A thought has occurred to me, it may be interesting to constrict the “cat’s” available moves to only those towards the “mouse” -along the shortest line between the two; similarly the mouse towards the “goal.” Moreover, let’s reconstruct the game so that both the cat and mouse exist on a graph. In particular let’s say a Cayley graph Γ(G,S) and restrict their movements to the edges of the graph. If we maintain that the goal of the mouse is to reach the vertex mapped by max{x|x∈G∩S*} where S* ⊂ S, and the cat’s goal is to “catch” the mouse; how then does this change the dynamics of the game? What is now the optimal strategy for a mouse starting on any given vertex, not max{x|x∈G∩S*}? What about the cat’s?

I will have to come back to this thought later.

Thoughts: …the more I reread this the more I see other interesting questions about this game. For example, there are several ways to turn this into a game of incomplete information. And what happens as we add or subtract vertices or add players? What if we change the rate of motions(e.g. cat 2 vertices to the mouse’s 1, or 3:1, 3:2… ect. -I sense a possible covering problem here)

Note to self: I suspect this may “reduce” nicely to a coloring problem. I should also go back and look at these older projects and expand on them… make them interesting.