On an \(n \times n\) chessboard, the total number of rectangles which are not squares is 350 . Then, the…
On an \(n \times n\) chessboard, the total number of rectangles which are not squares is 350 . Then, the number of white squares on the chessboard is .......
32
50
18
72
Solution
Number of rectangles in a chess board
(including squares) \(=\Sigma n^3\)
\(=\frac{n^2(n+1)^2}{4}\)
Number of squares in a chess board
\(=\Sigma n^2=\frac{n(n+1)(2 n+1)}{6}\)
So, Number of rectangles which are not squares
\(\begin{aligned}
&=\Sigma n^3-\Sigma n^2 \\
&=\frac{n^2(n+1)^2}{4}-n \frac{(n+1)(2 n+1)}{6}=350 \\
& \Rightarrow \quad n\left(\frac{n+1}{2}\right)\left(\frac{n(n+1)}{2}-\frac{2 n+1}{3}\right)=350
\end{aligned}\)
\(\begin{aligned}
\Rightarrow & & n\left(\frac{n+1}{2}\right)\left(\frac{3 n^2+3 n-4 n-2}{6}\right) & =350 \\
\Rightarrow & & n(n+1)\left(3 n^2-n-2\right) & =350 \times 6 \times 2 \\
\Rightarrow & & n(n+1)\left(3 n^2-3 n+2 n-2\right) & =350 \times 6 \times 2 \\
\Rightarrow & & n(n+1)(3 n(n-1)+2(n-1) & =350 \times 6 \times 2 \\
\Rightarrow & & n(n+1)(n-1)(3 n+2) & =350 \times 6 \times 2 \\
\Rightarrow & & (n-1) n(n+1)(3 n+2) & =5 \times 6 \times 7 \times 20
\end{aligned}\)
By comparison we get
\(\therefore \quad n=6\)
Hence, total number of black and white squares
\(=n \times n=6 \times 6=36\)
Half of there are
\(\text {white }=\frac{36}{2}=18 \text { squares. }\)