Question 45
Player Zero and player One play a game on an $n \times n$-board $(n \geq 1)$. The columns of this $n \times n$-board are numbered by powers of two, so we have column 1, column 2, column 4 up to and including column $2^{n-1}$. Taking turns, the players put their own number (so Zero a 0 and One a 1) in one of the empty squares. Player 0 starts. When the board is full, the game ends and in each row a (reverse binary) number is formed by adding up the values of the squares with a 1 in them. So if $n=4$, a row with 0101 corresponds to the number $0 \cdot 1+1 \cdot 2+0 \cdot 4+1 \cdot 8=10$.
a) For which natural numbers $n$ can player One always make sure that at least one of the rows is divisible by 4?
b) And for which natural numbers $n$ can player One always make sure that at least one of the rows is divisible by 3?
a) For which natural numbers $n$ can player One always make sure that at least one of the rows is divisible by 4?
b) And for which natural numbers $n$ can player One always make sure that at least one of the rows is divisible by 3?