Three, a certain football invitational tournament has sixteen cities participating, with each city sending two teams, Team A and Team B. According to the competition rules, each pair of teams plays at most one match, and the two teams from the same city do not play against each other; after several days of the tournament, a statistical review found that, except for Team A from City A, all other teams have played a different number of matches. How many matches has Team B from City A played? Please prove your conclusion.
Has played 15 games
Three, [Analysis and Solution] 16 cities have a total of 32 teams participating in the competition, but the two teams from the same city do not play against each other, so each team can play at most 30 games.
Since the number of games played by the 31 teams except for Team A from City A are all different, the number of games played by these teams are 0, 1, 2, ..., 30.
Let the team that has played $K$ games be $a_{k}(k=0,1 \cdots 30)$.
Obviously, $a_{30}$ and $a_{0}$ are from the same city, because $a_{30}$ has played 30 games, meaning they have played against all other city teams, and only $a_{0}$ has not played against $a_{30}$.

Similarly, $a_{29}$ and $a_{1}$ are from the same city, because $a_{30}$ has played against all other city teams except for $a_{0}$ and their own city's team, and $a_{1}$ has only played one game, which was against $a_{30}$.

By analogy, we can deduce that $a_{28}$ and $a_{2}$, $a_{27}$ and $a_{3}$, ..., $a_{16}$ and $a_{14}$ are from the same city. These teams are not Team B from City A, so Team B from City A must be $a_{15}$, which has played 15 games.