7.11. The Knight’s Tour Problem

Another classic problem that we can use to illustrate a second common graph algorithm is called the knight’s tour. The knight’s tour puzzle is played on a chess board with a single chess piece, the knight. The object of the puzzle is to find a sequence of moves that allow the knight to visit every square on the board exactly once. One such sequence is called a tour. The knight’s tour puzzle has fascinated chess players, mathematicians, and now, computer scientists, for over a thousand years. The upper bound on the number of possible legal tours for an $8\times 8$ chessboard is known to be $1.305\times {10}^{35}$; however, there are even more possible dead ends. Clearly this is a problem that requires some real brains, some real computing power, or both.

Although researchers have studied many different algorithms to solve the knight’s tour problem, a graph search is one of the easiest to understand and program. Once again we will solve the problem using two main steps:

• Represent the legal moves of a knight on a chessboard as a graph.

• Use a graph algorithm to find a path of length $rows\times columns-1$ where every vertex on the graph is visited exactly once.

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