KAPREKAR MAPPING

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

The algorithm is as follows:^[2]

Choose any natural number $n$ in a given number base $b$ . This is the first number of the sequence.
Create a new number $\alpha$ by sorting the digits of $n$ in descending order, and another number $\beta$ by sorting the digits of $n$ 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.

The sequence is called a Kaprekar sequence and the function $K_{b}(n)=\alpha -\beta$ 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 $b$ , and so is called a trivial Kaprekar's constant. All other Kaprekar's constant are nontrivial Kaprekar's constants.

For example, in base 10, starting with 3524,

K_{10}(3524)=5432-2345=3087

K_{10}(3087)=8730-378=8352

K_{10}(8352)=8532-2358=6174

K_{10}(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 $b$ 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)b^{2n+3}\left(\sum _{i=0}^{n-1}b^{i}\right)+(k-1)b^{2n+2}+(2k-1)b^{n+1}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b\left(\sum _{i=0}^{n-1}b^{i}\right)+(k)

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

Proof

$\alpha =(2k-1)b^{2n+2}\left(\sum _{i=0}^{n}b^{i}\right)+(k)b^{n+1}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)\left(\sum _{i=0}^{n}b^{i}\right)$

$\beta =(k-1)b^{2n+2}\left(\sum _{i=0}^{n}b^{i}\right)+(k)b^{n+1}\left(\sum _{i=0}^{n}b^{i}\right)+(2k-1)\left(\sum _{i=0}^{n}b^{i}\right)$

${\begin{aligned}K_{b}(m)&=\alpha -\beta \\&=((2k-1)-(k-1))b^{2n+2}\left(\sum _{i=0}^{n}b^{i}\right)+(k-k)b^{n+1}\left(\sum _{i=0}^{n}b^{i}\right)+((k-1)-(2k-1))\left(\sum _{i=0}^{n}b^{i}\right)\\&=kb^{2n+2}\left(\sum _{i=0}^{n}b^{i}\right)-k\left(\sum _{i=0}^{n}b^{i}\right)\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+b^{2n+2}-k\left(\sum _{i=0}^{n}b^{i}\right)\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+(2k)b^{2n+1}-k\left(\sum _{i=0}^{n}b^{i}\right)\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+(2k-1)b^{2n+1}+b^{2n+1}-k\left(\sum _{i=0}^{n}b^{i}\right)\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+(2k-1)b^{2n+1-1}\left(\sum _{i=0}^{1}b^{i}\right)+b^{2n+1-1}-k\left(\sum _{i=0}^{n}b^{i}\right)\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+(2k-1)b^{2n+1-n}\left(\sum _{i=0}^{n}b^{i}\right)+b^{2n+1-n}-k\left(\sum _{i=0}^{n}b^{i}\right)\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+(2k-1)b^{n+1}\left(\sum _{i=0}^{n}b^{i}\right)+b^{n+1}-k\left(\sum _{i=0}^{n}b^{i}\right)\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+(2k-1)b^{n+1}\left(\sum _{i=0}^{n}b^{i}\right)+(2k)b^{n}-k\left(\sum _{i=0}^{n}b^{i}\right)\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+(2k-1)b^{n+1}\left(\sum _{i=0}^{n}b^{i}\right)+kb^{n}-k\left(\sum _{i=0}^{n-1}b^{i}\right)\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+(2k-1)b^{n+1}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{n+1-1}+b^{n+1-1}-k\left(\sum _{i=0}^{n-n}b^{i}\right)\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+(2k-1)b^{n+1}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{n+1-n}\left(\sum _{i=0}^{n}b^{i}\right)+b^{n+1-n}-k\left(\sum _{i=0}^{n-n}b^{i}\right)\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+(2k-1)b^{n+1}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b\left(\sum _{i=0}^{n}b^{i}\right)+b-k\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+(2k-1)b^{n+1}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b\left(\sum _{i=0}^{n}b^{i}\right)+2k-k\\&=kb^{2n+3}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b^{2n+2}+(2k-1)b^{n+1}\left(\sum _{i=0}^{n}b^{i}\right)+(k-1)b\left(\sum _{i=0}^{n}b^{i}\right)+k\\&=m\\\end{aligned}}$

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

Kaprekar's constants and cycles of the Kaprekar mapping for specific base b

All numbers are expressed in base $b$ , using A−Z to represent digit values 10 to 35.

