If *n* is the smallest of the four integers, then the product is

(*n*)(*n*+1)(*n*+2)(*n*+3) = (*n*^{2} + 3*n*)(*n*^{2} + 3*n* + 2)

= (*n*^{2} + 3*n* + 1)^{2} – 1

This can’t be a perfect square, because two positive squares cannot differ by 1.

From Angela Dunn’s *Mathematical Bafflers*, via David Wells’ *Penguin Book of Curious and Interesting Puzzles*, 1992.