6,174

D. R. Kaprekar (1905–1986) taught at a school in Devlali, Maharashtra, and worked on recreational number theory largely outside the academic mathematical community, publishing in minor journals and presenting at conferences at his own expense. He described the routine now named after him in 1949. Several of his results — Kaprekar numbers, Harshad numbers, the Demlo numbers — entered the literature and remain in use, which makes him one of the clearest cases of amateur work becoming standard vocabulary.

Kaprekar's routine is one of the few named results a reader can verify unaided in a couple of minutes. Order the digits of any four-digit number downwards, order them upwards, subtract, and repeat with the answer; whatever you start from, provided the digits are not all identical, you land on 6174 within seven rounds and stay there. Because there are only 10,000 possible starting points, the claim has been checked exhaustively rather than argued.

The same routine run on three-digit numbers converges on 495 instead, and always within three steps. Two digits have no single fixed point at all — the routine falls into a cycle — and five digits has three separate cycles rather than one constant. So 6174 is not an instance of a general law: four digits is one of the few widths where the process happens to land on a single value, which is why the four-digit case is the one that is famous.

Kaprekar's name attaches to two unrelated things, which is a frequent source of confusion. A Kaprekar number is one whose square splits into two parts that sum back to it — 297² = 88,209, and 88 + 209 = 297 — and the set includes 9, 45, 55, 99 and 297. 6174 is not one of them. It is the fixed point of Kaprekar's routine, a separate result by the same author.

6174 factors as 2 × 3² × 7³, and it is a Harshad number: its digits sum to 18, and 6174 ÷ 18 = 343, which is 7³. None of this has any bearing on the routine that made it famous — the fixed point arises from digit rearrangement and subtraction, which are operations on the numeral rather than on the quantity, so its factorisation plays no part.