Alice has once again found herself in the court

愛麗絲又一次發現自己在法庭上

of the temperamental Queen of Hearts.

的喜怒無常的紅桃皇后。

She's about to pass through the garden undetected,

她就要穿過花園而不被發現了。

when she overhears the king and queen arguing.

當她無意中聽到國王和王后在爭吵時。

“It's quite simple,” says the queen. “64 is the same as 65, and that's that.”

"這很簡單，"女王說。"64和65是一樣的，就是這樣。"

Without thinking, Alice interjects. “Nonsense,” she says.

愛麗絲不假思索地插了一句。 "胡說八道。"她說。

“If 64 were the same as 65, then it would be 65 and not 64 at all.”

"如果64和65一樣，那就是65，根本就不是64。"

“What? How dare you!” the queen huffs.

"什麼，你怎麼敢！"女王氣急敗壞。

“I'll prove it right now, and then it's off with your head!”

"我現在就證明給你看，然後就把你的頭砍下來！"

Before she can protest,

在她抗議之前。

Alice is dragged toward a field with two chessboard patterns—

愛麗絲被拖向一個有兩個棋盤圖案的領域----------。

an 8 by 8 square and a 5 by 13 rectangle.

一個8乘8的正方形和一個5乘13的長方形。

As the queen claps her hands, four odd-looking soldiers approach

當王后拍手時，四個長相怪異的阿兵哥走了過來

and lie down next to each other, covering the first chessboard.

並挨著躺下，蓋住第一塊棋盤。

Alice sees that two of them are trapezoids with non-diagonal sides measuring 5x5x3,

愛麗絲看到，其中有兩個是非對角線邊長為5x5x3的梯形。

while the other two are long triangles with non-diagonal sides measuring 8x3.

而另外兩個是非對角線邊長為8x3的長三角形。

“See, this is 64.”

"看，這是64號。"

The queen claps her hands again.

女王又拍了拍手。

The card soldiers get up, rearrange themselves,

卡牌兵們起身，重新佈置。

and lie down atop the second chessboard.

並躺在第二張棋盤上。

“And that is 65."

"那就是65歲。"

Alice gasps. She's certain the soldiers didn't change size or shape

愛麗絲喘息著。她確定那些阿兵哥沒有改變大小和形狀。

moving from one board to the other.

從一個板塊移動到另一個板塊。

But it's a mathematical certainty that the queen must be cheating somehow.

但從數學上來說，皇后一定是在作弊，這是一種數學上的肯定。

Can Alice wrap her head around what's wrong— before she loses it?

愛麗絲能不能在她失去理智之前想明白什麼是錯的？

Pause the video to figure it out yourself. Answer in 3.

暫停視頻，自己想辦法。回答在3。

Answer in 2

答覆2

Answer in 1

回答1

Just as things aren't looking too good for Alice, she remembers her geometry,

就在愛麗絲的情況不怎麼好的時候，她想起了自己的幾何。

and looks again at the trapezoid and triangle soldier

又看了看梯形和三角形的戰士

lying next to each other.

躺在旁邊的。

They look like they cover exactly half of the rectangle,

它們看起來像正好覆蓋了長方形的一半。

their edges forming one long line running from corner to corner.

它們的邊緣形成一條長線，從角到角。

If that's true, then the slopes of their diagonal sides

如果是這樣，那麼它們的對角線的斜率為

should be the same.

應該是一樣的。

But when she calculates these slopes

但當她計算這些斜坡時

using the tried and true formula "rise over run,"

利用屢試不爽的公式 "漲過跑"

a most curious thing happens.

一件很奇怪的事情發生了。

The trapezoid soldier's diagonal side goes up 2 and over 5,

梯形兵的對角線面上升2，超過5。

giving it a slope of two fifths, or 0.4.

使其斜率達到五分之二，即0.4。

The triangle soldier's diagonal, however, goes up 3 and over 8,

而三角兵的對角線，則上升3個，超過8個。

making its slope three eights, or 0.375.

使其斜率達到了八分之三，即0.375。

They're not the same at all!

他們根本就不一樣!

Before the queen's guards can stop her,

在女王的衛兵阻止她之前

Alice drinks a bit of her shrinking potion to go in for a closer look.

愛麗絲喝了一點她的縮小藥水，要進去仔細看看。

Sure enough, there's a miniscule gap between the triangles and trapezoids,

果然，三角形和梯形之間的差距微乎其微。

forming a parallelogram that stretches the entire length of the board

形成一個平行四邊形，延伸整個板子的長度。

and accounts for the missing square.

並交代了缺失的方。

There's something even more curious about these numbers:

這些數字還有更奇怪的地方。

they're all part of the Fibonacci series,

它們都是斐波那契系列的一部分。

where each number is the sum of the two preceding ones.

其中每一個數字都是前面兩個數字之和。

Fibonacci numbers have two properties that factor in here:

斐波那契數有兩個特性，在這裡也是一個因素。

first, squaring a Fibonacci number gives you a value

首先，對斐波那契數進行平方化處理，就會得到一個值。

that's one more or one less

孰多孰少

than the product of the Fibonacci numbers on either side of it.

比它兩邊的斐波那契數的乘積。

In other words, 8 squared is one less than 5 times 13,

換句話說，8的平方是13的5次方少一個。

while 5 squared is one more than 3 times 8.

而5的平方則是8的3倍多1。

And second, the ratio between successive Fibonacci numbers is quite similar.

其次，連續的斐波那契數之間的比例也很相似。

So similar, in fact, that it eventually converges on the golden ratio.

如此相似，事實上，它最終趨於黃金比例。

That's what allows devious royals to construct slopes

這就是讓狡猾的王室成員建造斜坡的原因。

that look deceptively similar.

看起來很相似的。

In fact, the Queen of Hearts could cobble together an analogous conundrum

事實上，紅桃皇后可以湊出一個類似的難題

out of any four consecutive Fibonacci numbers.

在任何四個連續的斐波那契數中。

The higher they go, the more it seems like the impossible is true.

越是往高處走，越是覺得不可能的事情是真的。

But in the words of Lewis Carroll— author of Alice in Wonderland

但用《愛麗絲夢遊仙境》的作者劉易斯-卡羅爾的話說就是：

and an accomplished mathematician who studied this very puzzle—

和一位研究這個謎題的傑出數學家-----。

one can't believe impossible things.

人不能相信不可能的事情。