A Stroke of Genius

The English mathematician G. H. Hardy once visited his collaborator Srinivasa Ramanujan, who was ill in hospital. For want of an opening remark, Hardy said that he had come in taxicab number $1729$, and that it struck him as a rather dull number. Ramanujan replied at once that, on the contrary, it was a most interesting one: it was the smallest number expressible as the sum of two cubes in two different ways.

Complete Ramanujan’s sentence with four words, two of them the same, that add up to four: the smallest number that is the sum of two

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Solution

The missing words are cubes, in two ways: two cubes in two ways, and indeed $2 + 2 = 4$. The two ways are $$1729 = 1^3 + 12^3 = 9^3 + 10^3,$$ since $1 + 1728 = 1729$ and $729 + 1000 = 1729$. No smaller number is a sum of two positive cubes in two distinct ways, so $1729$ is the smallest. In Ramanujan’s honour, numbers with this kind of property are now called taxicab numbers.