Verified answer

Topic

Reasoning

Given

A grid of 8 × 8.

To Find

Number of rectangles and squares.

Solution

Counting Rectangles

A rectangle can be formed if we take a column of grid length 'n' and keep selecting rows of grid length 'm' to form a rectangle of m × n.

Number of ways to select a column or row from grid = 8.

Suppose we select a column of grid length '1' then there are 8(1 + 2 +3+. . . + 8) possible rectangles.

Similarly, if we select a column of grid length '2' then there are 7( 1 + 2 + 3+. . . + 8) possibe rectangles.

Similarly, if we select a column of grid length '3' then there are 6( 1 + 2 + 3+. . . + 8) possibe rectangles.

Similar process of selection will continue until we have selected a column of grid length '8'.

So, total number of rectangles will be

8(1 + 2 +3+. . . + 8) + 7(1 + 2 +3+. . . + 8) + . . . . . . . . +2(1 + 2 +3+. . . + 8) + 1(1 + 2 +3+. . . + 8)

Taking (1+2+3+4+5+6+7+8) as common,

(1+2+3+4+5+6+7+8)(1+2+3+4+5+6+7+8)

(1+2+3+4+5+6+7+8)²

(36)²

1296.

So, there are 1296 total rectangles.

Similarly,

Counting Squares

If we take a grid of a grid length '1' then there will be 8² possible number of squares.

If we take a grid of a grid length '2' then there will be 7² possible number of squares.

This process will continue until we take a grid of grid length '8' which gives 1² possible number of squares.

Total Squares = 8² + 7² + 6² +. . . .+ 2² + 1²

Total Squares = 64 + 49 + 36 +. . . + 4 + 1

Total Squares = 204

So, total 204 squares are there in given grid.

Answer

So, there are 1296 rectangles and 204 squares in given grid.

Method 2 ( Short Cut )

For counting number of rectangles in a n × n grid, we use formula of sum of cubes of first n natural numbers.

Sum of cubes of first n natural numbers =

[ n( n + 1 ) / 2 ]²

For counting number of squares in a n × n grid, we use formula of sum of squares of first n natural numbers.

Sum of squares of first n natural numbers =

[ n( n + 1 )( 2n + 1 ) / 6 ]

To check the formula, we can put n = 8, which will also give,

1296 rectangles and 204 squares.

Note : Total number of rectangles includes all squares as squares are also rectangles.

Why you ignore me

For a 2x2 square, we have a total of 4 possible rectangles, each 1x2 squares. For the 3x3 square, we can find 12 1x2 squares, 6 1x3 squares, and 4 2x3 squares for a total of 22 squares. For the 4x4 square, we can find 24 1x2's, 16 1x3's, 8 1x4's, 12 2x3's, 6 2x4's, and 4 3x4's for a total of 70 squares.