On the campus of

How Many Squares?

In this week’s puzzle students are asked to find out how many squares are in the figure shown. This is not a difficult puzzle, but it does require some careful observation, organization, and counting. I encourage you to turn to the puzzle right now and give it a try before reading on. This will help you gauge the puzzle’s appropriateness for your students.

The figure for this puzzle is constructed in a way that “hides” some of the squares. Most students quickly count 12-15 squares and think that they have found them all. If this is the case, they need to be told that there are still more squares in the figure than they have been able to find. (For those of you that haven’t done the puzzle yet, please do so now and see how many squares you can find.) As the teacher, you will have to determine if you want to tell your students the total number of squares (the square root of 289) in the figure or if you want to challenge them to determine this number for themselves and then support their answers.

As in any good mathematical activity, process is just as important as product. When students think they have found all the squares hidden in the figure, they are asked to describe the processes they used in finding the answer. This, and the sharing time at the end of the week, will give you important insights into how your students approached the problem. If you feel that some students need help developing a systematic approach to this puzzle, you want to ask a few leading questions such as “How many different sizes of squares are in the figure?” or “How can you make sure you don’t count the same square twice?” Questions like these may help some students approach the puzzle in a more organized way.

How many squares, of any size, are in the figure?


Describe the process you used to find your answer.


Click the arrow below to view the solution.

There are 17 squares of various sizes. The ones that are frequently missed are highlighted below.


One Response to How Many Squares?

  1. Finding all of the different sized squares is a powerful first step. Once you know all the possible squares it’s easy to find how many of each there are. I started out simply counting squares and I missed the second largest square.

Leave a reply