Editing Squaring the square (section)

== Cubing the cube ==

'''Cubing the cube''' is the analogue in three dimensions of squaring the square: that is, given a [[cube]] ''C'', the problem of dividing it into finitely many smaller cubes, no two congruent.

Unlike the case of squaring the square, a hard yet solvable problem, there is no perfect cubed cube and, more generally, no dissection of a [[rectangular cuboid]] ''C'' into a finite number of unequal cubes.

To prove this, we start with the following claim: for any perfect dissection of a ''rectangle'' in squares, the smallest square in this dissection does not lie on an edge of the rectangle. Indeed, each corner square has a smaller adjacent edge square, and the smallest edge square is adjacent to smaller squares not on the edge.

Now suppose that there is a perfect dissection of a rectangular cuboid in cubes. Make a face of ''C'' its horizontal base. The base is divided into a perfect squared rectangle ''R'' by the cubes which rest on it. The smallest square ''s''<sub>1</sub> in ''R'' is surrounded by ''larger'', and therefore ''higher'', cubes. Hence the upper face of the cube on ''s''<sub>1</sub> is divided into a perfect squared square by the cubes which rest on it. Let ''s''<sub>2</sub> be the smallest square in this dissection. By the claim above, this is surrounded on all 4 sides by squares which are larger than ''s''<sub>2</sub> and therefore higher.

The sequence of squares ''s''<sub>1</sub>, ''s''<sub>2</sub>, ... is infinite and the corresponding cubes are infinite in number. This contradicts our original supposition.{{r|bsst}}

If a 4-dimensional [[hypercube]] could be perfectly hypercubed then its 'faces' would be perfect cubed cubes; this is impossible. Similarly, there is no solution for all cubes of higher dimensions.