This problem exercises the basic concepts of game playing, using tic-tac-toe (noughts and crosses) as an example. We define Xn as the number of rows, columns, or diagonals with exactly n Xs and no Os. Similarly, On is the number of rows, columns, or diagonals with just n Os. The utility function assigns +1 to any position with X3=1 and -1 to any position with O3 =1. All other terminal positions have utility 0. For non-terminal position, we use a linear evaluation function defined as: Eval(s)=3X2(s) + X1(s) (3O2(s) + O1(s)).

a. Show the whole game tree starting from an empty board down to depth 2 (i.e., one X and one O on the board), taking symmetry into account.

b. Mark on your tree the evaluations of all the positions at depth 2.

c. Using the minimax algorithm, mark on your tree the backed-up values for the positions at depths 1 and 0, and use those values to choose the best starting move.