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Kaprekar's routine

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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:

9981 – 1899 = 8082
8820 – 0288 = 8532
8532 – 2358 = 6174
7641 – 1467 = 6174

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.[1] The algorithm runs on any natural number in any given number base.

Definition and properties

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The algorithm is as follows:[2]

  1. Choose any natural number in a given number base . This is the first number of the sequence.
  2. Create a new number by sorting the digits of in descending order, and another number by sorting the digits of in ascending order. These numbers may have leading zeros, which can be ignored. Subtract to produce the next number of the sequence.
  3. Repeat step 2.

The sequence is called a Kaprekar sequence and the function is the Kaprekar mapping. Some numbers map to themselves; these are the fixed points of the Kaprekar mapping,[3] 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,

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 and have the same digit sum and hence the same remainder modulo . Therefore, each number in a Kaprekar sequence of base numbers (other than possibly the first) is a multiple of .

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

Families of Kaprekar's constants

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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

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It can be shown that all natural numbers

are fixed points of the Kaprekar mapping in even base b = 2k for all natural numbers n.

Proof

Perfect digital invariants
k b m
1 2 011, 101101, 110111001, 111011110001...
2 4 132, 213312, 221333112, 222133331112...
3 6 253, 325523, 332555223, 333255552223...
4 8 374, 437734, 443777334, 444377773334...
5 10 495, 549945, 554999445, 555499994445...
6 12 5B6, 65BB56, 665BBB556, 6665BBBB5556...
7 14 6D7, 76DD67, 776DDD667, 7776DDDD6667...
8 16 7F8, 87FF78, 887FFF778, 8887FFeFF7778...
9 18 8H9, 98HH89, 998HHH889, 9998HHHH8889...

See also

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Citations

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  1. ^ Hanover 2017, p. 1, Overview.
  2. ^ Hanover 2017, p. 3, Methodology.
  3. ^ (sequence A099009 in the OEIS)

References

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