Question 17
Fix a positive integer $n$. Two players, Phil and Ellie, play the following game. First, Phil fills the squares of an $n \times n$ chessboard with nonnegative integers less than $n$. Then, Ellie chooses three squares making an L, as in any of the following pictures:
Ellie adds 1 to each of the three squares making the L, except that if the number $n$ appears in a square, it is immediately replaced by 0. Ellie wins if, after modifying finitely many L's in this way, she can change all the numbers on the board into 0's; otherwise Phil wins.
Which player has a winning strategy if
a. $n=12$ ?
b. $n=2012$ ?
Ellie adds 1 to each of the three squares making the L, except that if the number $n$ appears in a square, it is immediately replaced by 0. Ellie wins if, after modifying finitely many L's in this way, she can change all the numbers on the board into 0's; otherwise Phil wins.
Which player has a winning strategy if
a. $n=12$ ?
b. $n=2012$ ?