(May 12, 2016) We spent almost the whole hour examining the “total stopping time” of sequences of Hailstone Numbers, those formed by the Collatz Conjecture. (Pick a number. If it’s even divide it by 2, if it’s odd triple it and add 1.) Is there a starting number that doesn’t end up producing 1 as the final number of the sequence?
Start with 8. You end up with 4, then 2, then 1. So 8 has a total stopping time (TST) of 3 – it took 3 moves to get to 1. The students tried 88, got a TSP of 10. What happens when you start with 888?
By the time our hour was up, they had done 40+ moves starting with 888 and the sequence kept going. Many questions and conjectures arose:
> “What is the most moves this has taken?” According to Wikipedia, I explained, computers have checked starting numbers of up to 2 to the 60th power. This did not dissuade the kids from continuing their efforts on behalf of 888. Even though they realized that 888 is a small number that will not break the pattern, at this point they were more intrigued by the TSP of different numbers than by “solving” the open question. This is one of my favorite parts of math circles – when curiosity takes over as the driving force.
> “With 888, you get a lot of numbers that end in 9 or 8.” They tried to figure out why this was, but couldn’t. Then a bigger pattern emerged.
> “There’s a pattern in the last digit!” They noticed that starting with move #17, the last digit was 8 then 9 then 8 then 9… They decided to keep going to see how long this pattern lasted. It lasted until move #25.
> “Why is there this pattern?” they asked at first. (I don’t know, I answered. Even if I did know, I wouldn’t have told. This is a problem they will know forever, and maybe figure out its intricacies for themselves someday. Who am I to pre-empt that joy of discovery?)
> “Why did it just quit?” they asked at move #26. (Me: same response as above.)
> “Calculations go faster when you’re motivated.” I was surprised by how this problem held their interest for so long, considering the (IMHO) tedium of the calculations. One student, though, joyfully manned the calculations while the others watched her every step with great interest. After about half an hour, I asked if they wanted to switch focus and work on another problem, but only one student did. For that student’s benefit, I did insist we shift gears for the last 15 minutes.
OTHER THINGS WE TALKED ABOUT
We had fun talking about all the other names the Collatz Conjecture is known by - the Ulam conjecture, Kakutani’s problem, Thwaites conjecture, Hasse’s algorithm, the 3n+1 conjecture, and the Syracuse problem. Everyone had conjectures about the origins of these names. (Wikipedia)
It is becoming very apparent at this point where the students’ interests lie mathematically. You’ll get a sense of the divergence of their interests from how the students (sadly only three present this week) each chose to spend their last 15 minutes:
- J wanted to work on Multiplicative Persistence (MP), which we explored in the first 2 weeks. J was the student not interested in the Collatz calculations. I assumed that she was disinterested because of the tedium of the calculations – this student is pretty inexperienced in that area - but my motivation assumption was proved wrong by her choice of working on MP, another calculation-heavy question. Something about this problem has captured her interest in class and out for four solid weeks now, and I don’t know what it is.
- M, the student doing the Collatz calculations, wanted to spend her last bit of time continuing that sequence, so she did.
- The other M really wanted to watch those Collatz calculations, but also wanted to see J demonstrate the “No Three in a Line” problem that we had explored last week during M’s absence. So they spent a few minutes on that. I must confess that I had a secret agenda with this problem: I wanted them to “discover” the concept of slope by needing a way to determine which points were collinear on a large, hand-drawn grid. This did come up briefly but it was sorta forced and awkward because I was leading instead of following, and in a hurry. Next time I’ll do better with that, I hope. If you do work more on this problem with you kids, be sure to draw a sloppy grid so that they need a way to figure out which points are collinear.
That’s all for this week!