Three, a football invitational tournament involves sixteen cities, each sending two teams, Team A and Team B. According to the competition rules, each pair of teams plays at most one match. Additionally, the two teams from the same city do not play against each other. After a period of matches, it was 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.
Three, 16 cities have a total of 32 teams participating in the competition, but teams from the same city do not compete against each other, so each team can play at most 30 matches.

Except for Team A from City A, the 31 other teams have played different numbers of matches, so the number of matches these teams have played are $0, 1, 2, \cdots, 30$.
Let the team that has played $K$ matches 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 matches, meaning they have played against all other teams from different cities, and only $a_{0}$ has not played against $a_{30}$.

Similarly, $a_{29}$ and $a_{1}$ are from the same city, because $a_{29}$ has not played against $a_{0}$ and their own city's team, but has played against all other teams from different cities, and $a_{1}$ has only played one match, 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 matches.