Question:

Step 1: Determine the total number of $2 \times 2$ squares in the specified grid. The problem statement indicates that there are $7 \times 7$ such squares. $$ \text{Total number of } 2 \times 2 \text{ squares } (N) = 7 \times 7 = 49 $$ Step 2: Calculate the total number of ways to select two distinct squares from the available squares. This is a combination problem, selecting 2 squares out of $N=49$ squares, denoted as $\binom{N}{2}$. $$ \text{Total pairs of squares} = \binom{49}{2} = \frac{49 \times (49-1)}{2} = \frac{49 \times 48}{2} = 49 \times 24 = 1176 $$ Step 3: Identify the number of pairs of squares that share exactly one common vertex. The provided solution states this number directly. We assume this count refers to pairs that share a vertex but not a side, to allow for direct subtraction. $$ \text{Pairs sharing a common vertex (only)} = 7 \times 7 \times 2 = 98 $$ Step 4: Identify the number of pairs of squares that share at least one common side. The provided solution states this number directly. $$ \text{Pairs sharing a common side} = 7 \times 8 \times 2 = 112 $$ Step 5: Calculate the number of ways to select two squares that share neither a common vertex nor a common side. To find this, we subtract the number of pairs sharing a vertex (only) and the number of pairs sharing a side from the total number of pairs. $$ \text{Pairs sharing neither vertex nor side} = \text{Total pairs} - \text{Pairs sharing a vertex (only)} - \text{Pairs sharing a side} $$ $$ = 1176 - 98 - 112 $$ $$ = 1176 - (98 + 112) $$ $$ = 1176 - 210 $$ $$ = 966 $$ The final answer is $\boxed{966}$.