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Solving the $106$ years old $3^k$ points problem with the clockwise-algorithm

Marco Ripà

TL;DR

A general strategy that constructively produces minimum length covering trails, for any 𝑘 ∈ N−{0}, solving the NP-complete (3×3×⋯×3)-points problem inside a 3×3×⋯×3 hypercube.

Abstract

In this paper, we present the clockwise-algorithm that solves the extension in $k$-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 $k \in \mathbb{N}-\{0\}$, solving the NP-complete $(3 \times 3 \times \cdots \times 3)$-point problem inside $3 \times 3 \times \cdots \times 3$ hypercubes. In particular, using our algorithm, we explicitly draw different covering trails of minimal length $h(k)=\frac{3^k-1}{2}$, for $k=3, 4, 5$. Furthermore, we conjecture that, for every $k \geq 1$, it is possible to solve the $3^k$-point problem with $h(k)$ lines starting from any of the $3^k$ nodes, except from the central one. Finally, we cover a $3 \times 3 \times 3$ grid with a tree of size $12$.

Solving the $106$ years old $3^k$ points problem with the clockwise-algorithm

TL;DR

A general strategy that constructively produces minimum length covering trails, for any 𝑘 ∈ N−{0}, solving the NP-complete (3×3×⋯×3)-points problem inside a 3×3×⋯×3 hypercube.

Abstract

In this paper, we present the clockwise-algorithm that solves the extension in -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 , solving the NP-complete -point problem inside hypercubes. In particular, using our algorithm, we explicitly draw different covering trails of minimal length , for . Furthermore, we conjecture that, for every , it is possible to solve the -point problem with lines starting from any of the nodes, except from the central one. Finally, we cover a grid with a tree of size .
Paper Structure (6 sections, 3 theorems, 8 equations, 12 figures)

This paper contains 6 sections, 3 theorems, 8 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Solving the -point problem
  3. A tight lower bound
  4. The clockwise-algorithm
  5. Covering -points by trees
  6. Conclusion

Key Result

Theorem 2.1

For every positive integer $k$, $h(k) \geq \frac{3^k-1}{2}$.

Figures (12)

  • Figure 1: Solving the $3 \times 1$ puzzle inside the box ($3$ units of length), starting from one of the line segment endpoints. The puzzle is solvable with this $C(1)$ path starting from both the red points.
  • Figure 2: $C(2)$ is a path that consists of $h(2)=\frac{3^2-1}{2}$ lines. In order to solve the $3 \times 3$ puzzle with $4$ lines starting from one node of $G_2$, it is necessary to avoid starting from the central point of the grid.
  • Figure 3: $C(3)$ solves the $3 \times 3 \times 3$ puzzle inside a $3 \times 3 \times 3$ box ($27$ cubic units of volume), starting from face-centers or vertices, thanks to the clockwise-algorithm.
  • Figure 4: Solving the $3 \times 3 \times 3$ puzzle inside a $3 \times 3 \times 3$ box ($27$ cubic units of volume), starting from edges or vertices.
  • Figure 5: Solving the $3 \times 3 \times 3$ puzzle inside a $3 \times 3 \times 4$ box ($36$ cubic units of volume).
  • ...and 7 more figures

Theorems & Definitions (8)

  • Theorem 2.1
  • proof
  • Definition 2.1
  • Definition 3.1
  • Lemma 3.1
  • proof
  • Theorem 3.1
  • proof