Kaprekar's routine explained

In number theory, Kaprekar's routine is an iterative algorithm named after its inventor, Indian mathematician D. R. Kaprekar. Each iteration starts with a number, sorts the digits into descending and ascending order, and calculates the difference between the two new numbers.

As an example, starting with the number 8991 in base 10:

6174, known as Kaprekar's constant, is a fixed point of this algorithm. Any four-digit number (in base 10) with at least two distinct digits will reach 6174 within seven iterations. The algorithm runs on any natural number in any given number base.

Definition and properties

The algorithm is as follows:

in a given number base

. This is the first number of the sequence.

Create a new number

\alpha

by sorting the digits of

in descending order, and another number

\beta

by sorting the digits of

in ascending order. These numbers may have leading zeros, which can be ignored. Subtract

\alpha-\beta

to produce the next number of the sequence.

Repeat step 2.

K_b(n)=\alpha-\beta

is the Kaprekar mapping. Some numbers map to themselves; these are the fixed points of the Kaprekar mapping, and are called Kaprekar's constants. Zero is a Kaprekar's constant for all bases

, and so is called a trivial Kaprekar's constant. All other Kaprekar's constants are nontrivial Kaprekar's constants.

For example, in base 10, starting with 3524,

K₁₀(3524)=5432-2345=3087

K₁₀(3087)=8730-378=8352

K₁₀(8352)=8532-2358=6174

K₁₀(6174)=7641-1467=6174

with 6174 as a Kaprekar's constant.

All Kaprekar sequences will either reach one of these fixed points or will result in a repeating cycle. Either way, the end result is reached in a fairly small number of steps.

Note that the numbers

\alpha

and

\beta

have the same digit sum and hence the same remainder modulo

b-1

. Therefore, each number in a Kaprekar sequence of base

numbers (other than possibly the first) is a multiple of

b-1

When leading zeroes are retained, only repdigits lead to the trivial Kaprekar's constant.

Families of Kaprekar's constants

In base 4, it can easily be shown that all numbers of the form 3021, 310221, 31102221, 3...111...02...222...1 (where the length of the "1" sequence and the length of the "2" sequence are the same) are fixed points of the Kaprekar mapping.

In base 10, it can easily be shown that all numbers of the form 6174, 631764, 63317664, 6...333...17...666...4 (where the length of the "3" sequence and the length of the "6" sequence are the same) are fixed points of the Kaprekar mapping.

b = 2k

It can be shown that all natural numbers

m=(k)b²ⁿ

	n-1
\left(\sum
	i=0

b^i\right)+(k-1)b²ⁿ+(2k-1)bⁿ

	n
\left(\sum
	i=0

b^i\right)+(k-1)b

	n-1
\left(\sum
	i=0

b^i\right)+(k)

are fixed points of the Kaprekar mapping in even base for all natural numbers .

References

Hanover. Daniel . The Base Dependent Behavior of Kaprekar's Routine: A Theoretical and Computational Study Revealing New Regularities . International Journal of Pure and Applied Mathematics . 2017 . 1710.06308.

External links

Web site: Bowley. Roger. 6174 is Kaprekar's Constant. Numberphile. 5 December 2011 . Brady Haran. University of Nottingham. 2024-01-17.
Working link to YouTube
Sample (Perl) code to walk any four-digit number to Kaprekar's Constant
Sample (Python) code to walk any four-digit number to Kaprekar's Constant