University of Michigan Math and Science Scholars
Hex and the 4 Cs
After a very long night of homework, you finally finish your math assignment. While double-checking your work, you realize that you have done problems from page 221, not page 212 as your teacher requested. In disgust, you rip the paper out of your notebook, wad it up, and toss it back down on your notebook. Too frustrated to begin your assignment anew, your mind begins to wander. You wonder: Is there a point in the wadded up paper that lies exactly above the location from which it started? After you pour your parent’s morning cup of Joe, the coffee comes to rest while you sleepily (because of the whole homework thing) search in the fridge for the cream. After adding and stirring the cream into the cup, you watch the pretty patterns made by the swirling coffee and cream as the contents come to rest. You wonder: Is there a point in the coffee that lies at the same point both before and after the cream was stirred in? We shall use mathematics to model and answer the above questions. Initially, the above questions will motivate our study of four fundamental concepts in mathematics, all of which begin with the letter C: continuity (what sorts of wadding/stirring are allowed), completeness (what if our paper/coffee has “gaps”), compactness, and connectedness. Interestingly, these are also the concepts one needs in order to rigorously understand why Calculus works. Our modeling will lead us to the Brouwer fixed-point theorem; a very nice topological result. To show that the Brouwer fixed-point theorem is true, we shall also learn about the game of Hex. The game of Hex is an easy to describe board game for two players (Google “Hex game” to find a description). The game has many interesting features. For example: one of the two players must win, the first player to move should (theoretically) win, and nobody knows a strategy to guarantee that the first player wins. We will explore the mathematics required to understand why every game of Hex has a winner. Finally, we shall stitch all of the above together by showing that the fact that there are no ties in Hex implies that there is a point in your parent’s cup of Joe which lies at the same point both before and after the cream was stirred in.