You may also like

problem icon

Factoring a Million

In how many ways can the number 1 000 000 be expressed as the product of three positive integers?

problem icon

Thank Your Lucky Stars

A counter is placed in the bottom right hand corner of a grid. You toss a coin and move the star according to the following rules: ... What is the probability that you end up in the top left-hand corner of the grid?

problem icon

Card Game (a Simple Version of Clock Patience)

Four cards are shuffled and placed into two piles of two. Starting with the first pile of cards - turn a card over... You win if all your cards end up in the trays before you run out of cards in one pile. What is the probability that you win?

Sheffuls

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.