Matt Enlow’s Rectangle Division Puzzle

This week’s puzzle was suggested by Matt Enlow, a math teacher at the Dana Hall School in Wellesley, Mass.

Rectangle Division

What is the fewest number of squares into which you can divide an 11 x 12 rectangle? What about a 12 x 13 rectangle? What about an 11 x 13 rectangle?

A beautifully simple puzzle. I asked Matt how he came up with it. Here’s his response:

This one came about almost by accident. I teach an Advanced Topics course to seniors, and we do some number theory. I was trying to think of a way to get them to “discover” the Euclidean algorithm for themselves, so I just asked a similar question, such as, “What’s the fewest number of squares you could divide a 5 x 6 rectangle into?”

I had assumed that the Euclidean algorithm would always yield that minimal number, but a student rather quickly found an answer that was smaller than the one the E.A. gave! What started as a Euclidean algorithm lesson quickly became an exploration of this new idea. One of my favorite “mistakes” in lesson-planning ever!

Thank you, Matt. For more Matt Enlow, check out his tweets here. And with that we wrap up this week’s rectangle challenge. As always, once you’re able to read comments for this post, use Gary Hewitt’s Enhancer to correctly view formulas and graphics. (Click here for an intro.) And send your favorite puzzles to gary.antonick@NYTimes.com.

Solution

Here’s the solution by Matt Enlow:

Hi, everyone!

Here are my answers (all of which have been mentioned by at least one person here): 7 squares for 11 x 12 and 12 x 13, and 6 squares for the 11 x 13.

Photo

As many have discovered, it is much easier to be confident that one has used the fewest possible squares than it is to prove it! Personally, I am convinced that these are optimal, but I can’t yet prove it to my satisfaction.

I love so much of what I’ve seen here in the comments: The talk of recursion, the possibility of non-integer side lengths, the attempts at generalization… all great stuff! I hope that you enjoyed playing around with these ideas, and continue to do so. If you find anything interesting, feel free to share it with me at matt.enlow@danahall.org.

Thanks!

Thanks, Matt! And thanks as well to everyone who participated this week: Andrew Ciszewski, Dr W, Joe Fendel, LAN, Matt Enlow, Michael Josephy, PK, Ravi and sj.