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Painting Cubes

Imagine you have six different colours of paint. You paint a cube using a different colour for each of the six faces. How many different cubes can be painted using the same set of six colours?

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Permute It

Take the numbers 1, 2, 3, 4 and 5 and imagine them written down in every possible order to give 5 digit numbers. Find the sum of the resulting numbers.

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Flagging

How many tricolour flags are possible with 5 available colours such that two adjacent stripes must NOT be the same colour. What about 256 colours?

Bell Ringing

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

    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?