The Counting Challenge

Our first trip of 2019 to Dalhousie University Math Circles was last Wednesday, January 16th. We had two vans filled with students eager to explore the “Counting” workshop that was presented by Dr. Peter Selinger, a Dalhousie University professor in the Department of Mathematics and Statistics. He posed counting problems that seemed easy enough at first, like how many ways could you colour a 2 by 2 square using only two colours? Immediately, everyone started drawing squares, first shading top left, then a different square shading top right, another shading bottom left, and then one shading bottom right. Then the colouring shifted to 2 squares, first on the top, then one side, the bottom, other side, etc. I am sure you get the picture. While it was interesting to try to find all possibilities, a mathematical discussion revealed an easy formulaic way to figure it out. In the case of the 2 by 2 grid, there turned out to be 16 possibilities or simply 2x2x2x2. The 3 by 3 square proved more difficult to draw all possibilities, but a quick calculation (29) indicated there were 512 different ways. Of course the challenge did not stop there. The counting became more challenging when symmetries were considered. In other words, how many ways could you tile the 3x3 square with black and white tiles, if two tilings that differ by a rotation were considered equal? And then, moving into three dimensions, how many ways are there of coloring the 6 sides of a 3x3 cube with 3 colours, up to a rotation of the cube? (answer=24) The evening became a true counting challenge, and at the end of the seminar Dr. Selinger’s explanation of Pólya's counting method provided a quick and accurate way to do so (answer to the final challenge is 24). Our counting was interrupted when the variety of pizzas arrived. 

As always, it was an evening of collaborative problem solving, pizza and fun. I commend David Helyer, Luke Mainwaring, Athena Cox, Silas Fillmore, Yifan (Doreen) Xing, Aden O’Callaghan, Holden Hoover, Zhi (Angel) Li, Jiahuan (Edward) He, Yi (Edward) Chang, and Haichuan (River) Qi for their ongoing interest in Math Circles. It was also terrific to meet up with alumna Mona Mohamed (2015-18)  and former student Matvey Semenenko at the event. 

We look forward to the next Dalhousie Math Circles event: Linear Inequalities and Economic Problems on Wednesday, February 13th. Interested students can sign up on the whiteboard in my Math classroom. Special thanks to Mr. Glen Faucher for driving the second van.

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