TED科普：你能解开囚犯帽子谜题吗？

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囚犯帽子逻辑谜题，你听说过吗，我们来看看这些人怎么解决？

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You and nine other individuals have been captured by super intelligent alien overlords.

你和其他九个人被高智商的外星人统治者俘虏了。

The aliens think humans look quite tasty, but their civilization forbids eating highly logical and cooperative beings.

他们觉得地球人看起来很好吃，但是他们的文明禁止他们吃有很强逻辑性和合作性的生物。

Unfortunately, they're not sure whether you qualify, so they decide to give you all a test.

不幸的是，他们不确定你们是否合乎标准，所以他们决定给你们所有人一个测试。

Through its universal translator, the alien guarding you tells you the following:

通过他们的“全宇宙通翻译”软件，外星守卫告诉你以下信息：

You will be placed in a single-file line facing forward in size order so that each of you can see everyone lined up ahead of you.

你们会被从高到矮排成一条直线，这样每个人就可以看到站在前面的所有人。

You will not be able to look behind you or step out of line.

你不能往后看或者走到线外。

Each of you will have either a black or a white hat on your head assigned randomly, and I won't tell you how many of each color there are.

每个人的头上会有一顶白色或者黑色的帽子。帽子的颜色是随机分配的，而且我不会告诉你每种颜色的帽子总共有几个。

When I say to begin, each of you must guess the color of your hat starting with the person in the back and moving up the line.

当我说“开始”时，每个人必须猜测自己帽子的颜色，从最后一个人开始。

And don't even try saying words other than black or white or signaling some other way, like intonation or volume; you'll all be eaten immediately.

不要试图说除了黑或白以外的词，或者通过声调或音量等其他方式做出暗示，要不你们就会马上被吃掉。

If at least nine of you guess correctly, you'll all be spared.

如果至少九个人能猜对，你们就会被释放。

You have five minutes to discuss and come up with a plan, and then I'll line you up, assign your hats, and we'll begin.

你们有五分钟的时间来商量，想出一个方案，然后我会把你们排成一排，分发帽子，然后就开始。

Can you think of a strategy guaranteed to save everyone?

你能想到一个绝对能救大家的计划吗？

The key is that the person at the back of the line who can see everyone else's hats can use the words "black" or "white" to communicate some coded information.

其实重点在于排在队尾的人，他在看到其他所有人的帽子后可以用黑白来传递加密信息。

So what meaning can be assigned to those words that will allow everyone else to deduce their hat colors?

那么我们应当在这些词上附加什么含义，以使得其他人可以推测他们帽子的颜色呢？

It can't be the total number of black or white hats.

首先不能是黑帽子或白帽子的总数。

There are more than two possible values, but what does have two possible values is that number's parity, that is whether it's odd or even.

那样可能的值就会超过两种。但是数字的奇偶性恰好只有两种可能，那就是奇数或偶数。

So the solution is to agree that whoever goes first will, for example, say "black" if he sees an odd number of black hats and "white" if he sees an even number of black hats.

所以，解决方案就在于第一个说的人，举个例子，比如他看到了奇数个黑帽子，他就要说“黑色”，当他看到了偶数个黑帽子时就要说“白色”。

Let's see how it would play out if the hats were distributed like this.

我们看下如果帽子颜色是这样分配的话，这个策略执行起来如何。

The tallest captive sees three black hats in front of him, so he says "black," telling everyone else he sees an odd number of black hats.

最高的人看到前面有三个黑帽子，所以他说“黑色”，告诉其他所有人他看到的是奇数个黑帽子。

He gets his own hat color wrong, but that's okay since you're collectively allowed to have one wrong answer.

他没有说对自己帽子的颜色，但是没关系，因为所有被抓的人总共可以犯一个错误。

Prisoner two also sees an odd number of black hats, so she knows hers is white, and answers correctly.

第二高的人也看到奇数个黑帽子，她就会知道她的是白色的，就答对了。

Prisoner three sees an even number of black hats, so he knows that his must be one of the black hats the first two prisoners saw.

第三个人看到前面是偶数个黑帽子，所以他知道他的一定是前面两个人看到的其中一顶黑帽子。

Prisoner four hears that and knows that she should be looking for an even number of black hats since one was behind her.

第四个人听到后就知道她应当看到前面有偶数顶黑帽子，因为其中一顶在她身后。

But she only sees one, so she deduces that her hat is also black.

但是她只看到了一个，所以她推测出自己的也是黑帽子。

Prisoners five through nine are each looking for an odd number of black hats, which they see, so they figure out that their hats are white.

第五个人至第九个人每个都寻找奇数个黑帽子，他们找到了，所以他们推测自己的帽子都是白色的。

Now it all comes down to you at the front of the line.

现在到了站在最前面的你了。

If the ninth prisoner saw an odd number of black hats, that can only mean one thing.

假如第九个人看到的是奇数个黑帽子的话，那就只有一种可能（最后一个人是黑帽子）。

You'll find that this strategy works for any possible arrangement of the hats.

你会发现这个策略对所有的排列组合都是适用的。

The first prisoner has a 50% chance of giving a wrong answer about his own hat, but the parity information he conveys allows everyone else to guess theirs with absolute certainty.

最开始的那个人有50%的几率出错，但是他传达的奇偶性的信息让其他所有人都可以猜对自己帽子的颜色。

Each begins by expecting to see an odd or even number of hats of the specified color.

每个人在开始时都假定自己应当在身前看到奇数或偶数个特定颜色的帽子。

If what they count doesn't match, that means their own hat is that color.

如果他们的数字不对的话，意味着他们自己的帽子就是那种特定的颜色。

And everytime this happens, the next person in line will switch the parity they expect to see.

每次这样的情况发生后，下个人就知道他们需要看到奇数还是偶数个特定颜色的帽子。

So that's it, you're free to go.

好了成功了，你们可以走了。

It looks like these aliens will have to go hungry, or find some less logical organisms to abduct.

看起来这些外星人只好饿肚子了，或者去找些其他逻辑差的生物来绑架。

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