

A261547


The 3 X 3 X ... X 3 dots problem (3, n times): minimal number of straight lines (connected at their endpoints) required to pass through 3^n dots arranged in a 3 X 3 X ... X 3 grid.


4




OFFSET

0,3


COMMENTS

This is an ndimensional generalization of the wellknown "Nine Dots Problem".
Bounds for this problem, for n >= 5, are:
ceiling((3^n + n  3)/2) <= a(n) <= 42*3^(n  4)  1.
a(5) is 123, 124 or 125, since 123 is the lower bound calculated as above and 125 is the best solution found as of Aug 06 2018.
Except for n < 2, the a(n) represent "outside the box" solutions.


LINKS

Table of n, a(n) for n=0..4.
M. Ripà, nxnx...xn Dots Puzzle
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.
M. Ripà, The 3 X 3 X ... X 3 Points Problem solution, Notes on Number Theory and Discrete Mathematics, 2019, 25(2), 6875.
Marco Ripà, The n X n X n Points Problem Optimal Solution
Wikipedia, Nine dots puzzle


FORMULA

a(n) = ceiling((3^n + n  3)/2), for any n >= 2 (conjectured).


EXAMPLE

For n=4, a(4) = 41. You cannot touch (the centers of) the 3^4 = 81 dots using fewer than 41 straight lines, following the "Nine Dots Puzzle" basic rules.


CROSSREFS

Cf. A058992, A225227.
Sequence in context: A196480 A326008 A196713 * A237853 A132357 A262875
Adjacent sequences: A261544 A261545 A261546 * A261548 A261549 A261550


KEYWORD

nonn,more,hard


AUTHOR

Marco Ripà, Aug 24 2015


EXTENSIONS

a(4) added by Marco Ripà, Aug 06 2018


STATUS

approved



