Imagine an island where 100 people, all perfect logicians, are imprisoned by a mad dictator.
想像一下，有座島上住著100位居民，他們每一個都是優秀的邏輯學家，卻被一位邪惡的獨裁者所監禁。
There's no escape, except for one strange rule.
唯一能逃離島上的方法，必須符合一條特殊的規則。
Any prisoner can approach the guards at night and ask to leave.
所有人都可以在晚上去找守衛，並要求離開島上。
If they have green eyes, they'll be released.
只要這位居民擁有綠色眼睛，就可以被釋放。
If not, they'll be tossed into the volcano.
但如果不是綠色眼睛，就要被丟進火山。
As it happens, all 100 prisoners have green eyes,
碰巧的是，那100位居民都有綠色眼睛。
but they've lived there since birth, and the dictator has ensured they can't learn their own eye color.
但是自從出生以來，他們一直住在島上，獨裁者很確定，居民們都不知道自己眼睛的顏色。
There are no reflective surfaces, all water is in opaque containers,
島上沒有任何物品能夠反射出自己的影像，所有的水都裝在不透明的容器裡，
and most importantly, they're not allowed to communicate among themselves.
最重要的是，居民之間不準互相溝通。
Though they do see each other during each morning's head count.
雖然每天早上點名時，他們會看到彼此。
Nevertheless, they all know no one would ever risk trying to leave without absolute certainty of success.
不過他們都知道，從來沒有人敢冒險嘗試離開，因為無法確定自己一定能成功。
After much pressure from human rights groups,
在許多人權團體的壓力下，
the dictator reluctantly agrees to let you visit the island and speak to the prisoners under the following conditions:
獨裁者勉強同意，讓你拜訪那座島，你可以跟被監禁的居民談話，但是必須符合以下的條件：
you may only make one statement, and you cannot tell them any new information.
你只能說一件事，而且不能告訴他們任何新的消息。
What can you say to help free the prisoners without incurring the dictator's wrath?
你要怎麼說才能幫助居民獲得自由，而且不會激怒獨裁者？
After thinking long and hard, you tell the crowd, "At least one of you has green eyes."
經過一段時間的思考後，你告訴居民：「你們之中至少一位擁有綠色眼睛。」
The dictator is suspicious but reassures himself that your statement couldn't have changed anything.
獨裁者雖然有點懷疑，但依然告訴自己：你說的話不會改變任何事情。
You leave, and life on the island seems to go on as before.
然後你離開了，島上的生活又變得和之前一樣。
But on the hundredth morning after your visit, all the prisoners are gone, each having asked to leave the previous night.
但是在你拜訪之後的第一百個早上，所有的居民都離開了，每一位居民，都在前一天晚上要求離開。
So how did you outsmart the dictator?
你是用什麼方法智取獨裁者的？
It might help to realize that the amount of prisoners is arbitrary.
把居民的數目任意化也許可以有所幫助。
Let's simplify things by imagining just two, Adria and Bill.
讓我們先把情況簡化，想像只有兩個居民 Adria 和 Bill。
Each sees one person with green eyes, and for all they know, that could be the only one.
他們各自看到對方擁有綠色眼睛。所以他們都知道：對方是可以離開島上的那一位。
For the first night, each stays put.
第一天晚上，兩個居民都留下來了。
But when they see each other still there in the morning, they gain new information.
但是到了隔天早上，當他們看到彼此都還在，他們得到了新的訊息。
Adria realizes that if Bill had seen a non-green-eyed person next to him,
Adria的想法是：如果Bill看到的人沒有綠色眼睛，
he would have left the first night after concluding the statement could only refer to himself.
那Bill在第一天晚上就會要求離開。當他得到這個結論的時候，就會推論出：自己有綠色眼睛。
Bill simultaneously realizes the same thing about Adria.
而Bill也會得到相同的結論。
The fact that the other person waited tells each prisoner his or her own eyes must be green.
由於另一個人還在等待，沒有離開島嶼，這個事實告訴每位居民：他們自己的眼睛一定是綠色的。
And on the second morning, they're both gone.
根據第二天早上的結果，他們兩個都會要求離開。
Now imagine a third prisoner.
現在，假設島上有三個居民。
Adria, Bill and Carl each see two green-eyed people,
Adria，Bill和Carl都看到其他兩位擁有綠色眼睛，
but aren't sure if each of the others is also seeing two green-eyed people, or just one.
但無法確定其他兩位，也同樣看到兩個綠眼的人，或是只看到一個(綠眼的人)。
They wait out the first night as before, but the next morning, they still can't be sure.
他們像之前一樣度過第一個晚上，到了隔天早上，他們還是不能確定。
Carl thinks, "If I have non-green eyes, Adria and Bill were just watching each other,
Carl想：「如果我不是綠眼睛，Adria和Bill只會互看彼此，
and will now both leave on the second night."
然後兩人將會在第二個晚上離開。」
But when he sees both of them the third morning, he realizes they must have been watching him, too.
但當Carl第三天早上看到其他兩人還在，Carl了解到一件事：其他兩人知道Carl有綠色眼睛。
Adria and Bill have each been going through the same process, and they all leave on the third night.
而Adria和Bill也經過同樣的推理過程，然後他們全部在第三個晚上要求離開。
Using this sort of inductive reasoning, we can see that the pattern will repeat no matter how many prisoners you add.
用同樣的歸納推理方式，我們可以發現：不論島上有多少居民，這個模式會不斷重複。
The key is the concept of common knowledge, coined by philosopher David Lewis.
這個解答的關鍵就在於哲學家David Lewis所提出的公開知識這個哲學概念。
The new information was not contained in your statement itself, but in telling it to everyone simultaneously.
你的敘述本身，並不包含新的訊息，但是一旦告訴了每一個人，就會產生新的訊息。
Now, besides knowing at least one of them has green eyes, each prisoner also knows that
現在，除了知道居民當中至少一位有綠色眼睛，這些人也想著：
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.
所有人都在觀察著其他擁有綠色眼睛的人，他們每個人都知道這件事。
What any given prisoner doesn't know is whether they themselves are one of the green-eyed people the others are keeping track of
居民們唯一無法確定的是：他們在其他人眼中是不是擁有綠色眼睛的人，
until as many nights have passed as the number of prisoners on the island.
一直要等到他們等待的夜晚天數，和島上的居民數目一樣為止。
Of course, you could have spared the prisoners 98 days on the island by telling them at least 99 of you have green eyes,
當然，你可以幫這些居民省下98天觀察等待的時間。方法就是直接告訴他們：「你們當中至少有99位擁有綠色眼睛。」
but when mad dictators are involved, you're best off with a good headstart.
但是別忘了還有獨裁者正在監視，所以你還是小心一點比較好。