dear Warren,
Play "go" on an NxN grid (usual is N=19). In "H-handicap" go, the second player (white) generously passes his first H turns, before starting to play. For example, some pro could easily give me handicap H=9 and still destroy me.
PUZZLE: for the smallest H that you can, formulate a strategy such that the first player can assure a win, even against a perfect opponent.
In this configuration . . . . . . . . . @ . . . . . . . . . . . . . . . . . . @ . . . . . . . . . . . . . . . . . . @ . . . . . . . . . . . . . . . . . . @ . . . . . . . . . . . . . . . . . . @ . . . . . . . . . . . . . . . . . . @ . . . . . . . . . . . . . . . . . . @ . . . . . . . . . . . . . . . . . . @ . . . . . . . . . . . . . . . . . . @ . . . . . . . . . @ @ @ @ @ @ @ @ @ . @ @ @ @ @ @ @ @ @ . . . . . . . . . @ . . . . . . . . . . . . . . . . . . @ . . . . . . . . . . . . . . . . . . @ . . . . . . . . . . . . . . . . . . @ . . . . . . . . . . . . . . . . . . @ . . . . . . . . . . . . . . . . . . @ . . . . . . . . . . . . . . . . . . @ . . . . . . . . . . . . . . . . . . @ . . . . . . . . . . . . . . . . . . @ . . . . . . . . . with white to move, Black can be assured of having the first move in 2 quadrants. Playing on a (3,3) point denies white to chance to survive in that corner (even with white playing first on (3,3), life is questionable). Any of the top go programs should be able to attest to that, but turning that into a formal proof is still a big challenge; black has to play enough stones to deny white enough eye space, and connect them to his central chains, which are easily made unconditionally alive. regards, -John