Can you solve the famously difficult green

你能解決著名的綠色難題嗎

來源視頻

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

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

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

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

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

想象在一個島上，一個瘋狂的獨裁者囚禁了100個完美的邏輯家他們沒辦法逃離.

除了一個奇怪的規(guī)則.

在晚上任何囚犯都可以請求警衛(wèi)放他走.

若他有綠眼睛，他就會被釋放

若不是，他就會被扔進火山口。

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.?

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

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

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

實際上，那一百個邏輯家都有綠眼睛，但他們從出生起就住在那兒,而且獨裁者確保他們不會知道自己眼睛什么顏色.

那兒也沒有反光表面，所有水都裝在不透明的容器里，而且最重要的，他們不能互相交流.

盡管他們每天早上點人數(shù)時都能看到彼此。

不僅如此，他們都知道 若沒有絕對的把握會成功沒人會嘗試離開

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.

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

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

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

在人權組織的壓力下，那個獨裁者勉強允許你訪問那個島并且和犯人說話時 要遵從以下條件：你只可陳述一個聲明，你不能告訴他們新的信息。

你說什么才能幫助這些犯人獲得自由而不激怒這位獨裁者呢？

經(jīng)過深思熟慮你告訴人群：你們當中 至少有一個人有綠眼睛。

獨裁者持有懷疑的態(tài)度但還是告訴自己你的言論不會改變?nèi)魏问虑?/p>

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

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

So how did you outsmart the dictator??

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

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

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

你離開了，島上的生活依然照舊

但在你訪問后的第一百天早上，所有的犯人都不見了，每一個人都在前天晚上要求離開

所以你是如何智勝獨裁者的？

意識到犯人的數(shù)量不固定也許能幫到你

我們來把這個問題簡化一下， 假設只有兩個犯人，小紅和小明彼此都能看到對方的綠眼睛

他們心里清楚 也許只有對方有綠眼睛

For the first night, each stays put.?

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

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.

Bill simultaneously realizes the same thing about Adria.?

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

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

所以第一天，誰也沒輕舉妄動

但當他們第二天早上又看到對方時，他們得到了新的信息

小紅意識到如果小明看到了一個“非綠眼睛”的人在他旁邊，他就知道自己擁有綠眼睛，第一天就會離開

小明也同一時間想到了同樣的事情

兩個人都在等這個事實告訴她或他的眼睛一定是綠色的

之后的那天早上，他倆都走了

Now imagine a third prisoner.?

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.?

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

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."?

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

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

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

現(xiàn)在想象第三個犯人

小紅小明和小剛各看到兩個綠眼睛的人，但不確定另外兩人是不是也看到了一個或者兩個綠眼睛的人

他們第一天也和之前一樣等著到了第二天，他們還是不確定

小剛想：如果我沒有綠眼睛小紅和小明只是看著彼此，那他們第二天早上就會都走了

但當他第三天早上看到另外兩人時，他意識到那兩人肯定也在觀察他

小紅和小明在想同樣的事情，然后他們?nèi)嗽诘谌焱砩隙甲吡?/p>

用這種歸納性推理我們可以看出這種模式會一直重復， 不論犯人的數(shù)量多少

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

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

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.?

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.

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.

關鍵點是公識的概念來自哲學家David Lewis的創(chuàng)造

新知識不在你的陳述里，而在于你的聲明同時陳述給了所有人

現(xiàn)在呢，每個犯人除了知道他們中至少一個有綠眼睛，也知道其他人都在觀察記錄他們能看到的所有綠眼睛的人，而且所有人都知道每個人都是這么想的，每個犯人不知道的是他們自己是不是別人觀察記錄的綠眼睛的人中的一員直到和犯人數(shù)量同等的夜數(shù)過去了才能確定。

當然啦，你也可以讓犯人早98天離開這島告訴他們至少99個人有綠眼睛，但當瘋狂獨裁者在的時候，你最好做出明智的選擇。