Comments on: How Many Squares?

By: casey

casey — Thu, 13 Apr 2017 22:37:41 +0000

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.

By: Daniel Knox

Daniel Knox — Thu, 13 Apr 2017 16:33:24 +0000

https://www.desmos.com/calculator/sv7nav3zoe

By: Harry O'Malley

Harry O'Malley — Thu, 13 Apr 2017 07:15:47 +0000

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.

By: Harry O'Malley

Harry O'Malley — Thu, 13 Apr 2017 05:03:00 +0000

Here's a solution: Solution In summary, imagine an m x n rectangle where n is greater than or equal to m. This solution involves taking the solution for an m x m square and adding to it the number of additional squares that are created by adding the additional rows required to make it an m x n rectangle.

By: Michael Fenton

Michael Fenton — Thu, 13 Apr 2017 04:56:08 +0000

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.

Voilà: https://paper.dropbox.com/doc/How-Many-Squares-v8kjPTUcKVpDWlrw6y9Mo

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. 🙂

By: Daniel Knox

Daniel Knox — Thu, 13 Apr 2017 03:23:36 +0000

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.

By: Daniel Knox

Daniel Knox — Thu, 13 Apr 2017 03:20:27 +0000

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)?