Question

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 $A,$ $B,$ and $C$ start with $15,$ $14,$ and $13$ tokens, respectively. How many rounds will there be in the game?

$36$

$37$

$38$

$39$

$40$

Solution:

After the first three rounds the counts go from $(15, 14, 13)$ to $(14, 13, 12).$ In general, every three rounds each player loses exactly one token.

After $36$ rounds the counts are $(3, 2, 1).$ On the $37$ th round the leader gives away three tokens and drops to $0,$ ending the game.

Thus, the correct answer is B.

2004 AMC 10A Problem 8

Solution:

Problem 8 in Other Years