你解决了吗? The magic of the Borromean rings

Earlier today I set the following puzzle, inspired by the Borromean rings (剩下), which are three interlocking loops with the property that when you remove any one of them, the other two are no longer linked. In the puzzle everything falls apart when one element is removed.

Smash the picture

The picture below shows the conventional way to hang a picture on a wall with two nails. The two nails give each other back-up: if one fails, the picture will still hang (wonkily) on the other nail.

Can you think of a way to hang a picture on a wall using string and two nails, such that if either of the nails fails (and the other one doesn’t) then the picture will fall to the floor?

Solution

There are several ways to solve this puzzle. One way the Borromean rings. Just as they are three interconnected rings that fall apart when one is removed, the puzzle involves three interconnected elements (two nails and a piece of string), and when one is removed (the nail) the other two are no longer linked. Our task is thus to work out exactly how the puzzle models the rings. Here’s how you might go about it. Make a set of Borromean rings with two plastic rings and a piece of string as below:

下一个, separate the rings as if they are nails on the wall.

The way the string loops between the rings is the solution we are after, presented below. If either of the ‘nails’ are removed, we know that the string and the other nail cannot be linked, and thus the painting will crash to the floor.

There are other solutions too, 如:

You may have preferred a physics-style answer that uses force and friction. It may be possible to stick the nails in so close to each other that they clasp a string that holds up a painting. In this case, removing either nail will cause the painting to drop.

If you are interested in this area, the paper Picture-hanging Puzzles by E. Demaine, 中号. Demaine, Y. Minsky, Ĵ. Mitchell, 电阻. Rivest and M. Patrascu has many more examples and touchers on deeper ideas in topology and computer science.

A 3D version of the Borromean rings (剩下) is the logo of the International Mathematical Union , which is having its centennial on September 27 和 28. The schedule features talks by 17 of the world’s top mathematicians and will be broadcast live.

我每两个星期在这里设置一个难题. 我一直在寻找很棒的谜题. 如果您想提出一个建议, 给我发邮件.

我是几本难题的书的作者, 最近 语言爱好者的拼图书. 我还给学校讲数学和拼图 (限制允许). 如果您的学校有兴趣,请 保持联系.

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