The Knight's Tour Problem

The Knight's Tour problem is a combinatorial problem involving finding a route for a knight on an empty chessboard such that it visits each square exactly once. The problem can also be generalized to boards of any size and even to multi-dimensional boards. A closed knight's tour occurs when the knight's final square is one knight's move away from the starting square. Otherwise, the tour is considered open.

The first recorded solution is credited to the French mathematician Abraham de Moivre (1667 - 1754). However, his tour ended far from the starting position. Another French mathematician, Adrien-Marie Legendre (1707-1783), eventually found a closed knight's tour where the knight ends on a square that is only one move away from the starting square. Additionally, the Swiss mathematician Leonard Euler (1707-1783) discovered a tour where the knight traverses one half of the chessboard before moving to the other half. Euler was the first to write a mathematical essay analyzing the Knight's Tour.

The Knight's Tour problem is a special case of the Hamiltonian path problem, a well-known problem in graph theory where all nodes must be visited exactly once. If the last square in the sequence can be connected back to the first square, a Hamiltonian cycle is found. The Hamiltonian path problem is NP-complete (nondeterministic polynomial time). This implies that it is likely not efficiently solvable. However, efficient algorithms have been developed for the Knight's Tour problem, such as the backtracking method. In this approach, an initial route is chosen, and if a dead end is reached, the last move is undone, and an alternative move is selected.