Knight's Tour

The Knight's Tour asks for a sequence of legal knight moves that visits every square of a board exactly once.

On an $N \times N$ board, each square is a vertex and each legal knight move is an edge. So the puzzle becomes a graph problem:

• an open tour is a Hamiltonian path;
• a closed tour is a Hamiltonian cycle.

Why this puzzle is important

Knight's Tour combines:

• mathematics in chess (movement constraints, board geometry),
• graph theory (Hamiltonian paths/cycles),
• algorithmic thinking (heuristics and backtracking).

Heuristic idea: Warnsdorff's rule

A strong strategy is to move the knight to a square that has the fewest onward moves left. This usually avoids getting trapped too early.

In this library page and its linked interactives, auto-solve uses:

1. Warnsdorff ordering;
2. backtracking when a branch fails.

Board sizes in this set

• 5x5: compact introduction to the move geometry.
• 6x6: larger search space, still manageable by hand.
• 7x7: much richer branching.
• 8x8: the classical board.

Tips for solving manually

1. Start near a corner and pay attention to low-degree squares.
2. Avoid isolating corner/edge squares until you can still reach them.
3. If stuck, undo a few moves and try a different low-degree branch.

Use the linked problems below to practice each board size. You can play manually or let the solver complete from your current position.