Question

The number of non-congruent rectangles that can be formed on a chessboard, is
A.28
B.36
C.64
D.224

Answer 1

Hint: Here we will firstly write the number of boxes in the chess board. Then we will see the combination of possible rectangles that can be formed by taking sides 1 to 8. We will then add all the possible rectangles to get the total number of non-congruent rectangles that can be formed on a chessboard.

Complete step-by-step answer:
We know that a chess board is in the shape of a square divided into 64 smaller squares of equal dimensions i.e. \[8 \times 8\].
Now we will form the possible combination for the non-congruent rectangles that can be formed.
If a rectangle formed has one side length 1, then combinations possible will be
\[\left( {1 \times 1} \right),\left( {1 \times 2} \right),\left( {1 \times 3} \right),\left( {1 \times 4} \right),\left( {1 \times 5} \right),\left( {1 \times 6} \right),\left( {1 \times 7} \right),\left( {1 \times 8} \right)\]
Hence there are 8 possible non-congruent rectangles that can be formed with one side length as 1.
Now we will find the combinations possible if a rectangle formed has one side length 2, we get
\[\left( {2 \times 2} \right),\left( {2 \times 3} \right),\left( {2 \times 4} \right),\left( {2 \times 5} \right),\left( {2 \times 6} \right),\left( {2 \times 7} \right),\left( {2 \times 8} \right)\]
Hence there are 7 possible non-congruent rectangles that can be formed with one side length as 2.
Similarly, the possible non-congruent rectangles that can be formed with one side length as 3, 4, 5, 6, 7 and 8 will be 6, 5, 4, 3, 2 and 1 respectively.
Now we will add all the possible non-congruent rectangles. Therefore, we get
The total number of non-congruent rectangles \[ = 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1 = 36\].
Hence, 36 numbers of non-congruent rectangles that can be formed on a chessboard.
So, option B is the correct option.

Note: A chessboard has alternate black and white colour small square boxes. Square is the shape in which the length of the square is equal to the breadth of the square and they are known as the side of the square. Rectangle is the two dimensional shape which does not have the same length and the breadth.