You must randomly place a destroyer (a 1x2 sized ship) on a 5x5 grid such that if I searched in any single square, my probability of finding the destroyer there is exactly 2/25. Is such a probability distribution possible? You cannot simply choose randomly one of the 40 possible positions of a destroyer because corners would have a 1/20 chance to contain the destroyer, while the center would have 1/10 chance.
Generalize to a 1xN sized ship in a MxM grid. When is it possible to place the ship with an even probability distribution in each square?
The probability distribution is possible if and only if M is a multiple of N. If M is a multiple of N, divide the whole grid into a (M/N)^2 squares of NxN size, and place the ship randomly within any of these NxN-sized squares.
Puzzle Thoughts
