Research in Math

by Lizzy Doebel

Have you ever wondered what research in math looks like? You’re probably thinking it sounds like a ton of fun, crunching numbers and solving equations all day. Well, this semester I started working with a professor in the math department and two other undergraduate math majors on a research project. We are researching knight tours on Aztec diamonds. So, what exactly are knight tours and what do they have to do with Aztec diamonds or even math?

In chess there is the knight piece. It has the ability to move in an L shape: 2 moves in one direction and 1 move in the other direction. A knight tour involves the knight piece and any size board. This also includes any shape of board. The goal is to hit every square on the board once. There is a closed knight tour and an open knight tour. The closed knight tours involves starting and stopping on the same square while hitting every square once. An open tour starts and stops in two different squares but still hits every square once. So, again, what does this have to do with math?

Well, to figure out what size boards work for a knight tour and the order in which to make the tour, one uses the method of induction. A computer program can tell us that a knight tour works for some m x n board, but it cannot tell us the route to take to hit every square once. Part of our research involves trial and error to dissect the simple cases, such as a 5 x 5 or a 3 x 11 board.  Soon we will look at an Aztec diamond to see if a knight tour can be done on it. I guess when we discover that, I’ll have a leg up on my honors project 😉 But until then, I’ll work on my chess strategy with the knight.

DoebelDifferent m x n boards and knight tour paths

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