Did you solve it? The puzzle with a twist

On my puzzle blog earlier today, I set you the following puzzles:

1) What is numerically interesting about April 25, 1849 - the date of birth of the German mathematician Felix Klein - and why is this year’s anniversary particularly noteworthy?

Solution His birthday is 25/4/1849, which is 2²/5²/43², and he is celebrating his 169th = 13² anniversary. All the numbers are squares. Extra marks if you spotted they are all primes. Klein was destined to be a mathematician!

2) What happens when you draw two parallel lines down a Möbius strip as below and then cut along them?

(If you are reading this for the first time, check out the story where I explain in more detail what a Möbius strip is and why it is interesting.)

Solution

This is what you get: a Möbius strip interlocked with a longer loop with two twists:

Here’s how you might have thought this through. Cutting along the lines is like trimming the sides of a central band. This central band retains the form of a Möbius strip, just a thinner one.

The trimmed edges are both joined, and they become the longer loop with two twists. Remember, in the set up I told you that when you draw a single line down a Möbius strip and cut along it you get a single loop with two twists. In the case with two lines, imagine the two lines becoming closer and closer together. In the limit they are the single line.

If you want more fun with Möbius strips, first make two more bands. Mark a line down the middle of each of them, and stick them together as below:

I guarantee that you will get a huge surprise when you cut down the green lines. You will gasp! It will become your party piece!

Once you have done this connect two Möbius strips in the same way. You can connect two Möbius strips that twist the same way, or that twist in different ways. Now try cutting down the middle.

Get out the scissors pronto!

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 popular maths, including the puzzle books Can You Solve My Problems? and Puzzle Ninja. I also co-write the children’s book series Football School.

Thanks again to Simon Singh for today’s puzzles. Don’t forget to check out his Parallel maths project for 11 to 13-year olds.