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)

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**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)

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**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

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**30 squares Total**

** **

__Pattern / Formula__

Generalising

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

**=n ^{2} + (n-1)^{2} + (n-2)^{2} ………….3^{2} + 2^{2} + 1^{2}**

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**