Can you solve the prisoner hat riddle? - Alex Gendler

譯者: 庭芝 梁 審譯者: Chenxiao Qian

You and nine other individuals have been captured

你和其他九個人

被絕頂聰明的外星人首領綁架了。

by super intelligent alien overlords.

The aliens think humans look quite tasty,

外星人覺得地球人看起來很美味， 但是他們的文明，

but their civilization forbids eating highly logical and cooperative beings.

嚴禁食用具有高度邏輯 與團隊合作能力的種族。

很不幸的，他們並不知道 你是否符合條件，

Unfortunately, they're not sure whether you qualify,

so they decide to give you all a test.

所以他們決定進行一項測驗。

Through its universal translator,

透過他們的星際翻譯，

the alien guarding you tells you the following:

外星守衛告訴你下列內容：

You will be placed in a single-file line facing forward in size order

首先，你們所有人， 依照身高依序排成一列，

so that each of you can see everyone lined up ahead of you.

所以你們每個人都可以看見 排在你前面的所有人。

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

你們不許向後看，或是離開隊伍。

Each of you will have either a black or a white hat on your head

你們所有人都會被戴上一頂 黑色或白色的帽子，

assigned randomly,

顏色是隨機安排的，

and I won't tell you how many of each color there are.

而且我不會告訴你， 每種顏色的帽子有幾頂。

When I say to begin, each of you must guess the color of your hat

當我說開始，你們每個人必須 猜出頭上帽子的顏色，

starting with the person in the back and moving up the line.

順序從隊伍最後一個人開始， 依序往前回答。

And don't even try saying words other than black or white

除了黑色、白色之外， 你們不能說出其他任何字，

or signaling some other way, like intonation or volume;

或是用任何方式， 包括語調、音量來打暗號，

you'll all be eaten immediately.

否則你們所有人立刻會被吃掉。

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

如果你們之中，有九位以上回答正確， 所有人就能被釋放。

You have five minutes to discuss and come up with a plan,

你們有五分鐘進行討論， 並且提出計劃，

and then I'll line you up, assign your hats, and we'll begin.

然後我會把你們排成一列、 分配帽子，然後開始測驗。

Can you think of a strategy guaranteed to save everyone?

你是否能想出一個策略， 確保所有人都能夠被釋放？

Pause the video now to figure it out for yourself.

如果你想要自己找出答案， 請按下暫停鈕。

Answer in: 3

倒數 3 秒

Answer in: 2

2 秒

Answer in: 1

1秒

The key is that the person at the back of the line

關鍵就在於：

who can see everyone else's hats can use the words "black" or "white"

排在隊伍最後面的人， 他能夠看見其他人帽子的顏色，

to communicate some coded information.

並且使用 "黑色" 或 "白色" 來傳達一些經過加密的訊息。

So what meaning can be assigned to those words

到底有什麼訊息， 可以被隱藏在這兩個字詞當中，

that will allow everyone else to deduce their hat colors?

並且讓其他人， 能夠推論出帽子的顏色？

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

這個訊息不可能是 黑色或白色帽子的數量，

There are more than two possible values,

因為數量已經超過 兩個字詞所能代表的數字。

but what does have two possible values is that number's parity,

但是，剛好能用兩個數值來代表的， 就是數字的「奇偶性」，

that is whether it's odd or even.

也就是帽子的數量為 「奇數」或是「偶數」。

So the solution is to agree that whoever goes first will,

所以正確的解答是： 第一個回答的人，

for example, say "black" if he sees an odd number of black hats

如果他所看到的黑色帽子數量是「奇數」， 他就回答「黑色」；

and "white" if he sees an even number of black hats.

如果看到的黑色帽子數量是「偶數」， 就要回答「白色」。

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

我們來看看，像影片中的帽子顏色分布， 應該如何回答？

The tallest captive sees three black hats in front of him,

首先，身高最高的俘虜 可以看到前面共有三頂黑帽子，

so he says "black," telling everyone else he sees an odd number of black hats.

所以他回答「黑色」，告訴其他人 他看到的黑帽子數量是「奇數」。

He gets his own hat color wrong, but that's okay

他答錯了自己帽子的顏色， 但是那並不重要，

since you're collectively allowed to have one wrong answer.

因為你們可以有一次答錯的機會。

Prisoner two also sees an odd number of black hats,

第二位俘虜，同樣看到了 前面共有奇數頂黑帽子，

so she knows hers is white, and answers correctly.

所以她知道自己的帽子是白色的， 於是她答對了。

Prisoner three sees an even number of black hats,

第三位俘虜，看到了前面 黑帽子的數量是偶數，

so he knows that his must be one of the black hats

所以他知道頭上戴的帽子

the first two prisoners saw.

一定是前兩位俘虜 所看到的黑帽子其中之一。

Prisoner four hears that and knows

第四位俘虜 一聽之後就明白：

that she should be looking for an even number of black hats

她應該看到前面有偶數頂黑帽子，

since one was behind her.

因為她背後已經有一頂了。

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

但是她看到前面只有一頂黑帽子， 所以她推論出自己的帽子也是黑色的。

Prisoners five through nine are each looking for an odd number of black hats,

接下來第五位到第九位俘虜， 每一位都應該看到奇數頂黑帽子，

which they see, so they figure out that their hats are white.

也就是這一頂，所以他們都能回答 自己的帽子是白色的。

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

現在，輪到隊伍最前面的你。

If the ninth prisoner saw an odd number of black hats,

如果第九位俘虜看到 前面有奇數頂黑帽子，

that can only mean one thing.

那就代表只有一種可能 (最後一人的帽子是黑色)

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

你會發現這個策略， 適用於所有的排列組合。

The first prisoner has a 50% chance of giving a wrong answer about his own hat,

第一位俘虜有百分之五十的機率 會猜錯自己帽子的顏色，

but the parity information he conveys

但是他所傳佈的「奇偶性」訊息，

allows everyone else to guess theirs with absolute certainty.

能夠確保其他所有俘虜， 都能正確回答頭上帽子的顏色。

Each begins by expecting to see an odd or even number of hats

每個人原本期望能看到

of the specified color.

奇數 (或偶數) 頂特定顏色的帽子，

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

如果他們看到的和預期不同， 就代表自己頭上的帽子是特定顏色。

And everytime this happens,

每當有人答出「指定顏色」的帽子，

the next person in line will switch the parity they expect to see.

下一位俘虜就會改變 帽子數目的「奇偶性」。

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

所以現在，你們可以自由了！

It looks like these aliens will have to go hungry,

看來這些外星人 得繼續餓肚子了，

or find some less logical organisms to abduct.

或是去綁架 邏輯不好的其他生物。