[pzhon]

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I'd rather someone tell my why cubes add up to squares. Using geomety, not algebra.

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(1+2+...+n)^2 = 1^3 + 2^3 + .. + n^3.

Proof without Words: Sum of Cubes
Alan L. Fry
Mathematics Magazine, Vol. 58, No. 1 (Jan., 1985), p. 11

I don't have that in front of me, but typically these proofs are inductive. You find some way to illustrate why n^3 = (1+2+...+n)^2-(1+2+...+(n-1))^2. If you fit the smaller square inside the larger, the remainder is an L shape, the intersection of two strips that are n x (1+2+..+n), or a strip that is n x (1+2+...+n) plus a strip that is n x (1+2+...+(n-1)). Break the strips into subrectangles indicated by the sum and pair n x k and n x (n-k), forming n pairs (pair n x n with n x 0) fitting into nxn squares.