Kaprekar's Constant 6174 for Four Digits Number Reality to Other

International journal of Chemistry, Mathematics and Physics (IJCMP)

[Vol-4, Issue-5, Sep-Oct, 2020]

https://dx.doi.org/10.22161/ijcmp.4.5.1

ISSN: 2456-866X

http://www.aipublications.com/ijcmp/ Page | 92

Open Access

Kaprekar’s Constant 6174 for Four Digits

Number Reality to Other Digits Number

Dr. Pijush Kanti Bhattacharjee

Department of Electronics & Communication Engineering, Camellia Institute of Technology & Management, Bainchi, Hooghly, West

Bengal, Pin-712134, India.

Abstract— Indian mathematician Kaprekar discovered a constant or dead end 6174 for four digits decimal

number when the digits in the number are not repeated, ascending order digits number is subtracted from

descending order digits number, and the result is further processed likewise till constant 6174 reached. All

the subtraction results are divisible by 9 and 3 without any remainder. In the same way, constant or dead

end is discovered for two digits number as 9, three digits number as 495, ten digits number as 9753086421

and for binary number as 01 or 1.

Keywords— Kaprekar’s constant 6174; Two digits number constant 9; Three digits number constant

495; Ten digits number constant 9753086421; Binary number constant 01 or 1.

I. INTRODUCTION KAPREKAR’S CONSTANT

6174 is the number discovered by an Indian

mathematician, Dattatreya Ramchandra Kaprekar [1] – [2].

It is called magic number. By looking at the number you

will not feel anything weird, but has pleasantly mesmerized

several learned mathematicians no ends. From the year

1949 till now, this number has remained a puzzle for all

around the world.

Indian mathematician Dattatreya Ramchandra Kaprekar

loved experimenting with numbers. In the process of one

of his experiments, he discovered a bizarre coincidence and

during a ‘Mathematics Conference’ held in erstwhile

Madras in the year 1949, Kaprekar introduced this number

6174 to the world.

To understand why this number is so magical let’s look at

some interesting facts. For example, choose any decimal

number of four digits keeping in mind that no digit is

repeated. Let’s take 1234 for example. Write the number in

descending order: 4321. Now write in ascending order:

1234. Now subtract the smaller number from the large

number: 4321 – 1234 = 3087.

Now the result again is arranged in descending and

ascending order. For 3087, place the digits in decreasing

order: 8730. Now place them in increasing order: 0378.

Now subtract the smaller number from the large number,

we get, 8730 – 0378 = 8352. Repeat the above procedures

with the number found in the result. So, 8532 – 2358 =

6174. Let us repeat this process with 6174. The result is

given by, 7641 – 1467 = 6174.

We have reached a dead end and there is no point repeating

the process, since we will get the only one result 6174. If

you think that this is just a coincidence, repeat this process

with any other four digits decimal number. Voila! Your

final result will be 6174. This formula is called Kaprekar’s

constant. Also the subtraction results are completely

divisible by 9 and 3 having no remainder.

In a computer based experiment a gentleman named

Nishiyama discovered that the Kaprekar process reached

6174 in a maximum of seven stages [3]. According to

Nishiyama, ‘If you do not reach 6174 even after repeating

the process seven times, then you must have made a

mistake and you should try again’.

However, almost all renowned mathematicians of the era

mocked his discovery. Some Indian mathematicians

rejected his work and termed his theory childish. In time,

discussions of this discovery slowly started gaining

foothold both in India and abroad. Martin Garder,

America’s best-selling author with deep interest in

mathematics wrote an article about him in “Scientific

America”, a popular science magazine. Today, Kaprekar is

regarded a mathematical wizard and his discovery is

slowly gaining traction. Intrigued Mathematicians all over

the world are engrossed in researching this baffling reality.

International journal of Chemistry, Mathematics and Physics (IJCMP)

[Vol-4, Issue-5, Sep-Oct, 2020]

https://dx.doi.org/10.22161/ijcmp.4.5.1

ISSN: 2456-866X

http://www.aipublications.com/ijcmp/ Page | 93

Open Access

II. RESULTS BY KAPREKAR’S PROCESS FOR

OTHER DIGITS NUMBER

A decimal number, containing digits 0 to 9 only and having

the base 10, is chosen such that no digit is repeated as

taken in Kaprekar’s procedure. Then digits in the number

are arranged in ascending and descending order.

Thereafter, the ascending order digits number is subtracted

from the descending order digits number, it is seen that the

subtraction results are always divisible by 9 and 3 with no

remainder. We have seen that in any number, if the sum of

the digits is 9, then the number is completely divisible by 9

and 3 without any remainder.

