## MAT Exam Expository Papers

De Bruijn Cycles

#### Date of this Version

7-2008

In partial fulfillment of the requirements for the Master of Arts in Teaching with a Specialization in the Teaching of Middle Level Mathematics in the Department of Mathematics. David Fowler, Advisor. July 2008

#### Abstract

Problem 1. A robot is moving on a cyclic track. The track is marked at evenly spaced intervals with 0s and 1s, with a total of 8 marks. The robot can see the 3 marks closest to him. How should the 0s and 1s be put on the track so that the robot knows where on the track he is by just looking at the 3 closest marks?

Problem 2. The city of Konigsberg, Prussia is set on the Pregel River and includes two large islands, which are connected to each other and the mainland by seven bridges. Is it possible to walk a route that crosses each bridge exactly once?

Problem 3. Sally wants to make a necklace. She has several colors of beads to make her necklace. She wants to make a necklace in which there are no repeating patterns. How should she arrange her beads on the string?

The problems above seem to be very different, but mathematically they are the same. All of them can be solved using De Bruijn Cycles.

COinS