Question 36
Player Zero and player One play a game on an $n \times n$-board ($n \ge 1$). The columns of this $n \times n$-board are numbered by powers of two, so we have column 1, column 2, column 4 through column $2^{n-1}$. Alternately, players put their own number (i.e. Zero a 0 and One a 1) in one of the empty squares. Player Zero starts. When the board is full, the game ends and a (reverse binary) number is created in each row by adding the values of the boxes with a 1 in them. So if $n = 4$, then a row with 0101 contains 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 ensure that at least one of the rows is divisible by 4?
b) And for which natural numbers $n$ can player One always ensure that at least one of the rows is divisible by 3?
a) For which natural numbers $n$ can player One always ensure that at least one of the rows is divisible by 4?
b) And for which natural numbers $n$ can player One always ensure that at least one of the rows is divisible by 3?