Non optimal Tic Tac Toe
Source: AMC 12A #22
November 11, 2021
2021 AMC Fall
Problem Statement
Azar and Carl play a game of tic-tac-toe. Azar places an X in one of the boxes in the -by- array of boxes, then Carl places an O in one of the remaining boxes. After that, Azar places an X in one of the remaining boxes, and so on until all boxes are filled or one of the players has 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 O. How many ways can the board look after the game is over?