You have n cubical building blocks. You try to arrange them into the largest possible solid cube, but you find that don’t have quite enough blocks: One side of the large cube has exactly one row too few.

Prove that n is divisible by 6.

Let x be the number of blocks in a full row. Then

n = x^{3} – x

= x(x + 1)(x – 1).

These factors are three consecutive integers, which means that at least one of them is even and one is divisible by 3.

Since its factors include numbers that are divisible by both 2 and 3, n is divisible by 6.

(Thanks, Allan.)

May 20, 2013 | Puzzles