Can you solve the multiplying rabbits riddle? - Alex Gendler

翻译人员: Evina Wang 校对人员: Fandi Yi

After years of experiments,

经过数年的实验，

you’ve finally created the pets of the future–

你终于创造出了未来宠物——

nano-rabbits!

纳米兔子！

They’re tiny, they’re fuzzy…

它们很小且毛茸茸的.......

and they multiply faster than the eye can see.

它们繁殖的速度快到肉眼看不见。

In your lab there are 36 habitat cells,

在你的实验室里，有36间"孵化小屋"，

arranged in an inverted pyramid,

以倒金字塔形式排列。

with 8 cells in the top row.

最上面一层有8间。

The first has one rabbit,

第1间有一只兔子，

the second has two, and so on,

第2间有2只兔子，以此类推，

with eight rabbits in the last one.

最后一间有8只兔子。

The other rows of cells are empty…

其它层的"小屋"都是空的——

for now.

目前是空的。

The rabbits are hermaphroditic,

这些兔子雌雄同笼，

and each rabbit in a given cell will breed once

在每一间中的每一只兔子都会

with every rabbit in the horizontally adjacent cells,

与横向隔壁间的兔子交配一次，

producing exactly one offspring each time.

每次都生出一只新兔子。

The newborn rabbits will drop into the cell

新生出来的兔子会直接落到

directly below the two cells of its parents,

父母住处正下方的那件“小屋”，

and within minutes will mature and reproduce in turn.

在几分钟之内就会长大， 可以换它们开始交配。

Each cell can hold 10^80 nano-rabbits –

每一间“小屋”可以容纳 10^80只纳米兔子——

that’s a 1 followed by 80 zeros –

也就是是1后面加80个0——

before they break free and overrun the world.

超过这个数字的话，他们就会 逃出“小屋”，占领世界。

Your calculations have given you a 46-digit number

你通过计算，得出在最低下的“小屋”中

for the count of rabbits in the bottom cell–

兔子的个数只有46位数——

plenty of room to spare.

空间绰绰有余

But just as you pull the lever to start the experiment,

但当你拉动操作杆要开始实验时，

your assistant runs in with terrible news.

你的助理带着坏消息跑来。

A rival lab has sabotaged your code

你竞争对手的实验室 破坏了你的程序代码，

so that all the zeros at the end of your results got cut off.

所以你最后算出的结果 其实最后面的0都被切掉了。

That means you don’t actually know

那就表示你其实不知道

if the bottom cell will be able to hold all the rabbits –

最下面的“小屋”是否能 容纳所有的兔子——

and the reproduction is already underway!

但繁殖已经开始了！

To make matters worse,

更糟糕的是，

your devices and calculators are all malfunctioning,

你的设备和计算器都出故障了，

so you only have a few minutes to work it out by hand.

你只有几分钟的时间手动计算。

How many trailing zeros should there be

最底层小屋中

at the end of the count of rabbits in the bottom habitat?

兔子的数量应该含有多少个0？

And do you need to pull the emergency shut-down lever?

你需要去拉动紧急关闭的控制杆吗？

Pause the video now if you want to figure it out for yourself.

【如果你想要自己解题 请在这里将影片暂停】

Answer in 3

【答案即将公布：倒计时3】

Answer in 2

【答案即将公布：倒计时2】

Answer in 1

【答案即将公布：倒计时1】

There isn’t enough time to calculate the exact number of rabbits in the final cell.

我们没有足够的时间算出 最后一间小屋中的兔子数目。

The good news is we don’t need to.

好消息是，我们并不需要算出它。

All we need to figure out

我们只需要算出

is how many trailing zeros it has.

后面有多少个0。

But how can we know how many trailing zeros a number has

但我们怎么能在不计算出 这个数目的情况下

without calculating the number itself?

知道后面有几个0呢？

What we do know is that we arrive at the number of rabbits in the bottom cell

我们只知道，底层小屋的兔子数目

through a process of multiplication –

是经过相乘运算得到的——

literally.

字面意义。

The number of rabbits in each cell

每间小屋的兔子数目

is the product of the number of rabbits in each of the two cells above it.

是上面两间小屋中的兔子数目相乘。

And there are only two ways

只有两种方法

to get numbers with trailing zeros through multiplication:

可以透过乘法得到 后面有0的数字：

either multiplying a number ending in 5 by any even number,

把尾数是5的数字 和任何偶数相乘。

or by multiplying numbers that have trailing zeroes themselves.

或是将本身尾数就是0的数字相乘。

Let’s calculate the number of rabbits in the second row

让我们来算一下 第二层的兔子数目，

and see what patterns emerge.

看看会发现什么规律。

Two of the numbers have trailing zeros –

有两个数字的尾数有0——

20 rabbits in the fourth cell and 30 in the fifth cell.

第4间有20只兔子， 第5间有30只。

But there are no numbers ending in 5.

没有尾数是5的数字出现。

And since the only way to get a number ending in 5 through multiplication

因为若要透过乘法 得到尾数是5的数字，

is by starting with a number ending in 5,

一定要用尾数是5的数字来乘，

there won’t be any more down the line either.

因此可确定下面也不会 出现尾数是5的数字。

That means we only need to worry

那就表示: 我们只需担心

about the numbers that have trailing zeros themselves.

本身末尾是0的数字。

And a neat trick to figure out the amount of trailing zeros in a product

有个小妙招可以知道 乘积的尾数有几个0，

is to count and add the trailing zeros in each of the factors –

那就是算出每个因数尾数的 0有几个，再将再们相加——

for example, 10 x 100 = 1,000.

比如，10 × 100 = 1000。

So let’s take the numbers in the fourth and fifth cells

所以，我们从第二排第4、第5间的数目

and multiply down from there.

着手开始向下相乘。

20 and 30 each have one zero,

20和30尾数都有1个0，

so the product of both cells will have two trailing zeros,

所有这两间相乘尾数会有2个0，

while the product of either cell and an adjacent non-zero-ending cell

这两间小屋和其隔壁 数目尾数非0小屋相乘

will have only one.

得到的乘积尾数只有1个0。

When we continue all the way down,

我们继续向下计算，

we end up with 35 zeros in the bottom cell.

会算出在最底层小屋中的 兔子数目尾数有35个0。

And if you’re not too stressed about the potential nano-rabbit apocalypse,

如果你不太担心纳米兔 可能带来的灾难，

you might notice that counting the zeros this way

你可能会注意到用这种方式计算0

forms part of Pascal’s triangle.

会形成杨辉三角形。

Adding those 35 zeros to the 46 digit number we had before

将那35个0加在我们之前 算出的46位数字后面

yields an 81 digit number –

得到一个81位数数字——

too big for the habitat to contain!

超出小屋的容量！

You rush over and pull the emergency switch

你跑过去，拉下紧急开关，

just as the seventh generation of rabbits was about to mature –

此时第7代兔子刚要长大——

hare-raisingly close to disaster.

差一点点就要发生大灾难...