Question 46
Alice and Barbara play a game on a blackboard. At the start, zero is written on the board. Alice goes first, and the players alternate turns. On her turn, each player replaces $x$ – the number written on the board – with any real number $y$, subject to the constraint that $0 \lt y - x \lt 1$.
a. If the first player to write a number greater than or equal to $2010$ wins, determine, with proof, who has the winning strategy.
b. If the first player to write a number greater than or equal to $2010$ on her $2011$th turn or later wins (if a player writes a number greater than or equal to $2010$ on her $2010$th turn or earlier, she loses immediately), determine, with proof, who has the winning strategy.
a. If the first player to write a number greater than or equal to $2010$ wins, determine, with proof, who has the winning strategy.
b. If the first player to write a number greater than or equal to $2010$ on her $2011$th turn or later wins (if a player writes a number greater than or equal to $2010$ on her $2010$th turn or earlier, she loses immediately), determine, with proof, who has the winning strategy.