(April 7, 2015) We started today with child-produced function machines, then moved into our old standby, the Hadwiger-Nelson Problem.
INTRODUCING DOMAINS OF FUNCTIONS
M approached me before class with a slip of paper with her rule written on it: +10. She explained that she had only practiced with numbers up to 1,000. What to do if someone called out an “in” number over 1,000? I explained that you can make specifications about the in numbers; those limits are called domains. She decided to present her function as one with a domain of “0 < in < 1,000,” to prevent difficult subtraction or getting negative numbers as inputs. This led to a lot of math fun, with the kids successfully solving the machine, which had the appearance of a face.
Now it was A’s turn.
“Is he going to draw Big Ben?” asked someone? (When I had been drawing the machines in earlier sessions, every idea A contributed toward the drawing was related to Big Ben.)
“He’s definitely going to draw Big Ben,” replied someone else.
“Is it?” asked yet someone else, as the tension mounted. A knows how to work the crowd so A milked it by drawing slowly on the board. He surprised everyone by turning his tall narrow rectangle into something from Star Wars and not Big Ben.
“Mine has a domain too,” explained A. “The in number has to be zero.” Giggles from the group. On the board I wrote “in = 0.” Everyone took a turn calling out an “in” number. Every person said “zero!” on his or her turn. Deliciously fun. For every out number, A replied “Zero!” There were various conjectures about what he rule might be: adding zero? subtracting zero? adding then subtracting 65? A replied to each conjecture, “that may give you zero, but that’s not my rule.” Finally he had to declare his rule to the class: add zero then add zero again.
The kids noticed how different “rules” gave the same results. I then asked whether they were truly different rules then. No one had an answer to this, but it’s good food for thought, hopefully. Then we moved to the other, more private room, as is our custom, to work on our ongoing problem.*
CHROMATIC NUMBER, AKA HADWIGER NELSON
I added a new turn to our ongoing story:
The astronaut Miriam suggested to the group that Astronaut Blake’s plan might just work if only we didn’t use such a big grid. Start small, she said.
“What’s the smallest grid we could draw?” I asked the kids.
“One by one,” said someone.
“What does that look like?” I ask.
“A square,” came the reply.
“Does that help?”
“So what’s the next size we should try if we’re starting small?”
“Two by two.” So I drew a 2x2 grid on the board, and colored the top left square in it blue. Without going into the play-by-play, suffice it to say that soon everyone in the group agreed that you need four colors to color a square grid. In other words, the chromatic number is four.
“Using four colors would take a lot of energy,” said someone, remembering that the space scenario we were using described how/why we needed to minimize the number of colors to solve the problem.
“Actually,” said someone else, “ You don’t have to minimize the number of colors to save energy. “There’s a way to use the same paint color machine for more than one color.” (Here we go again with the engineering solutions!)
“There is a reason that in this case, you have to have separate paint manufacturing machines for each color. I don’t know the reason, but I know there is one. There is no engineering solution that would allow the astronauts to avoid having to minimize the number of colors.”
“My dad is an engineer,” said Z.
“Oh,” I replied, “maybe he would know why the different colors have to be produced by different machines.”
“Nah,” said Z, “he doesn’t know anything about Botso.”
While this discussion was going on, many of the kids were attempting their own coloring patterns.
N experimented with making grid of squares with 1/2 unit sides, instead of 1 unit.
M cut each square in her grid with a diagonal.
Both were convinced that their tiling patterns could solve the problem* with fewer than four colors. I reproduced their maps on the board (see photos) and the group discussed them. A few kids in the group, though, were getting antsy.
“When is math circle over?” asked one child. His question made me realize that these little kids had been struggling for 5 weeks with basically the same question, using the same approach. Maybe people’s attention for coloring tiling patterns was wearing thin. Time for another development in the story.
I drew 6 tiny circles on the board and connected them with line segments marked them “1 unit” apart, forming a hexagon.**
Astronaut Z thinks we’re going about this the wrong way. He says that we should color only enough land for one single plant, then plant the next plant one unit away. We’d be safe from the aliens as long as no plants one unit away are the same color.
“We are not going about this the wrong way,” said M. “We are making progress!”
A few other kids agreed. “You’re right, we are making progress,” I said, “but let’s just see about Z’s idea. If we color this plant (tiny circle) orange, what color would it’s neighbor be?” The kids dictated colors to me for a graph-theory type diagram (more mathematically and simply, a graph) to soon realize that a hexagonal graph required only 2 colors. (See photo!) Maybe Astronaut Z was onto something after all. Everyone was excited.
“But if only 6 plants are planted, would that be enough food for everyone?” I lamented.
“No,” said everyone,
“So the question really is whether this graph can be extended with only 2 colors.”
A raced up to the board, held out his hand for me to pass him the marker, climbed onto a chair, and began extending the diagram. Soon, S, M, and N were up there with him. This group of 4 kids began successfully extending the graph with two colors. Then just about everyone came up to the board. The kids had essentially taken over. (This is a goal of a math circle – for the leader to step aside while the kids explore the math as a leaderless group.) Sadly, though, we only had one very small board, and it was up high. Eventually, people were drawing their own conjectures on top of others’. And there weren’t enough chairs to stand on, so a few kids started to topple. (We caught them, though. Phew!) And not everyone could reach the board.
So the group split – some at the board, some working independently. Those working independently each went off in different directions. R, for example, diagrammed two perpendicular number lines. (I asked him to see if he could figure out how many units points on the different lines were from each other.) Z drew hexagonal graphs of his own. My helper J helped me take photos of individual kids’ work. (See photos) And I promised to do my best to bring more whiteboards next week.
*Click here for the reports of weeks 1-4 if you need to get caught up on the ongoing saga.
** Hint to parents (I have not mentioned this to the kids): you need to use both approaches to successfully answer the question – “colorings on embedded graphs with just a few points to make it possible to prove the lower bounds, and various tilings of the plane… to prove the upper bounds. They need to find colored tilings (usually regular ones such as squares, triangles, and hexagons), which show a repeating pattern that makes sure that no points exactly one unit apart are the same color.” (Thanks to mathematician Dr. Amanda Sereveny for this clear explanation.)