(April 14, 2015) In case you haven’t been following along, a bunch of 6-8 year olds have been tackling the Hadwiger-Nelson Problem (to determine the chromatic number of the plane) with the support of a narrative tale about aliens on a planet called Botso. This week (our final), I dropped all references to the story. The group engaged in an attempt to solve an unanswered (“open”) question in theoretical mathematics unaided by a narrative thread. Most of the kids rose to the occasion beautifully.

I spread whiteboards around the room and drew some leading diagrams on them:

- a tiling filled with regular hexagons
- a graph copied from a mathematical comic strip – see photos. This was a graph of a star within, but not overlapping, a pentagon.

I posited the question again, in pure mathematical terms, to make sure the kids understood their task. They did. I reminded them, “People have been working on this question for over 60 years, so I don’t expect you to solve it in 6 weeks, but I want you to have enough comprehension and experience with it that you can continue your work at home.” Then I armed them with dry-erase markers and asked them to get to work.

**WHAT THEY DID**

[NOTE: Go to the photo gallery to see actual photos of the work that these mathematicians did.]

M tackled the graph. She thought she could color it in only 2 colors. She called me over to check her work. I spotted a flaw, so she got back to work. When she called me over again, several students came with me to see her progress. One of them spotted a flaw. Back to work. She called more over again, this time having successfully colored the graph with only 3 colors. I asked her to see if she gets the same result if she extended the pattern further outward into the plane. Back to work. By the end of class she had continued the pattern one more layer outward and successfully colored it with 3 colors. “Could it continue forever with only 3 colors?” I asked. M wasn’t positive, but her conjecture was yes.

N also tackled the graph. He colored each vertex with 2 colors and made a convincing argument for how this could be a solution. I don’t want to quash kids’ enthusiasm when their solutions either don’t work, or conflict with the conventional solution, so, as always, I tried really hard to follow the students’ reasoning. (This is hard for me, since my mind has become limited over the years by conventional, expected, or convergent solutions.) I finally realized that (I think) his solution was a 3-D solution. I congratulated him on his deep thinking and creativity, and then ask him to look at it some more as a 2-D problem. As mathematicians often do, he took a break from the problem to check out what the other mathematicians were doing, and helped me give feedback to them.

R tackled the tiling. First she colored selected tiles in a particular pattern, then went and filled in the other colors. She eventually got a coloring that met the requirements of the problem and that gave a chromatic number of four. “Take a look at this,” I said, and handed her a printout of the Wikipedia article and diagram about this problem. “This diagram seems to show that a hexagonal tiling requires 7 colors. Why do you think you got 4?” She stared at her coloring. The kids who had been following me to study everyone’s work stared at her coloring. “I’ll come back in a bit to see what you think.” I moved on to look at other students’ work. When I returned, she had a conjecture: her coloring worked with only four colors because it had a small number of tiles, hence it wouldn’t work if the pattern extended outward into the plane. She decided to rearrange the way she colored while imagining the diagram extending. She ended up with another 4-color coloring, with an almost, but not-quite, repeating pattern. Her new conjecture is that this one will work with 4 colors if extended. I told her that I’ll post a photo of her work on the website so she can check this at home. (Sadly, I couldn’t give her the whiteboard to take home.)

Z also tackled the tiling. Unfortunately, by the time I got to him, we were running out of time. He had been experimenting with scale – smaller and larger. First he had taken a single hexagonal tile, split it into 6 equilateral triangles, and colored it. Then he took a whole section of tiles and consumed them into a larger hexagon. He split that one too into triangles. This time I wasn’t sure whether they were equilateral, whether he has simply experimenting with scale factor, coming up with a new way to look at the problem, or changing the original question. (Darn that clock!) I told all of the kids that photos of their whiteboard work would appear on the Talking Stick website so that they can continue their explorations at home.

R (the other one) and L worked on paper creating their own patterns and colorings. Some were graphs; some were tilings. They explained them to me briefly at the end. They also had been taking breaks to study other people’s work. See the photo gallery for images of their work.

A and S spent a good chunk of their time drawing diagrams connected to the alien narrative – space ships, etc. While they did not color a provided pattern or invent their own, they were not disengaged from the math. They spent time looking at and commenting on others’ mathematical explorations. I think the times that the kids walked around looking at each other’s work and commenting on it as fellow mathematicians was one of my favorite parts of this math circle session. They had become real, collaborative mathematicians.

Finally, I showed the comic that the graph I challenged them with came from. They marveled at the work they had done, since the comic strip tells the story of someone coming to graph colorings as an adult. At this point, I dismissed them with a request to keep me informed of their progress if they keep working on the problem.*

**CONTINUED WORK AT HOME**

A week later, I got a text from N’s mom. She said that in the car, N suddenly yelled out “Mom! I’ve got it! I know the answer to Rodi’s math question!” She said he’s very excited. I’m excited to connect with him (and anyone else who has be comtemplating) to see his progress.

**FUNCTION MACHINES**

We had run out of time so quickly because we spent the first 15 minutes of our session on student-created and student-presented function machines, leaving only 45 minutes for our other problems. But the function machines were well worth doing.

L told the group that the domain of his machine is “all numbers.” (The mathy part of me almost wrote on the board, out of habit, “all real numbers” – this near-slip made me realize all the fun we could have by playing more with domains and expanding into ranges.) The freedom implied by this domain made the group have a lot of fun coming up with interesting “in” numbers. The most interesting was “negative zero” because of the resulting brief number theory discussion. L’s rule, BTW, was f(x)= x + 11, or simply “plus eleven” in the vocabulary we were using.

R’s domain (0 < x < 1000) introduced a new mathematical concept to our circle – the difference between “less than” and “less than or equal to.” Everyone’s thinking deepened when he decided to allow an input of 1000 and therefore had to restate the domain as 0 < x ≤ 1000. (BTW in this report I’m being a bit mathier than I was in class. In class I wrote “in” on the board instead of x. If we had one more week I would have switched to using x with the kids.) The in numbers of 1000, 15, 10, and 1 produced out numbers of 4000, 60, 40, and 4. Conjectures for the rule included

- “conditional?”
- “+3000?”
- “+300?”
- “+3?”
- “added to itself?”
- “doubles twice?”

It wasn’t any of these, he said. Finally someone posited “multiplied by 4?” That was it, and a brief conversation ensued about whether that’s the same as doubling twice.

Now it was time for the other R. She declared a domain of 0 < in < 900. After putting in a few numbers, the group was pretty sure her rule was “plus nine.” It was not, she said. Was it decreasing the units digit by 1 each time? No. People were stumped pretty quickly, and asked her to reveal the rule. It was “plus ten minus 1.” She was the first to admit that this produces the same result as “plus 9,” so we continued our ongoing discussion about whether some rules are actually identical.

Thanks to all of you for sharing your kids with me for these 6 weeks. I’ve been hearing a lot about function machine activity going on at home – so much so that I’m going to include it in the schedule for our upcoming slightly older class too. Have fun continuing these math explorations.

Best,

Rodi

*BTW this comic strip isn’t the graph that’s the most relevant to this problem, but it is relevant to the big point of this math circle, that kids can be real research mathematicians.

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