Can you solve the Alice in Wonderland riddle? - Alex Gendler

譯者: Lilian Chiu 審譯者: Helen Chang

After many adventures in Wonderland,

在仙境經歷了數次冒險之後，

Alice has once again found herself in the court

愛麗絲再次來到

of the temperamental Queen of Hearts.

易怒的紅心皇后的宮庭。

She’s about to pass through the garden undetected,

她正要偷偷摸摸穿過花園時，

when she overhears the king and queen arguing.

不小心聽見了國王和皇后在爭論。

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

皇后說：「很簡單，」

「64 和 65 一樣，就這樣。」

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

想也沒想，愛麗絲 就插嘴：「胡說。」

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

「如果 64 和 65 一樣，

那它就是 65，而不是 64 了。」

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

皇后惱火：「什麼？ 你好大的膽子！」

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

「我現在就能證明， 接著就要砍掉你的頭！」

Before she can protest,

在能抗議之前，

愛麗絲就被拖向了 有兩個棋盤圖案的場地——

Alice is dragged toward a field with two chessboard patterns—

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

8 乘以 8 的正方向 與 5 乘以 13 的長方形。

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

皇后拍手之後便來了 四個外表奇怪的士兵，

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

他們相鄰躺下，把第一個棋盤蓋住。

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

愛麗絲看見其中兩個士兵是梯形的，

斜邊以外的三個邊長為 5x5x3，

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

另外兩個士兵是長三角形，

斜邊以外的兩個邊長為 8x3，

“See, this is 64.”

「看，這是 64。」

The queen claps her hands again.

皇后再次拍手。

The card soldiers get up, rearrange themselves,

撲克牌士兵爬起來，重新排列，

and lie down atop the second chessboard.

躺在第二個棋盤上。

“And that is 65."

「而這是 65。」

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

愛麗絲很驚訝。

她肯定士兵從一個棋盤 到另一個棋盤時，

moving from one board to the other.

形狀大小都沒有變。

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

但皇后一定有以某種方式作弊， 從數學上來看絕對是如此。

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

愛麗絲能否想通問題在哪裏 ——在她輸掉之前？

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

請在此暫停解題。 答案即將公佈：三

Answer in 2

答案即將公佈：二

Answer in 1

答案即將公佈：一

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

當情況看似對愛麗絲不利時，

她想起了她學過的幾何學，

and looks again at the trapezoid and triangle soldier

再次去觀察併排躺下的梯形

和三角形士兵。

lying next to each other.

They look like they cover exactly half of the rectangle,

看起來，他們剛好蓋住半個長方形，

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

他們的邊緣形成 連接兩個對角的長邊。

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

如果真是如此，那麼他們 斜邊的斜率應該相同。

should be the same.

But when she calculates these slopes

但當她使用數學上已證實的公式，

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

用高度除以長度 去計算這些斜率之後，

a most curious thing happens.

最妙的事情發生了。

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

梯形士兵的斜邊是 2 除以 5，

giving it a slope of two fifths, or 0.4.

也就是 2/5 或 0.4。

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

然而，三角形士兵則是 3 除以 8，

making its slope three eights, or 0.375.

斜率為 3/8 或 0.375。

They’re not the same at all!

兩個斜率完全不一樣！

Before the queen’s guards can stop her,

皇后的守衛還沒能阻止愛麗絲，

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

她就喝下了縮小藥水，去看個清楚。

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

在三角形和梯形之間 的確有個小小的縫，

forming a parallelogram that stretches the entire length of the board

形成一個平行四邊形， 從棋盤的一端到另一端，

and accounts for the missing square.

正好代表失蹤的那一塊 1x1 正方形。

There’s something even more curious about these numbers:

這些數字還有神奇之處，

they’re all part of the Fibonacci series,

它們都屬於費氏數列，

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

也就是說，每個數字 都是前兩個數字的加總。

Fibonacci numbers have two properties that factor in here:

在這裡，費氏數列的數字 有兩項特性十分重要：

first, squaring a Fibonacci number gives you a value

第一，費氏數列中 任何一個數字的平方，

會比它在數列中前後兩個數字相乘

that’s one more or one less

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

多 1 或少 1。

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

換言之，8 的平方 會比 5 乘以 13 少 1，

while 5 squared is one more than 3 times 8.

而 5 的平方會比 3 乘以 8 多 1。

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

第二，

費氏數列中，兩個相鄰 數字的比值會非常相近。

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

相近到它最終會收斂到黃金比率。

That’s what allows devious royals to construct slopes

因為如此， 狡詐的皇室成員才能

做出看起來相似到可以唬人的斜邊。

that look deceptively similar.

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

事實上，紅心皇后 可以用費氏數列中的

out of any four consecutive Fibonacci numbers.

任何四個連續數字 拼湊出類似的難題。

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

數字越大，不可能的事就越像真的。

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

但，引用路易斯卡羅的話—— 《愛麗絲夢遊仙境》的作者

and an accomplished mathematician who studied this very puzzle—

及研究了這個謎題的高超數學家——

one can’t believe impossible things.

我們不能去相信不可能的事。