Optimal solution of the Guarini puzzle extension using tripartite graphs

Abstract

In this paper there will be presented a software implemented in the C# programming language using. Net Frameworks platform which allows an optimal solution of a puzzle obtained through Guarini puzzle extension. The generalization of Guarini puzzle consists in considering 2·n knights placed on the chessboard of 4·n dimension. Half of these knights are white and are positioned on the first line of the chessboard, and the other half are black and are positioned on the last line of the chessboard. The aim of this puzzle is to move the knights through a minimum number of moves so on the first line we have all the black knights, and on the last line we have all the white knights. This puzzle belongs to a category of problems which can be efficiently resolved by use of tripartite graph. Numbering the chessboard squares these will represent the vertices of the graph, and the possible moves of the knights will represent the edges of the graph. Modelling this problem associated with this puzzle, it can be noticed that the corresponding vertices of the squares from the first and last line form the first vertices subset from the tripartite graph, the ones corresponding to the second line of squares form the second vertices subset from the tripartite graph, and the ones corresponding to the third line of squares form the third vertices subset from the tripartite graph.