# Splitting the gate money

Here is a variation on a classic puzzle called "The Monkey and the Coconuts". When it was first published in an American newspaper in 1926, the offices of the newspaper were showered with over two thousand letters in the first week after the problem appeared, with people demanding to know what the answer was. I have adapted it from the version given in Martin Gardner's excellent book "More Mathematical Puzzles and Diversions".

## The puzzle

After a football match it is too late to divide the takings so the five directors of the club decide to guard the money, which is in a heap of ten pound notes, through the night by sleeping alongside it in the boardroom. When they were all asleep one director woke up and he thought there might be a row about dividing up the money in the morning, so he decided to take his share. He divided up the money into five piles, but had one ten pound note left over, so he put it in the charity box on the bar. He hid his share and put the rest back together. By and by the next director woke up and did the same thing and he had one ten pound note left over, which he put in the charity box. All the five directors did the same thing, one after the other; each one taking a fifth of the ten pound notes in the pile when he woke up, and each one having one ten pound note left over for the charity box. In the morning when they all woke up, they divided the remaining ten pound notes and came out with five equal shares. Of course each one must have known that there were some ten pound notes missing, but each one was as guilty as the others, so they didn't say anything. How much money was there in the beginning?

## The solution

£31,210 is the smallest solution. The first director gets £6240, leaving £24960 in the pile and £10 in the charity box. The second director gets £4990, leaving £19960 in the pile and £10 in the charity box. The third director gets £3990, leaving £15960 in the pile and £10 in the charity box. The fourth director gets £3190, leaving £12760 in the pile and £10 in the charity box. The fifth director gets £2550, leaving £10200 in the pile and £10 in the charity box. Each director then gets £2040 in the five way split with nothing left over.