I need help with this problem:
$K$ is equal to $111\ldots111 - 22\ldots22$ where $1$ is used $2n$ times and $2$ is used $n$ times. Prove that $K$ is a perfect square.
It is true, but I can't find a way to prove it. The only interesting thing I've found is that when $n$ is $1, K=3^2$, when $n$ is $2, K=33^2$ and so on. I tried to prove it using induction but wasn't able to find a way to do it, so your help would be really appreciated.