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Counting Squares in a Square Grid

Logic, Pattern and Formula for counting number of squares in a square grid.

Number of squares in a 6×6, 5×5, 4×4, 3×3 grids.

Number of squares in a 1 X 1 grid

1 square of 1 X 1


Number of squares in a 2 x 2 grid

4 squares of 1 X 1     (sq. 1,2,3,4)

1 square of   2 x 2     (sq. 1234)

______________________

5 squares   Total


Number of squares in a 3 x 3 grid

9 squares of  1 X 1  (squares  1,2,3,4,5,6,7,8,9)

4 squares of 2 x 2   (squares made by combining squares 1245, 2356, 4578, 5689)

1 square of  3 x 3    (square made by joining all 9 squares 123456789)

______________________

14 squares        Total

 

Therefore, number of squares in a 4×4 grid

16 squares of   1 X 1

9 squares of    2 X 2

4 squares of    3 X 3

1 square of     4 X 4

______________________

30 squares  Total

 

Pattern / Formula

Generalising

Number of squares in a nxn square grid = sum of squares of n natural numbers

=n2 + (n-1)2 + (n-2)2 ………….32 + 22 + 12

                      or

n(n+1)(2n+1)/6 (sum of square of n natural numbers)

Using the formula, we can get number of squares in a 5 x 5 grid.

= 5(5+1)(2×5+1)/6

= 55

number of squares in a 6 x 6 grid

= 6(6+1)(2×6+1)/6

= 91