Now lets see how many $7 \times 7$ squares are possible
These two patterns can shift to right as well as follows:
So, $7\times 7$ squares possible is $4$
Now lets see how many $6 \times 6$ squares are possible
So, $6\times 6$ squares possible is $9$
Now lets see how many $5 \times 5$ squares are possible:
$4$ vertical moves $\times 4$ horizontal moves $=4^2$ possibilities.
Proceeding like this,
- $8\times 8$ squares possible $: 1\times 1=1$
- $7\times 7$ squares possible $:2\times 2=4$
- $6\times 6$ squares possible $:3\times 3=9$
- $5\times 5$ squares possible $:4\times4=16$
- $4\times 4$ squares possible $: 5\times5=25$
- $3\times 3$ squares possible $: 6\times6 =36$
- $2\times 2$ squares possible $: 7\times7 =49$
- $1\times 1$ squares possible $:8\times8=64$
Total squares $\quad: 204$
Now we can generalize like with $n \times n$ chess board total squares $=1^2+2^2+3^2+\ldots +n^2 = \frac{n(n+1)(2n+1)}{6}$
Correct Answer: $C$