Your uninterpreted syntax will be at least as hard as Egyptian hieroglyphics to crack, and I won't have a Rosetta stone unless you give me one . This is a logic course. I will give no grade higher than a 'C' to systems that do not conform to technical definition of a formal system. Living up to the definition is the minimum for this assignment. You may know that formal systems and programs are isomorphic, and you may know how to program your idea, but don't underestimate the difficulty of finding the formal system that is isomorphic to the program you have in mind.
Games are possible, but very difficult.
You should at least understand that games are interactive, while formal systems are not. There are many ways around this problem, but don't be surprised if they make the result look less like the game you intended than a mathematical study of the game. You can make each permissible move a permissible instantiation of axiom schemata. You can batch all the moves of a particular game and make them axioms of a system. You can make a series of systems corresponding to all possible legal moves and games.