This week I introduced the students to the finite field of seven elements via the following puzzle found at http://mathfair.com/puzzles.html:

Redraw the figure so that each of the seven circles is big enough to hold a penny. The problem is to shade each of the shaded circles. However, there’s a catch. Here’s the rules that you have to follow.

Starting with any circle, count clockwise three circles and shade the third circle. Continue in this way, each time *beginning at an empty circle *and counting clockwise three circles and then shading.

As there are seven circles, the students where able to change the problem from counting over three spaces to counting over n spaces. They where able to figure out that multiples of seven will not work and that 10 is congruent to 3 modulo 7.

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## About Branden

As a GK-12 Fellow, I meet with a group of middle school students at Rosedale Middle School in Kansas City, Kansas. There I talk to them about what a mathematician does and teach them ways of thinking about problems.
I am a student of Craig Huneke at the University of Kansas. As a far as my research is concerned, I am currently studying the theory of maximal Cohen-Macaulay modules. In particular, I am interested in determining if countable Cohen-Macaulay type implies finite Cohen-Macaulay type over a complete local Cohen-Macaulay ring with an isolated singularity. I also enjoy working with Macaulay2 .