512 has a party trick: add its digits, cube the sum, land back on 512
In one of Henry Dudeney's puzzles, a retired professor claims to have found a trick for extracting cube roots: add up a number's digits, cube that total, and you land back on the number you started with. It doesn't work for just any number — but it does for 512, since 5+1+2=8 and 8³=512. Besides 1 and 512, exactly four other whole numbers share this same strange property. Can you find them?
Reveal the answer
The complete list is 1, 512, 4913, 5832, 17576, and 19683 — only six numbers total, since digit sums grow far slower than cubes, so no larger example can exist (the largest, 19683, is 27³, and 1+9+6+8+3=27). These are now called Dudeney numbers, after Henry Dudeney's 1926 puzzle 'Root Extraction' in his collection Modern Puzzles, which first pointed out the pattern.
— Henry Dudeney, Modern Puzzles — Puzzle #69, 'Root Extraction,' 1926