Egyptian Fractions

I had a great time at the Julia Robinson Math Festival the weekend before last.

Hundreds of kids attended, most of them girls, it seemed to me. The setup: many, many tables; at each table, one or two adult guides, and a math problem that combines access and depth. Students choose a table, and work on the problem. They stay at a table until they feel ready to move on to a different problem. The adults provide support, largely in the form of questions. The atmosphere is informal, and has none of the intensity and pressure that accompanies math contests. Kudos to Joshua Zucker, who started this, and coordinates these festivals. And thanks to the American Institute of Mathematics, who provide the institutional home for them.

The problem at my table was called “Egyptian Mystery”. It involved a graphical and numerical representation of equations such asef2ef1

4/5 = 1/2 + 1/5 + 1/10

and more generally,

4/n = 1/a + 1/b + 1/c

Students were to figure out that this is what was being represented, and then try to write 4/n as a sum of three Egyptian fractions (i.e. numerator 1) for as many n’s as possible. (The Erdos-Straus conjecture states that this can be done for any whole number n.)

The problem lends itself to many approaches, and yields many partial results, so it was perfect for the festival. It did require some facility with fraction arithmetic, but I was impressed at how students at every level of competence beyond that stuck with the problem and took it further than they initially thought possible.

They were getting a taste of doing mathematics away from the pressures of school, and pursuing problems for their own sake. It was thrilling.


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