Dissecting a Square Into Squares

This figure shows a large square cut up into smaller squares, no two of which are the same size. It's trivial to cut up a square into smaller squares that are the same size, but not so easy if we require the smaller squares to all be different. This example was found in 1978 by A J W Duijvestijn and is known to use the smallest possible number of squares. A wonderful account of the search for these dissections can be found in Think of a Number, by Malcom E Lines.

Is it possible to dissect a cube into smaller cubes no two of which are the same size?