“There is a unicorn dying at the end of a bridge. He has 17 minutes left to live, unless 4 people can join hands around him and recite a magical spell. There are 4 people on the other side of the bridge, but it is very dark, they have one flashlight, and the bridge can only hold 2 people at a time. Ginny can cross in 1 minute, Ron in 2, Fred in 5, and Percy in 10. Can they save the unicorn?”
So began our younger Math Circle today (after playing Function Machines for a few minutes to make sure everyone was comfortable adding numbers). Immediately the participants began doing what mathematicians do: clarifying the question. (Can they throw the flashlight? Can they carry each other? How much does the unicorn weigh? Can they get across without the flashlight? Can one person go at a time? Did they bring any tools? Can the unicorn use magic? And so on.) They asked many questions beginning with those 2 magic words that spark mathematical discovery, “what if….?” (What if Ginny made return trips with the flashlight? What if Fred and Percy travelled together?) Some were looking forloopholes in the rules (a skill that will come in handy for proofs) while others were adding up the numbersto see what is possible. While stymied with every attempt, enthusiasm grew.
We struggled with the idea of combined rates (how long would it take Ginny and Ron to cross together?), so we shifted to a historical anecdote about the mathematician Gauss and some of his early childhood math-related experiences. I was ready to return to Function Machines to work on the rate issue, but the ideas, excitement, and questions about the problem just kept coming. With one minute remaining, 1 young mathematician jumped out of her seat and came up to the board to illustrate her solution idea. We left the question open to play with more at home, with the instruction to email me with questions and ideas. A hint to the parents in the room: question your assumptions.
I asked the older group, “Can you predict the number of pieces you get when you connect the dots on the edge of a circle?” I drew a circle with one dot on the edge and asked, “How many pieces?” We added a few more dots and connecting lines to realize that the question needed clarifying to specify the maximum number of pieces and what our explicit goal was. (In a Math Circle, initial questions are intentionally vague to foster skillful question-asking.) One person soon noticed that the number of pieces was always even. We added a fifth dot and line and another member of our group piped in the idea that “they are doubling.” I asked what is doubling to elicit accuracy of description, and the pattern was named: “The number of pieces doubles each time you add a dot.” I asked for a prediction for 6 dots, and they all agreed that there would be 32 pieces. I asked how we can be sure, and they said to draw and count. So we did. We counted 31 pieces. The group decided to redraw and recount several times, with the process becoming more systematic each time. Their claim of “the drawing is wrong” became “the rule is wrong.”
At this point I asked how many examples we need to prove a rule (“a lot!”). I mentioned empiricism(conclusions that arise initially from observation, as espoused by Aristotle) versus rationalism (theory first, then experiment, as espoused by Desartes). We discussed which they thought was better, and how both can be used in real life. We then talked about the Ptolemaic astronomers who struggled with making predictions based upon their observations and in light of more and more evidence of a sun-centered universe. We took a quick detour into the question of whether the astronomer Ptolemy might have been related to Cleopatra, then returned to our original question.
The group decided to count the pieces for 7 dots, and discussed which counting strategy is better: adding to the current diagram, or starting anew with each revision. We counted 57 pieces from 7 dots. We stared. We thought. We presented and then rejected conjectures about patterns. Frustration mounted. (One goal of a Math Circle is to be able to tolerate higher and higher levels of mathematical frustration and hence greater future rewards of discovery.) Time was almost up when one participant said, “Look! 3 dots, 3 lines, that’s times 1. 5 dots, 10 lines, that’s times 2. 7 dots, 21 lines, that’s times 3. Every other one is times another number!” When asked what we could do with that idea, they all agreed that we should draw a circle with 9 dots and see if we got the predicted 36 lines. Despite only having a few minutes left, the consensus was adamant that we start from scratch with our diagram to ensure accurate counting. We applied our most systematic drawing and counting strategies and discovered that the prediction held: 36 lines! Delight filled the room.
We ended with hope that with time and effort, we could probably test this pattern on larger numbers of dots, and also continue to look for a pattern in the number of pieces. We talked about how mathematicians play with problems like this for years, and take breaks, and come back again. I took a picture of the last diagram (attached), and look forward to hearing ideas from anyone who plays with this some more.
Below are some references if you are curious about any of the topics we explored today:
· List of notable astronomers with brief bios. This list includes Ptolemy, and also 2 women of more recent times: http://www.starteachastronomy.com/astronomers.html
· Geocentric view of the solar system: http://en.wikipedia.org/wiki/Geocentric
· Ptolemy, including discussion of his ancestry: http://en.wikipedia.org/wiki/Ptolemy
· Dark bridge problem: http://www.coolmath4kids.com/math_puzzles/b2-darkbridge.html (but please, play with this one first before looking here, and if parents can’t resist peeking at this, please give your kids the chance to work on this themselves over time! Remember, our goal is for them to think like mathematicians, so revisiting this problem after the passage of time is valuable. This link is reluctantly included for parents only.)