How Many Squares?

I had dinner with the kiddos at Round Table tonight. (Side note: yum.)

Must’ve had squares on the brain (thanks Dan and Anna!) because as soon as I sat down with the receipt I folded (and unfolded) it like this:

And then I asked Ainsley (4): “How many squares do you see?”

She counted the top row: “1… 2… 3… 4…” Then a long pause, followed by (pointing along the bottom row): “5… 6… 7… 8…”

We both stared at the receipt for a little while longer. Neither of us spoke for a bit. Then—for better or for worse—I broke the silence, tracing the perimeter of a larger square consisting of a 2-by-2 array of smaller squares. She helped me count this 9th square plus two more just like it, and we landed on a total of 11.

It’s hard to tell if those last three squares were lost on her, even after we traced them together. In all likelihood, they probably were. Anyway, that’s not the point of the post. (Nor is her original counting of eight, two rows of four at a time, though I’m confident there’s more than enough material for a blog post in the “how do you see it” conversation.)

A Familiar Problem

Here’s where I am headed with this.

I found myself recalling the classic problem “how many squares on a chessboard?” And then I wondered, “how many squares on an n-by-n board?” I’ve explored both of these questions before, and while I cannot recall the generalization off the top of my head, I’m confident I could find an explicit formula if I tried.

A New (to me) Problem

That sequence of thought led to a new question, or rather, a question that’s likely been posed many times by others but is brand new to me:

How many squares on an m-by-n board?

I’m sharing it here sans answer because I don’t yet have an answer (and it’d be no fun to spoil your fun in tracking one down). But I’m excited to start exploring. It may not be the sexiest or most challenging problem in the world, but it’s grabbed my attention nonetheless.

And if I’ve piqued your interest, I’d love if you gave it a try as well.

In which case, drop a line in the comments describing your approach, your answer, and (bonus!) another question about rectangles/squares/boards that interests you.

Comments 7

  1. Thanks for this distraction tonight. So here’s my approach: I just started drawing grids of mxn rectangles and looking for patterns. I started with 1xn, then moved to 2xn, 3xn, etc. These are the rectangles I drew:
    Squares Product difference
    1X2 = 2 S prod 2 0
    1X3 = 3 S prod 3 0
    1X4 = 4 S prod 4 0

    2X3=8 S prod 6 2
    2X4 = 11 S prod 8 3
    2X5 = 14 S prod 10 4

    3X4 = 20 S prod 12 8
    3X5 = 26 S prod 15 11
    3X6 = 32 S prod 18 14

    4X5 = 40 S prod 20 20
    4X6 = 50 S prod 24 26
    4X7 = 60 S prod 28 32

    I started looking for patterns and found the product of each mxn (second column) and then the difference between product and the number of sqares (3rd column). Then I noticed that the differences were the same as the number of squares in an (n-1)X(m-1) rectangle. So they just build on themselves.

    It’s like a little Russian doll situation!

    So, to find the number of squares in an mxn rectangle, you can find mxn + (m-1)*(n-1) + (m-2)*(m-2) + … until one of those factors is zero. That sum will give you the number of rectangles.

    So my question is: Is there a better way to generalize that formula, other than just finding the sum of a lot of products (if m and n are bigger numbers)?

  2. So the formatting didn’t do what I wanted and it looks jumbled in the middle.

    It should be three columns, the first being mxn = number of Squares (for example 1X2 = 2 S)
    The second column is the product (for example prod 2)
    The third column is the difference between product and number of squares.

  3. Post

    Daniel, you’re welcome for the distraction! And thanks for chiming in. It looks like we ended up in a similar place, though I took a bit longer to get there.

    I scribbled some notes into a Dropbox Paper document while I watched the Warriors game. 🙂

    Two objectives here: Give the Paper app a thorough test drive (I started using it yesterday) and generate something I could easily share in the comments of the blog.


    As for the test drive… I like this app. Super easy to use. I might just ditch my other note taking app for this one. We’ll see. Anyway, I assume no one actually cares about that except for me. 🙂

  4. OK. I’ve summarized the solution I linked to in my last comment and created a simulation that allows us to directly experience the number of squares that are created for a large number of different rectangles. I’ve made a blog post about it here:

    Experiencing Michael Fenton’s Rectangle World

    Check out the simulation. Trust me. It’s cool.

  5. Thanks for the problem. That was fun. I used the summation n^2 with the limit n= 1 to infinity. I think I did it right. It makes me want to teach Algebra 2 and not 8th grade math. Thanks.

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