1. Knight’s Tour Explanation and Algorithms

2. GROUP MEMBERS Hassan Tariq (2008-EE-180) ZairHussainWani (2008-EE-178)

3. Introduction

4. What is ‘Knight’s Tour’? Chess problem involving a knight Start on a random square Visit each square exactly ONCE according to rules Tour called closed, if ending square is same as the starting

5. Constraints A closed knight’s tour is always possible on an m x n chessboard, unless: m and n are both odd, but not 1 m is either 1, 2 or 4 m = 3, and n is either 4, 6 or 8

6. m and n are both odd, but not 1

7. Knight moves either from black square to white, or vice versa In closed tour knight visits even squares If m and n are odd i.e. 3x3, total squares are odd so tour doesn`t exist

8. m = 1, 2, or 4; m and n are not both 1

9. for m = 1 or 2, knight will not be able to reach every square for m = 4, the alternate pattern of white and black square is not followed so tour not closed

10. m = 3; n = 4, 6, or 8

11. Have to be verified for each case For n > 8, existence of closed tours can be proved by induction

12. Algorithms Neural Network Solutions Warnsdorff’s Algorithm

13. Neural Network Solutions Every move represented by neuron Each neuron initialized to be active or inactive ( 1 or 0 ) Each neuron having state function initialized to 0

14. Neural Network Solutions (contd.) Ut+1 (Ni,j) = Ut(Ni,j) +2 –  Vt(N) NG(Ni,j) 1 Ut+1(Ni,j) > 3 Vt+1(Ni,j) = 0 Ut+1(Ni,j) < 0 Vt(Ni,j) otherwise

18. Set P to be the position in S with minimum accessibility

19. Mark the board at P with the current move numberReturn the marked board – each square will be marked with the move number on which it is visited.

20. Comparison Neural networks Warnsdorff's Algorithm Complex algorithm (a lot of variables to be monitored) Longer run-time NOT always gives a complete tour Simple algorithm Linear run-time Always gives a CLOSED tour

21. Conclusion WARNSDORFF’S ALGORITHM IS BETTER