Solving the $106$ years old $3^k$ points problem with the clockwise-algorithm
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911760432234496 |
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| author | Ripà, Marco |
| author_facet | Ripà, Marco |
| contents | 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 $12$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_10163 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Solving the $106$ years old $3^k$ points problem with the clockwise-algorithm Ripà, Marco General Mathematics 05C85 (Primary) 05C57, 68R10 (Secondary) 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 $12$. |
| title | Solving the $106$ years old $3^k$ points problem with the clockwise-algorithm |
| topic | General Mathematics 05C85 (Primary) 05C57, 68R10 (Secondary) |
| url | https://arxiv.org/abs/2401.10163 |