Marco Ripà

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 22, 2016, Number 2, Pages 36—43

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## Details

### Authors and affiliations

Marco Ripà

*Economics − Institutions and Finance
Roma Tre University, Rome, Italy
*

### Abstract

We provide an optimal strategy to solve the *n* × *n* × *n* points problem inside the box, considering only 90° turns, and at the same time a pattern able to drastically lower down the known upper bound. We use a very simple spiral frame, especially if compared to the previous plane by plane approach that significantly reduces the number of straight lines connected at their end-points necessary to join all the *n*^{3} dots. In the end, we combine the square spiral frame with the rectangular spiral pattern in the most profitable way, in order to minimize the difference *h _{u}*(

*n*) −

*h*(

_{l}*n*) between the upper and the lower bound, proving that it is ≤ 0.5 ∙

*n*∙ (

*n*+ 3), for any

*n*> 1.

### Keywords

- Topology
- Inside the box
- Nine dots
- Straight line
- Outside the box
- Upper bound
- Graph theory
- Three-dimensional
- Segment
- Point

### AMS Classification

- Primary: 91A44
- Secondary: 37F20, 91A46

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## Related papers

- Ripà, Marco. “The Rectangular Spiral or the
*n*_{1}×*n*_{2}× … ×*n*Points Problem.” Notes on Number Theory and Discrete Mathematics 20, no. 1 (2014): 59-71._{k}

## Cite this paper

Ripà, M. (2016). The *n* × *n* × *n* Points Problem optimal solution. Notes on Number Theory and Discrete Mathematics, 22(2), 36-43.