This activity is an invitation for your students to take part in an original piece of mathematical research. We begin with a problem: there is a 6 x 6 apartment split up into 36 "tiles" that need to be arranged so that all 4 apartments use the same number of tiles (9 each) and where each apartment is the exact same shape. Oh, and the apartment has to include a bathroom and kitchen, as shown on the floor plan.
From here, the problem morphs into a larger exploration of how many ways this 6 x 6 tile grid can be split into 4 pieces that are the same size and shape. I've been working on this problem for the last 8 months and have discovered 41 different variations. Whether there are more, I have no idea.
I am calling upon teachers worldwide to contribute to this original piece of mathematical research. I have scoured the web and have never seen this particular problem proposed or described, so I'm going to speculate that it has never been investigated. I would like to submit these results to a mathematics research journal for publication at a later date.
In the meantime, I will be publishing this problem and solutions to it from your students at my blog at www.samizdatmath.com.