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?
Age
11 to 14
Challenge level
filled star filled star empty star
Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving
Being curious Being resourceful Being resilient Being collaborative

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.

    Bell Ringing
    By these rules NRICH can at first change to RNICH but not to RINCH. What are the other orders of the letters of the word NRICH that can be obtained in just one change of this sort?

    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?