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 A250000 Peaceable coexisting armies of queens: the maximum number m such that m white queens and m black queens can coexist on an n X n chessboard without attacking each other. 7

%I

%S 0,0,1,2,4,5,7,9,12,14,17,21,24,28,32

%N Peaceable coexisting armies of queens: the maximum number m such that m white queens and m black queens can coexist on an n X n chessboard without attacking each other.

%C Two solutions are regarded as equivalent if one can be obtained from the other by rotations, reflections, or interchanging the colors (a group of order 16). For n=1,2,3,4,5 the number of solutions is respectively 1,1,1,8,3. What is the number of solutions for n=6? - _Rob Pratt_ and _N. J. A. Sloane_, Jul 29 2015 Answer: 34, see A260680. - _Luca Petrone_, Mar 11 2016

%C From _Bob Selcoe_, Feb 09 2015: (Start)

%C For n = 4m, a generalized quasi-symmetric pattern of queen arrangements exists showing that a(n) >= ceiling((n+4)(n-2)/8) + floor((n-4)^2/64) == (m+1)(2m-1) + A002620(m-1).

%C For n = 4m-1, a slightly different pattern exists showing that a(n) >= m(2m-1) + A002620(m).

%C Both patterns are difficult to describe easily: as m increases, each depends on slight variations to standard arrangements of opposing queens in "blocks" on opposite corners of the chessboard, plus an additional block arrangement which is "forced" by virtue of the corner blocks. See below for examples of boards for n = {12,16,20,24} that show the pattern for n = 4m.

%C For all n >= 16, a(n) > ceiling(9n^2/64), which is the best asymptotic lower bound presently known.

%C It is likely that similar "block" patterns exist for n = {4m+1, 4m+2}.

%C (End)

%C From _Daniel Forgues_, Feb 27 2015: (Start)

%C Observation: Suppose n >= 2 (omitting the 1 X 1 board):

%C for n = 2k, k >= 1, the values of a(n) are

%C {0, 2, 5, 9, 14, 21, ...}

%C for n = 2k+1, k >= 1, the values are

%C {1, 4, 7, 12, 17, 24, ...}

%C and then a(2k+1) - a(2k), k >= 1, yields

%C {1, 2, 2, 3, 3, 3, ...}.

%C (End)

%C From _Peter Karpov_, Apr 03 2016: (Start)

%C It appears that the maximal asymptotic density of one color for a configuration consisting of two pentagonal regions and their antipodal counterparts (with respect to the center) is 7/48.

%C Empirical observation: except for two small cases (n = 5, 9), the known values are given by a(n) = floor(7*n^2/48) (see A286283).

%C (End)

%C On a board with a maximal set of coexisting armies of queens, is every cell not occupied by a queen attacked by at least one queen of either color? - _David A. Corneth_, Oct 16 2018

%D Knuth, Donald E., Satisfiability, Fascicle 6, volume 4 of The Art of Computer Programming. Addison-Wesley, 2015, page 180, Problem 488; see also pages 282-283.

%H Bosch, Robert A., <a href="http://www.mathopt.org/Optima-Issues/optima62.pdf">Peaceably coexisting armies of queens</a>, Optima (Newsletter of the Mathematical Programming Society) 62.6-9 (1999): 271.

%H Bosch, Robert A., <a href="/A250000/a250000_1.png">Peaceably coexisting armies of queens</a>, Optima (Newsletter of the Mathematical Programming Society) 62.6-9 (1999): 271. [Scanned copy of page containing the problem, with permission]

%H Bosch, Robert A., <a href="http://www.mathopt.org/Optima-Issues/optima64.pdf">Armies of Queens, Revisited</a>, Optima (Newsletter of the Mathematical Programming Society) 64 (2000): 15.

