<p><span style="font-size:20px;"><span style="color:#27ae60;">Solving the Problem</span></span></p>
<p>This is a pretty tricky one, it's best to do it slowly, step by step.</p>
<p><strong>Method 1 (Algebra)</strong></p>
<p>We can start by figuring out the number of total squares in the grid, in terms of $r$:</p>
<ul>
<li>There are $r$ rows in the grid</li>
<li>There are $r + 1$ columns in the grid</li>
<li>So, in total we have $r * (r+1) = r^2 + r$ squares in the grid</li>
</ul>
<p>If we take away the $4th$ row (which contains $r+1$ squares, one for each column in the row), we have $r^2 + r - (r+1)$ squares left.</p>
<p>If we take away the $7th$ column (which contains $r$ squares, one for each row in the column), we have $r^2 + r - (r+1) - r$ squares left.</p>
<p>We can simplify this to $r^2 - r - 1$</p>
<p>But there's also a square where the $4th$ row and the $7th$ column overlap, which we took away twice, so we need to add it back.</p>
<p>So in total, we have $r^2 - r - 1 + 1 = r^2 - r$ squares that are not in the $4th$ row or the $7th$ column.</p>
<p><strong>Method 2 (Choosing Numbers)</strong></p>
<p>Let's pretend $r$ is $14$, meaning there will be $14$ rows and $15$ columns.</p>
<p>So, we have a total of $14 * 15 = 210$ squares.</p>
<p>When we take away the $4th$ row (with $15$ squares for each column in the row), we have $210 - 15 = 195$ squares left.</p>
<p>Then, if we remove the $7th$ column (with $14$ squares for each row in the column, we have $195 - 14 = 181$ squares left.</p>
<p>Now we need to add back the square where the $4th$ row and the $7th$ column overlap, since we took this away twice, so we have $181 + 1 = 182$ squares.</p>
<p>Now, if we plug 14 into the equation $r^2 - r$, we can see that we would get $14^2 - 14 = 182$, which is what we are looking for.</p>
<p>So, the correct answer must be <span style="color:#27ae60;">A</span>.</p>