Two numbers $x$ and $y$ are each chosen at random from 1, 2, 3, 4, the two choices being independent so that a repeat is allowed. What is the probability that the product $xy$ is a perfect square?

There are $4\times 4=16$ equally likely pairs. The product is a perfect square for (1,1) = 1, (2,2) = 4, (3,3) = 9, (4,4) = 16 and also for the mixed pair (1,4) = 4 and (4,1) = 4 — six pairs in all — so the probability is 6/16 = 3/8. Keeping only the four pairs with $x=y$ gives 4/16 = 1/4 and overlooks that 1 and 4 multiply to a square as well; missing just one of the six gives 5/16; and 1/2 would need eight favourable pairs.