# Christmas Cross Country Quiz Puzzle Answer

Near Quiz Master Shop Towers is a small town with five different sports clubs; namely the Cricket Club, the Football Club, the Hockey Club, the Rugby Club and the Tennis Club. On Boxing Day these five clubs organise a cross country race, and it gets quite competitive, as each club likes to think its members are the fastest and fittest.

The format is the same each year. Each of the five clubs selects a cross country team of the same number of runners. All the runners in all the teams pay a £1 entry fee, and these entry fees are pooled to provide the prize fund. The winning team gets all of the prize fund to share between the runners in the team.

The winning team can do what it likes with its winnings, but as it happens all five teams share the money between the team members. All of the teams give all of their runners some money, and all give the runners a whole number of pounds.

Also, all of the teams give a decreasing amount of money to their runners based on the runners' individual finishing positions - the first runner in the team gets the most money, the second runner gets the second most, down to the last runner in the team to finish, who gets the smallest amount.

Oddly, each of the five teams has devised a different scheme to divide up the winnings. You would not have thought this was possible, but it can be done, just.

In the last running of this race, the Rugby Club won. Pugh was their first runner home, closely followed by Smith. Jones was a little way back as their third finisher, with Taylor hot on his heels.

Pugh was delighted to have won his prize money, as he got the same amount as Jones and Taylor combined.

How much did Pugh win?

Well, the obvious difficulty is that at no point in this puzzle is the size of the cross country teams stated explicitly. And finding the team size is key to solving the puzzle.

There is a clue to the team sizes "Oddly, each of the five teams has devised a different scheme to divide up the winnings. You would not have thought this was possible, but it can be done, just."

The team size allows the prize money to be divided in exactly five different ways among the number of team members.

There are five teams, so the prize fund must be a multiple of five, and you need to find 5n (where n is the team size) such that 5n can be divided up into n different numbers in five different ways.

Working through 20, 25, 30 etc. you will find the number that you want, namely 40. This can be partitioned into:

a) 1, 2, 3, 4, 5, 6, 7, 12

b) 1, 2, 3, 4, 5, 6, 8, 11

c) 1, 2, 3, 4, 5, 6, 9, 10

d) 1, 2, 3, 4, 5, 7, 8, 10

e) 1, 2, 3, 4, 6, 7, 8, 9

We also know that the first runner (Pugh) got the same amount as third and fourth runners (Jones and Taylor). This is only possible in (b) above, where 11 = 5 + 6.

So Pugh won £11.