2004 AMC 12A 第 7 题

Question

一个游戏按照以下规则使用筹码。每一轮中，筹码最多的玩家给其他每个玩家各一个筹码，并且还把一个筹码放入弃置堆。当某个玩家没有筹码时，游戏结束。玩家

A

、

B

和

C

分别以

15

、

14

和

13

个筹码开始。这个游戏会进行多少轮？

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

答案：B

知识点：过程模拟不变量

难度评级：1390

小提示：

追踪完整三轮循环中的筹码数。

Track the token counts over one full cycle of three rounds

大提示：

每三轮每个玩家都恰好少一个筹码，所以找出领先者何时变为

0

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

0

解答：

三轮后，玩家

A

、

B

和

C

分别有

14

、

13

和

12

个筹码。之后每三轮都会使每个玩家的筹码数减少一个。

经过

36

轮后，他们分别有

3

、

2

和

1

个筹码；第

37

轮中，玩家

A

把自己的三个筹码全部给出并放入弃置堆后归零，游戏结束。

所以正确答案是 B。

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.

2004 AMC 12A 第 7 题

答案：B

难度评级：1390

小提示：

大提示：

解答：

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