Please help me solve a GRE math problem!?

just remove one row and one column....the ans will be same if u remove any row or column... so the remaining will be r-1 rows and r columns so there will be r(r-1) squares...which is a. r^2-r

Ok.
Game board size is x = r(r+1), because r rows * (r+1) columns = # of squares.

or r² + r = x

You have one row, and one column that are not to be counted in the total.
So, you have r squares (in the column) and r+1 squares (in the row).
However, there is one square where the two cross. Therefore you have 2r squares that are not to be counted.

so the equation for counted squares is Cs = r² + r - 2r

Combine like terms and Cs = r² - r.

There are r(r+1) squares total. Each row contains r+1 squares (one for each column). Similarly, each column contains r squares. So it would be tempting to say that the answer is r(r+1) - (r+1) - r. But that would be wrong, because we've subtracted the square in the 4th row and 7th column twice; we only want to subtract it once, so we need to add 1 back into our answer for a final answer of

r(r+1) - (r+1) - r + 1 = r^2 - r.

If you want to know more, the fancy math term for this type of argument is the "Inclusion-exclusion principle."

wow its' too hard...