### Factoring a Million

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

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?

### 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:

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.