THE FOLD / LOOT / THE DROP / THE DUDENEY
THE DUDENEY
numbers equal to the cube of their own digit sum
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Dudeney numbers are a perfect little coincidence: a positive integer that is a cube whose cube root equals the sum of its own digits. That is, n = (digit sum of n)3. The example that started it: 512 = 83, and 5 + 1 + 2 = 8. There are exactly six of them: 1, 512, 4913 (= 173, digits sum to 17), 5832 (= 183), 17576 (= 263), and 19683 (= 273). After that, cubes simply grow faster than any digit sum can reach — so the list is complete and finite.
LIT verified live: scanning cubes k3 and keeping those whose digit sum equals k yields exactly {1, 512, 4913, 5832, 17576, 19683} (window.__dudeney). FIG no framing; exact cube and digit-sum arithmetic.
LIT verified live: scanning cubes k3 and keeping those whose digit sum equals k yields exactly {1, 512, 4913, 5832, 17576, 19683} (window.__dudeney). FIG no framing; exact cube and digit-sum arithmetic.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-drop — a rare drop: out of every cube, only six carry the exact coincidence that their digits sum back to their own cube root. Those six are the loot. AVAN (AI) built the instrument: the digit-sum, the cube test, and the exhaustive scan proving there are exactly six.
Credit as content: Dudeney numbers (named for Henry Ernest Dudeney, the English puzzlist, 1857–1930). The weave: David names the-drop; I cube each candidate root, add up the digits of the result, and collect the cases where the digit sum lands exactly back on the root — finding all six and no more.
Credit as content: Dudeney numbers (named for Henry Ernest Dudeney, the English puzzlist, 1857–1930). The weave: David names the-drop; I cube each candidate root, add up the digits of the result, and collect the cases where the digit sum lands exactly back on the root — finding all six and no more.
3 ONE DIMENSION
512 = 8³, 5+1+2 = 8 ✓. 4913 = 17³, 4+9+1+3 = 17 ✓. 19683 = 27³, 1+9+6+8+3 = 27 ✓. Only six exist: 1, 512, 4913, 5832, 17576, 19683.
4 TWO DIMENSIONS · INTERACTIVE
A cube root, its cube, and whether the cube’s digits sum back to it; the census of six checked.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a cube that sums back to its root.
AVAN’s addition (the inverse-companion): demand that cubing and digit-summing be inverse on a number — digit-sum then cube must return the number itself. The inverse of ‘cube the root’ is ‘sum the digits’, and Dudeney numbers are the fixed points where both agree. The inverse of ‘n → digit sum → cube’ closes the loop only six times. Magenta is an ordinary cube; green is the self-closing Dudeney cube. A number that cubes back to itself through its digits.
LIT Genuine Dudeney numbers (named for Henry Ernest Dudeney, the English puzzlist, 1857–1930). Verified live: cubing each candidate root k, summing the digits of k³, and keeping the cases where the digit sum equals k yields exactly the six numbers {1, 512, 4913, 5832, 17576, 19683} (window.__dudeney.census).
FIG No framing: the digit-sum, the cube test, and the exhaustive scan proving there are exactly six all run in-browser with exact arithmetic. The AVAN inverse is honest — demanding that cubing and digit-summing be mutually inverse on a number (digit-sum then cube returns the number) is a genuine fixed-point condition that closes only six times; magenta is an ordinary cube, green the self-closing Dudeney cube. A number that cubes back to itself through its digits.
FIG No framing: the digit-sum, the cube test, and the exhaustive scan proving there are exactly six all run in-browser with exact arithmetic. The AVAN inverse is honest — demanding that cubing and digit-summing be mutually inverse on a number (digit-sum then cube returns the number) is a genuine fixed-point condition that closes only six times; magenta is an ordinary cube, green the self-closing Dudeney cube. A number that cubes back to itself through its digits.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE DROP · David Lee Wise (ROOT0), with AVAN