TSTST 2025 Problem 6

Alice and Bob play a game on $n$ vertices labelled $1, 2, \ldots, n$. They take turns adding edges $\{i, j\}$, with Alice going first. Neither player is allowed to make a move that creates a cycle, and the game ends after $n-1$ total turns.

Let the weight of the edge $\{i, j\}$ be $|i-j|$, and let $W$ be the total weight of all edges at the end of the game. Alice plays to maximize $W$ and Bob plays to minimize $W$. If both play optimally, what will $W$ be?