Knight Tour Problem

The classic puzzle of finding a closed knight's tour on a chessboard consists of moving a knight from square to square in such a way that it lands on every square once and returns to its starting point. The 8 × 8 chessboard can easily be extended to rectangular boards, and in 1991, A. Schwenk characterized all rectangular boards that have a closed knight's tour. More recently, Demaio and Hippchen investigated the impossible boards and determined the fewest number of squares that must be removed from a rectangular board so that the remaining board has a closed knight's tour. In this paper we define an extended closed knight's tour for a rectangular chessboard as a closed knight's tour that includes all squares of the board and possibly additional squares beyond the boundaries of the board and answer the following question: how many squares must be added to a rectangular chessboard so that the new board has a closed knight's tour? Schwenk's Theorem. An m × n chessboard with m ≤ n has a closed knight's tour in all but the following cases:

A Knight Tour problem is an ancient puzzle which remains as a focus of current researcher. The objective of a Knight Tour algorithm is to find out a solution with construct a closed moving path by a knight for visiting each square of a chessboard by only once. This paper proposes a new structural Knight Tour algorithm by four knights moving all together with heuristic preset rules. This new proposed method fulfills any board size of 4n+2 where n>1 with lower execution time and without the needs of any brute force method and backtracking method.

A closed knight's tour of a chessboard uses legal moves of the knight to visit every square exactly once and return to its starting position. The closed knight’s tour is a recreational variation of the well-known Traveling Salesman Problem (TSP). Gerlach uses Warnsdorff's heuristic technique and SAS to produce closed knight's tours for the traditional 8 × 8 board. Generalizing the same approach, Gerlach and Gerlach produce closed knight’s tours for the n × 8 × 8 chessboard for n = 2, 3, 4, 5, 6 and theorize that the technique works for larger values of n as well. In 1991, Schwenk completely classified the n × m rectangular chessboards that admit a closed knight's tour. In 2011, DeMaio and Mathew completely classified the i × j × k rectangular prism chessboards that admit a closed knight's tour. This paper demonstrates how to utilize PROC OPTGRAPH with the TSP option to construct closed knight's for rectangles, rectangular prisms, hexagons and other variant ch...

A Knight Tour problem is an ancient puzzle which remains as a focus of current researcher. The objective of a Knight Tour algorithm is to find out a solution with construct a closed moving path by a knight for visiting each square of a chessboard by only once. This paper proposes a new structural Knight Tour algorithm by four knights moving all together with heuristic preset rules. This new proposed method fulfills any board size of 4n+2 where n>1 with lower execution time and without the needs of any brute force method and backtracking method.

The aim of the paper is to enumerate all closed knight paths of length n over a square board of size n+1. The closed knight paths of length 4, 6 and 8 are classified up to equivalence. We determine that there are exactly 3 equivalence classes of closed knight paths of length 4, exactly 25 equivalence classes of closed knight paths of length 6 and exactly 478 equivalence classes of closed knight paths of length 8.

The aim of the paper is to enumerate all closed knight paths of length n over a square board of size n+1. The closed knight paths of length 4, 6 and 8 are classified up to equivalence. We determine that there are exactly 3 equivalence classes of closed knight paths of length 4, exactly 25 equivalence classes of closed knight paths of length 6 and exactly 478 equivalence classes of closed knight paths of length 8.

The Firefly Algorithm (FA) has a few disadvantages in solving the constrained global optimization problem, including that it is difficult to produce initial population, the size of relative attractiveness has nothing to do with the absolute brightness of fireflies, the inertia weight does not take full advantage of the information of objective function, and it cannot better control and constrain the mobile distance of firefly. In this paper, we propose a novel method based on discrete firefly algorithm combining genetic algorithm for traveling salesman problem. We redefine the distance of firefly algorithm by introducing swap operator and swap sequence to avoid algorithm easily falling into local solution and accelerate convergence speed. In addition, we adopt dynamic mechanism based on neighborhood search algorithm. Finally, the comparison experiment results show that the novel algorithm can search perfect solution within a short time, and greatly improve the effectiveness of solving the traveling salesman problem, it also significantly improves computing speed and reduces iteration number.

—A Knight Tour problem is an ancient puzzle which remains as a focus of current researcher. The objective of a Knight Tour algorithm is to find out a solution with construct a closed moving path by a knight for visiting each square of a chessboard by only once. This paper proposes a new structural Knight Tour algorithm by four knights moving all together with heuristic preset rules. This new proposed method fulfills any board size of 4n+2 where n>1 with lower execution time and without the needs of any brute force method and backtracking method.

A Knight Tour problem is an ancient puzzle which remains as a focus of current researcher. The objective of a Knight Tour algorithm is to find out a solution with construct a closed moving path by a knight for visiting each square of a chessboard by only once. This paper proposes a new structural Knight Tour algorithm by four knights moving all together with heuristic preset rules. This new proposed method fulfills any board size of 4n+2 where n>1 with lower execution time and without the needs of any brute force method and backtracking method.

The knight's tour problem is an ancient puzzle where the objective of the puzzle is to construct a series of legal moves made by a knight so that it visits every square of a chessboard exactly once. This paper proposes a model using Firefly Algorithm to solve the problem. Each firefly represents a possible solution of the problem. Each dimensions of the search space represents n th move taken by the knight. The numerical value for each space can be in the range of 0 to 7 according to the 8 possible moves that can be taken by the knight. The fitness of a firefly is then calculated based on the number of legal moves it has taken. The proposed model has been tested using 8x8 chessboard setup. Result obtained shows that the proposed model has a potential to be applied to solve the general knight's tour problem.

The knight’s tour problem is a sub chess puzzle where the objective of the puzzle is to find combination moves made by a knight so that it visits every square of the chessboard exactly once. This paper proposes a model using Binary Magnetic Optimization Algorithm to solve the problem. Each particle represents a possible solution of the problem. A pair of 3 binary bits is used to represent a move made by the knight for each move. The numerical value for each move can be in the range of 0 to 7 according to the 8 possible moves that can be taken by the knight. The fitness of a particle is then calculated based on the number of legal moves it has taken starting from the first move. The proposed model has been tested using the standard 8x8 chessboard setup. A comparative performance study is done based on two updating rules of Binary Magnetic Optimization Algorithm: sigmoid & tanh functions. Results obtained show that the proposed model has a potential for further improvement.

The knight’s tour problem is an ancient puzzle where the objective of the puzzle is to construct a series of legal moves made by a knight so that it visits every square of a chessboard exactly once. This paper proposes a model using Firefly Algorithm to solve the problem. Each firefly represents a possible solution of the problem. Each dimensions of the search space represents nth move taken by the knight. The numerical value for each space can be in the range of 0 to 7 according to the 8 possible moves that can be taken by the knight. The fitness of a firefly is then calculated based on the number of legal moves it has taken. The proposed model has been tested using 8x8 chessboard setup. Result obtained shows that the proposed model has a potential to be applied to solve the general knight’s tour problem.