

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




OFFSET

1,2


COMMENTS

A generalization of the wellknown "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


LINKS

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), 5971.
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


EXAMPLE

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.


CROSSREFS

Cf. A058992, A261547, A318165.
Sequence in context: A130594 A227173 A178998 * A188929 A258767 A169845
Adjacent sequences: A225224 A225225 A225226 * A225228 A225229 A225230


KEYWORD

nonn,more,hard,bref


AUTHOR

Marco Ripà, May 02 2013


EXTENSIONS

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


STATUS

approved



