What's this Kaprekar Series?

Kaprekar Series and Kaprekar Constants are numbers generated on applying the Kaprekar Routine on a number!

OK, that sentence leaves a lot of things unanswered. :)

Take any number. Now sort the digits of this number in ascending or descending order. Subtract these two numbers. Keep repeating these steps. You have applied the Kaprekar Routine on the original number now!

On iterating, you'll either reach -

  1. 0 (a degenerate case - I told you not to choose those simple numbers, didn't I?),
  2. a constant number (Kaprekar Constant), or
  3. a cycle of numbers (Kaprekar Series).

These (no, not the zero) are the Kaprekar Series numbers.

If you had tried a 4-digit number, you will reach 6174 and for 3-digit numbers you'll reach 495!

Kaprekar Routine

Just consider a four digit number ... hmmm lets say 3241. Now order the digits in ascending and descending order: that gives us 4321 and 1234. Now subtract the two numbers - which gives us 3087. Now repeat this operation and notice what happens ...

- 1234

- 0378

- 2358

- 1467

- 1467

Notice that you are stuck with 6174, once you reach that number. The fun part is that all four digit numbers, except 9 of them (no awards for guessing that right!), lead to this very same number! The number of steps required to reach the number might be different, but sooner or later you'll land up with 6174.

You can try this with larger or smaller numbers. For three digit numbers you end up with 495.

Wonder what happens if you start with a number with more digits? See the Sample Output page for the results.

The above sequence of steps is called the Kaprekar Routine.

Is it the same as Kaprekar Number ...?

No, Kaprekar Number is different than Kaprekar Series. Lets start with an example - 297 is a kaprekar number. Why?

2972 = 88,209 and 88 + 209 = 297!!!

So, just square the number, split it in half (leave the larger portion to the right) and add the two halves - if you get the original number back, then its a Kaprekar Number.

A few more examples of Kaprekar Numbers are - 9, 45, 297, 4879, 17344, 538461, ...

Kaprekar Series Generator - what does it do?

The Kaprekar Series Generator is a java application to help find all the Kaprekar Series for a specified number of digits. So, say, you want to find all the Kaprekar Series for 5-digit numbers (there are three of them), just ask the generator to generate all the series for 5-digit numbers - and voila! out pops all the three series. You can do this for any arbitrarily large number of digits (make sure you have enuf cpu power before trying the larger digits!).

Can I see some examples, please!

Yep, right away! You can see some of the sample output on this site. I also looked for "patterns" in the series for 2-digit to 50-digit Kaprekar Series, you can check that out too!

Are we restricted only to base-10 numbers?

No, the field is far and wide! You can apply the Kaprekar Routine to numbers with any radix.

Where was all this published?

You can find the original papers here -

Kaprekar, D. R. "An Interesting Property of the Number 6174." Scripta Mathematica, v. 15, pp. 244-245, 1955.
Kaprekar, D. "On Kaprekar Numbers." Journal of Recreational Mathematics, v. 13, no. 2, pp. 81-82, 1980-1981.

Oh, Did no one else write on the Kaprekar Series?

Well, thats not the case. You can find a lot of papers/articles/websites on Kaprekar Series!

Is that all?

I had written a short blub when I had released the initial version of Kaprekar Series Generator. You can read it here - On the Kaprekar Series Generator

Where can I find the Kaprekar Series Generator?

The Kaprekar Series Generator project is located at http://kaprekar.sourceforge.net.

You can download the latest release from the Sourceforge download site.

Or you could browse the cvs repository.


Feedback is definitely welcome on any aspects of the Kaprekar Series Generator project. If you find any bugs in it, please report the bug and you can file Feature requests at its sourceforge site.

Where else to find the Kaprekar Series Generator?

I have registered the Kaprekar Series Generator project at these sites -