Question

Number of rectangles which are not square, on a chessboard.
A) 1296
B) 204
C) 1092
D) 580

Answer 1

Hint: Here to solve this question we have to count the number of rectangles in an chessboard, and there is standard formulae for that, on solving that formulae for our chessboard we can get the total number of rectangles and after simplification with the number of squares we can get the final required result.

Formulae Used:\[ {1^3} + {2^3} + ... + {n^3}\]is the number of rectangles for any “n*n” chess board.

Complete step by step solution:
The given question need to be solved for the number of rectangles in chessboard which is not a square, to solve this question we have to first find the number of rectangles in chess board, on solving we get:
For \[8 \times 8\]chessboard, the number of rectangles is equal to:
\[ \Rightarrow {1^3} + {2^3} + ... + {8^3}\]
We got this result by using the formulae of “n*n” chessboard which says:
\[ \Rightarrow {1^3} + {2^3} + ... + {n^3}\]
Now on simplifying the result we get:
\[ \Rightarrow {1^3} + {2^3} + ... + {8^3} = 1296\]
Now number of rectangles that are not square in an chessboard of eight into eight can be given by:
\[ \Rightarrow number\,of\,rectangles - number\,of\,squares = 1296 - 204 = 1092\]

Hence our final option is “c” which is the correct answer for the question.

Note: Here the above question can also be simplified by commutation property in which we have to see for the number of horizontal as well for vertical lines which is nine, and then we can use the commutation of rectangle number of sides, and after simplification we can obtain the final answer for the given question.