I want to solve this problem:
Let there be $n \ge 2$ points around a circle. Alice and Bob play a game on the circle. They take moves in turn with Alice beginning. At each move:
Alice takes one point that has not been colored before and colors it red.
Bob takes one point that has not been colored before and colors it blue.
When all $n$ points have been colored:
Alice finds the maximum number of consecutive red points on the circle and call this $R$.
Bob finds the maximum number of consecutive blue points on the circle and call this $B$.
If $R \gt B$, Alice wins. If $B \gt R$, Bob wins. If $R = B$, no one wins. Does any of the players have a winning strategy?
We still seem not to know for which odd $n$ Alice has a winning strategy. She does for $n=3$, and it seems also for $n=5$. But in general, I'm not sure. Could someone help?