TheronShifflett at 13:38, 1 May 2026

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{{Short description|Indian recreational mathematician (1905–1986)}}
{{EngvarB|date=September 2014}}
{{Use dmy dates|date=September 2014}}
{{Infobox person
| birth_name = D.R. Kaprekar
| image = D. R. Kaprekar.gif
| alt = D. R. Kaprekar
| caption =
| birth_date = {{Birth date|df=yes|1905|01|17}}
| birth_place = [[Dahanu]], [[Bombay Presidency]], [[British Raj|India]]
| death_date = {{death date and age|df=yes|1986|7|4|1905|1|17}}
| death_place = [[Nasik]], Maharashtra, India
| known_for = Contributions to [[recreational mathematics]]
| occupation = School teacher
}}

'''Dattatreya Ramchandra Kaprekar''' ({{langx|mr|दत्तात्रेय रामचंद्र कापरेकर}}; 17 January 1905{{snd}}4 July 1986) was an Indian [[recreational mathematician]] who  described several [[Template:Classes of natural numbers|classes of natural numbers]] including the [[Kaprekar number|Kaprekar]], [[harshad number|Harshad]] and [[self number|self]] numbers and discovered [[Kaprekar's constant]], named after him.<ref>{{Cite web |date=2023-01-17 |title=क्‍या आप जानते हैं जादुई नंबर 6174 की पहेली? इस भारतीय गणितज्ञ ने की खोज |url=https://www.aajtak.in/education/knowledge/story/indian-mathematician-dattatreya-ramchandra-kaprekar-who-invented-magical-6174-number-1616954-2023-01-17 |access-date=2024-10-13 |website=आज तक |language=hi}}</ref> Despite having no formal postgraduate training and working as a schoolteacher, he published extensively and became well known in recreational mathematics circles.<ref name="mactutor">{{MacTutor|id=Kaprekar}}</ref>

== Education and work ==

Kaprekar received his secondary school education in [[Thane]] and studied at [[Pune University|Fergusson College]] in [[Pune]]. In 1927, he won the Wrangler R. P. Paranjpye Mathematical Prize for an original piece of work in mathematics.<ref>{{cite web |title=Dattaraya Ramchandra Kaprekar |author=Dilip M. Salwi |url=http://www.4to40.com/legends/index.asp?id=142 |date=24 January 2005 |accessdate=30 November 2007 |url-status=dead |archiveurl=https://web.archive.org/web/20071116164833/http://www.4to40.com/legends/index.asp?id=142 |archivedate=16 November 2007 |df=dmy }}</ref>

He attended the [[University of Mumbai]], receiving his bachelor's degree in 1929. He was a schoolteacher at the government junior school in Devlali, [[Maharashtra]], India for his entire career (1930-1962). Cycling from place to place, he also tutored private students with unconventional methods, cheerfully sitting by a river and "thinking of theorems". He published extensively, writing about such topics as [[recurring decimal]]s, [[magic squares]], and integers with special properties.{{cn|date=September 2024}}

==Discoveries==
Working largely alone, Kaprekar discovered a number of results in number theory and described various properties of numbers.<ref>{{cite book|last1=Athmaraman|first1=R.|title=The Wonder World of Kaprekar Numbers|date=2004|publisher=The Association of Mathematics Teachers of India|location=Chennai (India)}}</ref> In addition to the [[Kaprekar's constant]] and the [[Kaprekar number]]s which were named after him, he also described [[self number]]s or ''Devlali numbers'', the [[harshad number]]s and [[Demlo number]]s. He also constructed certain types of magic squares related to the Copernicus magic square.<ref>{{cite journal |author=Kaprekar, D. R. |year=1974 |title=The Copernicus Magic Square |journal=Indian Journal of History of Science |volume=9 |issue=1}}</ref> Initially his ideas were not taken seriously by Indian mathematicians, and his results were published largely in low-level mathematics journals or privately published, but international fame arrived when [[Martin Gardner]] wrote about Kaprekar in his March 1975 column of ''Mathematical Games'' for ''[[Scientific American]]''. A description of [[Kaprekar's constant]], without mention of Kaprekar, appears in the children's book ''The I Hate Mathematics Book'', by [[Marilyn Burns (mathematics educator)|Marilyn Burns]],<ref>{{cite book |last1=Burns |first1=Marilyn |title=The I Hate Mathematics Book |date=1975 |publisher=Little Brown and Company |location=Boston |isbn=0-316-11741-2 |page=85}}</ref> published in 1975. Today his name is well-known and many other mathematicians have pursued the study of the properties he discovered.<ref name=mactutor/>

===Kaprekar's Constant===
{{main|Kaprekar's routine}}

In 1955, Kaprekar discovered an interesting property of the number 6174, which was subsequently named the Kaprekar constant.<ref>{{cite journal |author=Kaprekar |first=D.R. |year=1955 |title="An interesting property of the number 6174" |journal=[[Scripta Mathematica]] |volume=21 |pages=304 |via=Elsevier Science Direct}}</ref> He showed that 6174 is reached in the end as one repeatedly subtracts the highest and lowest numbers that can be constructed from a set of four digits that are not all identical. Thus, starting with 1234, we have:
:4321 − 1234 = 3087, then
:8730 − 0378 = 8352, and
:8532 − 2358 = 6174.
Repeating from this point onward leaves the same number (7641 − 1467 = 6174). In general, when the operation converges it does so in at most seven iterations.

