Question

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

Solution:

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 Problem 4

Solution:

Problem 4 in Other Years