%H Bosch, Robert A., <a href="/A250000/a250000_2.png">Armies of Queens, Revisited</a>, Optima (Newsletter of the Mathematical Programming Society) 64 (2000): 15. [Scanned copy of section of page containing the article, with permission]

%H Daniel M. Kane, <a href="https://arxiv.org/abs/1703.04538">Asymptotic Results for the Queen Packing Problem</a>, arXiv:1703.04538 [math.CO], Mar 16 2017

%H Michael De Vlieger, <a href="/A250000/a250000_MDV-1.png">"Peace to the Max" T-shirt illustrating a(11)=17</a>

%H Michael De Vlieger, <a href="/A250000/a250000-MDV-2.jpg">"Peace to the Max" T-shirt illustrating a(11)=17</a> [Version with no background, suitable for printing]

%H Michael De Vlieger, <a href="/A250000/a250000-MDV-2-Boards.pdf">Graphic illustrations of other solutions</a>

%H Peter Karpov, <a href="http://inversed.ru/InvMem.htm#InvMem_22">InvMem, see Item 22</a>

%H Peter Karpov, <a href="/A250000/a250000_3.png">InvMem, see Item 22</a> [Scanned copy of Item 22, with permission.]

%H Peter Karpov, <a href="/A250000/a250000.png">An asymptotic configuration with density 7/48</a> (Not known to be optimal.)

%H Steven Prestwich and J. Christopher Beck, <a href="http://tidel.mie.utoronto.ca/pubs/pseudo.pdf">Exploiting Dominance in Three Symmetric Problems</a>, in Proceedings Fourth International Workshop on Symmetry and Constraint Satisfaction Problems (SymCon'04), (2004) pp. 63-70; also available from http://zeynep.web.cs.unibo.it/SymCon04/proceedings.html

%H N. J. A. Sloane, <a href="/A195264/a195264.pdf">Confessions of a Sequence Addict (AofA2017)</a>, slides of invited talk given at AofA 2017, Jun 19 2017, Princeton. Mentions this sequence.

%H Barbara M. Smith, Karen E. Petrie, and Ian P. Gent, <a href="http://ipg.host.cs.st-andrews.ac.uk/papers/spgW9.pdf">Models and symmetry breaking for 'Peaceable armies of queens'</a>, Lecture Notes in Computer Science 3011 (2004), 271-286. [Version on St Andrews web site, 16 pages.]

%H Barbara M. Smith, Karen E. Petrie, and Ian P. Gent, <a href="https://www.researchgate.net/profile/Karen_Petrie/publication/221353569_Models_and_Symmetry_Breaking_for_%27Peaceable_Armies_of_Queens%27/links/0fcfd5111456cdbaa6000000.pdf">Models and symmetry breaking for 'Peaceable armies of queens'</a>, Lecture Notes in Computer Science 3011 (2004), 271-286. [Version on ResearchGate web site, 17 pages]

%H Barbara M. Smith, Karen E. Petrie, and Ian P. Gent, <a href="/A250000/a250000.pdf">Models and symmetry breaking for 'Peaceable armies of queens'</a>, Lecture Notes in Computer Science 3011 (2004), 271-286. [Cached copy, from ResearchGate]

%H Barbara M. Smith, Karen E. Petrie, and Ian P. Gent, <a href="/A245783/a245783.pdf">Equal sized armies of queens on an 11x11 board</a> (Fig. 2 from the reference)

%H Paul Tabatabai, <a href="/A250000/a250000.txt">Three illustrations for a(14) = 28</a>

%F There is an asymptotic lower bound of (9/64)*n^2. But see Comments for a better lower bound.

%e Some examples, in increasing order of size of board.

%e n=3: There is a unique solution (up to obvious symmetries):

%e +-------+

%e | W . . |

%e | . . . |

%e | . B . |

%e +-------+

%e n=4: From _Rob Pratt_, Jul 29 2015. There are eight inequivalent solutions (up to obvious symmetries):

%e +---------+ +---------+ +---------+ +---------+

%e | . . B . | | . B . . | | . B . . | | . . . . |

%e | . . . . | | . B . . | | . . . B | | . B . B |

%e | . . . B | | . . . . | | . . . . | | . . . . |

%e | W W . . | | W . W . | | W . W . | | W . W . |

%e +---------+ +---------+ +---------+ +---------+

%e .

