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你能解开这个囚犯搬箱子难题吗? – 译学馆
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你能解开这个囚犯搬箱子难题吗?

Can you solve the prisoner boxes riddle? - Yossi Elran

[音乐]
[Music]
你喜欢的乐队演奏音乐很棒
Your favorite band is great at playing music,
但在条理方面做的就没那么好了
but not so great at being organized.
他们在演出中经常乱放乐器
They keep misplacing their instruments on tour,
简直要把他们的经理逼疯了
and it’s driving their manager mad.
在一次大型音乐会的那天
On the day of the big concert,
乐队成员们醒来后发现他们都被绑在
the band wakes up to find themselves tied up
一个没有窗户的隔音的练习室里
in a windowless, soundproof practice room.
经理给他们解释发生了什么事:
Their manager explains what’s happening.
外面有十个大箱子
Outside, there are ten large boxes.
每个箱子里都有一件乐器
Each contains one of your instruments,
但不要被箱子外面的图片所迷惑
but don’t be fooled by the pictures
因为乐器是随机放置的
– they’ve been randomly placed.
一次只允许出去一个人
I’m going to let you out one at a time.
当你在外面的时候
While you’re outside,
在安全的把你带回演出巴士前
you can look inside any five boxes
你可以任意打开五个箱子
before security takes you back to the tour bus.
你不能触碰乐器
You can’t touch the instruments
也不能以任何方式与他人交流你获得的信息
or in any way communicate what you find to the others.
不要标记箱子 大喊大叫 什么都别做
No marking the boxes, shouting, nothing.
如果你们每个人都能找到自己的乐器
If each one of you can find your own instrument,
那么你们今晚就可以演出
then you can play tonight.
否则 就没你的份儿
Otherwise, the label is dropping you.
在我们开始之前 你们有三分钟思考时间
You have three minutes to think about it before we start.
乐队成员们都感到绝望了
The band is in despair.
毕竟 每位乐师打开五个随机的箱子
After all, each musician only has a 50% chance of finding their instrument
只有50%的机会找到他们的乐器
by picking five random boxes.
十个人全部成功的几率更低
And the chances that all ten will succeed are even lower –
只有1/1024
just 1 in 1024.
但突然间 鼓手提出了一个不错的策略
But suddenly, the drummer comes up with a valid strategy
这个策略的胜率高于35%
that has a better than 35% chance of working.
你能猜出这个策略吗?
Can you figure out what it was?
接下来 请暂停视频 独立思考!
Pause the video on the next screen if you want to figure it out for yourself!
答案倒计时:3
Answer in: 3
答案倒计时:2
Answer in: 2
答案倒计时:1
Answer in: 1
下面就揭晓鼓手的策略:
Here’s what the drummer said:
每个人先打开有你的乐器图片的箱子
Everyone first open the box with the picture of your instrument.
如果箱子里是你的乐器 你就成功了
If your instrument is inside, you’re done.
否则 看看箱子里面是什么乐器?
Otherwise, look at whatever’s in there,
然后打开有这个乐器图片的那个箱子
and then open the box with that picture on it.
按照这个方法直到找到你自己的乐器
Keep going that way until you find your instrument.
乐队成员都很怀疑
The bandmates are skeptical,
但令人惊奇的是
but amazingly enough,
他们都找到了自己需要的东西
they all find what they need.
几个小时后
And a few hours later,
他们给成千上万的粉丝进行了演奏
they’re playing to thousands of adoring fans.
那么鼓手的策略为什么起作用了呢?
So why did the drummer’s strategy work?
每位队员都按照这个连续的顺序
Each musician follows a linked sequence
从有与其乐器匹配图片的箱子开始
that starts with the box whose outside matches their instrument
并以实际装有其乐器的箱子结束
and ends with the box actually containing it.
注意到 如果他们一直这么做
Note that if they kept going,
就会被引导回到开始的箱子
that would lead them back to the start,
所以这是一个循环
so this is a loop.
例如 如果这些箱子是这样排列的
For example, if the boxes are arranged like so,
歌手会打开第一个箱子 然后去找鼓
the singer would open the first box to find the drums,
接着打开第八个盒子 去找贝斯
go to the eighth box to find the bass,
然后在第三个箱子里找到她的麦克风
and find her microphone in the third box,
这个箱子指向的是第一个箱子
which would point back to the first.
这比随机猜测要好得多
This works much better than random guessing
因为从有他们乐器图片的箱子开始
because by starting with the box with the picture of their instrument,
每位乐师都被限定在一条包含其乐器的循环路径上
each musician restricts their search to the loop that contains their instrument,
而且这个策略的胜率很不错 大约为35%
and there are decent odds, about 35%,
所有的循环路径长度为五个或五个以下的箱子
that all of the loops will be of length five or less.
我们如何计算这些概率呢?
How do we calculate those odds?
简单起见
For the sake of simplicity,
我们用一个简化的例子进行解释:
we’ll demonstrate with a simplified case,
四种乐器 每位乐师猜测不得超过两次
four instruments and no more than two guesses allowed for each musician.
我们先从失败的可能性开始
Let’s start by finding the odds of failure,
有人在找到他们的乐器之前
the chance that someone will need to open three or four boxes
可能要打开三到四个箱子
before they find their instrument.
有六条不同的“四箱”循环路径
There are six distinct four-box loops.
可以用一个有趣的方法来计数
One fun way to count them
画个正方形
is to make a square,
把乐器放在每个角上
put an instrument at each corner,
然后画出对角线
and draw the diagonals.
看看你能找到多少条独立的循环路径
See how many unique loops you can find,
要注意的是 这两条是一样的
and keep in mind that these two are considered the same,
他们只是始发处不同而已
they just start at different points.
然而 这两条是不同的
These two, however, are different.
我们可以用三角形可视化这八条不同的“三箱”循环路径
We can visualize the eight distinct three-box loops using triangles.
你会找到四种可能的三角形
You’ll find four possible triangles
每种三角形缺少一种乐器
depending on which instrument you leave out,
而且每种三角形有两条不同的路径
and two distinct paths on each.
所以在24种可能的箱子组合中
So of the 24 possible combinations of boxes,
有14种组合是不能成功的
there are 14 that lead to faliure,
有10种组合是可以成功的
and ten that result in success.
这种计算方法适用于任何含有偶数队员的情况
That computational strategy works for any even number of musicians,
但如果你想寻得捷径
but if you want a shortcut,
这儿推导出了一个简便的公式
it generalizes to a handy equation.
代入10位队员的数据
Plug in ten musicians,
我们可以算出胜率约为35%
and we get odds of about 35%.
如果有1000位队员呢? 1,000,000呢?
What if there were 1,000 musicians? 1,000,000?
随着n不断增加 胜率会趋近于30%
As n increases, the odds approach about 30%.
不敢保证 但是凭着队员们的运气
Not a guarantee, but with a bit of musician’s luck,
成功的几率还是不错的
it’s far from hopeless.
大家好 如果你喜欢这个谜题
Hi everybody, if you liked this riddle,
也可以试试解开这两个谜题
try solving these two.
[音乐]
[Music]

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视频概述

你喜欢的乐队演奏音乐很棒,但是他们太随意了,经常乱放乐器,简直要把他们的经理逼疯了,经理想了个办法……

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视频来源

https://www.youtube.com/watch?v=vIdStMTgNl0

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