AMC10 2004 10A, Problem 8

A game is played with tokens according to the following rule.  In each round, the player with the most tokens gives one token to each of the other players and also places one token in the discard pile.  The game ends when some player runs out of tokens.  Players $A$, $B$, and $C$ start with $15$, $14$, and $13$ tokens, respectively.  How many rounds will there be in the game?

Solution: We look at a set of three rounds, where the players begin with $x+1$, $x$, and $x-1$ tokens. After three rounds, there will be a net loss of $1$ token per player (they receive two tokens and lose three). Therefore, after $36$ rounds -- or $12$ three-round sets, $A,B$ and $C$ will have $3$, $2$, and $1$ tokens, respectively.  After $1$ more round, player $A$ will give away $3$ tokens, leaving them empty-handed, and thus the game will end. We then have there are $36+1=\fbox{37}$ rounds until the game ends.

Answer: 37