Question

The number of squares on a chess board is

• A. $1296$
• B. $204$
• C. $120$
• D. $130$

Answer 1

The correct option is B $204$

Number of squares in a grid of order $2\times 2$

$=4\text{(Number of small squares)}+1\text{(A square containing all small squares)}=2^2+1^2$

Number of squares in a grid of order $3\times 3$

$=9\text{(Number of small squares)}+4\text{(Number of squares containing 4 small squares)}+$

$1\text{(A square containing all components)}$

$=3^2+2^2+1$

Similarly, Number of squares in a grid of order $4\times 4=4^2+3^2+2^2+1^2$

We get,

Number of squares in a grid of order $n\times n=n^2+(n-1{)}^2+\cdot \cdot \cdot \cdot +1^2={\sum }_{r=1}^n{r}^2$

And $n^2+(n-1{)}^2+\cdot \cdot \cdot \cdot +1^2=\frac{n(n+1)(2n+1)}{6}$

Since, chess board is of order $8\times 8$

So, Number of squares on a chess board $={\sum }_{r=1}^8{r}^2=\frac{8(8+1)(2\times 8+1)}{6}=\frac{8\times 9\times 17}{6}=204$

Hence, option B is correct.