Kaprekar’s Routine

Starting number

Enter a number and press Run to begin.

Current run

Starting number —

Steps to converge —

Fixed point —

Mode 4-digit

Histogram stats

Valid numbers —

Max steps —

Average steps —

Most common —

Color key

Step count frequency

Current number’s steps

Type a number and press Run or Enter · Random picks a valid starting number · Switch between 4-digit and 3-digit modes · Show all histogram computes convergence steps for every valid number

What’s happening

Kaprekar’s routine

In 1949, Indian mathematician D.R. Kaprekar discovered a surprising property of 4-digit numbers. Take any 4-digit number where not all digits are the same. Sort its digits in descending order to get the largest possible arrangement, then sort ascending for the smallest. Subtract the smaller from the larger. Repeat with the result. Within at most 7 iterations, you always arrive at 6174 — and 6174 maps to itself: 7641 − 1467 = 6174.

The algorithm

function kaprekar(n):
    pad n to 4 digits
    desc = sort digits descending  (e.g. 3087 → 8730)
    asc  = sort digits ascending   (e.g. 3087 → 0378)
    return desc - asc              (8730 - 378 = 8352)

Why does it converge?

There is no simple algebraic proof — convergence was verified computationally for all 4-digit inputs. The routine acts as a contraction mapping on a finite set: 6174 is the unique fixed point. The 3-digit version converges to 495 (e.g. 954 − 459 = 495). Not all digit lengths have a single fixed point: 2-digit numbers cycle, and some bases or digit counts produce cycles instead of fixed points.

The histogram

The histogram shows how many valid starting numbers take each number of steps to reach the fixed point. For 4-digit numbers, no starting value takes more than 7 steps. The distribution is not uniform — most numbers converge in 3–5 steps, with very few taking 1 or 7.

Extensions

The 3-digit analogue converges to 495 in at most 6 steps. For other digit lengths the behavior varies: 5-digit numbers don’t have a single fixed point but instead reach cycles. Kaprekar’s routine has been studied in other number bases too — base 5, base 16 — each with its own constants and cycle structures.