Numeri di Kaprekar

Dattatreya Ramchandra Kaprekar was born in Dahanu, on the coast north of Bombay, on 17 January 1905. He studied at Fergusson College in Pune from 1923, won the Wrangler R. P. Paranjpe Mathematical Prize in 1927, took his B.Sc. in 1929, and that same year began work as a schoolmaster in Devlali, where he stayed until retiring at 58 in 1962. He never held a university post. For more than thirty years he investigated the decimal representations of integers in his spare time, publishing in local journals and pamphlets and lecturing to anyone who would listen.

The discovery he is remembered for came in 1946: take any four-digit number with at least two distinct digits, arrange its digits into the largest and smallest numbers they can form, subtract, and repeat. You always reach 6174, and never in more than seven steps. He announced this at the Madras Mathematical Conference in 1949 and wrote it up as "Problems involving reversal of digits" in Scripta Mathematica in 1953. The number 6174 has been called Kaprekar's constant ever since.

The Kaprekar numbers are a separate idea, defined by splitting a square rather than sorting digits, and he returned to them in the Journal of Recreational Mathematics in 1980–1981. He also gave his name to Harshad numbers and to the self or Devlali numbers.

For most of his life the professional community ignored him. That changed in 1975, when Martin Gardner devoted his Mathematical Games column in the March issue of Scientific American to Kaprekar and his numbers. Kaprekar died in Devlali in 1986, by then internationally known.

Proprietà principali

• The Kaprekar numbers below five million are 1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, 4879, … — 56 of them in all, ending at 4,927,941.
• 45² = 2025 and 20 + 25 = 45; 703² = 494,209 and 494 + 209 = 703.
• If n² splits as q·10^m + r with q + r = n, then n² − n = q·(10^m − 1), so 10^m − 1 always divides n(n − 1).
• That same identity forces 10^m > n, so the cut can never fall inside the last digits of n — but it can fall to the left of the square, as in 4879² = 23,804,641 = 238 + 04641.
• Powers of ten are excluded by convention: 10² = 100 splits as 10 + 0, which satisfies the equation only because r is allowed to be zero.
• Kaprekar's routine, a separate process, sends every four-digit number with at least two distinct digits to 6174 in at most seven subtractions, and every three-digit number with at least two distinct digits to 495 in at most six.
• Repdigits are the only four-digit starting values the routine fails on: 1111 and its siblings go straight to 0 and stay there.
• Three and four digits are the only lengths with a single non-zero fixed point. At 2, 5 and 7 digits there is none, and at 6, 8 and 9 digits there are two apiece — 549945 and 631764, 63317664 and 97508421, 554999445 and 864197532 — which is why 495 and 6174 get to be called constants and the others do not.

Dove si incontrano

• Kaprekar's routine is a staple of school mathematics clubs and outreach talks, because the result is startling and the verification needs nothing but subtraction.
• Martin Gardner's Mathematical Games column in Scientific American, March 1975, introduced Kaprekar's work to a general readership and is the reason the name travelled outside India.
• 6174 appears in puzzle collections, Project Euler-style programming challenges and countless "write a program that reaches 6174" exercises.
• Harshad numbers (divisible by their own digit sum) and the self or Devlali numbers are both Kaprekar's coinages, and Devlali numbers still carry the name of the town where he taught.
• Kaprekar numbers generalise to other bases and other powers, and the base-and-power table is a common worked example in computational number theory courses.