Kaprekar number checker

The Kaprekar number checker answers a classic recreational number-theory question: for a positive integer n, can the decimal digits of n squared be split into a left part and a right part that add back to n itself? Famous examples include forty-five (forty-five squared is two thousand twenty-five, and twenty plus twenty-five equals forty-five) and two hundred ninety-seven (eighty-eight thousand two hundred nine splits as eighty-eight plus two hundred nine). The engine uses deterministic BigInt arithmetic and an exhaustive scan of split points, so the result is free of floating-point drift and identical on every run. It never calls the network, never samples randomness, and never depends on the clock.

What a Kaprekar number is and what you see

A positive integer n is a Kaprekar number when the decimal representation of n squared can be divided into two contiguous parts — a left part and a right part — such that those two integers sum exactly to n. The right-hand part may begin with zeros when read as a digit string, but it must not be entirely zero; that rule correctly rejects numbers such as one hundred, whose square is ten thousand and only "works" if a zero right half is allowed.

By long-standing convention the number one is Kaprekar: one squared is one, read as zero plus one. Nine is Kaprekar because eighty-one splits as eight plus one. Forty-five and fifty-five are the best-known two-digit examples; ninety-nine, two hundred ninety-seven, and seven hundred three appear in every standard list.

Ten is composite and positive but not Kaprekar: one hundred cannot be split into non-zero right parts that sum to ten. This checker returns the parsed n, the exact square string, left and right when a valid split exists (otherwise null), and kaprekar true if and only if a valid split was found. Non-Kaprekar positive integers are valid successful responses with kaprekar false; they are not errors.

If you need only the square of an integer without the Kaprekar split test, square it yourself; this page answers the Kaprekar-number question with both parts of the winning split when one exists.

How inputs are read and how the square split stays exact

Optional leading plus signs are allowed for positive values; embedded spaces, hex prefixes, and exponential notation are rejected so a bad cell does not become a wrong flag. Negative integers and zero are rejected because this checker defines Kaprekar numbers on positive integers starting at one. The engine stays inside JavaScript safe integers for the input, which is more than enough for classroom problems, programming contests, and most applied checks on moderately large n.

The algorithm then walks every split index from one through the last digit exclusive: for each index it reads left as the integer prefix and right as the integer suffix, skips suffixes that are all zeros, and accepts the first pair whose sum equals n. There is no floating-point step that could round a near-match into a false Kaprekar claim.

Where Kaprekar checks fit in study, research demos, and code

Students meet Kaprekar numbers when exploring recreational number theory, digit-based properties, and curiosities named after D. R. Kaprekar alongside Kaprekar's constant six thousand one hundred seventy-four from his routine on four-digit rearrangements. Teachers can generate answer keys for worksheets without trusting opaque calculator apps that hide the square-and-split step.

Developers use the same contract to validate sample data for number-theory libraries, to gate property tests on random integer generators, or to pre-check that a seed sits in the Kaprekar class before a demonstration step. Contest practice often needs a fast offline source of Kaprekar answers so solutions can be checked without downloading OEIS tables.

Pair it with a Smith-number checker when you need digit sums of prime factors, with an abundant-number checker when you need divisor-sum classification, and with prime factorization when you need prime powers rather than square splits.