Mathematics Quiz II - How Many Squares Are In This Picture

Calculate the total number of squares in the following image.

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Answer : 40

Explanation :

If there is an n x n square (number of squares are in horizontal row should be equal to vertical row)

there is a way to calculate total number of squares in it.

Suppose there is a 2 x 2 square

How many squares are in it. If you say 4 then you are not absolutely right. Correct answer is 5. There is also a big square ACIG

Now how do you calculate this answer in mathematical terms.
Total number of squares in an n x n square =  n² + (n-1)² + (n-2)² + ......+ 1²
For 2 x 2 square = 2² + 1² = 4 + 1 =  5

Now suppose there is 3 x 3 square

How many squares are in it. Using above formula, we can easily calculate.

Total number of squares in an n x n square =  n² + (n-1)² + (n-2)² + ......+ 1²
For 3 x 3 square = 3² + 2²  + 1² = 9 + 4 + 1 =  14
So where are these 14 squares.
ABFE, BCGF, CDHG, EFJI, FGJK, GHLK, IJMN, JKON, KLPO, ACKI, BDLJ, EGOM, FHPN and ADPM

Now similar way we can calculate squares in Original question. There is one 4 x 4 square and two 2 x 2 squares in the image.
Total number of squares in a 4 x 4 square = 4² + 3² + 2²  + 1² = 16 + 9 + 4 + 1 =  30
Total number of squares in a 2 x 2 square = 2² + 1² = 4 + 1 =  5
There is another 2 x 2 square in image = 2² + 1² = 4 + 1 =  5
So Total number of squares = 30 + 5 + 5 = 40

Exercise for you : How many squares are in a Chess Board? :)