Bellringers have a special way to write down the patterns in
which their bells are rung. They give each bell a number, starting
with the bell with the highest pitch at number 1 and working down
so that the bell with the lowest pitch has the largest number. They
write down their patterns in rows. Each row is read from left to
right, and the rows are read from top to bottom.

Change ringing requires each bell to be played exactly once in
each row. To move from one row to the next, we swap round some of
the bells. We can only swap two bells if they are next to each
other (although we can swap several pairs at a time).

Here is an example of a pattern for four bells.

1234

2143

2413

4231

4321

3412

3142

1324

1234

Can you see which bells are swapped at each stage? You might
find it useful to write down the positions of any bells that don't
move in between two rows.

Hopefully you have seen that on alternate rows we swap all of
the bells (in pairs, of course - and there is only one way to do
this). On the remaining rows, we fix the bells on either end and
swap the rest. We repeat this until we get back to the starting
pattern (1234), which happens after 8 changes.

We can add coloured lines to show the paths of the different
bells. This is how bellringers remember what to do.

Now let's try the same pattern with six bells. (That is, swap
all the bells in pairs, then fix the end two and swap the rest in
pairs, then swap all the bells, then fix the end two and swap the
rest in pairs, etc.) How many changes does it take to get back to
123456? Write out the rows.

Draw on the paths of the different bells.You might need some
coloured pencils!

Can you draw the pattern for eight bells without writing out
the numbers?

This is the first of three related problems. The second
problem is You Owe Me Five Farthings, Say the Bells
of St Martin's , and the third is When Will You Pay Me? Say the Bells of Old
Bailey .