You are asked to arrange four tiles into a 2 x 2 square, according to the following rule:要求您根据以下规则将四块瓷砖拼成一个 2×2 的正方形：
- Tiles may only touch each other at sides that have exactly the same symbol.瓷砖只能在具有完全相同符号的边相互接触。
Example: 例子:
You are now given the five tiles below.现在给你下面这五块拼图。
You must arrange four of these five tiles into a 2 x 2 square that follows the rule above. (In this case, there is only one possible choice of four tiles that allows this.)您必须从这五块拼图中挑选四块，拼成一个符合上述规则的 2×2 正方形。（在这种情况下，只有唯一一组四块拼图能够做到这一点。）
Question:问题:
Which tile will you NOT use? 你不会用哪块瓷砖？
A. 
B. 
C. 
D.
E. 
Explanation 解释
The correct answer is: 正确答案是：
We assign letters to the tiles as follows:我们按如下方式给这些方块分配字母：
It is possible to create a 2 x 2 square that follows the rule using tiles A, B, D and E, as shown below:可以使用 A、B、D 和 E 这些瓷砖拼出一个符合规则的 2×2 正方形，如下图所示：
Trying the many possibilities until one happens to fit is an option, but there is a way to reduce the number of possibilities that have to be checked.尝试各种可能性直到恰好有一个合适是一种选择，但有一种方法可以减少需要检查的可能性的数量。
We do this in two separate steps.我们分两步来做这件事。
- We make a diagram of the five tiles and connect them with a line if they share at least one symbol.我们绘制出这五块拼图的示意图，并在它们至少有一个相同符号时用线将它们连接起来。
Diagram Explanation: C and E are connected by a line, because both tiles have a little black cloud on one of their sides. A and C are not connected, because none of the symbols on A occurs on C. (The simplified diagram on the right is called a graph.)图示说明：C 和 E 由一条线相连，因为这两块瓷砖中各有一面带有小黑云图案。A 和 C 不相连，因为 A 上的图案在 C 上一个都没有。 （右边的简化图称为图。）
Explanation of the term 4-cycle: a path from tile to tile that follows the lines and ends up at the start after 4 steps from one tile to the next. It is important to realize that the four tiles in a 2 x 2 square that follows the rule, correspond to a 4-cycle.术语“4 循环”的解释：沿着线条从一块瓷砖移动到另一块瓷砖，经过 4 步后回到起点的路径。重要的是要认识到，遵循规则的 2×2 方形中的四块瓷砖对应于一个 4 循环。
- With the diagram's help, we find out all the possible 4-cycles. In our diagram, there are three 4-cycles:借助该图，我们找出所有可能的 4 阶环。在我们的图中，有三个 4 阶环：
- The first goes from A to B to D to E and then back to A (That 4-cycle corresponds to our solution).第一个从 A 到 B 再到 D 到 E 然后回到 A（这个 4 个节点的循环对应于我们的解决方案）。
- The second is A-B-C-E-A. 第二个是 A-B-C-E-A。
- The third is B-C-E-D-B. 第三种是 B-C-E-D-B。
Of course, all cyclic permutations of the five letters (i.e. starting from any letter, but keeping the order) are equivalent, as well as their reverse. For example, A-B-C-E-A, B-C-E-A-B and E-C-B-A-E are equivalent.当然，这五个字母的所有循环排列（即从任何字母开始，但保持顺序）都是等价的，它们的逆序排列也等价。例如，A-B-C-E-A、B-C-E-A-B 和 E-C-B-A-E 都是等价的。
Is it possible that the second or third 4-cycle may lead to another solution of our puzzle?第二个或第三个四循环有可能会给我们这个谜题带来另一种解法吗？
Both cycles contain B-C-E, and there are only two possibilities to join these tiles in that order:这两个循环都包含 B-C-E，而且按照这个顺序将这些牌连接起来只有两种可能性：
Neither tile A nor D can be used to complete these to a 2 x 2 square (and still conform to the rule).无论是瓷砖 A 还是 D 都无法用来完成一个 2×2 的正方形（并且仍符合规则）。
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