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Why the sum of the first n cubes is always a perfect square

numberssumscubesproof

1 = 1. 1 + 8 = 9. 1 + 8 + 27 = 36. 1 + 8 + 27 + 64 = 100. 1 + 8 + 27 + 64 + 125 = 225.

The sums of the first few cubes are all perfect squares. Which squares, and why does this always happen? Give a proof that a non-mathematician could follow.

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The squares 1, 9, 36, 100, 225 are the squares of 1, 3, 6, 10, 15, which are the triangular numbers 1, 1 + 2, 1 + 2 + 3, and so on. Look for a picture: a square of side 1 + 2 + ... + n split into pieces.

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