Judy asked for my topic during #tiles week in the #MathPhoto15 challenge. That had provoked an in depth discussion on tiling vs tessellation, and whether aperiodic tilings were tessellations, and then somebody mentioned Wang Tiles. Wang Tiles? Oh are those cool. Digging around about those led to finding a very fun series of blogposts from Steve Natusiak on MacMahon tiles. (Here’s the first. One cool sequence. The whole schmegegge.) These tiles were introduced early in the 20th century by Percy MacMahon and are just a lovely construction. MacMahon's idea was squares with colored edges, that you could tile if the matching edges colors matched.
I read up on math circles protocols, and headed for Kalamazoo. (After some printing troubles, which had the Xerox actually spewing curled up sheets of paper into the air like in a sitcom.) I prepared a Google doc, mostly for follow up resources, and some blank tiles (pdf). Judy said they would have scissors and markers.
In the break before my session, LuAnn Murray posed a sticky not problem. How many of the numbers between 1 and 100 can you make with 4 9's and any operation found on a calculator. (So exponentiation and square roots - despite the implied 2 - are in.)
I showed a couple tiles (P,P,R,G & P,P,G,R), and asked what they noticed. First question: do all three colors need to be present? So I also showed all Green. The observed the properties and I asked what I was going to ask them. (So meta. But a room full of teachers, so...) Correct: how many tiles? I asked for estimates ahead of time, which ranged from 12 to 1296. They jumped to working right away and then the clarifying questions began. Biggest: if you rotate them and they match are they different? I asked what did they think? Unanimously, they thought those should be the same. What about flipping? Different. "So they're not colored on both sides!" Is no color an option? (No.) Is red, red, green, purple different than red, green, purple, red? (Yes.)
People worked on lists, making diagrams, a few trees and a couple purely combinatorial approaches. A few were actually coloring them out. I let them know that each table would need a set for the next part, which encouraged some more actual coloring. A couple times I polled the tables for how many they thought, and answers started to converge. When there was agreement but not yet unanimous, I brought them together to share. One teacher jumped up right away: these were all the ones with four red sections, three red sections and two red sections. She didn't do one red, because that would show up in the other tiles. The green, watching out for repeats then purple. Went down by two each time. One person brought up her list with less, and we worked together to figure out what was missing. "It's hard to figure out what's left out!" So how do we do it?
Then I asked: now what? We've got these tiles figured out. What should we do next?
I was really curious to see what kind of problems were posed. Here's what they suggested...
|Michael Tanoff takes off when I start... |
comes back and he's got the book!
Now they were playing!
They had several different approaches to this, as well, but it was much more collaborative in general. Maybe because most tables only had one set of tiles made from the first half. They posed conjectures pretty quickly. They gathered data about how many triangles of each color. They got close and tried small swaps, but also realized that some configurations were a dead end and required starting over.
Nobody was quite done when our time was up. I assured them there was a solution, then stole a couple minutes from working for a reflection and explained why reflection is so important to me. They'd been focused on the SMP, so I asked them to think about SMP1 - especially the perseverance. I asked them to share at table and then just a couple shared with the whole group. They pointed out how I asked more questions than told answers, but encouraged, too. They mentioned how working together helped with perseverance and the problem solving. They appreciated the different methods that people had.
All in all, I was pleased. I think this problem got at the spirit of the math circle, and had plenty of problem solving opportunity. The teachers were great and showed a lot of strong mathematical thinking and practice. And they continued to work on the puzzle while I was leaving.
P.S. Totally an aside & a plug: one of the other benefits of the tiling discussions was that I finally got around to making a sketch for all 17 wallpaper groups in GeoGebra.
I'm pretty happy with how it turned out, but am very open to suggestions.