2004 AMC 12A Problem 7

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

Answer: B

Concepts:process simulation invariant

Difficulty rating: 1390

Small hint:

Track the token counts over one full cycle of three rounds

Big hint:

Every three rounds each player loses exactly one token, so find when the leader hits

0

Solution:

After three rounds the players

A,

B,

and

C

have

14,

13,

and

12

tokens, respectively. Every subsequent three rounds reduces each player’s supply by one token.

After

36

rounds they have

3,

2,

and

1

tokens. In the

37

th round player

A,

who has the most, gives away all three of their tokens and runs out, ending the game.

Thus, the correct answer is B.