# Longest uncrossed knight's path

The **longest uncrossed** (or **nonintersecting**) **knight's path** is a mathematical problem involving a knight on the standard 8×8 chessboard or, more generally, on a square *n*×*n* board. The problem is to find the longest path the knight can take on the given board, such that the path does not intersect itself. A further distinction can be made between a **closed** path, which ends on the same field as where it begins, and an **open** path, which ends on a different field from where it begins.

## Known solutions

The longest open paths are known only for *n* ≤ 9. Their lengths for *n* = 1, 2, …, 9 are:

The longest closed paths are known only for *n* ≤ 10. Their lengths for *n* = 1, 2, …, 10 are:

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The longest closed path for n = 7of length 24. |
The longest open path for n = 8of length 35. |

## Generalizations

The problem can be further generalized to rectangular *n*×*m* boards, or even to boards in the shape of any polyomino. Other standard chess pieces than the knight are less interesting, but fairy chess pieces like the camel ((3,1)-leaper), giraffe ((4,1)-leaper) and zebra ((3,2)-leaper) lead to problems of comparable complexity.

## See also

- A knight's tour is a self-intersecting knight's path visiting all fields of the board.
- TwixT, a board game based on uncrossed knight's paths.

## References

- L. D. Yarbrough (1969). "Uncrossed knight's tours".
*Journal of Recreational Mathematics*.**1**(3): 140–142.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> - George Jelliss, Non-Intersecting Paths
- Non-crossing knight tours