Can you solve the multiplying rabbits riddle

你能解决倍增的兔子之谜吗

来源视频

After years of experiments, you’ve finally created the pets of the future– nano-rabbits!

They’re tiny, they’re fuzzy… and they multiply faster than the eye can see.

In your lab there are 36 habitat cells, arranged in an inverted pyramid, with 8 cells in the top row. 

The first has one rabbit, the second has two, and so on, with eight rabbits in the last one. 

The other rows of cells are empty…

经过数年的实验，你终于创造出了未来宠物——纳米兔子！

它们很小且毛茸茸的.......它们繁殖的速度快到肉眼看不见。在你的实验室里，有36间"孵化小屋"，以倒金字塔形式排列。最上面一层有8间。第1间有一只兔子，第2间有2只兔子，以此类推，最后一间有8只兔子。其它层的"小屋"都是空的——目前是空的。

for now. The rabbits are hermaphroditic, and each rabbit in a given cell will breed once with every rabbit in the horizontally adjacent cells, producing exactly one offspring each time. 

The newborn rabbits will drop into the cell directly below the two cells of its parents, and within minutes will mature and reproduce in turn. 

Each cell can hold 10^80 nano-rabbits – that’s a 1 followed by 80 zeros – before they break free and overrun the world.

这些兔子雌雄同笼，在每一间中的每一只兔子都会与横向隔壁间的兔子交配一次，每次都生出一只新兔子。

新生出来的兔子会直接落到父母住处正下方的那件“小屋”，在几分钟之内就会长大， 可以换它们开始交配。

每一间“小屋”可以容纳 10^80只纳米兔子——也就是是1后面加80个0——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 for the count of rabbits in the bottom cell– plenty of room to spare. 

But just as you pull the lever to start the experiment, your assistant runs in with terrible news.

A rival lab has sabotaged your code so that all the zeros at the end of your results got cut off.

That means you don’t actually know if the bottom cell will be able to hold all the rabbits – and the reproduction is already underway! 

To make matters worse, your devices and calculators are all malfunctioning, so you only have a few minutes to work it out by hand.

How many trailing zeros should there be at the end of the count of rabbits in the bottom habitat? 

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

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

Answer in 3 

Answer in 2

Answer in 1

你通过计算，得出在最低下的“小屋”中兔子的个数只有46位数——空间绰绰有余但当你拉动操作杆要开始实验时，你的助理带着坏消息跑来。

你竞争对手的实验室 破坏了你的程序代码，所以你最后算出的结果 其实最后面的0都被切掉了。

那就表示你其实不知道最下面的“小屋”是否能 容纳所有的兔子——但繁殖已经开始了！

更糟糕的是，你的设备和计算器都出故障了，你只有几分钟的时间手动计算。

最底层小屋中兔子的数量应该含有多少个0？

你需要去拉动紧急关闭的控制杆吗？

【如果你想要自己解题 请在这里将影片暂停】

【答案即将公布：倒计时3】

【答案即将公布：倒计时2】

【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. 

The good news is we don’t need to. 

All we need to figure out is how many trailing zeros it has. 

But how can we know how many trailing zeros a number has without calculating the number itself? 

What we do know is that we arrive at the number of rabbits in the bottom cell through a process of multiplication – literally. 

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

And there are only two ways to get numbers with trailing zeros through multiplication: either multiplying a number ending in 5 by any even number, or by multiplying numbers that have trailing zeroes themselves.

我们没有足够的时间算出 最后一间小屋中的兔子数目。

好消息是，我们并不需要算出它。

我们只需要算出后面有多少个0。

但我们怎么能在不计算出 这个数目的情况下知道后面有几个0呢？

我们只知道，底层小屋的兔子数目是经过相乘运算得到的——字面意义。

每间小屋的兔子数目是上面两间小屋中的兔子数目相乘。

只有两种方法可以透过乘法得到 后面有0的数字：把尾数是5的数字 和任何偶数相乘。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 and see what patterns emerge. 

Two of the numbers have trailing zeros – 20 rabbits in the fourth cell and 30 in the fifth cell.

But there are no numbers ending in 5. 

And since the only way to get a number ending in 5 through multiplication is by starting with a number ending in 5, there won’t be any more down the line either. 

That means we only need to worry about the numbers that have trailing zeros themselves.

And a neat trick to figure out the amount of trailing zeros in a product is to count and add the trailing zeros in each of the factors – for example, 10 x 100 = 1,000.

让我们来算一下 第二层的兔子数目，看看会发现什么规律。

有两个数字的尾数有0——第4间有20只兔子， 第5间有30只。

没有尾数是5的数字出现。

因为若要透过乘法 得到尾数是5的数字，一定要用尾数是5的数字来乘，因此可确定下面也不会 出现尾数是5的数字。

那就表示: 我们只需担心本身末尾是0的数字。

有个小妙招可以知道 乘积的尾数有几个0，那就是算出每个因数尾数的 0有几个，再将再们相加——比如，10 × 100 = 1000。

So let’s take the numbers in the fourth and fifth cells and multiply down from there. 

20 and 30 each have one zero, so the product of both cells will have two trailing zeros, while the product of either cell and an adjacent non-zero-ending cell will have only one. 

When we continue all the way down, we end up with 35 zeros in the bottom cell. 

And if you’re not too stressed about the potential nano-rabbit apocalypse, you might notice that counting the zeros this way forms part of Pascal’s triangle. 

Adding those 35 zeros to the 46 digit number we had before yields an 81 digit number – too big for the habitat to contain! 

You rush over and pull the emergency switch just as the seventh generation of rabbits was about to mature – hare-raisingly close to disaster.

所以，我们从第二排第4、第5间的数目着手开始向下相乘。

20和30尾数都有1个0，所有这两间相乘尾数会有2个0，这两间小屋和其隔壁 数目尾数非0小屋相乘得到的乘积尾数只有1个0。

我们继续向下计算，会算出在最底层小屋中的 兔子数目尾数有35个0。

如果你不太担心纳米兔 可能带来的灾难，你可能会注意到用这种方式计算0会形成杨辉三角形。

将那35个0加在我们之前 算出的46位数字后面得到一个81位数数字——超出小屋的容量！

你跑过去，拉下紧急开关，此时第7代兔子刚要长大——差一点点就要发生大灾难...