(A) Two Digits Number Subtracted Ascending Order

from Its Descending Order

The following experiments are done on two digits decimal

number, where no digit is repeated.

(i) If two digits number 30 is taken, then 30 – 03 = 27;

repeating the process, 72 – 27 = 45; 54 – 45 = 9; Dead or

last end is reached to 9; all the subtraction results are

completely divisible by 9.

(ii) If two digits number 23 is chosen, then 32 – 23 = 9,

dead end is reached and the result is divisible by 9.

(iii) If two digits number 48 is selected, then 84 – 48 = 36.

If continuing the process with the result, 63 – 36 = 27; 72 –

27 = 45; 54 – 45 = 9; i.e., ultimately it is merging to 9.

Therefore, 9 is the dead end for two digits number. All the

subtraction results are completely divisible by 9.

It is also verified that when all two digits number (not

repeating digits) are arranged in ascending and descending

order, thereafter subtracting the ascending order number

from the descending order number, the result must be

divisible by 9 without any remainder, if the process is

repeated, at last we will reach to number 9 which is the

dead end or constant for this system of two digits number

like Kaprekar’s constant 6174 for four digits number.

(B) Three Digits Number Subtracted Ascending Order

from Its Descending Order

Next if the decimal number is chosen for three digits where

no digit is repeated, following results are obtained and all

the subtraction results are completely divisible by 9.

(i) The number is 710, 710 – 17 = 693; 963 – 369 = 594;

954 – 459 = 495; If we repeat, we will get same result as

495; Therefore, 495 is dead end for three digits number.

Again, all the subtraction results are divisible by 9. Also, 4

+ 9 + 5 = 18; 1 + 8 = 9; Thus 495 is completely divisible

by 9.

(ii) The number is 123, 321 – 123 = 198; If the process is

repeated with the result, then 981 – 189 = 792; 972 – 279 =

693; 963 – 369 = 594; 954 – 459 = 495; Thus 495 is the

last stage or dead end for three digits number. All the

subtraction results are completely divisible by 9.

(iii) The number is 963, 963 – 369 = 594; 954 – 459 = 495;

Thus final stage 495 is reached.

All the subtracting results are completely divisible by 9.

Therefore, for three digits number, the dead end or constant

is 495, i.e., ultimately we will reach to 495 when this

process continues; hence it will merge to 495 as last stage.

from Its Descending Order

For four digits decimal number, Kaprekar’s constant 6174

is already discovered as a dead end or constant. Therefore,

mathematician Kaprekar revealed a great land mark by

identifying the constant 6174 for four digits number first

time.

(D) Five Digits to Nine Digits Number Subtracted

Ascending Order from Its Descending Order

For five digits to nine digits decimal number, when the

digits are not repeated, and it is seen that an ascending

order digits number is subtracted from the descending

order digits number, in most of the times the some

digit/digits are repeated in the subtraction result (only in

very few subtraction results, the digits are not repeated),

therefore it cannot be continued for further processing, but

all the subtraction results are divisible by 9 without any

remainder. Following examples are mentioned below.

(i) For five digits number 12345, 54321 – 12345 = 41976,

then continuing, 97641 – 14679 = 82962; digit 2 is

repeated.

(ii) For five digits number 12346, 64321 – 12346 = 51975;

digit 5 is repeated.

(iii) For five digits number 12349, 94321 – 12349 = 81972;

further continuing, 98721 – 12789 = 85932; 98532 – 23589

= 74943; digit 4 is repeated.

(iv) For six digits number 123458, 854321 – 123458 =

730863; digit 3 is repeated.

(v) For seven digits number 2345678, 8765432 – 2345678

= 6419754; digit 4 is repeated.

(vi) For eight digits number 12345678, 87654321 –

12345678 = 75308643; digit 3 is repeated.

(vii) For nine digits number 023456789, 987654320 –

23456789 = 964197531; digits 1 and 9 are repeated.

International journal of Chemistry, Mathematics and Physics (IJCMP)

[Vol-4, Issue-5, Sep-Oct, 2020]

https://dx.doi.org/10.22161/ijcmp.4.5.1

ISSN: 2456-866X

http://www.aipublications.com/ijcmp/ Page | 94

Open Access

Hence, it is observed that Kaprekar’s procedure cannot be

continued for numbers having digits from five to nine due

to some digits are repeated. It is a surprising thing that the

subtraction results are always divisible by 9 having no

remainder. Hence, there is no dead end or lock stage, i.e.,

constant found for five digits to nine digits numbers.

