The n × n × n Points Problem optimal solution

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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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 n3 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 hu(n) − hl(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

References

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  5. Ripà, M. (2014) The Rectangular Spiral or the n1 × n2 × … × nk Points Problem. Notes on Number Theory and Discrete Mathematics, 20(1), 59–71.
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Related papers

  1. Ripà, Marco. “The Rectangular Spiral or the n1 × n2 × … × nk Points Problem.” Notes on Number Theory and Discrete Mathematics 20, no. 1 (2014): 59-71.

Cite this paper

APA

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

Chicago

Ripà, Marco. “The n × n × n Points Problem Optimal Solution.” Notes on Number Theory and Discrete Mathematics 22, no. 2 (2016): 36-43.

MLA

Ripà, Marco. “The n × n × n Points Problem Optimal Solution.” Notes on Number Theory and Discrete Mathematics 22.2 (2016): 36-43. Print.

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