Question 43
Let $n \ge 2$ be an integer.
We draw an $n \times n$ grid on a board and label each box with either the number $-1$ or the number $1$. Then we calculate the sum of each of the $n$ rows and the sum of each of the $n$ columns and determine the sum $S$ of these $2n$ sums.
a. Show that there does not exist a labelling of the grid with $S = 0$ if $n$ is odd.
b. Show that there exist at least six different labellings with $S = 0$ if $n$ is even.
We draw an $n \times n$ grid on a board and label each box with either the number $-1$ or the number $1$. Then we calculate the sum of each of the $n$ rows and the sum of each of the $n$ columns and determine the sum $S$ of these $2n$ sums.
a. Show that there does not exist a labelling of the grid with $S = 0$ if $n$ is odd.
b. Show that there exist at least six different labellings with $S = 0$ if $n$ is even.