Kaprekar numbers: 45, 297 and split squares

The definition

Take a number n with d digits. Square it. Split the square so that the right part has exactly d digits and the left part holds whatever remains. If the two parts add up to n, and the right part is not zero, then n is a Kaprekar number.

The rule that the right part must be non-zero excludes trivial cases such as 100² = 10000, which would split into 100 + 00.

All Kaprekar numbers below 100,000

Checking every number up to 100,000 with the definition above gives this list:

Some references use a looser rule that allows the square to be split at any position. That version adds numbers such as 4879 (4879² = 23804641, and 238 + 04641 = 4879), 5292 and 38962. Both lists appear in the literature, so it is worth checking which definition a source uses.

Why 9, 99, 999 … always work

Numbers made only of nines are always Kaprekar numbers. For n = 10ᵏ − 1:

The left part is 10ᵏ − 2 and the right part is 1, and their sum is 10ᵏ − 1 = n. For example 99² = 9801, and 98 + 01 = 99.

Pairs that add up to a power of ten

Look at the list again: 45 and 55, 297 and 703, 2223 and 7777, 2728 and 7272, 4950 and 5050, 17344 and 82656. Each pair adds up to 100, 1,000, 10,000 or 100,000. This is not a coincidence. If n works, then 10ᵈ − n usually works too, because squaring the complement produces the same split with the parts swapped.

Behind this symmetry is a connection with modular arithmetic: Kaprekar numbers correspond to ways of splitting 10ᵈ − 1 into coprime factors. That is why numbers of the form 10ᵈ − 1 with many factors produce more Kaprekar numbers.

How to search for Kaprekar numbers

1. Choose a number n and count its digits d.
2. Compute n² and write it out in full.
3. Take the last d digits as the right part and everything before them as the left part.
4. Add the two parts. If the sum equals n and the right part is not zero, n is a Kaprekar number.

For example, n = 99 has two digits. 99² = 9801, the right part is 01 and the left part is 98, and 98 + 1 = 99. For n = 46 the square is 2116 and 21 + 16 = 37, so 46 is not a Kaprekar number.

Kaprekar numbers and the calendar

The year 2025 drew attention to the pattern because 2025 = 45² and 45 is a Kaprekar number. The next square year, 2116 = 46², does not work. The next year that is the square of a Kaprekar number is 3025 = 55², and indeed 30 + 25 = 55. Anyone hoping to see it will need considerable patience.

How is this related to 6174?

Mathematically, not much. The constant 6174 comes from sorting and subtracting digits; Kaprekar numbers come from squaring and splitting. They share a discoverer and an interest in base-ten digit patterns. Both are examples of how rearranging or cutting the decimal representation of a number can reveal structure that ordinary arithmetic hides.

Other digit-based families are described in harshad and self numbers and narcissistic numbers.