Base $b$	Digit length	Nontrivial (nonzero) Kaprekar's constants	Cycles
2	2	01^{[note 1]}	∅
	3	011^{[note 1]}	∅
	4	0111,^{[note 1]} 1001	∅
	5	01111,^{[note 1]} 10101	∅
	6	011111,^{[note 1]} 101101, 110001	∅
	7	0111111,^{[note 1]} 1011101, 1101001	∅
	8	01111111,^{[note 1]} 10111101, 11011001, 11100001	∅
	9	011111111,^{[note 1]} 101111101, 110111001, 111010001	∅
	10	0111111111,^{[note 1]} 1011111101, 1101111001, 1110110001, 1111000001	∅
3	2	∅	∅
	3	∅	022 → 121 → 022^{[note 1]}
	4	∅	1012 → 1221 → 1012
	5	20211	∅
	6	∅	102212 → 210111 → 122221 → 102212
	7	2202101	2022211 → 2102111 → 2022211
	8	21022111	∅
	9	222021001	220222101 → 221021101 → 220222101 202222211 → 210222111 → 211021111 → 202222211
	10	2210221101	1022222212 → 2102222111 → 2111011111 → 1222222221 → 1022222212
4	2	∅	03 → 21 → 03^{[note 1]}
	3	132	∅
	4	3021	1332 → 2022 → 1332
	5	∅	20322 → 23331 → 20322
	6	213312, 310221, 330201	∅
	7	3203211	∅
	8	31102221, 33102201, 33302001	22033212 → 31333311 → 22133112 → 22033212
	9	221333112, 321032211, 332032101	∅
	10	3111022221, 3220332111, 3311022201, 3331022001, 3333020001	∅
5	2	13	∅
	3	∅	143 → 242 → 143
	4	3032	∅
	5	∅	30432 → 40431 → 42411 → 32412 → 30432
	6	∅	214423 → 320322 → 304432 → 414421 → 331212 → 214423
	7	∅	3204322 → 4104331 → 4314211 → 3314212 → 3204322
	8	∅	30444432 → 42103321 → 42144221 → 33144212 → 33103312 → 32203222 → 30444432
	9	432043211	∅
	10	∅	3044444432 → 4210443321 → 4331033111 → 4222122221 → 3044444432
6	2	∅	05 → 41 → 23 → 05^{[note 1]}
	3	253	∅
	4	∅	1554 → 4042 → 4132 → 3043 → 3552 → 3133 → 1554
	5	41532	31533 → 35552 → 31533
	6	325523, 420432, 530421	205544 → 525521 → 432222 → 205544
	7	∅	4405412 → 5315321 → 4405412
	8	43155322, 55304201	31104443 → 43255222 → 33204323 → 41055442 → 54155311 → 44404112 → 43313222 → 31104443 42104432 → 43204322 → 42104432 53104421 → 53304221 → 53104421
	9	332555223	441054412 → 533153221 → 441054412
	10	4321044322, 4421553312, 5331044221, 5553042001	4211044432 → 4332043222 → 4211044432 5311044421 → 5333042221 → 5311044421 5531044201 → 5533042201 → 5531044201
7	2	∅	∅
	3	∅	264 → 363 → 264
	4	∅	3054 → 5052 → 5232 → 3054
	5	∅	40653 → 61641 → 54612 → 52632 → 42633 → 40653
	6	∅	320544 → 520542 → 531432 → 416643 → 526632 → 441423 → 320544
	7	∅	5306532 → 6316431 → 5506512 → 6426321 → 5416422 → 5316432 → 5306532
	8	∅	42166443 → 54066522 → 64305321 → 64166421 → 55366212 → 54314322 → 42166443
	9	∅	530666532 → 643164321 → 552065412 → 643263321 → 541666422 → 544065222 → 633164331 → 550666512 → 653666211 → 554263212 → 533164332 → 530666532
	10	∅	5316666432 → 5433053322 → 5316666432 5431055322 → 5431664322 → 5431055322 5440665222 → 6432663321 → 5440665222
8	2	25	07 → 61 → 43 → 07^{[note 1]}
	3	374	∅
	4	∅	1776 → 6062 → 6332 → 3774 → 4244 → 1776 3065 → 6152 → 5243 → 3065
	5	∅	42744 → 47773 → 42744 51753 → 61752 → 63732 → 52743 → 51753
	6	437734, 640632	310665 → 651522 → 532443 → 310665