A similar constant for 3 digits is [[495 (number)|495]].<ref>{{Cite web |title=Math Point: The mysterious 6174 revisited |url=https://mathpoint.blogspot.com/2006/12/mysterious-6174-revisited.html |access-date=2025-02-12 |language=en}}</ref> However, in base 10 a single such constant only exists for numbers of 3 or 4 digits; for other digit lengths or bases other than 10, the [[Kaprekar's routine]] algorithm described above may in general terminate in multiple different constants or repeated cycles, depending on the starting value.<ref>{{Cite web |date=2006-03-01 |title=Mysterious number 6174 {{!}} plus.maths.org |url=https://plus.maths.org/content/mysterious-number-6174 |access-date=2025-02-12 |website=plus.maths.org |language=en}}</ref>
For example, for 2-digit numbers, the numbers eventually enter a loop, for example:
:31 - 13 = 18
:81 - 18 = 63
:63 - 36 = 27
:72 - 27 = 45
:54 - 45 = 9
:90 - 9 = 81
:81 - 18 = 63
The loop in question is 63, 27, 45, 9, 81, and back to 63.

However, if in the above example 9 is not treated as a 2-digit number (09), all 2-digit numbers will end at 9. All differences between 2-digit number digital swaps are multiples of 9, and thus will immediately enter the loop above at some stage. (Notably, both 495 and 6,174 are multiples of 9.)

===Kaprekar number===
{{main|Kaprekar number}}

Another class of numbers Kaprekar described are Kaprekar numbers.<ref>{{MathWorld|urlname=KaprekarNumber|title=Kaprekar Number}}</ref> A Kaprekar number is a positive integer with the property that if it is squared, then its representation can be partitioned into two positive integer parts whose sum is equal to the original number (e.g. 45, since 45<sup>2</sup>=2025, and 20+25=45, also 9, 55, 99 etc.) However, note the restriction that the two numbers are positive; for example, 100 is not a Kaprekar number even though 100<sup>2</sup>=10000, and 100+00 = 100. This operation, of taking the rightmost digits of a square, and adding it to the integer formed by the leftmost digits, is known as the Kaprekar operation.

Some examples of Kaprekar numbers in base 10, besides the numbers 9, 99, 999, ..., are {{OEIS|id=A006886}}:
{| class="wikitable"
! Number !! Square !! Decomposition
|-
| 703 || 703² = 494209 || 494+209 = 703
|-
| 2728 || 2728² = 7441984 || 744+1984 = 2728
|}

===Devlali or self number===
{{main|Self number}}

In 1963, Kaprekar defined the property which has come to be known as self numbers,<ref name=devlali>Kaprekar, D. R. The Mathematics of New Self-Numbers Devalali (1963)nn: 19–20</ref> as the integers that cannot be generated by taking some other number and adding its own digits to it. For example, 21 is not a self number, since it can be generated from 15: 15 + 1 + 5 = 21. But 20 is a self number, since it cannot be generated from any other integer. He also gave a test for verifying this property in any number. These are sometimes referred to as Devlali numbers (after the town where he lived); though this appears to have been his preferred designation,<ref name=devlali/> the term "self number" is more widespread. Sometimes these are also designated ''Colombian number''s after a later designation.

===Harshad number===
{{main|Harshad number}}

Kaprekar also described the [[harshad number]]s which he named harshad, meaning "giving joy" ([[Sanskrit]] ''harsha'', joy ''+da'' taddhita pratyaya, [[causative]]); these are defined by the property that they are divisible by the sum of their digits. Thus 12, which is divisible by 1 + 2 = 3, is a harshad number. These were later also called ''Niven numbers'' after 1977 lecture on these by the Canadian mathematician [[Ivan M. Niven]]. Numbers which are harshad in all bases (only 1, 2, 4, and 6) are called ''all-harshad numbers''. Much work has been done on harshad numbers, and their distribution, frequency, etc. are a matter of considerable interest in number theory today.{{Citation needed|date=May 2020}}

=== Demlo number ===
Kaprekar also studied the [[Demlo number]]s,<ref>{{cite journal |author=Gunjikar, K. R.|author2= Kaprekar, D. R. |title=Theory of Demlo numbers |journal=J. Univ. Bombay |volume=VIII |issue=3 |pages=3–9 |year=1939 |url=http://OEIS.org/A249605/a249605.pdf}}</ref> name of which was derived from the name of a train station Demlo (now called [[Dombivli railway station|Dombivili]]) 30 miles from Bombay on the then [[G. I. P. Railway]] where he had the idea of studying them.<ref name=mactutor/> The best known of these are the Wonderful Demlo numbers 1, 121, 12321, 1234321, ..., which are the squares of the [[repunit]]s 1, 11, 111,1111, ....<ref>{{MathWorld |title= Demlo Number |urlname=DemloNumber}}</ref>

== See also ==
* [[Prahalad Chunnilal Vaidya]]

==References==
{{reflist|2}}

==External links==
* [http://plus.maths.org/issue38/features/nishiyama/index.html "Mysterious number 6174"]
* [https://www.youtube.com/watch?v=d8TRcZklX_Q Numberphile (Dec 5, 2011) 6174] a YouTube video by [[Numberphile]]
{{Authority control}}

{{DEFAULTSORT:Kaprekar, D. R.}}
[[Category:1905 births]]
[[Category:1986 deaths]]
[[Category:20th-century Indian mathematicians]]
[[Category:Indian number theorists]]
[[Category:Recreational mathematicians]]
[[Category:Magic squares]]
[[Category:People from Thane district]]
[[Category:Scientists from Maharashtra]]

D. R. Kaprekar - Revision history

TheronShifflett at 13:38, 1 May 2026