(April 21 and 28) So far, the questions I’m giving the kids to work on in Math Circle are leading them to ask very interesting questions on their own. The bullet points below are all questions and conjectures posited by the students, not me. The problems themselves are unanswered, or open, questions in mathematics. I used presentations similar to those of Gordon Hamilton from his most amazing website, Math Pickle. Thank you Gordon for doing so much of my prep work for me.
FINDING A PLANAR GRAPH*
“Consider the seven dwarves. Two are friends if their names share 2 or more letters. Draw a black line to connect pairs of friends. Two people are enemies if their names share no letters. Draw a red line to connect pairs of friends. What number of names do you need to insure success?”
- What are the names of the seven dwarves? I didn’t know, and had come to class assuming the kids would. They didn’t. So they made up their own list: Frisky, Sleepy, Toily, Happy, Lazy, Sleepy, Slappie, Grumpy, and Boldy. The list started with seven dwarves.
- Can the lines cross? No, so it didn’t work.
- It might work if we rearrange the positions of the names
- It depends on what spellings you use so let’s change Boldy’s spelling to Boldie. The kids did this to insure mathematical success. Once a successful graph was created, the students added another name to see if it’s possible with eight dwarves and so on.
- What topic in math is this question? Graph theory.
- What is graph theory? To quote Wikipedia, in mathematics and computer science, graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices, nodes, or points which are connected by edges, arcs, or lines.
- What happens if we include frenemies?
- Could we do it with our own names?
- What if we put all of us in a line?
- Could we try connecting our bodies with string?
- Is there a name we could add to the list that would definitely insure that you could make a graph? That you could not? (Quinn was their conjecture, since none of the students’ names has a Q.)
Note how vague the original question was. This was intentional. Through the students’ questions and conjectures, they refined the question. In week 2, I brought in a list of the actual names of Disney’s Seven Dwarves (as well as a very entertaining list of names that got rejected). The students tried and tried and tried, but couldn’t graph it without following the conventions of graph theory.
In week one, we had a beautiful day so sat outside in the grass around whiteboards on the ground. In week two it was rainy so we spread the whiteboards on tables indoors to create the graphs. It was fun to see students climbing on the table with dry erase markers to test their conjectures (see photos).
- If you put all the names in a circle, it might work, said S. (After a lot of work and repositioning, it didn’t.)
We summarized our attempts in a chart (see photo), and realized that we really know nothing about this question. “This is the world of a mathematician,” I said, “try try try fail try try try fail take a break from the problem then try again, maybe eventually solve it…” Some of the kdis were surprised to hear this. I’d like to think that it was encouraging for the kids to hear that struggle is normal and success is often possible.
“What is your favorite 2-digit number? Multiply its digits. If the result has more than one digit, multiply its digits. Repeat until you get to a one-digit number. How many times did you need to multiply? That number of moves is a number’s multiplicative persistence (MP). What’s the highest possible MP?”
I asked for each students’ number. The highest MP of their numbers (66) was three.
- Let’s try bigger numbers. 989 gave a MP of 4.
- What’s the MP of this number? J wrote a 78-digit number on the board (see photos). Someone quickly pointed out that since she had some zeroes in her number, the MP would be one. So she changed all the zeroes to one. No one wanted to multiply these digits. What to do? The number wouldn’t fit into a calculator. I promised to try at home. In class 2, I showed them an excel spreadsheet I had created with each digit in a different cell. I gave the column of numbers the command “PRODUCT.” It gave an answer in scientific notation. The students realized that we need more powerful computing tools. We did find out that J’s number resulted in a 50-digit number after the first round of digit multiplication. We also had super-brief discussions of exponents, scientific notation, and the beauty and usefulness of spreadsheets like excel.
- What if we try a fraction? We tried one half and seven eighths by converting them to decimals. There was nothing special there once people decided you can just ignore the decimal point.
