Cubic numbers

You know a cube from geometry: a solid whose edges are all equal and meet at right angles. How many 1 cm cubes make a cube of side 2 cm? Eight. Of side 3 cm? Twenty-seven.

A cube of edge 4 holds 64 unit cubes. The easy way to see it: each flat layer is 4 × 4 = 16 unit cubes, and there are 4 layers, so 4 × 4 × 4 = 64.

So 1, 8, 27, 64, 125, … are called perfect cubes: each is a number multiplied by itself three times. 1 = 1 × 1 × 1, 8 = 2 × 2 × 2, 27 = 3 × 3 × 3, 125 = 5 × 5 × 5.

Is 9 a cube? 2³ = 8 and 3³ = 27, so no, and nor is anything from 10 to 26. Cubes thin out fast.

We write n × n × n as n³. Like squares, you can cube fractions, decimals and negatives: (4/6)³ = 64/216, (13.08)³ = 2237.810112, (−6)³ = −216. Cubing a negative number keeps it negative, which squaring does not.

Can a cube end in exactly two zeros? No: a number ending in one zero has a cube ending in three.

A famous story. When G.H. Hardy visited Srinivasa Ramanujan in hospital in Cambridge, he said his taxi was numbered 1729, 'rather a dull number', and hoped it wasn't a bad sign. Ramanujan answered at once: 'No, Hardy, it is a very interesting number. It is the smallest number that can be expressed as the sum of two cubes in two different ways.' That is, 1729 = 1³ + 12³ = 9³ + 10³. It has been the Hardy–Ramanujan Number ever since, and numbers like it are called taxicab numbers; the next two are 4104 and 13832. His colleague John Littlewood said of him: 'Every positive integer was one of his personal friends.'