Congrats! 5 buttons is the lowest button count I've heard yet for the game! I'm glad you liked the button slicing.
Re: difficulty: I agree. I think if I were to fully flesh this game out, it would need a reset button and some sort of progression system so I could introduce the slice button later. As for joining buttons, the issue with that is that it introduces the possibility to create concave polygons. This particular polygon splitting algorithm only works with convex ones. I believe destroying polygons below a certain size might be the better option for reducing polygon count, but I'll need to test it out.
You're right about the math thing. At one point I looked at it and thought "Maybe there's a way to prove that it's always solvable. It's just modular arithmetic after all." But slicing and locking makes that math so much more difficult. So I didn't even try to prove it.
Thanks for your comment!