Working largely alone, Kaprekar discovered a number of results in number theory and described various properties of numbers. In addition to the
Kaprekar's constant and the
Kaprekar numbers which were named after him, he also described
self numbers or
Devlali numbers, the
harshad numbers and
Demlo numbers. He also constructed certain types of magic squares related to the Copernicus magic square. 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, published in 1975. Today his name is well-known and many other mathematicians have pursued the study of the properties he discovered. 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. 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. 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 Another class of numbers Kaprekar described are Kaprekar numbers. 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 452=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 1002=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 :
Devlali or self number In 1963, Kaprekar defined the property which has come to be known as self numbers, 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, name of which was derived from the name of a train station Demlo (now called
Dombivili) 30 miles from Bombay on the then
G. I. P. Railway where he had the idea of studying them. == See also ==