One of our colleagues went on holiday with a group of longstanding friends. They do it every year and the holiday has a certain competitive element.
Each person in the group has to organise a competition for all the others. We understand that these competitions can take any form, and each of them strive to come up with something inventive - even setting the competitions is competitive!
To add further spice each person in each competition (and no one is allowed not to compete) puts a pound into a kitty for that individual event. The organiser of that event then splits the kitty into three different amounts for first, second and third prizes, each an exact number of pounds.
When she returned she was exhausted, but seemed happy. Knowing the holiday's format we asked how much she had won, and were informed that she had come third in every event, except the one she had organised, of course.
She smiled and said that no organiser had split the kitty in the same way as any of the others, that there was only one way this could be done, and that the third prize in her event was £5.
So, how much did she win?
The solution is to find a number n where (n - 1) can be split into three numbers in n different ways.
We know that one of the third prizes was £5 so the smallest value for (n - 1) is 18, and in fact it is 18. There were 19 competitions, each with 18 people taking part, with prize funds of:
(15 2 1), (14 3 1), (13 4 1), (13 3 2), (12 5 1), (12 4 2), (11 6 1), (11 5 2), (11 4 3), (10 7 1), (10 6 2), (10 5 3), (9 8 1), (9 7 2), (9 6 3), (9 5 4), (8 6 4), (8 5 3) and (7 6 5)
Adding together all the third place prizes except the £5 gives a total of £37 won, and subtracting the £18 entry fees the winnings must be £19.