Question

一个坛子最初装有 $100$ 颗黑弹珠和 $100$ 颗白弹珠。反复从坛中取出三颗弹珠，再按照下表用坛外的弹珠替换：

取出的弹珠	放回的弹珠
$3$ 颗黑弹珠	$1$ 颗黑弹珠
$2$ 颗黑弹珠、 $1$ 颗白弹珠	$1$ 颗黑弹珠、 $1$ 颗白弹珠
$1$ 颗黑弹珠、 $2$ 颗白弹珠	$2$ 颗白弹珠
$3$ 颗白弹珠	$1$ 颗黑弹珠、 $1$ 颗白弹珠

反复执行这一过程后，坛中可能剩下下列哪一组弹珠？

Initially an urn contains $100$ black marbles and $100$ white marbles. Repeatedly, three marbles are removed from the urn and replaced from a pile outside the urn as follows:

Marbles removed	Replaced with
$3$ black	$1$ black
$2$ black, $1$ white	$1$ black, $1$ white
$1$ black, $2$ white	$2$ white
$3$ white	$1$ black, $1$ white

Which of the following sets of marbles could be the contents of the urn after repeated applications of this procedure?

$2$ 颗黑弹珠

$2$ black marbles

$2$ 颗白弹珠

$2$ white marbles

$1$ 颗黑弹珠

$1$ black marble

$1$ 颗黑弹珠和 $1$ 颗白弹珠

$1$ black and $1$ white marble

$1$ 颗白弹珠

$1$ white marble

答案：B

知识点：不变量 constructive process 奇偶性

难度评级：2330

小提示：

白弹珠的数量每次只会改变 $0$ 或 $2$

The number of white marbles always changes by $0$ or $2$

大提示：

在某次操作后第一次只剩至多两颗弹珠时，检查最后一次操作的可能输出

When an operation first leaves at most two marbles, inspect the possible outputs of that final operation

解答：

白弹珠数量的奇偶性始终不变，所以含一颗白弹珠的选项不可能出现。此外，最后一次操作从三颗弹珠开始时，可能留下 $1$ 颗黑弹珠、 $1$ 颗黑弹珠和 $1$ 颗白弹珠，或 $2$ 颗白弹珠，但绝不会留下 $2$ 颗黑弹珠。留下 $2$ 颗白弹珠的状态确实可以达到：不断把 $3$ 颗黑弹珠替换为 $1$ 颗黑弹珠，直到剩下 $2$ 颗黑弹珠；再两次使用“ $1$ 黑 $2$ 白”的规则，留下 $100$ 颗白弹珠；然后交替使用“ $3$ 白”规则和“ $1$ 黑 $2$ 白”规则，每两次操作把白弹珠数减少 $2$ ，直到剩下 $2$ 颗。

因此正确答案为 B。

The parity of the number of white marbles never changes, so choices with one white marble are impossible. Also a final operation starting with three marbles can leave $1$ black, $1$ black and $1$ white, or $2$ white, but never $2$ black. The state with $2$ white is attainable: repeatedly replace $3$ black by $1$ black until $2$ black remain; twice use the $1$ -black- $2$ -white rule, leaving $100$ white; then alternate the $3$ -white rule with the $1$ -black- $2$ -white rule, reducing the number of white marbles by $2$ per pair until $2$ remain.

Thus the correct answer is B.

1991 AMC 12 第 28 题

答案：B

难度评级：2330

小提示：

大提示：

解答：

其他年份的第 28 题