Mathematician engages students in solving the ostensibly insoluble

Postcards from the Classroom: Solving Puzzles, Equations, and Problems (also known as X)

Solving Puzzles, Equations, and Problems
Jason Zimba
This is a course about solving. Take this course with me and we will solve puzzles, transcendental equations, linear and nonlinear systems of equations, Diophantine equations, differential equations, integral equations, integro-differential equations, recurrence relations, and functional equations... use awesome tricks people have invented over the centuries for solving the seemingly insoluble...excavate the psychology and pathologies of solving...analyze the impact of the drive to solve on the history of mathematics and science... wrestle with the teaching and learning of problem solving (still a primitive field in math education)... and, after you have earned the right to do so, deconstruct solving itself. Opportunities for advanced work.

In the early 1700s, the secret of making porcelain was highly prized, carefully guarded, and known only in China. So what do you do if you're a 18th-century Saxon king who wants to get that formula? Lock an alchemist in prison, and tell him he won't be set free until he figures it out.

And what do you do if you're a mathematician who wants your students to solve a difficult problem they've never encountered before--a problem which includes so many digits that even powerful computer programs can't handle them all--and get it done in short order?

Lock them in prison. That is: tell them class will not meet again until they find the answer.

For the curious, this is the calculator-defying problem Jason Zimba presented to his students on the first day of class: "Consider an exponential tower consisting of three thousand sevens: (7^(7^(7^( . . . )^ 7))) What is the remainder when you divide the tower by 11?"

Sound simple? Liz Yenidjeian '07 thought so. "I think we all thought to ourselves, 'Ooooh, calculator time.' I pulled out my handy TI-89 in class, before Jason had even left the room, and started punching in sevens. After the third one, I snapped out of my delusion, realized this number was astronomical, and that a calculator wasn't going to cover it."

Yenidjeian found herself fixating on the problem. She got in the shower that night still thinking about it, and ended up rubbing facial scrub into her hair instead of shampoo. She posted queries on online math forums, and found herself learning a whole new vocabulary: modulo, Fermat's Little Theorem, modular exponentiation, Euler's Theorem.

One of the rules was that the class had to work out the problem together, and submit the answer with all their names signed at the bottom. Together, the students searched for patterns; consulted (and sometimes begged) other faculty, and scoured their books; delved into PhD-level math; learned about modular arithmetic.

And they found the answer. Class resumed one week after the problem had been first assigned. It was just the beginning.

All this obsession and desperation was, in fact, exactly what Zimba was looking for: "I could see it had really gotten under their skin the way I wanted it to." Among his many goals for the class, there lurks his own personal curiosity: he wants to figure out exactly what makes a good problem solver. "I'm being drawn to think more about the role of love or excitement," he says. "This Terminator-like tenacity, killing yourself for ten years to write a proof, has to come from somewhere--and it doesn't come from your boss saying, 'Do this.' How is it that one person looks for a pattern and finds it, and another doesn't?

"I think it's about building expertise and habits of patience. It's a discipline that's teachable, like yoga. When we watch someone solve these amazing equations, it's like watching someone wrap their leg around their head. You think, 'I could never do that!' But possibly you could."

Glen Heinrich-Wallace '09 has been investing some serious time into that discipline. After the initial "tower of sevens" problem, he became intrigued by modular arithmetic, which led to his current endeavor: working his way through a book on number theory and meeting with Zimba every week or two to discuss what he has come up with. One of the problems he's working on now first crossed his path years ago, but at the time, he didn't even know where to begin. Now he is learning how to get a foothold into problems that at first glance seem impregnable.

"In class we've been doing differential equations, functional equations, number theory, a huge amount of mathematics that I've never been exposed to before. But what the course is really teaching me is how to engage a problem that you don't know how to do, which is very tricky. It's about independent thinking and learning how to approach the unknown.

"One of the major themes of the class is The only thing you can't do is nothing. If you sit down and don't do anything, if you tell yourself it's too hard, it's not going to get done. So if you don't know where to begin, just start somewhere, and it might get done--which is infinitely better."

What's with the parrot? So you've read the whole article and you still don't know why there's a picture of a parrot accompanying this piece. It is relevant to the problem-solving class--but you won't discover the reason why until you find the parrot's mate. Your only clue: It's somewhere on this website, and you'll see its eye before you learn its secret. Happy hunting....

More:

• Science and Mathematics at Bennington
• Postcards from the Classroom: Consumerism
• Postcards from the Classroom: Borges and Mathematics
• Ask a Student
• Applying to Bennington

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