Can you solve the prisoner boxes riddle? - Yossi Elran

譯者: Priscilla Liu 審譯者: 添翼 徐

Your favorite band is great at playing music,

你最喜愛的樂隊演奏的時候很出色，

but not so great at being organized.

但並不怎麼擅長組織與編排工作。

They keep misplacing their instruments on tour,

他們老是在巡演的時候放錯樂器，

and it's driving their manager mad.

這讓他們的經理十分惱火。

On the day of the big concert,

就在一場大型音樂會的當天，

the band wakes up to find themselves tied up

這個樂隊醒來發現他們被綁在

in a windowless, soundproof practice room.

一個沒有窗戶並且隔音的練習室。

Their manager explains what's happening.

他們的經理解釋了這是怎麼回事。

Outside, there are ten large boxes.

在外面，一共有十個大盒子。

Each contains one of your instruments,

每一個裝著一件你們的樂器，

but don't be fooled by the pictures - they've been randomly placed.

但別被上面的圖片迷惑了， 他們都是隨機放置的。

I'm going to let you out one at a time.

每次我會讓你們其中的一人出去。

While you're outside, you can look inside any five boxes

這時候你們可以查看任意五個盒子。

before security takes you back to the tour bus.

然後保安會把你們帶回巴士。

You can't touch the instruments

你們不能觸碰樂器，

or in any way communicate what you find to the others.

也不能以任何形式 同他人交流你發現了什麼。

No marking the boxes, shouting, nothing.

不能在盒子上做記號， 不能喊叫，什麼都不能。

If each one of you can find your own instrument,

如果你們每人都能找對你們的樂器，

then you can play tonight.

你們今晚就能演出。

Otherwise, the label is dropping you.

否則演出將被取消。

You have three minutes to think about it before we start.

你們有三分鐘的思考時間， 然後我們就開始。

The band is in despair.

整個樂團都十分絕望。

After all, each musician only has a 50% chance of finding their instrument

畢竟，如果只是隨機看五個盒子，

by picking five random boxes.

每個人只有 50% 的機會 找到自己的樂器。

And the chances that all ten will succeed are even lower -

十個人都找對的幾率就更低了——

just 1 in 1024.

只有 1024 分之 1。

But suddenly, the drummer comes up with a valid strategy

但是突然，

鼓手想出了一個有效的策略，

that has a better than 35% chance of working.

有大於 35% 的機率能成功。

Can you figure out what it was?

你能想出來這個策略是什麼嗎？

Pause the video on the next screen if you want to figure it out for yourself!

如果你想自行思考該策略的話， 請在下一畫面按暫停！

Answer in: 3

3

Answer in: 2

2

Answer in: 1

1

Here's what the drummer said:

鼓手是這麼說的：

Everyone first open the box with the picture of your instrument.

每人首先打開 有自己樂器圖片的盒子。

If your instrument is inside, you're done.

如果裡面就是你們的樂器， 那就完成了。

Otherwise, look at whatever's in there,

如果不是，就看看裡面是什麼，

and then open the box with that picture on it.

再打開有該樂器圖片的盒子。

Keep going that way until you find your instrument.

以此類推，直到找到你們各自的樂器。

The bandmates are skeptical,

其他樂隊成員們都表示懷疑，

but amazingly enough, they all find what they need.

但令人驚奇的是， 他們最終都找到了各自的樂器。

And a few hours later, they're playing to thousands of adoring fans.

幾小時後，他們為成千上萬 喜愛他們的粉絲帶來了精彩演出。

So why did the drummer's strategy work?

為什麼鼓手的策略能得以成功呢？

Each musician follows a linked sequence

每個樂手的選擇 都是一個相聯結的序列，

that starts with the box whose outside matches their instrument

該序列始於有各自樂器圖片的盒子，

and ends with the box actually containing it.

終於實際裝有各自樂器的盒子。

Note that if they kept going, that would lead them back to the start,

值得注意的是，繼續以此類推的話， 他們的選擇將回到最初的盒子。

so this is a loop.

所以這其實是一個循環。

For example, if the boxes are arranged like so,

比如說，如果盒子是按此法擺放，

the singer would open the first box to find the drums,

歌手會先打開第一個盒子找到鼓，

go to the eighth box to find the bass,

然後打開第八個盒子找到貝斯，

and find her microphone in the third box,

然後打開第三個盒子找到麥克風，

which would point back to the first.

這也將指引他回到第一個盒子。

This works much better than random guessing

這樣比隨機猜五個盒子有效多了。

because by starting with the box with the picture of their instrument,

因為從有他們各自樂器 圖片的盒子開始找，

each musician restricts their search to the loop that contains their instrument,

每位樂手都將選擇限制在 包含他們各自樂器的循環裡，

and there are decent odds, about 35%,

並且有不錯的機率，大概有 35%，

that all of the loops will be of length five or less.

所有這樣的循環只要五步或者更少。

How do we calculate those odds?

我們如何算出這個機率的呢？

For the sake of simplicity, we'll demonstrate with a simplified case,

為了簡便，我們舉一個簡化的例子，

four instruments and no more than two guesses allowed for each musician.

只有四種樂器， 每個樂手最多只能打開兩個盒子。

Let's start by finding the odds of failure,

我們先來看看失敗的概率，

the chance that someone will need to open three or four boxes

也就是說需要打開三個或者四個盒子

before they find their instrument.

才能猜對自己的樂器的概率。

There are six distinct four-box loops.

總共有六種不同的四個步驟循環。

One fun way to count them is to make a square,

計算這個的一個有趣的辦法 是去畫一個方形。

put an instrument at each corner,

把四個樂器分別擺到四個角，

and draw the diagonals.

然後畫出對角線。

See how many unique loops you can find,

看看你究竟能找出多少獨特的循環，

and keep in mind that these two are considered the same,

注意這兩種循環其實是同一種，

they just start at different points.

只是始於不同的起點而已。

These two, however, are different.

然而，這兩種就不相同啦。

We can visualize the eight distinct three-box loops using triangles.

我們用三角形就能觀察到 八種不同的三個步驟的循環。

You'll find four possible triangles

總共有四種三角形，

depending on which instrument you leave out,

取決於哪一種樂器不含在其中，

and two distinct paths on each.

而每種三角形有兩種不同的路徑。

So of the 24 possible combinations of boxes,

所以在總共 24 種盒子的 排列組合中，

there are 14 that lead to faliure,

有 14 種會失敗，

and ten that result in success.

還有 10 種會成功。

That computational strategy works for any even number of musicians,

這種計算方法適用於 任何有偶數個樂手的情況，

but if you want a shortcut,

但是如果你想要一個簡便辦法，

it generalizes to a handy equation.

可以概括成一個方便的方程式。

Plug in ten musicians, and we get odds of about 35%.

代入 10 位樂手， 得出約 35% 的概率。

What if there were 1,000 musicians?

如果有 1,000 位樂手呢？

1,000,000?

1,000,000 位呢？

As n increases, the odds approach about 30%.

隨著 n 的增大，概率接近 30%

Not a guarantee, but with a bit of musician's luck, it's far from hopeless.

並不能完全保證成功，

但如果樂手運氣不錯，

總不至於是毫無希望的。

Hi everybody, if you liked this riddle, try solving these two.

如果大家喜歡這這個謎題， 就再來試試這兩個吧。