Holt claims “vivid, vital, pleasurable experiences are the easiest to remember” – I’d add interpersonal to that list and be ready to go. All of my favorite math problems sit nestled inside the memory of the who-what-when-and-where they were solved.
I’ve had a post saved in drafts for over a year now, detailing favorite problems. Perhaps this should be a series. Let’s start at the beginning.
The first summer I helped Dawson teach at VSA [Vanderbilt Summer Academy], I walked in each morning fresh to the puzzlers he planned to pose. Realistically, I could write an entire post about Dawson’s puzzlers – he’s got a treasure-trove of good stuff for high schoolers. Instead, I’ll write about the first and most memorable time I felt the thrill of simultaneous discovery – solving a good problem, and instead of stopping, demanding more of the problem than it offered at first glance.
Here’s the pirate problem. If you’re Dawson, a meticulously arranged treasure box featuring exactly 100 “gold” chocolate bars sets the stage. ARRRRRRRRR!
There are five pirates who steal exactly 100 gold coins. They go to a safe haven to distribute the gold among themselves. Being democratic pirates, they decide to vote on how to do so. Each pirate in turn submits a proposal for parceling out the booty. Immediately after the first proposal, they vote. If the proposal wins a majority of votes (MORE THAN HALF), they distribute the gold according to the proposal. If the proposal does not get a majority, they kill the pirate for suggesting it and move on to the next pirate, who makes his proposal. The process continues until a proposal receives a majority of votes cast.
You are the FIRST pirate to make a proposal. What should you suggest to maximize your share and, of course, remain alive? Explain why this proposal would win more than half the votes. (Minimally, the explanation requires saying who would vote for the proposal and explaining why they would do so). You may only presume that a pirate will vote for a given proposal if it is definitively better than another proposal that he may receive.
Each pirate knows his own and everyone else’s position-first, second, third, etc.-in the order of giving proposals. The pirates are entirely “logical” and unemotional people. All they care about is maximizing their share while remaining alive.
Solving the problem isn’t so tough. I set to work. I’m a terrible collaborator during problem solving – I want to hide in the corner and try all of my own ideas without sharing, then return the group and get louder and louder as I share my insights in half-formed, broken sentences or SHOUT my questions at whoever is unfortunate enough to be working with me. This was nearly 10 years ago, and Dawson and I were coming off of a spring of student teaching together – 6 am carpooling commutes from campus to class forge a bond that sticks. Dawson had solved the problem before, so I did my thing quickly. I watched the students in the class navigate their way through the problem after I figured it out, and poked and prodded at those who needed a push over some of the sticky places.
The day ended; the students left. I was still puzzling over the problem. “What if…?”
What if… there were 6 pirates? 7? How long does that pattern continue? When does it break down? Does anything interesting happen after that?
This was the first time I felt like I had stumbled on something new and my own. I’m not really sure how to express the sense of ownership I feel over this particular problem, this particular piece of mathematics that we carved out together. We set to work, we filled up a whiteboard, we invented ridiculous notation for pirates who won’t propose. I shouted at Dawson, he shouted back. We followed the problem where it lead us, and stumbled upon a surprising and interesting pattern. Apparently, we made some stranger who happened to be wandering the halls of the Blair School of Music come in and take a picture of us in front of our work.
Afterwards, I did some poking around the internet and found that Ian Stewart had written an article about the patterns that emerge in the extension – OUR extension – for Scientific American in 1999.
Postscipt: When I think about problem solving now, ideas like “solve a simpler problem,” and “extend the problem” seem so obvious, but I think this was one of the first (or at least one of the best) examples of me doing just that, before I had the framework to understand what I was doing. It’s not uncommon for people to ask me if “I always knew” I wanted to be a math teacher, and the honest answer is yes. But the longer answer is that the story of me falling in love with math, of seeing the richness and vastness of this subject, of understanding the way it excites, ignites, and brings people together… that story is still being written. And the pirates problem is a major chapter in me seeing myself as a mathematician – someone with agency to tinker, play, invent, and discover.
Update 4/23/16: As usual, Dawson says it better than me:
“WHOA! I’ve actually been thinking about that very afternoon a lot in the last week […] this was also my first such day, and it changed my teaching career and what I value about the mathematical classroom experience more than anything else I can think of. THANK YOU for shouting at me!”