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After many adventures in Wonderland, Alice has once again found herself in the court of the temperamental Queen of Hearts.
在仙境经历了许多冒险后，爱丽丝发现自己再一次来到了喜怒无常的红皇后的庭院。
She's about to pass through the garden undetected, when she overhears the king and queen arguing.
正当她要悄悄溜过花园时，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. "If 64 were the same as 65, then it would be 65 and not 64 at all."
爱丽丝不假思索就插话了。“胡说，”她说，“64要是和65一样，它就是65，根本不是64了。”
"What? How dare you!" the queen huffs. "I'll prove it right now, and then it's off with your head!"
皇后发怒道，“什么？你好大的胆子！我现在就来证明，你就等着掉脑袋吧！”
Before she can protest, Alice is dragged toward a field with two chessboard patterns -- an 8 by 8 square and a 5 by 13 rectangle.
还没来得及反抗，爱丽丝就被拽到了一块空地，那里有两个棋盘图案--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 and lie down next to each other, covering the first chessboard.
皇后拍了拍手后，来了四个外形古怪的士兵他们相邻躺下，把第一个棋盘盖住了。
Alice sees that two of them are trapezoids with non-diagonal sides measuring 5x5x3, while the other two are long triangles with non-diagonal sides measuring 8x3.
爱丽丝看见其中两个士兵是梯形的，斜边以外的边长是5x5x3，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." The queen claps her hands again.
“看，这就是64。”皇后又拍了拍手。
The card soldiers get up, rearrange themselves, and lie down atop the second chessboard. "And that is 65."
纸牌士兵们站了起来，重新排列，然后躺下，盖住了第二个棋盘。“而这就是65。”
Alice gasps. She's certain the soldiers didn't change size or shape moving from one board to the other.
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？
Just as things aren't looking too good for Alice, she remembers her geometry, and looks again at the trapezoid and triangle soldier lying next to each other.
就在情况看起来对爱丽丝很不利时，她想到了几何。她又看了看相邻躺下的梯形和三角形士兵。
They look like they cover exactly half of the rectangle, their edges forming one long line running from corner to corner.
他们貌似正好盖住了半个长方形，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 should be the same.
如果这是真的，他们斜边的斜率就应该是一样的。
But when she calculates these slopes using the tried and true formula "rise over run," a most curious thing happens.
但是当她用斜率公式“竖直位移比水平位移”计算斜率时，神奇的事情发生了。
The trapezoid soldier's diagonal side goes up 2 and over 5, giving it a slope of two fifths, or 0.4.
梯形士兵的斜边是竖直2，水平5，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, making its slope three eights, or 0.375. They're not the same at all!
但三角形士兵的斜边是竖直3，水平8，斜率是3/8，或者说0.375。它们根本就不一样！
Before the queen's guards can stop her, Alice drinks a bit of her shrinking potion to go in for a closer look.
在皇后的守卫阻止她之前，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, forming a parallelogram that stretches the entire length of the board and accounts for the missing square.
的确，在三角形和梯形之间存在一个微小的间隙，形成了一个从棋盘的一角延伸到对角的平行四边形，这也解释了少掉的方格去了哪里。
There's something even more curious about these numbers: they're all part of the Fibonacci series, where each number is the sum of the two preceding ones.
这些数字还有更奇妙的特征：它们都是斐波那契数列的一部分，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 that's one more or one less than the product of the Fibonacci numbers on either side of it.
首先，一个斐波那契数的平方比相邻它的两个数的乘积多1或者少1。
In other words, 8 squared is one less than 5 times 13, while 5 squared is one more than 3 times 8.
换句话说，8的平方比5乘13少1。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. That's what allows devious royals to construct slopes that look deceptively similar.
实际上是非常相近，以至于最后收敛到了黄金比例的数值。这也是为什么阴险的皇室能构建出看似一致的斜率。
In fact, the Queen of Hearts could cobble together an analogous conundrum out of any four consecutive Fibonacci numbers.
实际上，红皇后用任意四个连续的斐波那契数，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 and an accomplished mathematician who studied this very puzzle -- one can't believe impossible things.
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刘易斯·卡罗尔所说的，“一个人不能相信不可能的事情。”
