2013 AIME I 第 4 题

A coloring unchanged by the rotation is determined by one L-shaped arm: the four arms must be colored identically

To get $8$ red and $5$ blue, the center must be blue and each arm needs $2$ red and $1$ blue; compare with $\binom{13}{5}$ total colorings

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 $n = 429.$