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?

In the 1x2, 5x5 case, by assigning either 1/26 or 0 as the probability for each unoriented position, I get 1/13 for the probability of finding the destroyer lying on any given square.  The 0 probabilities make the destroyer lie in just two different possible positions over any given square.

Added in edit: The last sentence above is false, I now see.  I am beginning to think that the 1x2, 5x5 case is impossible for 2/25 or any other probability that is independent of the chosen square.

Edited on February 16, 2006, 8:04 pm

Posted by Richard on 2006-02-16 15:41:42