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Pigeon
Joined: Thu Mar 31, 2011 4:00 pm Posts: 10071

Kaprekar's Constant
6174 is known as Kaprekar's constant after the Indian mathematician D. R. Kaprekar. This number is notable for the following property:
1. Take any fourdigit number, using at least two different digits. (Leading zeros are allowed.) 2. Arrange the digits in ascending and then in descending order to get two fourdigit numbers, adding leading zeros if necessary. 3. Subtract the smaller number from the bigger number. 4. Go back to step 2.
The above process, known as Kaprekar's routine, will always reach its fixed point, 6174, in at most 7 iterations. Once 6174 is reached, the process will continue yielding 7641  1467 = 6174 Check it out Here

Thu Nov 07, 2013 2:59 am 


Royal
Joined: Mon Apr 11, 2011 6:55 pm Posts: 6494

Re: Kaprekar's Constant
Quote: D R Kaprekar was born in Dahanu, a town on the west coast of India about 100 km north of Mumbai. He was brought up by his father after his mother died when he was eight years old. His father was a clerk who was fascinated by astrology. Although astrology requires no deep mathematics, it does require a considerable ability to calculate with numbers, and Kaprekar's father certainly gave his son a love of calculating.
Kaprekar attended secondary school in Thane (sometime written Thana), which is northeast of Mumbai but so close that it is essentially a suburb. There, as he had from the time he was young, he spent many happy hours solving mathematical puzzles. He began his tertiary studies at Fergusson College in Pune in 1923. There he excelled, winning the Wrangler R P Paranjpe Mathematical Prize in 1927. This prize was awarded for the best original mathematics produced by a student and it is certainly fitting that Kaprekar won this prize as he always showed great originality in the number theoretic questions he thought up. He graduated with a B.Sc. from the College in 1929 and in the same year he was appointed as a school teacher of mathematics in Devlali, a town very close to Nashik which is about 100 km due east of Dahanu, the town of his birth. He spent his whole career teaching in Devlali until he retired at the age of 58 in 1962.
The fascination for numbers which Kaprekar had as a child continued throughout his life. He was a good school teacher, using his own love of numbers to motivate his pupils, and was often invited to speak at local colleges about his unique methods. He realised that he was addicted to number theory and he would say of himself:
A drunkard wants to go on drinking wine to remain in that pleasurable state. The same is the case with me in so far as numbers are concerned.
Many Indian mathematicians laughed at Kaprekar's number theoretic ideas thinking them to be trivial and unimportant. He did manage to publish some of his ideas in low level mathematics journals, but other papers were privately published as pamphlets with inscriptions such as Privately printed, Devlali or Published by the author, Khareswada, Devlali, India. Kaprekar's name today is wellknown and many mathematicians have found themselves intrigued by the ideas about numbers which Kaprekar found so addictive. Let us look at some of the ideas which he introduced.
Perhaps the best known of Kaprekar's results is the following which relates to the number 6174, today called Kaprekar's constant. One starts with any fourdigit number, not all the digits being equal. Suppose we choose 4637 (which is the first four digits of EFR's telephone number!). Rearrange the digits to form the largest and smallest numbers with these digits, namely 7643 and 3467, and subtract the smaller from the larger to obtain 4167. Continue the process with this number  subtract 1467 from 7641 and we obtain 6174, Kaprekar's constant. Lets try again. Choose 3743 (which is the last four digits of EFR's telephone number!).
He published the result in the paper Problems involving reversal of digits in Scripta Mathematica in 1953. Clearly starting with 1111 will yield 0 from Kaprekar's process. In fact the Kaprekar process will yield either 0 or 6174. Exactly 77 four digit numbers stabilize to 0 under the Kaprekar process, the remainder will stabilize to 6174. Anyone interested could experiment with numbers with more than 4 digits and see if they stabilise to a single number (other than 0).

Thu Nov 07, 2013 6:07 am 


Pigeon
Joined: Thu Mar 31, 2011 4:00 pm Posts: 10071

Re: Kaprekar's Constant
It was found that 495 is the constant for 3 digits.

Thu Nov 07, 2013 6:46 am 


Pigeon
Joined: Thu Mar 31, 2011 4:00 pm Posts: 10071

Re: Kaprekar's Constant
Was looking at this again today.
Numbers, for the lol of it.

Sun Nov 27, 2016 4:07 am 




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