Difference between revisions of "2004 AMC 12A Problems/Problem 7"
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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 <math>A</math>, <math>B</math>, and <math>C</math> start with 15, 14, and 13 tokens, respectively. How many rounds will there be in the game? | 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 <math>A</math>, <math>B</math>, and <math>C</math> start with 15, 14, and 13 tokens, respectively. How many rounds will there be in the game? | ||
| − | <math> \mathrm{(A) \ } 36 \qquad \mathrm{(B) \ } 37 \qquad \mathrm{(C) \ } 38 \qquad \mathrm{(D) \ } 39 | + | <math> \mathrm{(A) \ } 36 \qquad \mathrm{(B) \ } 37 \qquad \mathrm{(C) \ } 38 \qquad \mathrm{(D) \ } 39 \qquad \mathrm{(E) \ } 40 </math> |
==Solution== | ==Solution== | ||
Revision as of 11:05, 5 November 2006
Problem
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 
, 
, and 
start with 15, 14, and 13 tokens, respectively. How many rounds will there be in the game?
Solution
Look at a set of 3 rounds, where the players have 
, 
, and 
tokens. Each of the players will gain two tokens from the others and give away 3 tokens, so overall, each player will lose 1 token.
Therefore, after 12 sets of 3 rounds, or 36 rounds, the players will have 3, 2, and 1 tokens, repectively. After 1 more round, player will give away his last 3 tokens and the game will stop 
.
