Azar and Carl play a game of tic-tac-toe. Azar places an in one of the boxes in a 3-by-3 array of boxes, then Carl places an in one of the remaining boxes. After that, Azar places an in one of the remaining boxes, and so on until all 9 boxes are filled or one of the players has 3 of their symbols in a row—horizontal, vertical, or diagonal—whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third . How many ways can the board look after the game is over?
Pick one
Solution
Solution:
Answer (D): For Carl to win at his third turn, his 3 s must lie in one of the 8 winning configurations and Azar's 3 s must not (because that would have resulted in her winning after her third move). There are 6 vertical or horizontal rows for Carl's s, and in each case there are ways for Azar's s to not align. There are also 2 diagonal winning lines for Carl, and in those cases Azar could not have won first. The number of ways the board can look after the game is over is therefore
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