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How Many Squares?

How Many Squares?

How many squares do you see within the 4×4 grid? Today we’ll challenge readers and offer our solution.

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Count the Squares

A spoiler is given in this article as we demonstrate our strategy for solving this square puzzle. If you would like to solve the puzzle before viewing our solution, it is recommended that you do not scroll down past the square graphic below.

For the sake of clarity, it is probably easiest to count the squares from smallest in size to largest. It should be noted that some have argued that the two “extra” squares are intersected, and therefore should would change the solution. This of course boils down to how you choose to view this puzzle. We’re opting for solution that counts intersected squares.

First, let’s take a look at the unsolved puzzle as circulated online:

Squares6

Now we will break down the solution which we believe is the best answer of 40, beginning with the smallest sizes and working our way up to the largest.

Size #1: 8 Squares

There are a total of 8 squares of this size, as shown below. This is achieved by the intersections which occur to the two squares on on top of (or perhaps behind?) the grid.

Squares1

 Size #2: 18 Squares

Next, we have the 2nd largest square size. Of these we have a total of 18 more squares. We have shown squares 25 and 26 to be solid for illustrative purposes.

Running total: 8+18 = 26.

Squares2

Size #3: 9 Squares

Here we count squares which consist of 4 size #2 squares (2×2 squares). There are a total of 9 of these squares. Below you can see how these nine squares are created (we have displayed 9 small versions of the puzzle). The current total now reaches 35.

Running total: 8+18+9=35

Squares3

Size #4: 4 Squares

Here we have squares that consist of 9 size #2 squares (3×3 grid).

Running total: 8+18+9+4=39

Squares4

This leads us to the final square which outlines the entire puzzle, as shown below.

Running total: 8+18+9+4+1=40

Squares5

Bottom Line

We believe that the correct answer to this square counting puzzle is 40.

Did you come up with 40 for an answer? Let us hear your thoughts in the comments below.

Updated January 27, 2015
Originally published April 2013

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