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, and 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.
Three, 16 cities have a total of 32 teams participating in the competition, but teams from the same city do not compete, so each team can play at most 30 matches.

Except for Team A from City A, the 31 other teams have all 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 city teams, while only $a_{0}$ has not played against $a_{30}$.

Similarly, $a_{29}$ and $a_{1}$ are from the same city, because $a_{29}$, except for not playing against $a_{0}$ and their own city's team, has played against all other city teams, while $a_{1}$ has only played once, which was against $a_{30}$.

By analogy, we can see 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 can only be $a_{15}$, which has played 15 matches.