Exhibits › Mathematical Curiosities › 6174

The number almost every road leads to

Mathematical Curiosities

Think of any four-digit number. Your birth year. Your PIN. The last four digits of your phone number.

Now follow a very simple rule, over and over.

Within seven steps at most, you will land on the same number as everyone else who tries it:

6174.

And once you arrive, you can never leave.

Exhibit 01

Try it yourself

Here is the whole rule. It takes one line:

Arrange the digits from largest to smallest, then from smallest to largest, and subtract.

Then do the same thing with the answer. And again.

Let's try it with 3524:

1 5432 − 2345 = 3087

2 8730 − 0378 = 8352

3 8532 − 2358 = 6174

Three steps, and there it is. Now keep going, just to see what happens:

7641 − 1467 = 6174

↳ it gives back itself. Forever.

6174 is a place you can arrive at, but never leave.

Step-by-step illustration of Scrabble-like digit tiles rearranging through the Kaprekar process from 3524 to 6174, with a loop arrow at the end

Two small rules

Your number needs at least two different digits - 1111 or 7777 will just give you zero. And keep any zeros as zeros: if you get 999, treat it as 0999.

For example, starting from 2111:

2111 − 1112 = 0999

9990 − 0999 = 8991

9981 − 1899 = 8082

8820 − 0288 = 8532

8532 − 2358 = 6174

Digit tiles in a row with a zero tile holding its place as a placeholder

Exhibit 02

Warm ink etching of Kaprekar, an older Indian schoolteacher, seated at a small desk on a veranda with sheets of four-digit number chains around him and a gentle, contented expression

D. R. Kaprekar, Devlali, India

A schoolteacher in Devlali

The man who found this number was not a famous professor.

D. R. Kaprekar - Dattatreya Ramchandra Kaprekar - was a schoolteacher in Devlali, a small town near Nashik in western India. He studied at Fergusson College in Pune, then spent decades teaching at a local school.

He never earned an advanced degree in mathematics. He did not need one to fall in love with it.

Kaprekar played with numbers the way other people do crosswords - constantly, happily, and for their own sake. He once said that, like a drunkard who wants to keep drinking to stay in that pleasant state, he kept going with numbers for the joy of it.

Somewhere in all that play, he noticed something strange about four-digit numbers. He announced it at a mathematics conference in 1949, and published it a few years later.

The number 18 with 1 plus 8 equals 9 shown below it and a small spark motif

He also named numbers for joy

Kaprekar gave the name Harshad numbers to numbers that can be divided evenly by the sum of their own digits - like 18, because 1 + 8 = 9, and 9 divides 18. "Harshad" comes from Sanskrit and means roughly joy-giver.

Exhibit 03

Not everyone was impressed

At first, many professional mathematicians in India shrugged.

To them, Kaprekar's discoveries looked like puzzles - "recreational" mathematics, fun but not serious. Kaprekar kept going anyway, publishing his findings in small journals and pamphlets.

Then, in 1975, the famous puzzle writer Martin Gardner described Kaprekar's work in his column in Scientific American, one of the most widely read science magazines in the world.

Suddenly, readers everywhere were scribbling four-digit numbers in the margins of their magazines - and watching them all fall into 6174.

The small-town schoolteacher's number had gone global.

Two-part ink illustration: a quiet desk in Devlali with handwritten number chains on one side, and 1970s readers testing numbers in magazine margins on the other, connected by a trail of digits leading to 6174

Exhibit 04

Funnel illustration of thousands of number tiles pouring in at the top, narrowing step by step, until a single tile reading 6174 rests at the bottom like a still pool

Why does it always end at 6174?

It feels like magic. It is actually a funnel.

Here is the first clue: when you subtract a number from a rearrangement of its own digits, the answer is always a multiple of 9. So right away, huge numbers of possibilities disappear.

Here is the second: the answer only depends on how far apart the digits are - not on the digits themselves. That means very different starting numbers give exactly the same result.

There are 9,990 possible starting numbers. After just one step, they have collapsed into only 54 different numbers. After another step, even fewer. And after a few more, only one is left standing.

So why 6174 in particular?

