The last 60 bees of the species are in your terrarium.

這種蜜蜂的最後六十隻都在你的養育箱中。

You've already constructed wire frames of the appropriate size and shape.

你已經做好了尺寸和形狀都很適當的線框稿。

Now you need to turn them into working beehives by helping the bees fill every hex with wax.

現在為了將它們轉變為可使用的蜂巢， 你需要協助蜜蜂將六角形裝滿蜂蠟。

There are two ways to fill a given hex.

給定一個六角形，有兩種方式可以裝滿它。

The first is to place a bee into it.

第一：將一隻蜜蜂放在裡面。

Once placed, a bee cannot be removed without killing it.

一旦把蜜蜂放進去，需要殺死牠才能移除牠。

The second: if at any point an unfilled hex has three or more neighboring wax-filled hexes, the bees already in the hive will move in and transform it.

第二：如果在任何時點未裝滿的六角形，或至少有三邊鄰接到裝滿蠟的六角形，已經在蜂巢中的蜜蜂會移入，將之裝滿。

Once the bees have transformed every hex in a hive, you can place an additional bee inside and it'll specialize into a queen.

一旦蜜蜂將蜂巢中所有的六角形都裝滿，你便可以再放一隻額外的蜜蜂進去，牠會進化成為女王。

The hive, if well cared for, will eventually produce new bees and continue the species.

如果好好照顧蜂巢，最終它就會產生出新蜜蜂來，讓這個物種延續。

If there are no hexes with three or more transformed neighbors, the bees will just sit and wait.

如果沒有任何六角形邊鄰接到三個以上裝滿的六角形，蜜蜂就會坐著等待。

And once a bee transforms a hex, it can never become a queen.

當一隻蜜蜂把一個六角形裝滿，牠就不可能成為女王。

You could put 59 bees in one wire hive, wait till they transform all the hexes, and then create a queen.

你可以把五十九隻蜜蜂放入蜂巢原型中，等牠們將所有的六角形裝滿，接著再創造一隻女王。

But then just one collapse would end the species.

但只要一出狀況，整個物種就會滅亡。

The more viable hives you can make now, the better.

最好盡量做出更多可用的蜂巢。

So how many can you make with 60 bees?

所以，你能用六十隻蜜蜂做出幾個蜂巢？

Pause the video to figure it out yourself.

請在此暫停以解題，答案即將公佈：

Answer in 3

三

Answer in 2

二

Answer in 1

一

Answer in 0

零

What you're looking for here is some kind of self-sustaining chain reaction, where a small number of bees will transform an entire hive.

在這裡，你要找的是一種自我維持的連鎖反應，讓少量的蜜蜂將整個蜂巢裝滿。

The lower the number of bees needed, the better.

需要的蜜蜂數量越少越好。

So how low can we go, and how can we engineer a chain reaction?

我們能把數量壓到多低？又要如何設計連鎖反應？

Let's start with the first question.

我們先從第一個問題開始。

There's a really clever approach to this, which involves counting the sides of the filled-in hexes, and examining their total perimeter.

有一個相當聰明的方法可以用，這個方法需要計算裝滿的六角形有多少個邊，再計算它們的總邊長。

Let's suppose we put bees in these three hexes.

假設我們把蜜蜂放在這三個六角形中。

The total transformed perimeter has 18 sides.

被裝滿的六角形總共有十八個邊。

But the middle hex has three transformed neighbors, so the bees will transform it too.

但中間的六角形有三邊鄰接到裝滿的六角形，所以蜜蜂也會把這個六角形裝滿。

What happens to the perimeter?

周長會如何？

It's still 18!

仍然是十八！

And even after the bees transform the next sets of hexes with three neighbors, it still won't change.

就算蜜蜂把後續其他有三邊鄰接的六角形也裝滿，總邊長仍然不會變。

What's going on here?

這是怎麼回事？

Each hex that has at least three sides touching the bee-friendly space will remove those sides from the perimeter when it transforms.

任何六角形，只要至少有三邊接觸到對蜜蜂友善的空間，在這些六角形被裝滿時，就會從總邊長中除去那幾個邊。

Then it adds at most three new sides to the perimeter.

接著最多會增加三條新的邊到總邊長當中。

So the perimeter of the transformed hexes will either stay the same or shrink.

所以被裝滿的六角形的總邊長，若不是不變，就是變少。

The final perimeter of the entire hive is 54, so the total perimeter of the hexes we place bees in at the start must be at least 54 as well.

整個蜂巢的最終總邊長是五十四，所以一開始，需要我們放蜜蜂進去的那些六角形，總邊長必須至少是五十四。

Dividing that fifty-four by the six sides on each non-adjacent hex tells us it'll take at least 9 bees to transform the entire hive.

將五十四除以六個邊（每個不相鄰的六角形有六個邊）就可以知道我們至少需要九隻蜜蜂才能將整個蜂巢裝滿。

That's a great start, but we still have the tough question of where the nine bees should go, and if we'll need more.

那是個好的開始，但我們仍然要解決一個難題：這九隻蜜蜂要放在哪裡？還是我們需要不只九隻？

Let's think smaller.

我們先想小一點。

We already know that three bees could completely transform a hive this big.

我們已經知道三隻蜜蜂可以把這麼大的蜂巢完全裝滿。

What about a slightly bigger one?

那稍微大一點的呢？

The perimeter of this hive is 30, which means we'll need at least 5 bees to fill it in.

這個蜂巢的邊長是三十，那就表示我們至少需要五隻蜜蜂才能把它裝滿。

With 6 it'd be easy.

有六隻就很容易。

Placing them like this would fill out the whole hive in just three steps.

把蜜蜂像這樣放置，就能在三個步驟之後裝滿整個蜂巢。

But we can do better!

但我們還能做得更好！

We don't actually need to place a bee on this hex, since the other bees will transform that spot on their own.

我們並不需要在這個六角形當中放置蜜蜂，因為其他蜜蜂會靠牠們自己把這個六角形裝滿。

It looks like we have the beginning of a pattern.

看起來，我們已經有了一種模式的開端。

Can we extend it to our full hive?

我們能把它延伸到整個蜂巢嗎？

That would mean placing our 9 bees like so.

這表示我們要將九隻蜜蜂這樣排列。

Once they get to work, they'll create a chain reaction that fills in the center of the hive and extend it to its edges.

當他們開始工作後，會製造一連串連鎖反應將蜂巢中央填滿，再延伸到邊緣。

Add a 10th bee to the completed hive and it becomes a queen.

加入第十隻蜜蜂讓蜂巢完美，這只會變成女王。

Repeat that process five more times and you've helped the last 60 members of Apis trifecta create 6 producing hives.

重複這個過程五次，你就能幫助最後六十隻 Apis trifecta 創造六個有生產力的蜂巢。

All in all, it's a pretty good bee-ginning.

總之，這會是很好的開端。