We consider a Maker-Breaker type game on the plane, in which each player
takes t points on their tth turn. Maker wins if he obtains n
points on a line (in any direction) without any of Breaker's points between
them. We show that, despite Maker's apparent advantage, Breaker can prevent
Maker from winning until about his nth turn. We actually prove a
stronger result: that Breaker only needs to play ω(logt) points on his
tth turn to prevent Maker from winning until this time. We also
consider the situation when the number of points claimed by Maker grows at
other speeds, in particular, when Maker claims tα points on his
tth turn.Comment: 17 page