There is a **church** which insists on **ringing its bells** every 15 minutes, and for hours every Saturday next to my new apartment. I remember hearing that **there is some maths** behind the music they play, and indeed there is! Say they have 4 bells, then ‘*ringing the changes*’ involves ringing **all 24 possible sequences** of these four bells one after the other- with some rules. The first rule is that you must **start and end with** the sequence **1, 2, 3, 4**. The second rule is that each sequence must only be obtained from the previous one **by swapping bells that were played next to each other**. So the second one could be 2, 1, 3, 4- but not 4, 2, 3, 1.

The **animation** above **shows how this can be done**. Possible sequences are shown as rings, which are played clockwise from the top. Two rings are **joined together if the transition is allowed**, and the **thick red line shows the solution**. This song is synthesised in the recording above. There is clearly a lot of **symmetry** and mathematical **structure** in the problem and its solution, but it doesn’t make it any less *annoying*. [more] [code]