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结节@TED | 你能解决时间旅行难题吗？( 二 )


This question is so rich that an entirebranch of mathematics known as Ramsey Theory developed from it. Ramsey Theory is home to somefamously difficult problems. This one isn’t easy, but it can be handled if you approach it systematically. Imagine you brought just three nodules.
这个问题内涵丰富 ，知名的拉姆齐理论的整个数学分支都由此发展而来 。拉姆齐理论包含了许多著名难题 。本题虽不简单 ， 但通过系统性地分析 ，还是可以解决的 。假设只带三个结节 。
Would that be enough? No - for example,you might have two blue and one red connection,and be stuck in the past forever. Would four nodules be enough?No - there are many arrangements here that don’t give a blue or red triangle. What about five? It turns out there is an arrangement ofconnections that avoids creating a blue or red triangle. These smaller triangles don’t count becausethey don’t have a nodule at each corner.
三个够吗？不够 。 举个例子 ， 可能会出现两蓝一红的连接 ，那么你会被永远地困在过去 。四个够不够？不够 。 因为存在多种排列方式 可以使蓝色或红色三角不成立 。那么五个呢？ 还是会有一种排列方式 ，会使蓝色或红色三角不成立 。 中间这些小三角形不算 ， 因为它们的各顶点不是结节 。
However, six nodules will always create ablue triangle or a red triangle. Here’s how we can prove that withoutsorting through every possible case.
但是 ， 六个结节将始终可以构成蓝色或红色三角形 。下面是不用排除法 ， 我们证明这个结论的方法 。
Imagine activating the sixth nodule, and consider how it might connectto the other five. It could do so in one of six ways: with five red connections, five blueconnections, or some mix of red and blue. Notice that every possibility has at leastthree connections of the same color coming from this nodule.
试想 ， 激活第六个结节时 ， 它将如何与其它五个连接 。一定会是下列六种情况中的一种： 五红 ， 五蓝 ， 或红蓝混合 。可以发现 ， 所有可能性中该结节的连接至少有三个是同色的 。
Let’s look at just the noduleson the other end of those same three color connections. If the connections were blue, then any additional blue connection betweenthose three would give us a blue triangle. So the only way we could get in trouble is if all the connectionsbetween them were red. But those three red connectionswould give us a red triangle. No matter what happens,we’ll get a red or a blue triangle, and open our doorway. On the other hand, if the original three connections were all red instead of blue, the same argument still works, with all the colors flipped. In other words, no matter how theconnections are colored, six nodules will always create a red or blue triangle and a doorway leading home.
只看这三条同色连接的另一端的结节 ，如果连接为蓝色线 ，那么在这三个结节之间任意位置添加蓝色连接都会构成蓝色三角形 。所以唯一会让我们遇到麻烦的情形是 ，结节间的连接全部是红色 。但是三个红色连接将会构成一个红色三角形 。无论怎样 ， 我们都会得到一个红色或蓝色三角形 ，打开传送门 。另一方面 ，如果原来的三个连接均为红色 ， 而非蓝色 ，所有颜色翻个样 ， 同样的结论依然成立 。换句话说 ， 不论连接是什么颜色 ，六个结节始终能组成一个红色或蓝色三角 ， 打开回家的大门 。


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