Knight

1
Knights Tour using Graph Theory
By Drew Moen
2
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

3
Applications

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

4
Knights Tour

• Hamilton Path
• A path that visits every vertex on a graph one
  time
• Knights 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

5
Purpose

• Finding new ways to solve for a knights 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

6
Graph
7
Matrix
Four by FourB0 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 ThreeC0 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
8
Matrix Application

• A2All 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
9
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
10
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
11
Works 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 lthttp//aix1.uottawa.ca/jkhoury/graph.ht
  mgt.
• Ramas, Amy. "Art of Knight Graph." knight_tour.
  04 July 2007. 16 Dec 2008 lthttp//wiki.phiepsilon.
  org/doku.php?idknight_tourgt.
• "Graph Theory Knight's Tour." 18 Dec 2008
  lthttp//en.wikipedia.orggt.
• Farmer, Jesse. "Graph Theory." 31 July 2007. 15
  Dec 2008 lthttp//20bits.com/articles/graph-theorygt
  .
• Hickethier, Don. QA interview. 17 Dec 2008.