9. A pawn is initially on the central square of a $5 \times 5$ chessboard. A move of the pawn consists of moving to a square chosen at random from those that share exactly one vertex with the square it is on. What is the probability that after 12 moves the pawn is on any one of the corners of the chessboard?
(A) $\frac{1}{3}$
(B) $\frac{4}{25}$
(C) $\frac{1}{6}$
(D) $\frac{4}{13}$
(E) $\frac{1}{4}$
9. The answer is (E). Consider the chessboard colored as in the figure; a move of the pawn can only take it from a white square to a gray square, and vice versa: it follows that, after an odd number of moves, the pawn can only be on one of the four white squares. No matter which white square the pawn is on after 11 moves, there are 4 squares it can reach with the twelfth move, only one of which is a corner square. The probability that the pawn is in a corner after 12 moves (as well as after any positive even number of moves) is therefore $1 / 4$.

![](https://cdn.mathpix.com/cropped/2024_04_17_8e3f264af5d5452f36c1g-10.jpg?height=311&width=323&top_left_y=1609&top_left_x=1552)