Kaprekar Series

Ha! Finally I published the pages I had created on the Kaprekar routine/series/numbers; also made the code/software for Kaprekar series generation available for public download. Check it out at Kaprekar Series Generator.

Saturday, September 13, 2003

The Kaprekar Series!

Discovered by an Indian Mathematician in the 1950s, this series bears his name. He found that, given any 4-digit number, with distinct digits; on applying the Kaprekar process it will distill down to a single number - 6174 - called the Kaprekar constant.

The process is very simple.

Take any 4-digit number, say 1234.
Now write its digits in ascending and descending orders - 1234 and 4321 and subtract the two numbers so formed -

- 1234

Repeat the above process and you get these numbers after each step:

1234 -> 3087 -> 8352 -> 6174 -> 6174 -> 6174

hmm. You see the pattern?

Similarly, if you take up a 3-digit number (with distinct digits), you'll end up with 495 on applying the Kaprekar process.

For 2-digit number, it gets interesting. You don't end up with one specific number, but a sequence of numbers!

09 -> 81 -> 63 -> 27 -> 45 -> 09

For higher number of digits, you typically end up in one of many sequences!

For 5-digits - it will be one of three sequences
For 6-digits - its either 631764, or 549945 or a sequence
For 7-digits - you'll end up in a sequence of 8 numbers
and so on ...

I have put the java code I used to generate the series, for public download at