(E) Ten Digits Number Subtracted Ascending Order

from Its Descending Order

It is a beautiful thing that ten digits number, in which the

digits are not repeated, dead end or lock stage is obtained

by first subtraction. In decimal numbering system, the ten

digits decimal number is 0123456789 where the digits are

not repeated. Now arranging the ten digits number in

ascending and descending order and then subtracting the

ascending order digits number (smaller) from the

descending order digits number (larger), we get,

9876543210 – 0123456789 = 9753086421. If we further

continue, the same result will be obtained. Thus the dead

end or constant for ten digits number is 9753086421. It is

amazing fact that adding all digits, 0 + 1 + 2 + 3 + 4 + 5 +

6 + 7 + 8 + 9 = 45; 4 + 5 = 9; Hence, it is completely

divisible by 9. Also the ten digits number without repeating

the same digit is the last or end number in decimal

numbering system.

(F) Octal and Hexadecimal Numbering System

Octal number system is taken digits from 0 to 7, and the

base is 8. Hexadecimal number system is same as decimal

number system, only the digits in a number is extended up

to fifteen like 0 to F and the base is 16, where A = 10, B =

11, C = 12, D = 13, E = 14, F = 15. Therefore, for octal and

hexadecimal numbering systems like Kaprekar’s procedure

are adopted as similar to decimal numbering system.

(G) Binary Numbering System Subtracted Ascending

Order from Its Descending Order

A binary number is represented by two digits such as 0 and

1, and the base is 2. If the digits are not repeated, then the

binary number is expressed as descending order 10 and

ascending order 01, then subtracting ascending order

binary number from descending order binary number, we

get, 10 – 01 = 01, thus 01, i.e., 1 is the dead end or constant

in case of binary numbering system. Moreover, 01 is the

1’s complement of 10. Since, 1 is the highest digit in

binary numbering system, the dead end 1 is divisible by 1

also.

III. CONCLUSION

It is an astonishing fact that in decimal number system, the

ascending order digits number are subtracted from the

descending order digits number (where the digits are not

repeated), the subtraction results are always divisible by 9

having no remainder which is the highest digit in decimal

system, and if the process continues like this ultimately we

arrive a dead end or lock stage for two digits to four digits

number and ten digits number. For four digits number, the

dead end is already discovered by mathematician Kaprekar

and it is called Kaprekar’s constant 6174. In this paper, the

dead ends for all other digits numbers are discovered with

proper explanation.

Therefore, it is concluded that like Kaprekar’s constant

6174 for four digits decimal number, the dead end or

constant for two digits decimal number is 9, for three digits

decimal number is 495 and for ten digits decimal number is

9753086421. In binary numbering system, the dead end or

constant is 01 or 1.

Now-a-days for computerised algorithm and manipulation

of huge or big data, this inherent knowledge for decimal

and other numbering systems like binary, octal,

hexadecimal etc. will be useful, and identify a precise way

for mathematical computation.

REFERENCE

[1] Kaprekar DR, “An Interesting Property of the Number

6174”, Scripta Mathematica 15: 244-245, 1955.

[2] Bowley Roger, “6174 is Kaprekar's Constant”, Numberphile.

University of Nottingham: Brady Haran.

[3] Nishiyama Yutaka, “Mysterious Number 6174”, Plus

Magazine, 2006.

Dr. Pijush Kanti Bhattacharjee is

associated with the study in Engineering,

Management, Law, Indo-Allopathy, Herbal,

Homeopathic, and Yogic Medicines. He is

having qualifications Ph.D (Engg.), M.E,

MBA, MDCTech, A.M.I.E (B.E or B.Tech),

LLB, B.Sc, B.A, BIASM, CMS, PET, EDT,

FWT, DATHRY, KOVID, DH, ACE, FDCI

etc. He worked in Department of Telecommunications (DoT),

Government of India as a Telecom Engineer from 1981 to 2007,

then worked in different Engineering Colleges and Assam

University [Central University], Silchar, India as Assistant and

Associate Professor from 2007 to 2020. He has written fourteen

books and more than hundred research papers. He is a Member of

IACSIT, Singapore; CSTA, UACEE, USA; IAENG, IETI,

Hongkong; and IE, ISTE, IAPQR, IIM, India. His research

interests are in Telecommunications including Mobile

Communications, Image Processing, VLSI, Nanotechnology,

Electrical Power Systems, Power Electronics Circuits,

Environmental Pollution, Medicine and Mathematics.