Poker Players' Problem
A group of 2014 friends (a huge friend circle) discuss the number of times they've played poker with each other. They always play poker in a one on one (1:1) style. Each person has played every other person at least one time and at most \(n\) times. As they were talking, they noticed that there existed three people \(A\), \(B\), and \(C\) who pairwise played each other the same number of times.
\(X\), the mathematician of the group, noticed that no matter how they played each other, there always will exist three people who pairwise played the same number of games.
Find the maximum possible value of \(n\).
Details and Assumptions
A one on one style means that exactly two people play each other in each game.
Suppose \(A\) and \(B\) played 3 games together, \(B\) and \(C\) played 3 games together, and \(A\) and \(C\) played 3 games together. Then \(A\), \(B\), and \(C\) pairwise played the same number of games together: 3 games.