There's a math problem I've been working on for some time that looks like it's finally been solved. To explain it to people I usually start with a warm-up:

Cut a square into indentical pieces so
that they all touch the center point.

As I say, this is a warm up. It's fairly easy to do, and serves to introduce the ideas involved. So we do that, and we move on.

Recently, though, I wondered how many solutions there are to the above problem. I rapidly came up with a small number, and was reasonably content.

I was wrong. I now have what I think is the complete answer, but I was wondering if (a) I might have missed more, and (b) how can I prove that my current answer is complete.

I'd be interested in the musings and approaches taken here.

Thanks.