"You have 21 points on a circle and each turn a player may connect any two points with a chord so that it doesn't intersect any other chords (not even at the endpoints!). The player who can't move loses."
I want to solve this problem for a general odd number.
Progress: It is clear that P1 wins if there is an even number of points - draw a diameter and use symmetry. However, it is unclear what happens with an odd number of points.
Note that a move splits the game into two independent games. However, since the game is impartial, splitting a game into two winning positions does not guarantee a win. I tried working backwards and I tried considering the cases in which P1 or P2 can force a win OR a loss. For example, for $n = 4$, P1 can both force a win and a loss.
My stumbling block with this approach is that a player may choose to start playing a winning game on one of the games but then change his mind and start playing a losing game and it might happen that at that point he can actually force both. Any help is appreciated.