Non-crossing Knight's Tour in 3-Dimension

TL;DR

This paper explores the existence of non-crossing knight's tours in 3D cubes up to 8x8x8, demonstrating their feasibility and extending the concept to larger sizes with significant coverage percentages.

Contribution

It introduces the concept of non-crossing knight's tours in 3D and provides the first known constructions up to 8x8x8 cubes, including coverage metrics and potential for larger cubes.

Findings

Feasible non-crossing knight's tours in small 3D cubes.

Achieved coverage of up to 77% in 8x8x8 cubes.

Extended the concept to larger 3D cuboids.

Abstract

Non-crossing knight's tours in 3-dimension is a new field of research. The author has shown its possibility in small cuboids and in cubes up to 8x8x8 size. It can also be extended to larger size cubes and cuboids. The author has achieved jumps of length 15, 46, 88, 159, 258 and 395 in cubes of size 3x3x3, 4x4x4, 5x5x5, 6x6x6, 7x7x7 and 8x8x8 respectively. This amounts to covering 59%, 73%, 71%, 74%, 76% and 77% cells in these cubes.