The Better Living Through Mathematics Library

Every Thursday morning for the past 5 years I’ve been meeting with a 4th grade class to work on a variety of “puzzlas” to stretch their mathematical thinking. I pull these puzzles from a variety of sources, which is not hard as I have a library of math materials that I’ve been collecting for over 30 years. Truthfully, I don’t know if I’ve had an original mathematical thought in my life, as my library provides more than enough inspiration to cover me for several lifetimes.

Every once in a while, an interesting puzzle comes across my desk and this puzzle inevitably leads me to start looking at the problem in a bigger way. That is, the specific puzzle leads to a much bigger question. Such was the case of the problem below:

On the face of it, the problem has a single unintuitive answer which requires constructing a set of floor plans for 4 apartments that no one in their right mind would want to live in. However, it does beg a bigger question: suppose we got rid of the kitchens and bathrooms, and just had to divide up the square into 4 equal sections of 9 tiles each where each shape was congruent?

I started with the simplest solution possible: 4 squares of 9 squares each. I then started by moving one tile over from each of the squares:

Moving a single tile, as is done to create tesselations, creates new solutions to this puzzle.

I’ve been working on this puzzle for the past 8 months and have come up with over 40 different solutions, but my fear is that I’ve missed out on some. Here’s what the results of my research looks like:

….so here’s where you come in, dear teacher: visit my online store and you can download this activity for a (very) small payment! Print it up, try it out with your kids and let’s see if they can come up with variations that I have yet to discover. I’ll publish them here and then submit the results to some mathematics journals to see if we can get it published. Let’s get kids involved in making mathematical discoveries and show them that far from being a dead subject where everything is known, mathematics is alive and quite well.

## About rmberkman

This blog is the sole musings of one Robert M. Berkman, an educator who has taught math, science and technology for the past 30 years in New York. You can react to all his posts by emailing him at rants@bltm.com.