Two very good solutions were sent in from Madras College, St Andrews by Dorothy Winn and Hannah Bredin.


Imagine cubes on each of the squares marked with a blue lattice, a diagonal line of cubes of increasing size. Then take away all the layers except the bottom ones and spread them out, dividing them into two groups and arranging them to cover the whole square.

Start with one cube. Then make a 3 x 3 x 1 layer by breaking the top layer of a 2 x 2 x 2 cube in half and putting the halves on the edges. This shows that 1 3 + 2 3 = (1 + 2) 2.

Then make a 6 x 6 x 1 layer by separating the layers of a 3 x 3 x 3 cube and putting the top two layers along the edges. This shows that 1 3 + 2 3 + 3 3 = (1 + 2 + 3) 2.

The pattern should continue. The diagram shows how

1 3 + 2 3 + 3 3 + ... + 6 3 = ( 1+ 2 + 3 + ... + 6) 2.