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In the following picture, the formula for sum of squares of first $n$ natural numbers is derived using a clever construction of 3 triangles. This can be seen as a generalization to the legendary Gauss's trick, or perhaps as an application of group theory techniques. My question is if one can further generalize this idea and use a picture to prove that the sum of the cubes of the first $n$ natural numbers is the square of the sum of the first $n$ natural numbers.

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    $\begingroup$ I would think that the picture for $n=3$ would take an appropriate tetrahedron of numbers, draw out all 12 orientations of it, then add them together as above and divide by 6. (The fact that the symmetry group of the tetrahedron under rotations is the group-theoretic connection.) $\endgroup$ Sep 12 '14 at 16:22

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