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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

%I

%S 1,1,4,14,41

%N 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.

%C This is an n-dimensional generalization of the well-known "Nine Dots Problem".

%C Bounds for this problem, for n >= 5, are:

%C ceiling((3^n + n - 3)/2) <= a(n) <= 42*3^(n - 4) - 1.

%C 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.

%C Except for n < 2, the a(n) represent "outside the box" solutions.

%H M. Ripà, <a href="http://www.scribd.com/doc/138937268/Extended-9-Dots-Puzzle-to-nxnx-xn-Dots-General-Solving-Method">nxnx...xn Dots Puzzle</a>

%H M. Ripà, <a href="http://nntdm.net/volume-20-2014/number-1/59-71/">The rectangular spiral or the n1 X n2 X ... X nk Points Problem</a>, Notes on Number Theory and Discrete Mathematics, 2014, 20(1), 59-71.

%H M. Ripà, <a href="http://nntdm.net/volume-25-2019/number-2/68-75/">The 3 X 3 X ... X 3 Points Problem solution</a>, Notes on Number Theory and Discrete Mathematics, 2019, 25(2), 68-75.

%H Marco Ripà, <a href="/A261547/a261547_1.pdf">The n X n X n Points Problem Optimal Solution</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Thinking_outside_the_box#Nine_dots_puzzle">Nine dots puzzle</a>

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

%e 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.

%Y Cf. A058992, A225227.

%K nonn,more,hard

%O 0,3

%A _Marco Ripà_, Aug 24 2015

%E a(4) added by _Marco Ripà_, Aug 06 2018

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Last modified November 12 17:06 EST 2019. Contains 329058 sequences. (Running on oeis4.)