	7	6417532	5407633 → 7317541 → 6617512 → 6537322 → 5417533 → 6217552 → 6427432 → 5407633 6407632 → 7427431 → 6507622 → 7437331 → 6407632
	8	75306421	31106665 → 65515222 → 53324443 → 31106665 64106632 → 65406322 → 64306432 → 64106632
	9	443777334	543076433 → 732076541 → 764175311 → 665175212 → 654274322 → 543076433 644076332 → 743076431 → 763076411 → 765274211 → 664274312 → 644076332 651076622 → 754373321 → 652076522 → 744274331 → 651076622
	10	7753064201	3111066665 → 6555152222 → 5333244443 → 3111066665 6411066632 → 6554063222 → 6433064432 → 6411066632 6431066432 → 6541066322 → 6543064322 → 6431066432 7531066421 → 7553064221 → 7533064421 → 7531066421
9	2	∅	17 → 53 → 17
	3	∅	385 → 484 → 385
	4	∅	3076 → 7252 → 5254 → 3076 5074 → 7072 → 7432 → 5074
	5	62853	52854 → 60873 → 83841 → 74832 → 63843 → 52854
	6	∅	308876 → 850731 → 861621 → 753432 → 520764 → 740742 → 748832 → 652533 → 421665 → 540744 → 708872 → 858821 → 762522 → 542544 → 308876
	7	∅	6518633 → 7328552 → 6518633
	8	65288533	65407433 → 73188652 → 76407422 → 75388432 → 65407433
	9	753186532	641888643 → 754186432 → 753087532 → 854186431 → 773087512 → 865384321 → 763186522 → 754285432 → 652087633 → 853285531 → 762186622 → 754384432 → 641888643
	10	6632885523	5288888854 → 6432885543 → 6531077533 → 7632166522 → 6444164443 → 5288888854 6441886443 → 7521886632 → 7653075322 → 7542166432 → 6441886443
10^[4]	2	∅	09 → 81 → 63 → 27 → 45 → 09^{[note 1]}
	3	495	∅
	4	6174	∅
	5	∅	53955 → 59994 → 53955 61974 → 82962 → 75933 → 63954 → 61974 62964 → 71973 → 83952 → 74943 → 62964
	6	549945, 631764	420876 → 851742 → 750843 → 840852 → 860832 → 862632 → 642654 → 420876
	7	∅	7509843 → 9529641 → 8719722 → 8649432 → 7519743 → 8429652 → 7619733 → 8439552 → 7509843
	8	63317664, 97508421	43208766 → 85317642 → 75308643 → 84308652 → 86308632 → 86326632 → 64326654 → 43208766 64308654 → 83208762 → 86526432 → 64308654
	9	554999445, 864197532	865296432 → 763197633 → 844296552 → 762098733 → 964395531 → 863098632 → 965296431 → 873197622 → 865395432 →753098643 → 954197541 → 883098612 → 976494321 → 874197522 → 865296432
	10	6333176664, 9753086421, 9975084201	8655264432 → 6431088654 → 8732087622 → 8655264432 8653266432 → 6433086654 → 8332087662 → 8653266432 8765264322 → 6543086544 → 8321088762 → 8765264322 8633086632 → 8633266632 → 6433266654 → 4332087666 → 8533176642 → 7533086643 → 8433086652 → 8633086632 9775084221 → 9755084421 → 9751088421 → 9775084221
11	2	37	∅
	3	∅	4A6 → 5A5 → 4A6
	4	∅	3098 → 9452 → 7094 → 9272 → 7454 → 3098 5096 → 9092 → 9632 → 7274 → 5276 → 5096
	5	∅	72A74 → 82A73 → 84A53 → 73A64 → 72A74
	6	∅	531876 → 740964 → 931872 → 863643 → 531876
	7	∅	631A875 → 951A852 → 972A732 → 873A633 → 753A654 → 730A974 → A62A741 → 982A722 → 875A433 → 752A754 → 831A873 → 954A552 → 84AAA53 → 764A544 → 631A875 751A854 → 941A862 → 973A632 → 863A643 → 751A854
	8	74318764	53218876 → 76409644 → 93218872 → 86636443 → 53218876 75218854 → 762AA744 → 86309743 → 95418652 → 861AA843 → 97418632 → 86418643 → 75218854
