Sum of Squares

Let’s take a square num­ber (a square of some inte­ger) of cubes. They can be arranged in a square. Let’s make five such squares for first five nat­ural num­bers. Put them one upon the other and glue them together. We get a stairs-like piece.

Three such pieces can be put together to form a fig­ure look­ing like a par­al­lelepiped with a ledge. Other three pieces form an alike shape. Putting the two together, we get a solid par­al­lelepiped.

The vol­ume of this par­al­lelepiped, mea­sured in num­ber of cubes, equals, on the one hand, the prod­uct of each side’s lengths mea­sured in cubes, and on the other hand, six times the sum of first five squares. From here one can guess the gen­eral for­mula for the sum of first n squares.