This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A225227 #131 May 15 2025 22:11:42 %S A225227 1,7,13 %N A225227 The n X n X n dots problem: minimal number of straight lines (connected at their endpoints) required to pass through n^3 dots arranged in an n X n X n grid, without exiting from the box [0, n] X [0, n] X [0, n]. %C A225227 A generalization of the well-known "Nine Dots Problem", where the regular axis-aligned bounding box (RAABB:=[0, n] X [0, n] X [0, n]) has been declared. %C A225227 From _Marco Ripà_, Aug 10 2020: (Start) %C A225227 In particular, if we loosen the constraint on the allowed AABB, covering paths characterized by a shorter link-length can be found, such as 5 <= a(2) <= 6, where the aforementioned upper bound has been discovered by Koki Goma in August 2021, providing the self-crossing covering path (0,0,0)-(2,2,0)-(1/2,1/2,3/2)-(2,-1,0)-(0,1,0)-(0,1,1)-(0,0,1). %C A225227 Moreover, the above pattern suggests different uncrossing covering paths of the same link-length, such as (1,0,0)-(0,0,0)-(2,2,2)-(1/2,-1,1/2)-(-1/2,1,3/2)-(1,1,0)-(1,1,0) and also the (self-crossing) covering path (1,0,0)-(0,0,0)-(0,1,0)-(3/2,1,3/2)-(1/2,-1,1/2)-(-1/2,1,3/2)-(1,1,0) which is entirely contained inside a box of 1.5 X 2 X 2 units^3 but which does not match the RAABB. (End) %H A225227 Marco Ripà, <a href="http://dx.doi.org/10.13140/RG.2.2.12199.57769/1">Solving the n_1 <= n_2 <= n_3 Points Problem for n_3 < 6</a>, ResearchGate, 2020. %H A225227 Marco Ripà, <a href="https://doi.org/10.14710/jfma.v3i2.8551">Solving the 106 years old 3^k points problem with the clockwise-algorithm</a>, Journal of Fundamental Mathematics and Applications, 2020, 3(2), 84-97. %H A225227 Marco Ripà, <a href="https://doi.org/10.14710/jfma.v4i2.12053">General uncrossing covering paths inside the Axis-Aligned Bounding Box</a>, Journal of Fundamental Mathematics and Applications, 2021, 4(2), 154-166. %H A225227 Marco Ripà, <a href="https://hal.science/hal-03841209">General conjecture on the optimal covering trails in a k-dimensional cubic lattice</a>, hal-03841209v3, 2023. %H A225227 Wikipedia, <a href="http://en.wikipedia.org/wiki/Thinking_outside_the_box#Nine_dots_puzzle">Nine dots puzzle</a> %F A225227 For any n >= 3, (n^3 - 1)/(n - 1) <= a(n) <= floor((3*n^2)/2) - floor((n - 1)/4) + floor((n + 1)/4) - floor((n + 2)/4) + floor(n/4) + n - 2. - _Marco Ripà_, Oct 25 2024 %e A225227 For n = 2, a(2) = 7. You cannot touch the 8 vertices of a cube using fewer than 7 straight lines and without exiting from the box [0, 2] X [0, 2] X [0, 2], following the "Nine Dots Puzzle" basic rules. %Y A225227 Cf. A058992, A261547, A318165, A363755. %K A225227 nonn,more,hard,bref %O A225227 1,2 %A A225227 _Marco Ripà_, May 02 2013 %E A225227 Entry revised by _N. J. A. Sloane_, May 02 2013 %E A225227 a(3) corrected by _Marco Ripà_, Jul 19 2020