	9	9751A8532	871AAA833 → 9770A9332 → A763A6431 → 9741A8632 → 9752A7532 → 8741A8633 → 9552A7552 → 871AAA833
	10	∅	5322188876 → 7664096444 → 9322188872 → 8666364443 → 5322188876 7432188764 → 7643187644 → 7432188764 7631AA8744 → 9743097632 → 9744186632 → 8642188643 → 7652188544 → 7631AA8744 8621AA8843 → 9854186522 → 8661AA8443 → 9743AA6632 → 8762AA7433 → 8753097533 → 9543AA6652 → 8751099533 → 9853AA6522 → 8863097423 → 9654186542 → 8621AA8843
12	2	∅	0B → A1 → 83 → 47 → 29 → 65 → 0B^{[note 1]}
	3	5B6	∅
	4	∅	3BB8 → 8284 → 6376 → 3BB8 4198 → 8374 → 5287 → 6196 → 7BB4 → 7375 → 4198
	5	83B74	64B66 → 6BBB5 → 64B66
	6	65BB56	420A98 → A73742 → 842874 → 642876 → 62BB86 → 951963 → 860A54 → A40A72 → A82832 → 864654 → 420A98
	7	962B853	841B974 → A53B762 → 971B943 → A64B652 → 960BA53 → B73B741 → A82B832 → 984B633 → 863B754 → 841B974
	8	873BB744, A850A632	4210AA98 → A9737422 → 87428744 → 64328876 → 652BB866 → 961BB953 → A8428732 → 86528654 → 6410AA76 → A92BB822 → 9980A323 → A7646542 → 8320A984 → A7537642 → 8430A874 → A5428762 → 8630A854 → A540X762 → A830A832 → A8546632 → 8520A964 → A740A742 → A8328832 → 86546654
	9	665BBB556	8620BA954 → B852B8631 → A952B8622 → 9872B8433 → 9653B7653 → 8620BA954 8640BA754 → B641B9751 → AA51B9612 → A983B7322 → 9864B6533 → 8640BA754 9630BA853 → B762B8541 → A941B9722 → A874B6432 → 9742B8743 → 9642B8753 → 9641B9753 → A651B9652 → A840BA732 → B873B7431 → A852B8632 → 9852B8633 → 9652B8653 → 9630BA853
	10	8843BB7734	64310AA876 → A952BB8622 → 9984197323 → 8765286544 → 64310AA876 87650A6544 → A4310AA872 → A985286322 → 87650A6544 A8510AA632 → A9850A6322 → A8750A6432 → A8530A8632 → A8550A6632 → A8510AA632
13	2	∅	1B → 93 → 57 → 1B
	3	∅	5C7 → 6C6 → 5C7
	4	∅	30BA → B652 → 90B4 → B472 → 9294 → 7476 → 30BA 50B8 → B292 → 9654 → 50B8 5298 → 7296 → 70B6 → B0B2 → B832 → 9474 → 5298
	5	∅	83C85 → 92C94 → A4C73 → 95C64 → 83C85
	6	951A74	∅
	7	∅	951CA74 → B63C862 → A81CA43 → B75C652 → A61CA63 → B73C852 → A82C943 → A74C753 → 961CA64 → B62C962 → A92C933 → A75C653 → 951CA74 972C954 → A53C873 → 972C954 A71CA53 → B74C752 → A71CA53
	8	9541A874	641CCA87 → B840B842 → B9438832 → 96538764 → 641CCA87
	9	∅	9620CBA64 → C962C9631 → BA62C9622 → A982C9433 → A764C7653 → 9620CBA64 A751CA753 → B751CA752 → B951CA732 → B973C8532 → A862C9643 → A751CA753 A861CA643 → B761CA652 → B950CB732 → C983C8431 → B963C8632 → A861CA643
	10	95441A8874	86661A6665 → 92CCCCCC94 → A832CC9943 → A9750B7533 → B7621AA652 → A881CCA443 → B965CC6632 → A962CC9633 → A973299533 → 86661A6665 99650B7634 → B651CCA762 → BA640B8622 → B983CC8432 → A984CC7433 → 99650B7634 A8630B9643 → B763CC8652 → A9620BA633 → B875CC6542 → A8630B9643
14	2	49	2B → 85 → 2B 0D → C1 → A3 → 67 → 0D^{[note 1]}
	3	6D7	∅
	4	∅	61B8 → A1B4 → A574 → 61B8
	5	∅	75D77 → 7DDD6 → 75D77 93D95 → A3D94 → A5D74 → 94D85 → 93D95
