Can you solve the multiplying rabbits riddle

你能解決倍增的兔子之謎嗎

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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…

經(jīng)過數(shù)年的實(shí)驗(yàn)，你終于創(chuàng)造出了未來寵物——納米兔子！

它們很小且毛茸茸的.......它們繁殖的速度快到肉眼看不見。在你的實(shí)驗(yàn)室里，有36間"孵化小屋"，以倒金字塔形式排列。最上面一層有8間。第1間有一只兔子，第2間有2只兔子，以此類推，最后一間有8只兔子。其它層的"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.

這些兔子雌雄同籠，在每一間中的每一只兔子都會(huì)與橫向隔壁間的兔子交配一次，每次都生出一只新兔子。

新生出來的兔子會(huì)直接落到父母住處正下方的那件“小屋”，在幾分鐘之內(nèi)就會(huì)長(zhǎng)大， 可以換它們開始交配。

每一間“小屋”可以容納?10^80只納米兔子——也就是是1后面加80個(gè)0——超過這個(gè)數(shù)字的話，他們就會(huì) 逃出“小屋”，占領(lǐng)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

你通過計(jì)算，得出在最低下的“小屋”中兔子的個(gè)數(shù)只有46位數(shù)——空間綽綽有余但當(dāng)你拉動(dòng)操作桿要開始實(shí)驗(yàn)時(shí)，你的助理帶著壞消息跑來。

你競(jìng)爭(zhēng)對(duì)手的實(shí)驗(yàn)室 破壞了你的程序代碼，所以你最后算出的結(jié)果 其實(shí)最后面的0都被切掉了。

那就表示你其實(shí)不知道最下面的“小屋”是否能 容納所有的兔子——但繁殖已經(jīng)開始了！

更糟糕的是，你的設(shè)備和計(jì)算器都出故障了，你只有幾分鐘的時(shí)間手動(dòng)計(jì)算。

最底層小屋中兔子的數(shù)量應(yīng)該含有多少個(gè)0？

你需要去拉動(dòng)緊急關(guān)閉的控制桿嗎？

【如果你想要自己解題 請(qǐng)?jiān)谶@里將影片暫?！?/p>

【答案即將公布：倒計(jì)時(shí)3】

【答案即將公布：倒計(jì)時(shí)2】

【答案即將公布：倒計(jì)時(shí)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.

我們沒有足夠的時(shí)間算出 最后一間小屋中的兔子數(shù)目。

好消息是，我們并不需要算出它。

我們只需要算出后面有多少個(gè)0。

但我們?cè)趺茨茉诓挥?jì)算出 這個(gè)數(shù)目的情況下知道后面有幾個(gè)0呢？

我們只知道，底層小屋的兔子數(shù)目是經(jīng)過相乘運(yùn)算得到的——字面意義。

每間小屋的兔子數(shù)目是上面兩間小屋中的兔子數(shù)目相乘。

只有兩種方法可以透過乘法得到 后面有0的數(shù)字：把尾數(shù)是5的數(shù)字 和任何偶數(shù)相乘?；蚴菍⒈旧砦矓?shù)就是0的數(shù)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.

讓我們來算一下 第二層的兔子數(shù)目，看看會(huì)發(fā)現(xiàn)什么規(guī)律。

有兩個(gè)數(shù)字的尾數(shù)有0——第4間有20只兔子， 第5間有30只。

沒有尾數(shù)是5的數(shù)字出現(xiàn)。

因?yàn)槿粢高^乘法 得到尾數(shù)是5的數(shù)字，一定要用尾數(shù)是5的數(shù)字來乘，因此可確定下面也不會(huì) 出現(xiàn)尾數(shù)是5的數(shù)字。

那就表示:?我們只需擔(dān)心本身末尾是0的數(shù)字。

有個(gè)小妙招可以知道 乘積的尾數(shù)有幾個(gè)0，那就是算出每個(gè)因數(shù)尾數(shù)的?0有幾個(gè)，再將再們相加——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，我們從第二排第4、第5間的數(shù)目著手開始向下相乘。

20和30尾數(shù)都有1個(gè)0，所有這兩間相乘尾數(shù)會(huì)有2個(gè)0，這兩間小屋和其隔壁 數(shù)目尾數(shù)非0小屋相乘得到的乘積尾數(shù)只有1個(gè)0。

我們繼續(xù)向下計(jì)算，會(huì)算出在最底層小屋中的 兔子數(shù)目尾數(shù)有35個(gè)0。

如果你不太擔(dān)心納米兔 可能帶來的災(zāi)難，你可能會(huì)注意到用這種方式計(jì)算0會(huì)形成楊輝三角形。

將那35個(gè)0加在我們之前 算出的46位數(shù)字后面得到一個(gè)81位數(shù)數(shù)字——超出小屋的容量！

你跑過去，拉下緊急開關(guān)，此時(shí)第7代兔子剛要長(zhǎng)大——差一點(diǎn)點(diǎn)就要發(fā)生大災(zāi)難...