Given integer $n \geq 3$. There are $n$ dots marked $1$ to $n$ clockwise on a big circle. And between every two neighboring dots, there is a light. At first, every light were dark. A and B are playing a game, A pick up $n$ pairs from $\{ (i,j)|1 \leq i < j \leq n \}$ and for every pairs $(i,j)$. B starts from the point marked $i$ and choose to walk clockwise or counterclockwise to the point marked $j$. And B invert the status of all passing lights (bright $\leftrightarrow$ dark) A hopes the number of dark light can be as much as possible while B hopes the number of bright light can be as much as possible. Suppose A, B are both smart, how many lights are bright in the end? Proposed by BlessingOfHeaven