Question 105
Let $m, n$ be two integers greater than or equal to $2$. About a table with $m$ rows and $n$ columns it is known that every cell contains either the number $1$ or the number $-1$, and that the total sum of all the cells is greater than or equal to zero. Genoveffa considers the paths that join a cell of the first column (of her choice) to a cell of the last column (again of her choice) and that always move from a cell to an adjacent one horizontally or vertically, without passing twice through the same cell. The value of a path is the sum of the numbers in the cells it passes through.
a. Prove that for every $m, n \geq 2$ there exist tables with $m$ rows and $n$ columns with no paths of value $2$ or more.
b. Prove that it is always possible to find a path of value greater than or equal to $1$.
a. Prove that for every $m, n \geq 2$ there exist tables with $m$ rows and $n$ columns with no paths of value $2$ or more.
b. Prove that it is always possible to find a path of value greater than or equal to $1$.