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Knights tour
- 1. KNIGHT’S TOUR By SasankP P V Sasank 13CS01007 Under the guidance of Dr P L Bera
- 2. WHAT IS AKNIGHT’S TOUR PROBLEM? A knight’s tour problem is a mathematical problem involving a knight on a chess board . The knight on the chess board is moved according to the rules of chess. A knight's tour is a sequence of moves of a knight on a chessboard such that the knight visits every square only once.
- 3. A KNIGHT’S TOURLEGAL MOVES
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- 5. Knight’s tourcan be represented as a graph. The vertices -Represent the squares of the board. The edges -Represent a knight’s legal moves between squares.
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- 7. TYPES OF KNIGHT’STOUR PROBLEMS There are two types of problems: 1. Closed 2. Open
- 8. Knight’s tour • Closed •Open If knight ends on a square from which the starting square can be reached by the knight , Then that tour is a closed one. If the beginning square cannot be reached , Then that tour is open.
- 9. OPEN KNIGHT’S TOURPROBLEM
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- 11. VARIOUS ALGORITHMS Bruteforce search Divide and Conquer Warnsdorff’s rule
- 12. BRUTE FORCE SEARCH Very general problem solving technique. Iterates through all possible move sequences. For a regular 8x8 chess board, there are approximately 4×1051possible move sequences.
- 14. DIVIDE AND CONQUER Divide the board into smaller pieces. Construct tours on each piece. Patch all the pieces together.
- 15. WARNSDORFF’S ALGORITHM Introducedby H. C. Warnsdorff in 1823. Can be solved in linear time. Very efficient algorithm for solving knight’s tour.
- 16. WARNSDORFF’S ALGORITHM Somedefinitions: A position Q is accessible from a position P if P can move to Q by a single knight's move, and Q has not yet been visited. The accessibility of a position P is the number of positions accessible from P. Algorithm: 1. set P to be a random initial position on the board 2. mark the board at P with the move number "1" 3. for each move number from 2 to the number of squares on the board: I. let S be the set of positions accessible from the input position II. set P to be the position in S with minimum accessibility III. mark the board at P with the current move number 4.return the marked board :each square will be marked with the move number on which it is visited.
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- 18. SOLVING KNIGHT’S TOUR 2-D array of 8x8 is created to represent a chessboard. Each element in the array is assumed to be a square on the board. Initially all the elements are assigned to 0 , to indicate that knight has not visited any of these squares. Shift the knight from the initial input position to anyone of the possible positions and assign the number of that move to element in that position. If we could not find a tour , then shift the knight to any other of the possible solutions. In this way check all the positions till you find a solution.
- 19. C PROGRAM FORSOLVING KNIGHT’S TOUR ..DownloadsstudiesSeminarknights mainnew.docx
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- 21. SOLVING KNIGHT’S TOURUSING WARNSDROFF’S RULE ..DownloadsstudiesSeminarknights wandmainnew.docx
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- 23. INTERESTING FACTS ONKNIGHT’S TOUR 26,534,728,821,064 closed directed knight's tours are possible on 8x8 board. The exact number of open knight’s tours is not found yet. It’s estimated to be about 1015 to (2*1016).
- 24. MAGIC KNIGHT’S TOUR The squares of the chess board are numbered in the order of the knight’s moves. Full magic knight’s tour: Each column, row, and diagonal must sum to the same number. Magic knight’s tour: Each column and row must sum to the same number.
- 26. Existence offull magic knight’s tour on 8x8 was a 150-year-old unsolved problem. In August 5, 2003, after nearly 62 computation-days, a search showed that no 8x8 fully magic knight’s tour is possible
- 27. KNIGHT’S TOURS ANDCRYPTOGRAPHY A cryptotour is a puzzle in which the 64 words or syllables of a verse are printed on the squares of a chessboard and are to be read in the sequence of a knight’s tour.
- 28. Knight’s touris simply an instance of Hamiltonian path. A closed tour is a Hamiltonian cycle. Knight's tour problem can be solved in linear time.
- 29. CONCLUSION Warnsdroff’s ruleis the best and efficient method to solve a knight’s tour. Warnsdroff’s rule gives always a closed tour. Proficient usage of Data structures and the user interface help us to code and understand the tour easily.
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