ALGORITHMS FOR CONSTRUCTION, CLASSIFICATION AND ENUMERATION OF CLOSED KNIGHT’S PATHS

Abstract

Two algorithms for constructing all closed knight’s paths of lengths up to 16 are presented. An approach for classification (up to equivalence) of all such paths is considered. Two closed knight’s paths are called equivalent if one can be obtained from the other by applying one or more of the equivalences: translation, rotation, symmetry, or when the corresponding polygons (whose vertices are the cells visited by the knight), are geometrically congruent. By applying the construction algorithms and classification approach, we enumerate both nonequivalent and non-self-intersecting knight’s paths and show the obtained results. Some pedagogical aspects related to the problems under consideration and the teaching of subjects such as “Programming”, “Algorithms and Data Structures”, “Graph Algorithms” and “Competitive Programming” are also discussed.