	6	76DD67	530CA9 → C73962 → A60C74 → C60C72 → CA0C32 → CA6632 → A6DD64 → 973965 → 640C98 → C51B82 → B92A43 → 974865 → 530CA9 A40C94 → C64872 → A40C94 A80C54 → C62A72 → A80C54
	7	∅	A62DA74 → B63D973 → A82DA54 → B64D873 → A71DB64 → C73D962 → B92DA43 → B85D753 → A62DA74
	8	CA70C632	7550C887 → C32DDAA2 → BB8DD423 → BA72A633 → 9770C665 → C410CC92 → CBA48322 → A9739644 → 7550C887 A410CC94 → CB648722 → A940C944 → C6548872 → A430CA94 → C7648762 → A410CC94 A810CC54 → CB62A722 → A980C544 → C652A872 → A830CA54 → C762A762 → A810CC54
	9	776DDD667, B852DA853	A731DBA64 → C863D9752 → B941DB943 → C874D8652 → B831DBA53 → C884D8552 → B832DAA53 → B874D8653 → A731DBA64
15	2	∅	∅
	3	∅	6E8 → 7E7 → 6E8
	5	A4E95	∅
	6	∅	41EECB → DA0D42 → DB5832 → B82B64 → 971C76 → B2EEB4 → C9EE43 → BA2B44 → 975876 → 41EECB 960D86 → D31CB2 → CA7643 → 960D86
	7	∅	A62EB85 → C63EA83 → B93EA54 → B74E974 → A71EC75 → D72EB72 → CB3EA33 → B97E654 → A62EB85 B70ED74 → E93EA51 → DB4E932 → CA6E743 → B83EA64 → B73EA74 → B72EB74 → C73EA73 → B92EB54 → C75E873 → B70ED74
	8	A94EE955	8640DA87 → D620DC82 → 8640DA87
	9	C962EB853	A742EBA75 → C752EB973 → C961EC853 → D972EB752 → CB61EC833 → D994E9552 → C943EAA53 → B964E9854 → A742EBA75 B862EB864 → C751EC973 → D971EC752 → DB71EC732 → DB93EA532 → CA84E9643 → B862EB864
	10	AA54EE9945	96660D8886 → D3221CCCB2 → CAAA764443 → 96660D8886 B761EEC874 → DA640DA842 → DB661C8832 → CA821CC643 → BA961C8544 → B7641CA874 → B761EEC874
16^[5]	2	∅	2D → A5 → 4B → 69 → 2D 0F → E1 → C3 → 87 → 0F^{[note 1]}
	3	7F8	∅
	4	∅	3FFC → C2C4 → A776 → 3FFC A596 → 52CB → A596 E0E2 → EB32 → C774 → 7FF8 → 8688 → 1FFE → E0E2 E952 → C3B4 → 9687 → 30ED → E952
	5	∅	86F88 → 8FFF7 → 86F88 A3FB6 → C4FA4 → B7F75 → A3FB6 A4FA6 → B3FB5 → C5F94 → B6F85 → A4FA6
	6	87FF78	310EED → ED9522 → CB3B44 → 976887 → 310EED 532CCB → A95966 → 532CCB 840EB8 → E6FF82 → D95963 → A42CB6 → A73B86 → 840EB8 A80E76 → E40EB2 → EC6832 → C91D64 → C82C74 → A80E76 C60E94 → E82C72 → CA0E54 → E84A72 → C60E94
	7	C83FB74	B62FC95 → D74FA83 → C92FC64 → D85F973 → C81FD74 → E94FA62 → DA3FB53 → CA5F954 → B74FA85 → B62FC95 B71FD85 → E83FB72 → DB3FB43 → CA6F854 → B73FB85 → C63FB94 → C84FA74 → B82FC75 → D73FB83 → CA3FB54 → C85F974 → B71FD85
	8	∅	3110EEED → EDD95222 → CBB3B444 → 97768887 → 3110EEED 5332CCCB → A9959666 → 5332CCCB 7530ECA9 → E951DA62 → DB52CA43 → B974A865 → 7530ECA9 A832CC76 → A940EB66 → E742CB82 → CA70E854 → E850EA72 → EC50EA32 → EC94A632 → C962C964 → A832CC76 C610EE94 → ED82C722 → CBA0E544 → E874A872 → C610EE94 C630EC94 → E982C762 → CA30EC54 → E984A762 → C630EC94 C650EA94 → E852CA72 → CA50EA54 → E854AA72 → C650EA94 CA10EE54 → ED84A722 → CB60E944 → E872C872 → CA10EE54
	9	887FFF778, DA72FC853	C851FDA74 → E972FC862 → DC61FD933 → EAA5F9552 → D954FAA63 → C953FBA64 → C863FB974 → C851FDA74 C862FC974 → D861FD973 → EA71FD852 → EC82FC732 → DC94FA633 → CA83FB754 → C862FC974 CA40FEB54 → FA85F9751 → EA51FDA52 → EC84FA732 → DB82FC743 → DA83FB753 → CA62FC954 → D873FB873 → CA40FEB54

