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Take any number that has four digits different from each other. In other words, no digit appears more than once. Now form a new number by rearranging the digits in ascending order. Likewise, form another number by rearranging the digits in descending order. Subtract the smaller number from the greater one. The subtraction would yield a different four digit number. Carry out the same operation on this one. Keep doing it. Sooner or later, in fact, sooner than later, you would end up with a number 6174. Thereafter if you subject this to the same operation you would always end up with the same number 6174.For example, let us take the number 2873. The two numbers obtained by arranging the digits in ascending or descending order are 8732 and 2378. Subtracting the latter from the former gives us gives us 6354. Let us carry out the same operation on this number. We obtain 6543 and 3456. Subtracting one from the other yields the number 3087. Rearranging the digits leads to 8730 and 0378. The subtraction results in the number 8352. Rearrangement gives us the numbers, 8532 and 2358. If we now subtract the second from the first we get 6174. Repeating the same operations on this we keep getting the same number, 6174.
let us take the number 2873. The two numbers obtained by arranging the digits in ascending or descending order are 8732 and 2378. Subtracting the latter from the former gives us gives us 6354. Let us carry out the same operation on this number. We obtain 6543 and 3456. Subtracting one from the other yields the number 3087. Rearranging the digits leads to 8730 and 0378. The subtraction results in the number 8352. Rearrangement gives us the numbers, 8532 and 2358. If we now subtract the second from the first we get 6174. Repeating the same operations on this we keep getting the same number, 6174.
Once we reach this particular number we are stuck with it. But the more amazing thing is this: every four digit number whose digits are not, all the same, will eventually hit 6174, at most 7 steps and then stay there!
Kaprekar Routine & Kaprekar
This routine of operations is called the Kaprekar routine and the number Kaprekar constant. The discoverer of this intriguing property of numbers was Dattatray Ramchandra Kaprekar. He was born in Dahanu, a small 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 for calculating.
Kaprekar attended secondary school in Thane, northeast of Mumbai, but so close that it is essentially a suburb. There 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 formulated by a student and Kaprekar perfectly’ fit the bill. 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. 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’snumber theoretic ideas considering them as trivial.He did
Manage to publish some of his ideas in mathematics journals known internationally. That brought him to the attention of mathematicians, in particular, those interested in the theory of numbers, all over the world. Kaprekar’s name today is well known and many mathematicians have found themselves intrigued by the ideas about numbers which Kaprekar found so addictive. They gave him the sobriquet, Wizard of Numbers.
The constant 6174 now named after him is not the only contribution he made to number theory. He discovered many fascinating properties of numbers. Some of these are now called Kaprekar numbers. A Kaprekar number n is such that n^2 can be split into two so that the two parts sum to n. For example (703)^2 = 494209. But 494 + 209=703. Notice that when the square is split we can start, the right-hand part with 0s. For example (9999)^2 = 99980001. But 9998 + 0001=9999. Of course from this observation we see that there are infinitely many Kaprekar numbers (certainly 9,99,999,9999, .. .are all Kaprekar numbers).
The first few Kaprekar numbers are: 1,9,45,55,99,297,703, 999, 2223, 2728, 4879, 4950, 5050, 5292, 7272, 7777,9999, 17344, 22222, 38962, 77778, 82656, 95121, 99999,142857, 148149, 181819, 187110, 208495,……
Then there are what Kaprekar called ‘self or Swayambhu numbers’. Take a look at the number 23. The addition of its digits leads to the number 5. Adding this to 23 gives us 28. The sum of its digits is 10. Adding that to 28 yields 38. Repeating the same operation we obtain a sequence 23, 28, 38, 49, 62, 70… and so. But the number that started this whole string itself can be obtained from 16 which in turn is given by 8.As a matter of fact, the sequence we looked at really starts at 1; 1,2,4, 8, 16, 23, 28, 38, 49, 62, 70,…..Now let us look at the number 29. It gives the Sequence 29, 40, 44, 52, 59, 73… and so on. But 29 is generated by 19, which in turn is generated by 14, which is generated by 7.However, nothing generates 7 because it is a self-number.Few self-numbers are: 1, 3, 5, 7, 9, 20, 31, 42, 53,64,75,86,97,108, 110, 1211 132, 143, 154, 165, 176, 187, 198,209,211,222, 233, 244, 255, 266, 277, 288, 299, 310, 312,323,334,345…
Several other captivating numbers were discovered by him.For example, there are Harshad numbers. Harshad, a Sanskrit word, means one which gives joy. A Harshad number is defined as one which is divisible by the sum of its digits. Apart from the trivial single digit number, real Harshad numbers are 10,12,18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 70,72, 80, 81, 84, 90, 100, 102, 108, 110, 111, 112, 114, 117,120, 126, 132, 133, 135, 140, 144, 150, 152, 153, 156, 162,171, 180, 190, 192, 195, 198, 200, ….It will be noticed that 80, 81 are a pair of Consecutive numbers which are both Harshad, while the three Consecutive numbers 110, 111, 112 are all Harshad. It was proven in 1994 that no 21 consecutive numbers can all be Harshad numbers. It is possible to have 20 consecutive Harshad numbers but one has to go to numbers greater than (10)^44363342786 before such a sequence is found. One further intriguing property is that 2!, 3!, 4!, 5!, … are all Harshad numbers. One would be tempted to conjecture that n! is a Harshad number for every n—this, however, would be incorrect. The smallest factorial which is not a Harshad number is 432!.
Kaprekar was obsessed with numbers throughout his life. After his retirement from the school, he totally immersed himself in this work. But after the death of his wife in 1966 he was depressed. Moreover, he struggled to make a living on his meagre pension. This was despite the fact that Kaprekar lived in the cheapest possible way, being only interested in spending his waking hours experimenting with numbers. He was forced to give private tuition in mathematics and science to make enough money to survive.
Kaprekar invented different number properties through out his life. Yet, he was not well known beyond the limited circle of recreational mathematicians. This was certainly callous of the society at large since many of his papers were reviewed in Mathematical Reviews. International fame only came in 1975 when Martin Gardener wrote about Kaprekar and his numbers in his ‘Mathematical Games’ column in the March issue of Scientific American. He died, mostly unsung in his own country in 1986.
So who was the wizard of numbers? He was a modest school teacher, D R Kaprekar who discovered several beguiling properties of numbers some of which are now named after him.