Did you solve it? The magic of the Borromean rings

Earlier today I set the following puzzle, inspired by the Borromean rings (left), 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 is 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:

Next, 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, such as:

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, M. Demaine, Y. Minsky, J. Mitchell, R. 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 (left) is the logo of the International Mathematical Union , which is having its centennial on September 27 and 28. The schedule features talks by 17 of the world’s top mathematicians and will be broadcast live.

I set a puzzle here every two weeks on a Monday. I’m always on the look-out for great puzzles. If you would like to suggest one, email me.

I’m the author of several books of puzzles, most recently the Language Lover’s Puzzle Book. I also give school talks about maths and puzzles (restrictions allowing). If your school is interested please get in touch.

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