(May 5, 2016) We only had 4 kids today due to some illness going around. I had a lot more to say about the Collatz Conjecture from last week, and the kids wanted to talk about it too. But shouldn’t we wait until next week, when everyone who participated in this problem is back? The kids who had something to say didn’t want to wait, so I decided to talk a bit about it today and more next week.
R told me that she talked about the problem with her dad between sessions, and he had shown her how to graph number of steps it took to get to one for each original number tried. She was very excited about this, and applied this strategy to a problem we got to later in the session. Others had a few questions too.
“I want to talk about bunnies now,” I told the class. I started to tell a story about a male/female pair of bunnies who grew up for a month and then another month later begat a male/female pair of offspring. “How many bunnies are there at the end of each month?” I asked.
“Is this Fibonacci?” asked D. I was relieved that someone had seen this before since I have great difficulty telling this story correctly. (I think he had seen it in a math circle at Talking Stick years ago.)
“Yes, it is,” I said to D. “I’m glad you know it because I often tell it wrong. You can help me.” He didn’t know it well enough to tell it for me, but the other kids got the hang of it too so we all figured it out together. We ended up with a bunch of bunnies and a list of numbers on the board (1, 1, 2, 3, 5, 8, 13, ….), and several ways to explain what number comes next.*
“What’s the unanswered question here?” asked someone. Nothing, I explained. D asked whether Fibonacci really saw bunnies behaving like this, or if it was fictional.
“Several things are not realistic in this case,” I explained, “and that’s where the unanswered question comes from. What isn’t realistic in this bunny example?”
“They might not have exactly 2 offspring each time,” said M. “Or have them exactly a month apart,” added someone else. Everyone had something to contribute. R mentioned that if they kept breeding within the same family, there would be birth defects.
I explained that famed mathematician John Horton Conroy thinks the same thing.
THE KILLER BUNNY SEQUENCE
Suppose bunnies start breeding in the same pattern as Fibonacci’s bunnies, suggested Conroy, but after a certain amount of time, not all the bunnies survive. In fact, when the number of pairs of bunnies in a certain month turns out to be a composite (versus prime) number, that number is divided by its least prime factor and the result is the actual number of bunnies that survive that month. What happens?
For example, Fibonacci goes 1, 1, 2, 3, 5, 8, but 8 is composite. It’s least prime factor is 2, so divide it by 2, and you get that term in the sequence to be 8/2 = 4. So the Killer Bunny Sequence actually starts out 1, 1, 2, 3, 5, 4. The next term would be 5 + 4 but no, that’s 9, so you divide it by 3 and get 3 for that term. (I had the kids figure this out what this means without spoon-feeding them an example, but I want you readers to understand this report.)
The kids spent a looooooong time doing these calculations to figure out the next bunch of terms. One student had not yet encountered factors or primes in math so had an initiation by fire. This student, who unsurprisingly had the most trouble with the factoring part, had the easiest time following the addition part of sequence. By this time, 3 of the kids were getting weary, while another – R- was loving it. She starting zooming ahead on her own generating the sequence. She wanted to graph it in the manner she and her dad had done with Collatz. She saw that I had a printout of the sequence, and borrowed it the check her work. The others were still calculating, getting a bit distracted. I wanted them to lose their stress over the minutiae of the problem and get interested in the big picture.
“What do you think happens eventually in the Fibonacci sequence?” I asked. It grows infinitely, they said. “Does the same thing happen with the Killer Bunny sequence?” I asked. Hmmm… no one was sure, but they suspected the bunnies would not grow infinitely. At this point R had calculated (and looked at the printout) enough to see that eventually the numbers go into a self-perpetuating loop in the manner that 20 does in the Collatz Conjecture. I made a rough sketch on this on the board (see photos).
“Here’s where a bet comes in,” I said. This statement got everyone’s interest. “Conway and mathematician Richard Guy disagreed about what happens with this sequence when you start with numbers other than 1.”
“You don’t have to start with 1,” said someone, surprised.
“No. You can start with any two numbers less than ten. If you do, will they all loop?” The kids proposed starting with other pairs, such as 2,3 and 8,9 and 5,5 and 3,3. But no one had the energy to attack these pairs with calculations at this point. I wonder whether anyone will at home during the week. Each week I hear reports from most of the kids about their personal attempts with these problems.
While working this problem, the kids had an interesting discussion about prime numbers – not only what are they, but why isn’t one prime, etc.
NO THREE IN A LINE
What is the maximum number of points in an nxn grid that you can graph so that no three points are collinear?
It took some time to decipher that mouthful. But I wanted to present it without a story or translation because I wanted to kids to feel powerful, that they could figure out some technical-sounding questions. And they did. One had only used x as a variable, so using n threw her off. They had to collaborate to figure out what “grid,” “graph,” and “collinear” meant. The quickly then determined that you can graph 8 points on a 4×4 grid, and 10 points on a 5×5 grid. Then J asked the key question: Does is matter how big the grid is? That is the unsolved part of the problem. I told them so far, people have been able to graph 2n points on grids up to 52×52. “Let’s try it,” they said, and set to work drawing this grid. Then sadly I realized we had gone overtime, so we had to shelve this problem for now. See the links pasted below for more info on this problem.
NOTE – forgot to include link to last week’s photos https://talkingsticklearningcenter.org/open-qs-1-graph-theory-and-number-theory/
*The kids and I realized at the end of the discussion that the problem in my telling of the story was that I didn’t make the adult bunnies and baby bunnies look different from each other in the diagram. Had a drawn the adult bunnies bigger, it would have been much easier to see the pattern. It’s hard to draw bunnies fast!