12. Two people are playing a table tennis match, a best-of-three games, where the first to 11 points wins a game, provided the opponent has fewer than 10 points; if the score is 10-all, the first to gain a 2-point lead wins. The total points scored by both players are 31, one player won the first game and the match, how many possible scores are there for the second game?
【Answer】 8
【Solution】
Let the winner be A, and the loser be B. A wins the first game. If A wins again in the second game, A's total score will definitely be higher than B's, which does not meet the problem's condition. Therefore, A loses the second game and wins the third game.
A wins the first and third games, so A scores at least 4 points more than B in these two games. Therefore, B must win A by at least 4 points in the second game, so B can only win with 11 points.
Thus, the score of the second game can be: $0: 11, 1: 11, 2: 11 \cdots 7: 11$, a total of 8 possibilities. (No matter how many points B wins in the second game, A can make up the points in the first and third games to make the total scores equal, so A's score in the second game can range from $0 \sim 7$; for example, the scores of the three games can be $20: 18, 0: 11, 11: 2$)