Let m, n \geq 2 be positive integers. Each entry of an m \times n grid contains a real number in the range [-1,1], i.e. between -1 and 1 inclusively. The grid also has the property that the sum of the four entries in every 2 \times 2 subgrid is equal to 0. (A 2 \times 2 subgrid is the intersection of two adjacent rows and two adjacent columns of the original grid.)
Let S be the sum of all of the entries in the grid.
a. Suppose m=6 and n=6. Explain why S=0.
b. Suppose m=3 and n=3. If the elements of the grid are
\begin{array}{|c|c|c|}
\hline
a & b & c \\
\hline
d & e & f \\
\hline
g & h & i \\
\hline
\end{array}
show that S+e=a+i=c+g.
c. Suppose m=7 and n=7. Determine the maximum possible value of S.