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Bell Ringing

Age 11 to 14
Challenge Level Yellow starYellow star
  • Problem
  • Student Solutions

    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.

    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?

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    The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice.

    NRICH is part of the family of activities in the Millennium Mathematics Project.

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