# Bell Ringing

Suppose you are a bellringer. Can you find the changes so that,
starting and ending with a round, all the 24 possible permutations
are rung once each and only once?

## Problem

Suppose you are a bellringer holding a rope and you look
around the church tower and see the faces of 3 friends, all about
to start change ringing. To ring a 'round' each bell is rung in
turn (123412341234....). The bells can be rung in any order and
changing the order is known as a 'change'. As your bell goes round
on its wheel you can slow it down, or speed it up, just a little
but not much, so you can only change places in the ringing order
with the bell just before you or just after you.

Image

The following example shows very simple 'bell music' starting with a round and ending with a round of 4 bells, showing 8 of the 24 possible permutations, or orders.

1234

2143

2413

4231

4321

3412

3142

1324

1234

Can you find the changes so that, starting and ending with a round, all the 24 possible permutations are rung once each and only once?

## Student Solutions

We liked the solutions from Jake and Polly of West Flegg Middle School, Great Yarmouth.

NRICH can change to RNICH, RNCIH, RNIHC, NIRCH, NIRHC, NRCIH and NRIHC.

A peal of bells in which the 24 permutations of 4 bells are rung once each is given by

1234

2143

2413

4231

4321

3412

3142

1324

3124

1342

1432

4123

4213

2431

2341

3214

2314

3241

3421

4312

4132

1423

1243

2134

1234

For other solutions see the article Ding Dong Bell .