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A Visual Proof of Nicomachus's Theorem

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Nicomachus's theorem states that , where is a positive integer. In words, the sum of the cubes from 1 to is equal to the square of the sum from 1 to .

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For a visual proof, calculate the total area in the figure in two different ways: First, count the unit squares from the center to an edge to get , so that the total area is . Second, consider that each square ring consists of squares of side , with area .

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Contributed by: Michael Schreiber (March 2011)
Open content licensed under CC BY-NC-SA

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