In Chapter 3, we discussed the application of Boolean logic to AI-based approaches to playing…

 

In Chapter 3, we discussed the application of Boolean logic to AI-based approaches to playing games like Tic-Tac-Toe. (See p. 344, or Figure 9.27 for a 2-by-2 version of the game [Tic-Tac; the 3-by-3 version is Tic-Tac-Toe].) Specifically, recall the Tic-Tac-Toe game tree: the root of the tree is the empty board, and the children of any node in the tree are the boards that result from any move made in any of the empty squares. We talked briefly about why chess is hard to solve using an approach like this. (In brief: it’s huge.) The next few problems will explore why a little bit of cleverness helps a lot in solving even something as simple as Tic-Tac-Toe.

Tic-Tac-Toe ends when either player completes a row, column, or diagonal. But for this question, assume that even after somebody wins the game, the board is completely filled in before the game ends. (That is, every leaf of the game tree has a completely filled board.) How many leaves are in the game tree?

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