AMC12 2004 A

AMC12 2004 A · Q7

AMC12 2004 A · Q7. It mainly tests Recursion & DP style counting (basic), Patterns & sequences (misc).

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

一个游戏使用代币按照以下规则进行。每轮中，代币最多的玩家给每个其他玩家一个代币，并将一个代币放入弃牌堆。游戏在某个玩家代币用尽时结束。玩家 $A$、$B$ 和 $C$ 分别起始有 $15$、$14$ 和 $13$ 个代币。游戏有多少轮？

(A) 36 36

(B) 37 37

(D) 39 39

(E) 40 40

Answer

Correct choice: (B)

正确答案：（B）

Solution

We look at a set of three rounds, where the players begin with $x+1$, $x$, and $x-1$ tokens. After three rounds, there will be a net loss of $1$ token per player (they receive two tokens and lose three). Therefore, after $36$ rounds -- or $12$ three-round sets, $A,B$ and $C$ will have $3$, $2$, and $1$ tokens, respectively. After $1$ more round, player $A$ will give away $3$ tokens, leaving them empty-handed, and thus the game will end. We then have there are $36+1=\boxed{\mathrm{(B)}\ 37}$ rounds until the game ends.

我们考察一组连续三轮的情况，设三位玩家开始时分别有 $x+1$、$x$、$x-1$ 个代币。三轮之后，每位玩家净减少 $1$ 个代币（他们收到两个代币并失去三个）。因此经过 $36$ 轮——也就是 $12$ 组三轮——$A,B$ 和 $C$ 分别有 $3$、$2$ 和 $1$ 个代币。再进行 $1$ 轮后，玩家 $A$ 将送出 $3$ 个代币，从而手中为零，游戏结束。因此游戏结束前共有 $36+1=\boxed{\mathrm{(B)}\ 37}$ 轮。

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