How many squares do you see?

If you enjoyed the Facebook math problem about PEMDAS, I’m sure you will love this second math problem about counting squares which I also got from Facebook.

Problem

How many squares do you see?

Answers given in Facebook include 24, 25, 26, 27, 38, 34, 36, 40, 42.

Solution

Here’s how you can systematically count the squares.

# of 4 by 4 squares = 1

# of 3 by 3 squares = 4

# of 2 by 2 squares = 9

# of 1 by 1 squares = 16 + 2 (in the middle)

# of 1/2 by 1/2 squares = 8

Answer: 40

Extending the problem:

Suppose you have a 100 by 100 square grid. How many distinct squares are formed?

Challenge: How many squares with whole number area can you form with 4×4 grid? Click the link for some hints.