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The number equal to the sum of the cubes of its digits

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Find a three-digit number that is equal to the sum of the cubes of its digits. (For example, 123 does not work, because 1 + 8 + 27 = 36.)

Then explain why such numbers, whatever the number of digits, can never have more than a handful of digits.

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Start by cubing the digits: 0, 1, 8, 27, 64, 125, 216, 343, 512, 729. Three of them must add to a number between 100 and 999 whose own digits are the three you chose.

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