Question

A rectangular paper with dimensions 1" x 5" is cut in to 5 pieces such that it is possible to arrange these 5 pieces in to a square of area of 5 sq.inches

Answer 1

Answer:

We usually think the answer is 8*8=64, right?

But in this case we have just counted 1*1 squares. What about the 2*2 squares, 3*3 squares, 4*4 squares and so on?

Number of 1*1 squares= 8*8=64

Number of 2*2 squares= 7*7=49

Number of 3*3 squares= 6*6=36

Number of 4*4 squares= 5*5=25

Number of 5*5 squares= 4*4=16

Number of 6*6 squares= 3*3=9

Number of 7*7 squares= 2*2=4

Number of 8*8 squares= 1*1=1

Total number of Squares= 82+72+62+...+22+12= 204

Can you see a pattern?

In a 8*8 chessboard, the total number of squares is ∑82

We can generalize this in the following way:

Total number of squares in a n*n chessboard will be = ∑n2; n varying from 1 to n.

Now let us calculate the number of rectangles in 8*8 chessboard.

A rectangle can have the following dimensions: 1*1, 1*2, 1*3, 1*4… 1*8, 2*2, 2*3, 2*4, …2*8, 3*3, 3*4, ….7*8, 8*8.