It is known that for the Young diagram of any partition of an integer n, the sum of squares of the hook lengths of its cells is exactly n(2) morethan that of the contents of its cells. That is, for any partition lambda of an integer n, & sum;(u is an element of lambda )h(u)(2 )= n(2 )+ & sum;(u is an element of lambda )c(u)(2).We provide a bijective proof of this fact, thus solving a problem posed by Stanley. Along the way, we obtain a formula for the number of rectangles in the Young diagram of a partition. We also mention a result for sums of other powers of hook lengths and contents.