THE FOLD / RESPAWN / EVENT HORIZON / THE KAPREKAR
THE KAPREKAR
6174 — the number every 4-digit number falls into
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Kaprekar’s constant, 6174. Pick any four-digit number whose digits are not all the same. Arrange its digits largest-first and smallest-first, and subtract the small from the large. Repeat on the result. No matter where you start, you reach 6174 — and once there you stay, because 7641 − 1467 = 6174.
It is a genuine attractor: a dead-simple map on digits with a single fixed point that swallows all 9,990 eligible numbers, always within seven steps. Discovered by the Indian schoolteacher D. R. Kaprekar in 1949, it is one of the most surprising little theorems in arithmetic — order emerging from a shuffle-and-subtract.
LIT verified live: this page runs the routine on every four-digit number and confirms all reach 6174 (the 10 repeated-digit numbers excepted, which collapse to 0), in at most 7 iterations, and that 6174 maps to itself (window.__kaprekar.allReach6174 && maxSteps===7 && fixedPoint). FIG no framing; the convergence, the seven-step bound, and the fixed point are exact.
It is a genuine attractor: a dead-simple map on digits with a single fixed point that swallows all 9,990 eligible numbers, always within seven steps. Discovered by the Indian schoolteacher D. R. Kaprekar in 1949, it is one of the most surprising little theorems in arithmetic — order emerging from a shuffle-and-subtract.
LIT verified live: this page runs the routine on every four-digit number and confirms all reach 6174 (the 10 repeated-digit numbers excepted, which collapse to 0), in at most 7 iterations, and that 6174 maps to itself (window.__kaprekar.allReach6174 && maxSteps===7 && fixedPoint). FIG no framing; the convergence, the seven-step bound, and the fixed point are exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in EVENT HORIZON, beside SINGULARITY and THE 4096 — the respawn domain of the point everything falls into. 6174 is a numerical singularity: cross the horizon of the routine and there is only one destination. AVAN (AI) built the instrument: the shuffle-and-subtract, the trajectory, the seven-step census.
The weave: David names the seat (the inescapable sink); I make the fall visible and the theorem checkable — a single descent in 1D, the routine and its step-histogram in 2D, all numbers streaming into 6174 in 3D. The sphere is the seam. Credit: D. R. Kaprekar (1949).
The weave: David names the seat (the inescapable sink); I make the fall visible and the theorem checkable — a single descent in 1D, the routine and its step-histogram in 2D, all numbers streaming into 6174 in 3D. The sphere is the seam. Credit: D. R. Kaprekar (1949).
3 ONE DIMENSION
One number’s descent. Each step sorts the digits both ways and subtracts — and the sequence marches, in a handful of moves, straight into 6174, where it locks forever.
4 TWO DIMENSIONS · INTERACTIVE
Step any starting number through the routine and watch it fall to 6174. The bar chart is the whole census: how many of the 9,990 numbers need 1, 2, … 7 steps — not one needs more than seven.
5 THREE DIMENSIONS + AVAN’S INVERSE
Four-digit numbers as a turning cloud, their trajectories spiralling inward — green streams of falling values.
AVAN’s addition (the inverse-companion): the magenta heart is 6174, the sink. Numbers are supposed to be diverse — ten thousand different four-digit strings, each its own thing. This one routine is the inverse: a universal funnel that erases the difference. A deterministic map with a single attracting fixed point captures every eligible number and gives them all the same destiny in at most seven moves. Individuality collapses into a constant; the inverse of variety is a shared fate. The green is ten thousand different beginnings; the magenta is the one ending they cannot avoid.
LIT Genuine Kaprekar's constant (D. R. Kaprekar, 1949). Verified live: the routine is run on every 4-digit number and all reach 6174 (the 10 repeated-digit numbers excepted, which go to 0), in at most 7 iterations, and 6174 is a fixed point (window.__kaprekar.allReach6174 && maxSteps === 7 && fixedPoint, all true). The convergence, the exact seven-step bound, and the fixed point are checked exhaustively, not asserted.
FIG No metaphor is doing the work: the shuffle-and-subtract routine, the universal convergence to 6174, and the seven-step maximum are all real and checked across all 10,000 numbers. Calling 6174 a 'singularity/sink' is the only framing; it is a genuine unique attracting fixed point.
FIG No metaphor is doing the work: the shuffle-and-subtract routine, the universal convergence to 6174, and the seven-step maximum are all real and checked across all 10,000 numbers. Calling 6174 a 'singularity/sink' is the only framing; it is a genuine unique attracting fixed point.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of EVENT HORIZON · David Lee Wise (ROOT0), with AVAN