^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l ^m ⁿ ^o ^p ^q Leading zeroes retained.

Kaprekar's constants in base 10

Numbers of length four digits

In 1949 D. R. Kaprekar discovered^[6] that if the above process is applied to four-digit numbers in base 10, the sequence converges to 6174 within seven iterations or, more rarely, converges to 0. The number 6174 is the first Kaprekar's constant to be discovered, and thus is sometimes known as Kaprekar's constant.^[7]^[8]^[9]

The set of numbers that converge to zero depends on whether leading zeros are discarded (the usual formulation) or are retained (as in Kaprekar's original formulation). In the usual formulation, there are 77 four-digit numbers that converge to zero,^[10] for example 2111. However, in Kaprekar's original formulation the leading zeros are retained, and only repdigits such as 1111 or 2222 map to zero. This contrast is illustrated below:

discard leading zeros	retain leading zeros
2111 − 1112 = 999 999 − 999 = 0	2111 − 1112 = 0999 9990 − 0999 = 8991 9981 − 1899 = 8082 8820 − 0288 = 8532 8532 − 2358 = 6174

Below is a flowchart. Leading zeros are retained, however the only difference when leading zeros are discarded is that instead of 0999 connecting to 8991, we get 999 connecting to 0.

Sequence of Kaprekar transformations ending in 6174

Numbers of length three digits

If the Kaprekar routine is applied to numbers of three digits in base 10, the resulting sequence will almost always converge to the value 495 in at most six iterations, except for a small set of initial numbers which converge instead to 0.^[7]

The set of numbers that converge to zero depends on whether leading zeros are discarded (the usual formulation) or are retained (as in Kaprekar's original formulation). In the usual formulation, there are 60 three-digit numbers that converge to zero,^[11] for example 211. However, in Kaprekar's original formulation the leading zeros are retained, and only repdigits such as 111 or 222 map to zero.

Below is a flowchart. Leading zeros are retained, however the only difference when leading zeros are discarded is that instead of 099 connecting to 891, we get 99 connecting to 0.

Sequence of three digit Kaprekar transformations ending in 495

Other digit lengths

For digit lengths other than three or four (in base 10), the routine may terminate at one of several fixed points or may enter one of several cycles instead, depending on the starting value of the sequence.^[7] See the table in the section above for base 10 fixed points and cycles.

The number of cycles increases rapidly with larger digit lengths, and all but a small handful of these cycles are of length three. For example, for 20-digit numbers in base 10, there are fourteen constants (cycles of length one) and ninety-six cycles of length greater than one, all but two of which are of length three. Odd digit lengths produce fewer different end results than even digit lengths.^[12]^[13]

Sometimes these numbers (495, 6174, and their counterparts in other digit lengths or in bases other than 10) are called "Peyush constants" named after Peyush Dixit who solved this routine as a part of his IMO 2000 (International Mathematical Olympiad, Year 2000) thesis. ^[14]

Programming example

The example below implements the Kaprekar mapping described in the definition above to search for Kaprekar's constants and cycles in Python.

import string
from collections import deque
from collections.abc import Sequence, Generator


