Exercise 5.10

Consider the family of generalized tic-tac-toe games, defined as follows. Each particular game is specified by a set $\mathcal S$ of squares and a collection $\mathcal W$ of winning positions. Each winning position is a subset of $\mathcal S$. For example, in standard tic-tac-toe, $\mathcal S$ is a set of 9 squares and $\mathcal W$ is a collection of 8 subsets of $\cal W$: the three rows, the three columns, and the two diagonals. In other respects, the game is identical to standard tic-tac-toe. Starting from an empty board, players alternate placing their marks on an empty square. A player who marks every square in a winning position wins the game. It is a tie if all squares are marked and neither player has won.

  1. Let $N= |{\mathcal S}|$, the number of squares. Give an upper bound on the number of nodes in the complete game tree for generalized tic-tac-toe as a function of $N$.

  2. Give a lower bound on the size of the game tree for the worst case, where ${\mathcal W} = {{\,}}$.

  3. Propose a plausible evaluation function that can be used for any instance of generalized tic-tac-toe. The function may depend on $\mathcal S$ and $\mathcal W$.

  4. Assume that it is possible to generate a new board and check whether it is a winning position in 100$N$ machine instructions and assume a 2 gigahertz processor. Ignore memory limitations. Using your estimate in (a), roughly how large a game tree can be completely solved by alpha–beta in a second of CPU time? a minute? an hour?

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