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What is the smallest positive whole number that can be written as the sum of two positive cubes in two different ways?

Find the two ways, and explain how you would search for such numbers efficiently.

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List the cubes 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, 1331, 1728 and look for two pairs with the same sum.

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