> Can someone please turn this into a Zachtronics style game?!
That game is called math :) Partially joking, but I do think a game version would be fun.
> Is this what pure math is about?
More or less yes for an undergrad, but when you get to research it feels different.
> I badly, direly want this
Consider checking out an abstract algebra book. Maybe Dummit and Foote or something else popular. Proofs in algebra often have a satisfying game-like quality. The more popular books will have solutions you can find online if you get stuck.