Kaprekar's Symmetry

Theorem
Let $n$ be a Kaprekar number with $D$ digits.

Then $10^D - n$ is also a Kaprekar number.

Proof
Since $n$ is a Kaprekar number of $D$ digits:
 * $\begin {cases} n^2 = a \times 10^D + b \\ n = a + b \end {cases}$

for some positive integers $a$ and $b$, $b < 10^D$.

Hence:

and we have:
 * $\paren {10^D - a - 2 b} + b = 10^D - a - b = 10^D - n$

Finally we check that $10^D - a - 2 b \ge 0$:

$10^D - a - 2 b \le -1$.

Then:

but squares are positive, a contradiction.

Hence $10^D - n$ is also a Kaprekar number of $D$ digits.