Question

如下图所示的 $13$ 个方格组成的图形中， $8$ 个方格涂红色，其余 $5$ 个方格涂蓝色。在所有可能的这种涂色中随机选一种。若所选涂色绕中心方格旋转 $90^\circ$ 后看起来不变的概率为 $\frac{1}{n}$ ，其中 $n$ 是正整数，求 $n$ 。

In the array of $13$ squares shown below, $8$ squares are colored red, and the remaining $5$ squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated $90^\circ$ around the central square is $\frac{1}{n},$ where $n$ is a positive integer. Find $n.$

答案

答案：429

知识点：基本概率组合对称性

难度评级：2300

解答：

这个旋转会循环置换四条 L 形臂，因此对称涂色必须让四条臂完全相同，外侧 $12$ 个方格就是某一条臂样式的 $4$ 份副本。于是外侧红格数必须是 $4$ 的倍数。总共有 $8$ 个红格，所以中心格必须是蓝色，并且每条臂必须恰有 $2$ 个红格和 $1$ 个蓝格。

一条臂中蓝格的位置有 $3$ 种选择，所以在 $\binom{13}{5} = 1287$ 种等可能涂色中，恰有 $3$ 种满足对称。概率为 $\frac{3}{1287} = \frac{1}{429}$ ，因此 $n = 429$ 。

The rotation cycles the four L-shaped arms, so a symmetric coloring colors all four arms identically, and the $12$ outer squares contain $4$ copies of whatever the arm shows. The number of red squares among the outer twelve is therefore a multiple of $4.$ Since there are $8$ red squares in all, the center must be blue and each arm must contain exactly $2$ red squares and $1$ blue square.

The blue square within the arm can be chosen in $3$ ways, so exactly $3$ of the $\binom{13}{5} = 1287$ equally likely colorings are symmetric. The probability is $\frac{3}{1287} = \frac{1}{429},$ so $n = 429.$

2013 AIME I 第 4 题

答案：429

难度评级：2300

解答：

其他年份的第 4 题