2. (1985 National High School League Question) 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.
2. Consider teams as vertices, and if two teams have played against each other, connect the corresponding two vertices to form a graph $G$. Let the vertex corresponding to Team A of City A be $V^{*}$. Then the original problem is equivalent to finding the degree of the vertex corresponding to Team B of City A. Since each team can play at most 30 matches, the degree of any vertex can only be one of the 31 numbers: $0,1,2, \cdots, 30$. According to the problem, the degrees of the vertices in $V-\left\{v^{*}\right\}$ are exactly $0,1,2, \cdots, 30$. Let the vertex with degree $i$ be $V_{i}(i=0$, $1, \cdots, 30)$, then $V=\left\{v_{0}, v_{1}, v_{2}, \cdots, v_{30}, v^{*}\right\}$. Clearly, the vertices not adjacent to $v_{30}$ are only $v_{0}$, so $v_{30}, v_{0}$ are the two teams from the same city. Removing $v_{0}, v_{30}$, the degrees of the remaining vertices each decrease by 1. Thus, by the same reasoning, $v_{29}, v_{1}$ are the two teams from the same city, and so on. Generally, $v_{i}$ and $v_{30-i}$ are the two teams from the same city.
Therefore, the team in the same city as $v^{*}$ can only be $v_{15}$, so the desired number of matches is 15. Q.E.D.