Which boards have a closed knight's tour?

Unsolved Hard Archive

archive posted 23 Jun 2026

A closed knight's tour on an $m \times n$ board is a sequence of knight moves that visits every square exactly once and ends a knight's move away from the starting square. For which $m \leq n$ does a closed tour exist? As a first step, prove that none exists when $m$ and $n$ are both odd.

Posed by folklore (studied by Euler around 1759)

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History and references

Solved in 1991 by Allen J. Schwenk.

Knight's tours appear in medieval Indian texts, and Euler studied them systematically around 1759. In 1823 H. C. von Warnsdorff described a simple procedure that always moves to the square with the fewest onward moves, which finds tours on many boards. In 1991 Allen Schwenk proved that an $m \times n$ board with $m \leq n$ has a closed tour unless $m$ and $n$ are both odd, or $m$ is 1, 2 or 4, or $m = 3$ and $n$ is 4, 6 or 8. The odd case is a coloring argument, since every knight move changes the color of the square, so a closed tour alternates colors and needs an even number of squares. The ordinary $8 \times 8$ board has 26,534,728,821,064 directed closed tours.

References

Knight's tour (Wikipedia)

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