- What’s the MP of pi … of tau … of phi … of the square root of 2, 5, or 7? “D, does pi have a zero in it?” asked R of D. Her confirmed that it does. (In week 2, I confirmed that all of these numbers have zeroes as digits. An interesting MC question would be, “How likely is it that an irrational number would not contain zero?” but I didn’t want to lose the focus of this question, and no one asked.)
I put a 29-digit number on the board that someone with a more powerful computer had calculated a MP of 11. I explained that no one has yet found a number with a higher MP than 11. Is there one?
That question led to a new discussion:
THE NATURE OF PROOF
What would it take to prove that 11 is the greatest MP? What would it take to prove that 11 is not the greatest MP? Drat drat drat I’m having trouble remembering the details of this conversation. It was so deep. The question from the class was something like “does disproving the opposite of a claim prove that claim?” We were starting to get into different philosophies of proof. I said to the class, “I must be able to fly because you can’t prove that I can’t fly.”
One that discussion petered out, I asked, “Would this question would be as compelling if we did it in binary?” Most students were not familiar with binary, so we had a brief discussion (not enough to really make binary clear), with those who had seen it trying to explain it. Once the kids saw a list of binary numbers on the board, they realized that binary makes the question moot.
THE COLLATZ CONJECTURE*
“Icarus and Dedalus both have a dream. In Icarus’ dream, you put a number on a rock. It it’s even, you cut it in half. If it’s odd, you triple it and add one. Do the same with the result again and again. In Dedalus’ dream, you do the same thing except that if it’s odd, you triple it and subtract one. In either case, if you eventually get to one, you die. Would you rather choose Icarus’ pattern or Dedalus’?”
The kids immediately started testing numbers. The number 3 failed in both. Eventually, the class realized that the number 20 does not result in failure. It created a loop that brings you back to 20 again and again. (We did this quickly and verbally only, so it would be good to review.) Then questions started popping up:
- What exactly is the unanswered question here? Turns out that no one knows if there is a number you can plug into the Icarus rule and not end up at one.
- What happens if you use a decimal? The kids who knew what decimals/fractions are tried it and found that 0.25 works, but most couldn’t follow the calculations easily. I ended up drawing a pizza cut into quarters on the board.
- If .25 works, have we solved the problem?
- Who first asked this question and when? (Lothar Collatz, 1937)
- If we could solve it in five minutes, and people have been working on it for nearly 80 years, we probably are not allowed to use decimals. Are they?
- Is there a different base we could use where you couldn’t get down to a single digit? Maybe something like base .5? I was reluctant to answer this because (1) only the student who posed the question really understood different bases, and (2) as far as I know there are no bases below 2. I could have proven with the students why there is no base 1, but that would be a detour that we didn’t have the time for. I don’t know for sure, though, whether there is some new, abstract, or unconventional area of mathematics that involves fractional bases. (Mathematicians reading this – can you help?) I told that student we could explore it outside of class.
I had worked on the Collatz conjecture in a math circle about a year ago with some of the same students. They didn’t remember it. I had presented it in a discussion of infinite series back then. Gordon’s way of presenting – with Icarus and Dedalus - was much more engaging.
I was worried that the questions I’ve chosen are too arithmetic-centric, but so far this doesn’t seem to be a problem. There’s a wide age range in the group (some are still 10, others are almost 14, and the rest somewhere between.) The kids had fun doing arithmetic, even testing decimals/fractions into the Collatz Conjecture.
At one point I had to give the kids a 3-minute break to photograph and clean the boards. Student R (not my assistant R, who you often read about in these reports) entertained the group with a math puzzle she found online (see photos).
Toward the end of the session we talked about prize money offered for solutions to these and other open questions. Many of the kids want more details about that, so I’ll research it.
PS When viewing the photo gallery, note that the brightly colored images are from session 1 (outdoors!) and the yellowish images are from session 2 (indoors).
*I used variations of these questions from Gordon Hamilton’s website, Math Pickle.