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

翻译人员: Chuqi Hu 校对人员: Yolanda Zhang

After many adventures in Wonderland,

在仙境经历了许多冒险后，

Alice has once again found herself in the court

爱丽丝发现自己再一次来到了

of the temperamental Queen of Hearts.

喜怒无常的红皇后的庭院。

She’s about to pass through the garden undetected,

正当她要悄悄溜过花园时，

when she overhears the king and queen arguing.

她听到了国王和皇后在争论

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

“显而易见，” 皇后说， “64 和 65 是一样的，就这样。”

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

爱丽丝不假思索就插话了。 “胡说，”她说，

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

“64 要是和 65 一样， 它就是 65，根本不是 64 了。”

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

皇后发怒道， “什么？你好大的胆子！”

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

“我现在就来证明， 你就等着掉脑袋吧！”

Before she can protest,

还没来得及反抗，

Alice is dragged toward a field with two chessboard patterns—

爱丽丝就被拽到了一块空地， 那里有两个棋盘图案——

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

一个是 8x8 的正方形， 另一个是 5x13 的长方形。

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

皇后拍了拍手后， 来了四个外形古怪的士兵

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

他们相邻躺下， 把第一个棋盘盖住了。

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

爱丽丝看见其中两个士兵是梯形的， 斜边以外的边长是 5x5x3，

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

另外两个士兵是三角形， 斜边之外的两个边长是 8x3。

“See, this is 64.”

“看，这就是64。”

The queen claps her hands again.

皇后又拍了拍手，

The card soldiers get up, rearrange themselves,

纸牌士兵们站了起来， 重新排列，

and lie down atop the second chessboard.

然后躺下，盖住了第二个棋盘。

“And that is 65."

“而这就是65。”

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

爱丽丝一惊。她敢肯定士兵们 从一个棋盘移到另一个棋盘时

moving from one board to the other.

没有改变大小和形状。

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

但从数学角度出发， 皇后肯定以某种方式作弊了。

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

在丢掉脑袋之前， 爱丽丝能想出问题出在哪里吗——

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

【请暂停视频来思考答案】 【答案即将公布：3】

Answer in 2

【2】

Answer in 1

【1】

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

就在情况看起来对爱丽丝很不利时， 她想到了几何。

and looks again at the trapezoid and triangle soldier

她又看了看相邻躺下的

lying next to each other.

梯形和三角形士兵。

They look like they cover exactly half of the rectangle,

他们貌似正好盖住了半个长方形，

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

他们的边缘形成了一条 从一个端点到对角端点的长线。

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

如果这是真的， 他们斜边的斜率

should be the same.

就应该是一样的。

But when she calculates these slopes

但是当她用斜率公式

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

“竖直位移比水平位移” 计算斜率时，

a most curious thing happens.

神奇的事情发生了。

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

梯形士兵的斜边是 竖直 2，水平 5，

giving it a slope of two fifths, or 0.4.

也就是说斜率是 2/5， 或者说 0.4。

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

但三角形士兵的斜边是 竖直 3，水平 8，

making its slope three eights, or 0.375.

斜率是 3/8， 或者说 0.375。

They’re not the same at all!

它们根本就不一样！

Before the queen’s guards can stop her,

在皇后的守卫阻止她之前，

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

爱丽丝喝了点缩小药水， 走近瞧了瞧。

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

的确，在三角形和梯形之间 存在一个微小的间隙。

forming a parallelogram that stretches the entire length of the board

形成了一个从棋盘的一角 延伸到对角的平行四边形。

and accounts for the missing square.

这也解释了少掉的方格去了哪里。

There’s something even more curious about these numbers:

这些数字还有更奇妙的特征：

they’re all part of the Fibonacci series,

它们都是斐波那契数列的一部分，

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

也就是说，每个数字 都是之前两个数字的和。

Fibonacci numbers have two properties that factor in here:

斐波那契数列有两个特性 在这里起到了作用：

first, squaring a Fibonacci number gives you a value

首先，一个斐波那契数的平方

that’s one more or one less

比相邻它的两个数的乘积

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

多 1 或者少 1。

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

换句话说，8 的平方 比 5 乘 13 少 1。

while 5 squared is one more than 3 times 8.

而 5 的平方比 3 乘 8 多 1。

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

其次，连续的两个 斐波那契数的比率很相近。

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

实际上是非常相近，以至于 最后收敛到了黄金比例的数值。

That’s what allows devious royals to construct slopes

这也是为什么阴险的皇室

that look deceptively similar.

能构建出看似一致的斜率。

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

实际上，红皇后用任意四个 连续的斐波那契数，

out of any four consecutive Fibonacci numbers.

都可以设计出类似的谜题。

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

数字越大，不可能的情况 越看起来是真的。

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

但正如爱丽丝梦游仙境的作者， 以及研究了这个谜题的杰出数学家

and an accomplished mathematician who studied this very puzzle—

刘易斯·卡罗尔 （ Lewis Carroll）所说，

one can’t believe impossible things.

“一个人不能相信不可能的事情。”