I am sure this two-parameter family of games has been analyzed up the wazoo, so I am hoping somebody can just tell me what's known. X and O take turns claiming an integer from [1,n]. The winner is the first to have claimed an arithmetic sequence of length k. The classic strategy-stealing argument shows that this game is always either a win for the first player or a draw. What values of (n,k) have known outcomes? If k = 3, it's a draw for k < 6 and a win for k > 6; I haven't quite analyzed k = 6. Higher k's are more interesting. Is there a complete theory out there?