MathDB

Given integer

n \geq 3

. There are

n

dots marked

1

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
https://pbs.twimg.com/profile_images/1014932415201120256/u9KAaMZ4_400x400.jpg

Problem Statement