We wish to store tic$-$tac$-$toe boards in a hash table similar to the "Exercise to Try" in the recorded hash table lecture, except here we will stick to conventional $3$ by $3$ boards.

a$)$ How big should the hash table be if we consider all possible boards including ones which would not occur in a normal game $($e$.$g$.4$ X$\text{'}$s and $2$ O$\text{'}$s$)?$

b$)$ How big should the hash table be if we only consider boards that would actually occur in a real game, meaning either the same number of X$\text{'}$s and O$\text{'}$s or one more X than O$?$ That includes the empty board, boards with just one X$,$ boards with one X and one O$,$ boards with two X$\text{'}$s and one O etc.