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2004 AMC 10A Problem 8

Problem 8 of 25Easier

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 into a discard pile. The game ends when some player runs out of tokens. Players A,A, B,B, and CC start with 15,15, 14,14, and 1313 tokens, respectively. How many rounds will there be in the game?

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Solution

After the first three rounds the counts go from (15,14,13)(15, 14, 13) to (14,13,12).(14, 13, 12). In general, every three rounds each player loses exactly one token. After 3636 rounds the counts are (3,2,1).(3, 2, 1). On the 3737th round the leader gives away three tokens and drops to 0,0, ending the game. Thus, the correct answer is B.

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Concepts: process simulation · pattern recognition

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.