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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. 6
1, 7, 14 (list; graph; refs; listen; history; text; internal format)



A generalization of the well-known "Nine Dots Problem".

Bounds for this problem, for n>3, are n^2 + ceiling((3*n^2 - 4*n + 2)/(2*(n - 1)) <= a(n) <= floor(3/2*n^2) + n - 1. The current proved upper bound for n = 4 is 23 and, as for n = 3, it is an "outside the box" solution, while the aforementioned upper bound refers to "inside the box" patterns. - Marco Ripà, Aug 07 2018


Table of n, a(n) for n=1..3.

M. Ripà, nxnx...xn Dots Puzzle

M. Ripà, The Rectangular Spiral Solution for the n1 X n2 X ... X nk Points Problem

M. Ripà, The rectangular spiral or the n1 X n2 X ... X nk Points Problem, Notes on Number Theory and Discrete Mathematics, 2014, 20(1), 59-71.

Marco Ripà, Solving the n_1 x n_2 x n_3 Points Problem for n_3 < 6, viXra.org.

Marco Ripà, The n X n X n Points Problem Optimal Solution

Wikipedia, Nine dots puzzle


For n=3, a(3)=14. You cannot touch (the centers of) the 27 dots using fewer than 14 straight lines, following the "Nine Dots Puzzle" basic rules.


Cf. A058992, A261547, A318165.

Sequence in context: A130594 A227173 A178998 * A188929 A258767 A169845

Adjacent sequences:  A225224 A225225 A225226 * A225228 A225229 A225230




Marco Ripà, May 02 2013


Entry revised by N. J. A. Sloane, May 02 2013



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Last modified October 15 18:11 EDT 2019. Contains 328037 sequences. (Running on oeis4.)