On an (n n ) chessboard, the total number of rectangles which are not squares is 350 . Then, the number of white squares on the chessboard is .......

Step-by-step solution

Number of rectangles in a chess board (including squares) (= n^3 ) (= n^2(n+1)^2 4 ) Number of squares in a chess board (= n^2= n(n+1)(2 n+1) 6 ) So, Number of rectangles which are not squares ( aligned &= n^3- n^2 &= n^2(n+1)^2 4 -n (n+1)(2 n+1) 6 =350 & n ( n+1 2 ) ( n(n+1) 2 - 2 n+1 3 )=350 aligned ) ( aligned & & n ( n+1 2 ) ( 3 n^2+3 n-4 n-2 6 ) & =350 & & n(n+1) (3 n^2-n-2 ) & =350 6 2 & & n(n+1) (3 n^2-3 n+2 n-2 ) & =350 6 2 & & n(n+1)(3 n(n-1)+2(n-1) & =350 6 2 & & n(n+1)(n-1)(3 n+2) & =350 6 2 & & (n-1) n(