Math Problem Solving

The  purpose of this column is to talk about the weekly math puzzle night dinners run by myself and Professor Palsson, and to invite you to join us for this and other related activities. Here’s a quick summary of problem solving activities here; for more information or to be added to the email lists, contact me at sjm1@williams.edu.

  • We have a weekly problem solving dinner at 5:30pm in Dennett Private Dining Room at Mission on Wednesdays. There’s no prep work; feel free to drop in any time (and if you’re not on the meal plan we’ll provide a swipe). The way it works is we print out a math competition from somewhere in the world, and then brainstorm and attack the problems  together.
  • We’ll also meet for lunch on Wednesdays (in Mission at noon, room TBD) to do Project Euler problems. 
  • There are several math competitions each year. Some time in October or November we’ll defend the Green Chicken when Middlebury travels here. There’s also the Virginia Tech math competition (we’ll do this remotely Saturday October 25th), and the Putnam exam (which will probably be Sat Dec 6th).
  • We also frequently field teams for the Mathematical Contest in Modeling.
  • I’m also teaching a class on math puzzles and problem solving, Math 331: The little Questions. Feel free to check out the homepage for resources, as well as lectures (each class is recorded and uploaded to YouTube).

Many people love math puzzles or riddles. They’re often fun, frequently illustrate a beautiful concept or perspective, and unlike real world research problems they typically have an elegant answer. Below is one of my favorites. It can be solved by brute force but only at great cost (and a high probability of forgetting a case); however, if you  look at it the right way it’s just one line (and this proper perspective illustrates a powerful technique which is of use in research mathematics).

Problem: Imagine you have 10 distinct cookies but 5 distinguishable, hungry graduate students (so they’re not engineering students working away in a lab!). How many ways can you divide all the cookies among the people? In other words, we only care about how many cookies each person gets, not which ones.

I run a math riddles page (see http://mathriddles.williams.edu/ ), and this was the first problem I added; the problems and resources posted there for students and teachers are used in schools throughout the world, and if you’re interested in helping with the site drop me a line at sjm1@williams.edu. It turns out that this problem is equivalent to Waring’s Problem with exponent 1; I’ve used this idea with SMALL students in research problems related to Fibonacci numbers (this is discussed in another post, To Bead or Not To Bead). Enjoy!