BASE36_DIGITS = f"{string.digits}{string.ascii_uppercase}"


def digit_count(x: int, /, base: int = 10) -> int:
    count = 0
    while x > 0:
        count += 1
        x //= base
    return count


def get_digits(x: int, /, base: int = 10, init_k: int = 0) -> str:
    if init_k > 0:
        k = digit_count(x, base)
    digits = deque()
    while x > 0:
        digits.appendleft(BASE36_DIGITSx % base])
        x //= base
    if init_k > 0:
        for _ in range(k, init_k):
            digits.appendleft("0")
    return "".join(digits)


def kaprekar_map(x: int, /, base: int = 10, init_k: int = 0) -> int:
    digits = "".join(sorted(get_digits(x, base, init_k)))
    descending = int("".join(reversed(digits)), base)
    ascending = int(digits, base)
    return descending - ascending


def kaprekar_cycle(
    x: int | str | bytes | bytearray, /, base: int = 10
) -> listint | str]:
    """
    Return Kaprekar's cycles as list

    >>> kaprekar_cycle(8991)
    [6174]
    >>> kaprekar_cycle(865296432)
    [865296432, 763197633, 844296552, 762098733, 964395531, 863098632, 965296431, 873197622, 865395432, 753098643, 954197541, 883098612, 976494321, 874197522]
    >>> kaprekar_cycle('09')
    [9, 81, 63, 27, 45]
    >>> kaprekar_cycle('0F', 16)
    ['0F', 'E1', 'C3', '87']
    >>> kaprekar_cycle('B71FD85', 16)
    ['B71FD85', 'E83FB72', 'DB3FB43', 'CA6F854', 'B73FB85', 'C63FB94', 'C84FA74', 'B82FC75', 'D73FB83', 'CA3FB54', 'C85F974']

    """
    leading_zeroes_retained = not isinstance(x, int)
    init_k = len(x) if leading_zeroes_retained else 0
    x = int(x) if base == 10 else int(x, base)
    seen = []
    while x not in seen:
        seen.append(x)
        x = kaprekar_map(x, base, init_k)
    cycle = []
    while x not in cycle:
        cycle.append(x)
        x = kaprekar_map(x, base, init_k)
    return cycle if base == 10 else get_digits(x, base, init_k) for x in cycle


if __name__ == "__main__":
    import doctest

    doctest.testmod()

Citations

^ Hanover 2017, p. 1, Overview.
^ Hanover 2017, p. 3, Methodology.
^ (sequence A099009 in the OEIS)
^ "Sample Kaprekar Series".
^ "Sample Kaprekar Series for hexadecimal numbers".
^ Kaprekar DR (1955). "An Interesting Property of the Number 6174". Scripta Mathematica. 15: 244–245.
^ ^a ^b ^c Weisstein, Eric W. "Kaprekar Routine". MathWorld.
^ Yutaka Nishiyama, Mysterious number 6174
^ Kaprekar DR (1980). "On Kaprekar Numbers". Journal of Recreational Mathematics. 13 (2): 81–82.
^ (sequence A069746 in the OEIS)
^ (sequence A090429 in the OEIS)
^ "Sample Kaprekar Series".
^ "Playing with Numbers".
^ Hanover 2017, p. 14, Operations.

References

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.

External links

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

[retained-4] ^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k ^l ^m ⁿ ^o ^p ^q Leading zeroes retained.

[FOOTNOTEHanover20171Overview-1] Hanover 2017, p. 1, Overview.

[FOOTNOTEHanover20173Methodology-2] Hanover 2017, p. 3, Methodology.

[A099009-3] (sequence A099009 in the OEIS)

[sourceforge_dec-5] "Sample Kaprekar Series".

[sourceforge_hex-6] "Sample Kaprekar Series for hexadecimal numbers".

[Kaprekar1955-7] Kaprekar DR (1955). "An Interesting Property of the Number 6174". Scripta Mathematica. 15: 244–245.

[mathworld-8] Weisstein, Eric W. "Kaprekar Routine". MathWorld.

[9] Yutaka Nishiyama, Mysterious number 6174

[Kaprekar1980-10] Kaprekar DR (1980). "On Kaprekar Numbers". Journal of Recreational Mathematics. 13 (2): 81–82.

[11] (sequence A069746 in the OEIS)

[12] (sequence A090429 in the OEIS)

[13] "Sample Kaprekar Series".

[14] "Playing with Numbers".

[IMO2000-15] Hanover 2017, p. 14, Operations.

[1]

[2]

[3]

[note 1]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

Definition and properties