Can you solve the multiplying rabbits riddle? - Alex Gendler

譯者: Lilian Chiu 審譯者: Helen Chang

After years of experiments,

經過數年的實驗，

you’ve finally created the pets of the future–

你終於創造出了未來的寵物——

nano-rabbits!

奈米兔子！

They’re tiny, they’re fuzzy…

牠們很小且毛絨絨的……

and they multiply faster than the eye can see.

牠們繁殖的速度快到肉眼看不見。

In your lab there are 36 habitat cells,

在你的實驗室中，有 36 間住窩，

arranged in an inverted pyramid,

用倒金字塔的形式排列，

with 8 cells in the top row.

最上面一層有 8 間，

The first has one rabbit,

第 1 間有 1 隻兔子，

the second has two, and so on,

第 2 間有 2 隻兔子，以此類推，

with eight rabbits in the last one.

最後一間有 8 隻兔子。

The other rows of cells are empty…

其他層的住窩是空的……

for now.

目前是空的。

The rabbits are hermaphroditic,

這些兔子雌雄同體，

and each rabbit in a given cell will breed once

在每一間當中的每一隻兔子都會

與橫向隔壁間中的兔子交配一次，

with every rabbit in the horizontally adjacent cells,

producing exactly one offspring each time.

每次都是生出一隻後代。

The newborn rabbits will drop into the cell

新生出來的兔子會直接落到

directly below the two cells of its parents,

父母住窩正下方的那間住窩，

and within minutes will mature and reproduce in turn.

在幾分鐘之內就會長大成熟， 可以換牠們開始交配。

Each cell can hold 10^80 nano-rabbits –

每一間住窩可以容納 10^80 隻奈米兔子——

that’s a 1 followed by 80 zeros –

10^80 是 1 後面加 80 個 0 ——

before they break free and overrun the world.

超過的話，牠們就會 逃出住窩竄犯全世界。

Your calculations have given you a 46-digit number

你的計算算出來在最底層住窩中的

for the count of rabbits in the bottom cell–

兔子數目只有 46 位數——

plenty of room to spare.

空間綽綽有餘。

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

但當你拉動操縱桿要開始實驗時，

your assistant runs in with terrible news.

你的助理帶著壞消息跑來。

A rival lab has sabotaged your code

你競爭對手的實驗室 破壞了你的程式碼，

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

所以你最後算出的結果， 其實最後面的 0 都被切掉了。

That means you don’t actually know

那就表示你其實不知道

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

最下面一層的住窩是否 能容納所有的兔子——

and the reproduction is already underway!

但繁殖已經開始了！

To make matters worse,

更糟糕的是，

your devices and calculators are all malfunctioning,

你的裝置和計算機都故障了，

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

你只有幾分鐘的時間用手算出來。

How many trailing zeros should there be

底層住窩中的兔子數目，

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

後面應該要有多少個 0？

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

你需要去拉緊急關閉的控制桿嗎？

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

如果你想要自己解題， 請在這裡將影片暫停。

Answer in 3

答案即將公佈：3

Answer in 2

答案即將公佈：2

Answer in 1

答案即將公佈：1 沒有足夠的時間算出

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

在最後一間住窩中的兔子數目。

The good news is we don’t need to.

好消息是，我們並不需要算出它。

All we need to figure out

我們只需要算出後面有多少個 0。

is how many trailing zeros it has.

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

但我們要怎麼在不計算出 這個數目的情況下，

without calculating the number itself?

知道後面有幾個零？

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

我們只知道，底層住窩的兔子數目

through a process of multiplication –

是透過乘法（與「繁殖」同字）的 過程得到的——真的。

literally.

The number of rabbits in each cell

每間住窩中的兔子數目

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

是上面兩間住窩中的 兔子數目相乘。

And there are only two ways

只有兩種方式

to get numbers with trailing zeros through multiplication:

可以透過乘法得到 後面有 0 的數字：

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

把尾數是 5 的數字 和任何偶數相乘，

or by multiplying numbers that have trailing zeroes themselves.

或是將本身尾數 就是 0 的數字相乘。

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

咱們來計算上面數下來 第二層的兔子數目，

and see what patterns emerge.

看看會發現什麼模式。

Two of the numbers have trailing zeros –

兩個數字的尾數有 0 ——

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

第 4 間有 20 隻兔子， 第 5 間有 30 隻。

But there are no numbers ending in 5.

但沒有尾數是 5 的數目出現。

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

因為若要透過乘法 得到尾數是 5 的數字，

is by starting with a number ending in 5,

一定要用尾數是 5 的數字來乘，

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

因此可確定下面也不會 再出現尾數是 5 的數字。

That means we only need to worry

那就表示我們只需要擔心

about the numbers that have trailing zeros themselves.

本身尾數就是 0 的數字。

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

有招絕招可以知道乘法 乘積的尾數有幾個 0，

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

那就是算出每個因數尾數的 0 有幾個，再將它們加總——

for example, 10 x 100 = 1,000.

比如，10 x 100 = 1,000。

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

所以，咱們從第 4、5 間的數目

and multiply down from there.

著手開始向下相乘。

20 and 30 each have one zero,

20 和 30 尾數都有 1 個 0，

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

所以這兩間的乘積， 尾數會有 2 個 0，

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

這兩間住窩和其隔壁 數目尾數非 0 的住窩相乘

will have only one.

得到的乘積尾數只有 1 個 0。

When we continue all the way down,

我們繼續向下計算，

we end up with 35 zeros in the bottom cell.

會算出在最底層住窩中的 兔子數尾數有 35 個 0。

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

若在計算這可能會發生的奈米兔子 世界末日時你還有餘力的話，

you might notice that counting the zeros this way

你可能會注意到， 用這種方式計算 0，

forms part of Pascal’s triangle.

會形成一部分的楊輝三角形。

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

將那 35 個 0 加在我們之前 算出的 46 位數數字後面，

yields an 81 digit number –

得到一個 81 位數數字——

too big for the habitat to contain!

超出住窩的容量！

You rush over and pull the emergency switch

你跑過去，拉下緊急開關，

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

此時第 7 代的兔子正要長大——

hare-raisingly close to disaster.

差一點點就要發生繁殖災難。