An

integer which, when

multiplied by itself, yields a number whose

decimal representation, when cut in half down the middle, yields two integers whose

sum is the original

number.

*"What?!"*

OK, a simple example should clarify the muddy explanation above: if we multiply 45 by itself, we get 2025; cut 2025 in half to get 20 and 25, whose sum is the number we began with. Similarly, 99 times 99 = 9801, and 98+01 = 99 -- indeed, any integer containing only 9's is Kaprekar.

More generally, an integer is *n*-Kaprekar if when we split its square just left of its rightmost *n* digits and add the "halves", we obtain the original integer. Thus 4879 is 5-Kaprekar since 4879^{2} is 23804641 and 238+04641=4879. The set of *n*-Kaprekar integers is in one-to-one correspondence with the set of unitary divisors of 10^{n}-1.

If instead we work in binary, it turns out that every even perfect number is *n*-Kaprekar for some *n*, for example: 110^{2}=100100
and 10+0100=110.