惊倒朋友的数学魔术

数学可不总是板着一副脸孔。为证明此，这里介绍一个纸牌戏法，它能在酒吧里让你的朋友啧啧称奇。戏法的发明者是阿瑟.本杰明（Arthur Benjamin），他是加利福尼亚哈维玛德学院的数学家兼魔术师。

Take a pack of cards and deals the top 16 face down in four rows of four. Turn four of these cards face up. Call for a volunteer from the audience, who will repeatedly ‘fold up’ the square of cards, like folding a sheet of stamps along the perforations, until it ends up as a single pile of 16 cards.

拿一副纸牌，将顶上的16张面朝下分4排发牌。再将其中4张翻过来。从观众中叫一个志愿者，让他如同沿着齿孔折叠一大张邮票一样，反复“折叠”纸牌方阵，直至得到16张纸牌的单撂为止。

When the cards have been folded into one pile, the volunteer spreads them out on the table. Either 12 cards are face down and 4 face up, or 4 cards are face down and 12 face up. In the first case, the face-up cards miraculously turn out to be the four aces. In the second case, the volunteer takes the four face-down cards, and turnsthem over to reveal... the four aces.

由观众来决定从哪里折起。例如，第一次可以沿纸牌方阵中三条水平线中任意一条折叠，或者沿三条垂直线中任意一条折叠。

当纸牌折叠成一撂时，志愿者将其摊开在桌子上。此时可能出现两种情形：12张牌面朝下，4张牌面朝上；或者4张牌面朝下，12张牌面朝上。在第一种情形下，面朝上的纸牌不可思议地是4张A。在第二种情形下，志愿者可将4张面朝下的纸牌翻过来揭示结果……还是4张A。

Magic!

太奇妙了！

To prepare the pack, the magician arranges the four aces in positions 1, 6, 11, and 16 from the top down. After dealing out the four rows of cards, the aces lie along the diagonal from top left to bottom right. But they’re face down, so the audience doesn’t see that. He also has to turn over the right four cards, to make the squarelook like this:

为了准备那副纸牌，魔术师将4张A分别安排在从顶上数第1、6、11和16的位置。在分出4行纸牌之后，A正好从左上至右下沿对角线排列。不过因为它们是面朝下的，所以观众不会看到。魔术师还必须翻对4张牌，使得纸牌方阵如下图这样：

The trick then works automatically, no matter which sequence of folds the audience chooses. Magic again!

不管观众选择怎样的折叠序列，戏法之后就自动变出来了。奇妙再现！

Why does the trick work?

为什么戏法能变出来？

Imagine turning the cards along the diagonal the other way up. Then the square has a pattern like a chessboard, and the aces run along the diagonal:

假若沿另一方向的对角线将牌翻过来面朝上（如下图），那么该方阵就类似于棋盘格，且A都在对角线上：

However you fold the square, the cards that end up in a given position will all face the same way: either all up or all down. For instance, suppose you fold along the central vertical line, and think of the top row. The third card (up) turns over (down) and is placed on top of the second card --- also down. And the fourth card (down)turns over (up), and is placed on top of the first card --- also up.

无论怎么折叠这个方阵，最终落在某个指定位置上的纸牌朝向也都相同：要不都面朝上，要不都面朝下。例如，假定你沿垂直中线折叠，且看看第一行的变化。第三张牌（面朝上的）翻过来（面朝下），盖在第二张牌之上——第二张牌也是面朝下的。第四张牌（面朝下的）翻过来（面朝上），盖在第一张牌之上——第一张牌也是面朝上的。

Now you have a rectangle, made from cards or small piles of cards, and it has the same chessboard pattern of ups and downs. So the same thing happens for the next fold, and the next... By the time you reach a single pile, all of the cards in the pile face the same way.

现在得到的是一个矩形阵，由纸牌或小撂纸牌组成。它是朝向一致的棋盘格。所以下次折叠时同样的事还会发生，以此类推……当你得到单撂牌时，所有纸牌的朝向都是相同的。

However, when you started, the cards on the diagonal were the wrong way up compared to the chessboard pattern. After folding, they will again be the wrong way up. So instead of a pile of 16 cards all facing the same way, you will have a pile with 12 cards facing one way, and the four aces facing the other.

不过，在开始时，对角线上的纸牌是棋盘格的相反朝向。折叠后，它们也会朝向相反。所以，这撂牌中的16张不是朝向同一方向的，而是12张朝向一个方向，4张A朝向另一个方向。2

Mathematically, the chessboard pattern has ‘colour symmetry’. The fold lines act like mirrors, and the mirror-image of each card sits on top of a card that faces the other way. This idea is used to study how the atoms in crystals are arranged. The cunning bit is to turn the maths into an effective card trick.

数学上，棋盘格具有“着色对称性”。折叠线就类似于镜子，而每张纸牌的镜像置于另一张朝向不同的纸牌之上。该种思想用来研究晶体的原子排列。有点巧妙的地方是怎样让数学变成奏效的纸牌戏法。

Ian Stewart FRS is an Emeritus Professor of Mathematics at Warwick University.

作者Ian Stewart FRS是华威大学数学名誉教授。