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Card Game (a Simple Version of Clock Patience)

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Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Stephen from Framwellgate School sent us his reasoning about this problem, which gives us one of the solutions:

It is easy to see that the word sheffuls needs to have the e (which is in the third place) moved to the position the l (the seventh place) is, which needs to go to where the u (the sixth place) is and simply this u needs to go where the e was, so it can be represented on the cycle (3 7 6)

Mr Skipper from Smithden High has extended this to find all possible solutions:

The double f can stay where it is or switch around (making the first f into the second and vice versa). So either (4)(5) or (4 5). In the same way we have a choice with the first and second s. So either (1)(8) or (1 8).

So the options are (omitting 1-cycles for readability)

(3 7 6)
(1 8) (3 7 6)
(3 7 6) (4 5)
(1 8) (3 7 6) (4 5)

These can be written in different ways e.g. the last one could be written as (5 4) (6 3 7) (1 8), but this is effectively the same reordering, just written differently.