Consider the problem of solving two $8-$puzzles: there are two $8-$puzzle boards, you can make a move on either one, and the goal is to solve both.

a$.$ Give a complete problem formulation.

b$.$ How large is the reachable state space? Give an exact numerical expression.

c$.$ Suppose we make the problem adversarial as follows: two players take turns moving; a coin is flipped to determine the puzzle on which to make a move in that turn: heads puzzle one; tails puzzle two. The winner is the first to solve either puzzle. Which algorithm can be used to choose a move in this setting?

d$.$ In the adversarial problem, will one player eventually win if both play perfectly?