SOLVING THE 106 YEARS OLD 3^k POINTS PROBLEM WITH THE CLOCKWISE-ALGORITHM
Marco RipΓ Β· Journal of Fundamental Mathematics and Applications (JFMA) Β· 2020
In this paper, we present the clockwise-algorithm that solves the extension in π-dimensions of the infamous nine-dot problem, the well known two-dimensional thinking outside the box puzzle. We describe a general strategy that constructively produces minimum length covering trails, for any π β Nβ{0}, solving the NP-complete (3Γ3Γβ―Γ3)-points problem inside a 3Γ3Γβ―Γ3 hypercube. In particular, using our algorithm, we explicitly draw different covering trails of minimal length h(π) = (3^π β 1)/2, for π = 3, 4, 5. Furthermore, we conjecture that, for every π β₯ 1, it is possible to solve the 3^π-points problem with h(π) lines starting from any of the 3^π nodes, except from the central one. Finally, we cover 3Γ3Γ3 points with a tree of size 12.