TED科普： 著名的绿眼逻辑难题

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一名疯狂的独裁者将一百名绿眼的逻辑学家囚禁在一个岛上。他们获得自由的唯一希望在于解决一个著名的难题。你能够做到吗?

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Imagine an island where 100 people, all perfect logicians, are imprisoned by a mad dictator.

想像一下，有座岛上住着100位居民，他们每一个都是优秀的逻辑学家，却被一位邪恶的独裁者所监禁。

There's no escape, except for one strange rule.

唯一能逃离岛上的方法，必须符合一条特殊的规则。

Any prisoner can approach the guards at night and ask to leave.

所有人都可以在晚上去找守卫，并要求离开岛上。

If they have green eyes, they'll be released.

只要这位居民拥有绿色眼睛，就可以被释放。

If not, they'll be tossed into the volcano.

但如果不是绿色眼睛，就要被丢进火山。

As it happens, all 100 prisoners have green eyes, but they've lived there since birth, and the dictator has ensured they can't learn their own eye color.

碰巧的是，那100位居民都有绿色眼睛。但是自从出生以来，他们一直住在岛上，独裁者很确定，居民们都不知道自己眼睛的颜色。

There are no reflective surfaces, all water is in opaque containers, and most importantly, they're not allowed to communicate among themselves.

岛上没有任何物品能够反射出自己的影像，所有的水都装在不透明的容器里，最重要的是，居民之间不准互相沟通。

Though they do see each other during each morning's head count.

虽然每天早上点名时，他们会看到彼此。

Nevertheless, they all know no one would ever risk trying to leave without absolute certainty of success.

不过他们都知道，从来没有人敢冒险尝试离开，因为无法确定自己一定能成功。

After much pressure from human rights groups, the dictator reluctantly agrees to let you visit the island and speak to the prisoners under the following conditions: you may only make one statement, and you cannot tell them any new information.

在许多人权团体的压力下，独裁者勉强同意，让你拜访那座岛，你可以跟被监禁的居民谈话，但是必须符合以下的条件：你只能说一件事，而且不能告诉他们任何新的消息。

What can you say to help free the prisoners without incurring the dictator's wrath?

你要怎么说才能帮助居民获得自由，而且不会激怒独裁者？

After thinking long and hard, you tell the crowd, "At least one of you has green eyes."

经过一段时间的思考后，你告诉居民：“你们之中至少一位拥有绿色眼睛。”

The dictator is suspicious but reassures himself that your statement couldn't have changed anything.

独裁者虽然有点怀疑，但依然告诉自己：你说的话不会改变任何事情。

You leave, and life on the island seems to go on as before.

然后你离开了，岛上的生活又变得和之前一样。

But on the hundredth morning after your visit, all the prisoners are gone, each having asked to leave the previous night.

但是在你拜访之后的第一百个早上，所有的居民都离开了，每一位居民，都在前一天晚上要求离开。

So how did you outsmart the dictator?

你是用什么方法智取独裁者的？

It might help to realize that the amount of prisoners is arbitrary.

把居民的数目任意化也许可以有所帮助。

Let's simplify things by imagining just two, Adria and Bill.

让我们先把情况简化，想像只有两个居民 Adria 和 Bill。

Each sees one person with green eyes, and for all they know, that could be the only one.

他们各自看到对方拥有绿色眼睛。所以他们都知道：对方是可以离开岛上的那一位。

For the first night, each stays put.

第一天晚上，两个居民都留下来了。

But when they see each other still there in the morning, they gain new information.

但是到了隔天早上，当他们看到彼此都还在，他们得到了新的讯息。

Adria realizes that if Bill had seen a non-green-eyed person next to him, he would have left the first night after concluding the statement could only refer to himself.

Adria的想法是：如果Bill看到的人没有绿色眼睛，那Bill在第一天晚上就会要求离开。当他得到这个结论的时候，就会推论出：自己有绿色眼睛。

Bill simultaneously realizes the same thing about Adria.

而Bill也会得到相同的结论。

The fact that the other person waited tells each prisoner his or her own eyes must be green.

由于另一个人还在等待，没有离开岛屿，这个事实告诉每位居民：他们自己的眼睛一定是绿色的。

And on the second morning, they're both gone.

根据第二天早上的结果，他们两个都会要求离开。

Now imagine a third prisoner. 

现在，假设岛上有三个居民。

Adria, Bill and Carl each see two green-eyed people, but aren't sure if each of the others is also seeing two green-eyed people, or just one.

Adria，Bill和Carl都看到其他两位拥有绿色眼睛，但无法确定其他两位，也同样看到两个绿眼的人，或是只看到一个(绿眼的人)。

They wait out the first night as before, but the next morning, they still can't be sure.

他们像之前一样度过第一个晚上，到了隔天早上，他们还是不能确定。

Carl thinks, "If I have non-green eyes, Adria and Bill were just watching each other, and will now both leave on the second night."

Carl想：“如果我不是绿眼睛，Adria和Bill只会互看彼此，然后两人将会在第二个晚上离开。”

But when he sees both of them the third morning, he realizes they must have been watching him, too.

但当Carl第三天早上看到其他两人还在，Carl了解到一件事：其他两人知道Carl有绿色眼睛。

Adria and Bill have each been going through the same process, and they all leave on the third night.

而Adria和Bill也经过同样的推理过程，然后他们全部在第三个晚上要求离开。

Using this sort of inductive reasoning, we can see that the pattern will repeat no matter how many prisoners you add.

用同样的归纳推理方式，我们可以发现：不论岛上有多少居民，这个模式会不断重复。

The key is the concept of common knowledge, coined by philosopher David Lewis.

这个解答的关键就在于哲学家David Lewis所提出的公开知识这个哲学概念。

The new information was not contained in your statement itself, but in telling it to everyone simultaneously.

你的叙述本身，并不包含新的讯息，但是一旦告诉了每一个人，就会产生新的讯息。

Now, besides knowing at least one of them has green eyes, each prisoner also knows that everyone else is keeping track of all the green-eyed people they can see, and that each of them also knows this, and so on.

现在，除了知道居民当中至少一位有绿色眼睛，这些人也想着：所有人都在观察着其他拥有绿色眼睛的人，他们每个人都知道这件事。

What any given prisoner doesn't know is whether they themselves are one of the green-eyed people the others are keeping track of until as many nights have passed as the number of prisoners on the island.

居民们唯一无法确定的是：他们在其他人眼中是不是拥有绿色眼睛的人，一直要等到他们等待的夜晚天数，和岛上的居民数目一样为止。

Of course, you could have spared the prisoners 98 days on the island by telling them at least 99 of you have green eyes, but when mad dictators are involved, you're best off with a good headstart.

当然，你可以帮这些居民省下98天观察等待的时间。方法就是直接告诉他们：“你们当中至少有99位拥有绿色眼睛。”但是别忘了还有独裁者正在监视，所以你还是小心一点比较好。

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