Because it is the only four-digit number that turns back into itself:

7641 − 1467 = 6174

Every other number moves on. 6174 is the only one that stays still - so it is where everything eventually settles, like water running downhill into the one lake at the bottom of the valley.

Exhibit 05

Seven steps, never more

Try as many four-digit numbers as you like. You will never need more than seven steps.

Here is how all 9,990 possible starting numbers behave:

Steps to 6174

StepsNumbersNote

0 1 already home

1 383 –

2 576 –

3 2,400 largest group

4 1,272 –

5 1,518 5050 arrives here

6 1,656 –

7 2,184 1729 ends here

Some familiar faces from this museum make the journey too. 5050 gets there in five steps. And 1729 - Ramanujan's famous taxi number - takes the longest possible route, all seven:

1729's seven-stop journey

1729 → 8442

8442 → 5994

5994 → 5355

5355 → 1998

1998 → 8082

8082 → 8532

8532 → 6174

↳ even the most famous number in mathematics cannot escape.

Illustrated chart of travellers at seven gates and a vintage taxi cab marked 1729 driving a seven-stop road to 6174

Exhibit 06

Other numbers, other traps

Is there something special about four digits?

Try the same rule with three-digit numbers, and everything falls into a different trap: 495. (954 − 459 = 495. It turns into itself, just like 6174.)

With two digits, there is no single resting place. Instead, the numbers go round and round in a loop: 09 → 81 → 63 → 27 → 45 → 09…

With five digits, the numbers fall into loops too. With six digits, there is more than one place to land.

Only the three- and four-digit worlds each have one single destination that every number is drawn toward.

That is part of what makes 6174 feel so uncanny: with four digits, the numbers could have wandered anywhere. Instead, they all go home to the same address.

Cartographic map of digit worlds: a two-digit circular loop, a three-digit world centred on 495, a four-digit world centred on 6174, and larger tangled worlds fading into the distance

Exhibit 07

Elegant ink illustration of a marble rolling into a bowl, a pendulum coming to rest, and water flowing into a valley lake - each with 6174 at the resting point

The pull of a single point

Mathematicians have a word for a place that everything is drawn toward: an attractor.

Attractors are everywhere once you start looking.

○

A marble in a bowl always rolls to the bottom.

A swinging pendulum eventually hangs still.

Water finds the lowest point in a valley.

In each case, it does not matter where you start. The rules of the system pull everything toward the same place.

6174 is an attractor made of nothing but digits and subtraction.

That is the real surprise of Kaprekar's discovery. You do not need gravity, or water, or physics to get this kind of behaviour. A simple rule, repeated, is enough.

Exhibit 08

Why we still remember it

Kaprekar died in 1986. His name lives on in the constant, and in a whole family of number ideas that bear his name.

6174 has become one of the best-loved numbers in recreational mathematics - the kind of thing people show a friend on a napkin, or a teacher shows a class to make them gasp.

Partly because anyone can check it with a pencil.

Partly because it feels impossible until you see why.

And partly because of who found it: not a famous university professor, but a schoolteacher in a small Indian town who simply loved numbers, and kept playing with them long after others would have stopped.

Warm ink illustration of Kaprekar in his later years, smiling gently, surrounded by chains of four-digit numbers all flowing toward 6174

Exhibit 09

6174 in a nutshell

Here is why 6174 matters:

1.
Take almost any four-digit number (with at least two different digits).
2.
Arrange its digits largest-to-smallest and smallest-to-largest, and subtract.
3.
Repeat - and you will reach 6174 in at most seven steps.
4.
Once there, it stays: 7641 − 1467 = 6174.
5.
Discovered by D. R. Kaprekar, a schoolteacher from Devlali, India, and announced in 1949.
6.
Became world-famous after Martin Gardner wrote about it in 1975.
7.
It is an attractor: a single point that pulls every starting number toward it.

Explore more numbers Suggest a number

Recap illustration combining digit tiles rearranging, Kaprekar at his desk, the funnel narrowing to 6174, the 1729 taxi on its seven-step road, and 6174 glowing at the centre

The whole story, in one panel.

Start anywhere you like.

6174 is a number you can escape from almost nowhere.

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Collection

Mathematical Curiosities

Announced

1949, Pune

Published widely

1975, Scientific American

Entry added

2026