Uncrossed Knight's Tour

We conclude our exploration of the ancient challenge of knight’s tours with the uncrossed tour or non-intersecting tour.

Your task is to find the longest possible path of a knight (using standard chess knight moves) in such a way 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.

The first recording of this variant on knight’s tours was in the Journal of Recreational Mathematic of July 1968 by L. D. Yarbrough. What I love about this challenge is that Yarbrough only found an open uncrossed path of 16 on a 6x6 tour but another mathematician, Martin Gardner in his 1979 book, “Mathematical Circus,” explains that the 6x6 tour found by Yarbrough can be improved from 16 to 17 jumps.

On the worksheets I provided to the children, they were told the maximum solutions for the open and closed tours on boards of 3x3, 4x4, 5x5, 6x6, 8x8, and 10x10. For example, on the traditional chess board of 8x8, the maximum uncrossed knight’s tour is 35 on an open tour and 32 on a closed tour. For years, I have only been able to accomplish a solution of 28 on a closed tour so I challenged the children to beat my solution of 28 on an 8x8 closed tour. They may even get the help of their parents (no internet). If they do exceed my 28, they will get their picture on a mathlete dollar.

The key strategy is to explore the board in a way that hits as many squares as possible. Try not to miss an opportunity to reach areas of the board that can increase the number of moves. I have found that creating acute angles vs. obtuse angles will increase the number of moves.

I also suggested to the children that for each closed tour that they accomplish on any board, they should color in the polygon. A polygon is named by its number of sides. For example, on a 5x5 board, the maximum uncrossed closed knight’s tour is 8 moves and forms an octagon.