Two fair six-sided dice are thrown together. What is the probability that the product of the numbers shown is a perfect square?

List the (a, b) pairs among the 36 equally likely outcomes whose product a×b is a perfect square: (1,1), (1,4), (4,1), (2,2), (1,9)→invalid since dice max 6, (4,4), (3,3), (5,5), (6,6), (2,8)→invalid.

Valid pairs: (1,1), (1,4), (4,1), (2,2), (3,3), (4,4), (5,5), (6,6) — that is 8 outcomes.

Probability = 8/36 = 2/9.

Tip: Perfect-square products occur either when both dice match (a² is always a square) or when the two values multiply to another square (like 1×4); check both cases systematically.