Problem:
A piece is initially located on the central square of a chessboard. One move of the piece consists in moving to a square chosen at random among those that have exactly one vertex in common with the square on which it stands. What is the probability that after 12 moves the piece is located on any one of the corners of the chessboard?
Pick one
Solution:
The answer is (E). Consider the chessboard colored as in the figure; one move of the piece 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 piece can only be located on one of the four white squares. Whatever white square the piece is on after 11 moves, there are 4 squares that it can reach with the twelfth move, only one of which is a corner square. The probability that the piece is located on a corner after 12 moves (as well as after any
positive even number of moves) is therefore .