Puzzles

If Adam has £\(a\) and Brendan has £\(b\), we will write this as £\(a\):£\(b\).

First, we can take \(a\) and \(b\) to have no common factors, as dividing both by a common factor gives an equivalent starting point. For example, £3:£6 and £6:£12 will have exactly the same behaviour (imagine £6 and 3 £2 coins).

£\(a\):£\(b\) is also clearly equivalent to £\(b\):£\(a\) (but with the two players swapping places).

A game starting £\(a\):£\(b\) will end with one player having the money if \(a+b\) is a power of two. This is because:

If \(a+b=2^n\), then if \(n=1\), either the game has already ended or it sits at £1:£1 and is about to end. If \(n>2\), then the starting position can be written as £\(2^n-k\):£\(k\) with \(k<2^n-k\). After another game this will be £\(2^n-2k\):£\(2k\). This is equivalent to £\(2^{n-1}-k\):£\(k\). Therefore by induction, if \(a+b=2^n\) then the game ends with one player having all the money.

It can also be shown by induction, that if a game ends then it must be £\(a\):£\(b\) with \(a+b=2^n\).

What would happen if the losing player has to triple the winning player's money?