:

Alter: Nate, let’s play a game. I’ll pick an integer between 1 and 10 (inclusive), then you’ll pick an integer between 1 and 10 (inclusive), and then I’ll go again, then you’ll go again, and so on and so forth. We’ll keep adding our numbers together to make a running total. And whoever makes the running total be greater than or equal to 100 loses. You go first.

Nate: That’s not fair! Whenever I pick a number X, you’ll just pick 11-X, and then I’ll always get stuck with 99 and I’ll make the total go greater than 100.

Alter: Ok fine. New rule then, no one can pick a number that would make the sum of that number and the previous number equal to 11. You still go first. Now can we play?

Nate: Um…sure.

Who wins, and what is their strategy?

解析

Nate（第一位玩家）在这场游戏中总是可以获胜，只要开始 数字 3。在第一轮之后，Nate 可以强制跑步 总计以 12 为单位递增。这可能以 2 种不同的方式发生：

  1. $1

  2. $1

这样Nate就可以在运行时强制Alter选择数字 总计等于 3、15、27、39、51、63、75、87 和 99。此时 点，Alter 被迫将总数达到 100 或更大。

恭喜所有解决本月难题的人！

Original Explanation

Nate (the first player) can always win in this game, by starting with the number 3. After this first turn, Nate can force the running total to increment by units of 12. This could happen 2 different ways:

  1. If Alter picks some number X between 2 and 10, Nate chooses 12-X
  2. If Alter picks 1, Nate responds by picking 1 as well. Now Alter cannot pick 10 (since this would force the sum of the previous two numbers to be 11), and must pick some other number Y. Nate then picks 10-Y.

In this way, Nate can force Alter to choose numbers when the running total is equal to 3, 15, 27, 39, 51, 63, 75, 87, and 99. At this point, Alter is forced to take the total to 100 or greater.

Congratulations to everyone who solved this month’s puzzle!