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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
1, 1, 4, 14, 41 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

This is an n-dimensional generalization of the well-known "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), 59-71.

M. Ripà, The 3 X 3 X ... X 3 Points Problem solution, Notes on Number Theory and Discrete Mathematics, 2019, 25(2), 68-75.

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

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Last modified October 21 11:42 EDT 2019. Contains 328296 sequences. (Running on oeis4.)