Lattice paths and creative counting techniques were the center of attention at the October 2022 Southwest MTC session with co-leaders Beth DeWitt (Trinity Christian College) and Marie Meyer (Lewis University). Participants were tasked with enumerating ways to efficiently drive from home to work on a rectangular grid. It turns out the number of lattice paths can be found using Pascal’s Triangle!
In the first variation of the problem, participants had to determine the binomial coefficients that counted efficient paths from their home to their office for different office locations on the grid. Next, donut shops and potholes were introduced to complicate the home to work commute. Participants counted paths which visited one or two donut shops as well as paths which avoided a street with a pothole. In true Halloween fashion, zombies were introduced in the final variation. Participants had to navigate a zombie invasion in order to count safe paths, that is, paths that avoided contact with the intersections containing zombies.
Complementary counting and the Principle of Inclusion/Exclusion were some of the mathematical tools developed along the way in order to solve the variation of counting problems. Students at the middle school, high school, and college level would enjoy this activity.
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