# 6174 — Kaprekar's Black Hole

> An animated demonstration of Kaprekar's routine: pick any four digits, and watch them
> fall into 6174 in at most seven steps. Free, no account, runs entirely in the browser.

The site is a single page. You compose a four-digit number on a keypad that refuses only
four of a kind, and the routine then runs as an animation: the digits sort themselves into the high
and low numbers, the subtraction resolves column by column with its borrows, and each result
becomes the next number until 6174 appears. A counter records how many falls the site has
shown. Below the app, the same page explains the routine in text.

## The routine

1. Write the number with four digits, keeping any leading zeros.
2. Sort the digits from largest to smallest — the **high** number.
3. Sort the same digits from smallest to largest — the **low** number.
4. Subtract low from high, then start again with the result.

Worked example, starting from 3524:

| Step | High | Low  | Difference |
| ---- | ---- | ---- | ---------- |
| 1    | 5432 | 2345 | 3087       |
| 2    | 8730 | 0378 | 8352       |
| 3    | 8532 | 2358 | **6174**   |

## Key facts about 6174

- 6174 is called **Kaprekar's constant**, after D. R. Kaprekar, an Indian schoolteacher from
  Devlali who described the routine in 1949.
- It is the only four-digit **fixed point** of the routine: 7641 − 1467 = 6174.
- Every four-digit number reaches 6174 **except the ten repdigits** (1111, 2222, … 9999),
  whose high and low numbers are equal, so the first subtraction gives 0000.
- That leaves 9990 of the 10 000 four-digit strings, counting leading zeros, and **none of
  them needs more than seven steps**. The bound is exact and verifiable by exhaustion:
  only 715 digit multisets exist.
- **Repeated digits are fine.** Only four of a kind fails: 1122 reaches 6174 in four steps,
  1112 in five, 1111 in none. 4950 of the working numbers contain a repeat.
- Among the 9990 numbers that converge, the step counts are:
  0 steps → 1 number (6174 itself), 1 → 383, 2 → 576, 3 → 2400, 4 → 1272, 5 → 1518,
  6 → 1656, 7 → 2184. Three steps is the most common; more than a fifth take the maximum.
- The three-digit analogue is **495**, reached in at most six steps. Two, five and six-digit
  numbers have no single constant — they fall into cycles instead.

Built by imarkahann.

## Pages

- [6174 — Kaprekar's Black Hole](https://kaprekar.xyz/): the animated routine, plus a written
  explanation of how it works, why it always terminates, and how many steps it takes.
- [How it works](https://kaprekar.xyz/#how): the four operations, with a worked example.
- [Why it always ends at 6174](https://kaprekar.xyz/#why): the fixed point, the repdigit
  exception, and the distribution of step counts.
- [Common questions](https://kaprekar.xyz/#faq): short answers about the constant, its
  discoverer, and the three-digit version.

## Optional

- [6174 on Wikipedia](https://en.wikipedia.org/wiki/6174_(number)): background and history.
- [Kaprekar routine on MathWorld](https://mathworld.wolfram.com/KaprekarRoutine.html):
  the routine generalised to other digit lengths and bases.