Knight’s Tour using Graph Theory

Knight’s Tour using Graph Theory By: Drew Moen

Graph Theory History • Leonhard Euler - founder • The Seven Bridges of Königsberg • Cross every Bridge once • Change the city into a graph • Change the graph into a matrix

Applications • Programming • Engineering • Communications • Circuitry • Social Networks • Shortest Path

Knight’s Tour • Hamilton Path • A path that visits every vertex on a graph one time • Knight’s Tour • A path that a knight takes on a nxn or nxm checkerboard to visit every vertex once • Setup • Create a graph • Model graph with a matrix

Purpose • Finding new ways to solve for a knight’s tour • Figuring out where a knight can arrive with a restricted amount of moves • Finding out how many moves a knight needs to get anywhere on the board

Graph

Matrix Four by FourB=[0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0] [0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0] [0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0] [0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0] [0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0] [0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0] [1 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1] [0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0] [0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0] [1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1] [0 1 0 1 1 0 0 0 0 0 0 0 1 0 0 0] [0 0 1 0 0 1 0 0 0 0 0 0 0 1 0 0] [0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0] [0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0] [0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0] [0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0] Three by ThreeC=[0 0 0 0 0 1 0 1 0] [0 0 0 0 0 0 1 0 1] [0 0 0 1 0 0 0 1 0] [0 0 1 0 0 0 0 0 1] [0 0 0 0 0 0 0 0 0] [1 0 0 0 0 0 1 0 0] [0 1 0 0 0 1 0 0 0] [1 0 1 0 0 0 0 0 0] [0 1 0 1 0 0 0 0 0]

Matrix Application • A2=All locations a knight can travel in two moves • A3= three moves, A4, A5, A6… C2= [2 0 1 0 0 0 1 0 0] [0 2 0 1 0 1 0 0 0] [1 0 2 0 0 0 0 0 1] [0 1 0 2 0 0 0 1 0] [0 0 0 0 0 0 0 0 0] [0 1 0 0 0 2 0 1 0] [1 0 0 0 0 0 2 0 1] [0 0 0 1 0 1 0 2 0] [0 0 1 0 0 0 1 0 2]

More Moves C5= [0 6 0 6 0 10 0 10 0 ] [6 0 6 0 0 0 10 0 10] [0 6 0 10 0 6 0 10 0 ] [6 0 10 0 0 0 6 0 10] [0 0 0 0 0 0 0 0 0 ] [10 0 6 0 0 0 10 0 6 ] [ 0 10 0 6 0 10 0 6 0 ] [10 0 10 0 0 0 6 0 6 ] [ 0 10 0 10 0 6 0 6 0 ] C3= [0 1 0 1 0 3 0 3 0] [1 0 1 0 0 0 3 0 3] [0 1 0 3 0 1 0 3 0] [1 0 3 0 0 0 1 0 3] [0 0 0 0 0 0 0 0 0] [3 0 1 0 0 0 3 0 1] [0 3 0 1 0 3 0 1 0] [3 0 3 0 0 0 1 0 1] [0 3 0 3 0 1 0 1 0] C4= [6 0 4 0 0 0 4 0 2] [0 6 0 4 0 4 0 2 0] [4 0 6 0 0 0 2 0 4] [0 4 0 6 0 2 0 4 0] [0 0 0 0 0 0 0 0 0] [0 4 0 2 0 6 0 4 0] [4 0 2 0 0 0 6 0 4] [0 2 0 4 0 4 0 6 0] [2 0 4 0 0 0 4 0 6]

Patterns [0 496 0 496 0 528 0 528 0 ] [496 0 496 0 0 0 528 0 528] [0 496 0 528 0 496 0 528 0 ] [496 0 528 0 0 0 496 0 528] [0 0 0 0 0 0 0 0 0 ] [528 0 496 0 0 0 528 0 496] [0 528 0 496 0 528 0 496 0 ] [528 0 528 0 0 0 496 0 496] [0 528 0 528 0 496 0 496 0 ] C11= C10 = [272 0 256 0 0 0 256 0 240] [0 272 0 256 0 256 0 240 0 ] [256 0 272 0 0 0 240 0 256] [0 256 0 272 0 240 0 256 0 ] [0 0 0 0 0 0 0 0 0 ] [0 256 0 240 0 272 0 256 0 ] [256 0 240 0 0 0 272 0 256] [0 240 0 256 0 256 0 272 0 ] [240 0 256 0 0 0 256 0 272]

Work’s Cited • Rosen, Kenneth H.. Discrete Mathematics and Its Applications. Fifth. New York, NY: McGraw-Hill, 2003. • Strang, Gilbert. Introduction to Linear Algebra. Third. Wellesley MA: Wellesley-Cambridge Press, 2003. • Houry, J K.. "Application to Graph theory." 11 Nov 2008 <http://aix1.uottawa.ca/~jkhoury/graph.htm>. • Ramas, Amy. "Art of Knight Graph." knight_tour. 04 July 2007. 16 Dec 2008 <http://wiki.phiepsilon.org/doku.php?id=knight_tour>. • "Graph Theory & Knight's Tour." 18 Dec 2008 <http://en.wikipedia.org>. • Farmer, Jesse. "Graph Theory." 31 July 2007. 15 Dec 2008 <http://20bits.com/articles/graph-theory>. • Hickethier, Don. Q&A interview. 17 Dec 2008.

Knight’s Tour using Graph Theory