The Professor's Cube-Root Shortcut
A retired professor claims to have found a magic shortcut for cube roots: just add up a number's digits, and that sum is its cube root. Test it on 512 — its digits sum to 8, and 8 cubed is indeed 512. Besides the trivial case of 1, how many other whole numbers does this 'shortcut' actually work for, and what are they?
Reveal the answer
Exactly five more, for six in total: 1, 512 (8³), 4913 (17³), 5832 (18³), 17576 (26³) and 19683 (27³) — and then it stops forever; no larger number works. Henry Dudeney posed this as a joke about a professor who'd stumbled on a coincidence, in his puzzle 'Root Extraction.' Mathematicians now call these six values Dudeney numbers.
— Henry Dudeney, 536 Puzzles and Curious Problems — Puzzle 'Root Extraction', posthumous collection edited by Martin Gardner, 1967
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