%e +---------+ +---------+ +---------+ +---------+

%e | . B B . | | . . B . | | . . . W | | . . B . |

%e | . . . . | | . B . . | | . B . . | | . . . B |

%e | . . . . | | . . . W | | . . B . | | . W . . |

%e | W . . W | | W . . . | | W . . . | | W . . . |

%e +---------+ +---------+ +---------+ +---------+

%e n=5: One of the three solutions for n=5 puts one set of four queens in the corners and the other set in the squares a knight's move away, as follows:

%e +-----------+

%e | W . . . W |

%e | . . B . . |

%e | . B . B . |

%e | . . B . . |

%e | W . . . W |

%e +-----------+

%e There are two other solutions (up to symmetry) for n=5 (found by _Rob Pratt_, circa Sep 2014):

%e +-----------+

%e | . . B . B |

%e | W . . . . |

%e | . . B . B |

%e | W . . . . |

%e | . W . W . |

%e +-----------+

%e .

%e +-----------+

%e | . W . W . |

%e | . . W . . |

%e | B . . . B |

%e | . . W . . |

%e | B . . . B |

%e +-----------+

%e n=6: A solution for n=6:

%e +-------------+

%e | . W W . . . |

%e | . . W . . W |

%e | . . . . . W |

%e | . . . . . . |

%e | B . . . B . |

%e | B . . B B . |

%e +-------------+

%e n=8: a(8) = 9:

%e +-----------------+

%e | . . W W . . . . |

%e | . . W W . . . W |

%e | . . W . . . W W |

%e | . . . . . . W W |

%e | . B . . . . . . |

%e | B B . . . . . . |

%e | B B . . . B . . |

%e | B . . . B B . . | - _Rob Pratt_, Jul 29 2015

%e +-----------------+

%e n=9: A solution from _Bob Selcoe_, Feb 07 2015:

%e +-------------------+

%e | . B . B . B . B . |

%e | . . B . . . B . . |

%e | W . . . W . . . W |

%e | . . B . . . B . . |

%e | W . . . W . . . W |

%e | . . B . . . B . . |

%e | W . . . W . . . W |

%e | . . B . . . B . . |

%e | W . . . W . . . W |

%e +-------------------+

%e A solution for n=12 (from Prestwich/Beck paper):

%e +-------------------------+

%e | . . . B B B . . . . . B |

%e | . . . B B B . . . . B . |

%e | . . . B B B . . . B . B |

%e | . . . . B . . . . . B B |

%e | . . . . . . . . . B B B |

%e | . . . . . . . . . B B . |

%e | . . W . . . W . . . . . |

%e | . W W . . . . . . . . . |

%e | W W W . . . . . W . . . |

%e | W W . . . . . W W . . . |

%e | W . W . . . W W W . . . |

%e | . W . . . . W W W . . . |

%e +-------------------------+

%e A solution for n=13 (from Prestwich/Beck paper):

%e +---------------------------+

%e | B . . . B . B . . . B . B |

%e | . . W . . . . . W . . . . |

%e | . W . W . W . W . W . W . |

%e | . . W . . . . . W . . . . |

%e | B . . . B . B . . . B . B |

%e | . . W . . . . . W . . . . |

%e | B . . . B . B . . . B . B |

%e | . . W . . . . . W . . . . |

%e | . W . W . W . W . W . W . |

%e | . . W . . . . . W . . . . |

%e | B . . . B . B . . . B . B |

%e | . . W . . . . . W . . . . |

%e | B . . . B . B . . . B . B |

%e +---------------------------+

%e From _Bob Selcoe_, Feb 07 2015 (Start):

%e An alternative solution for n=13:

%e +---------------------------+

%e | . B . B . B . B . B . B . |

%e | . . B . . . B . . . B . . |

%e | W . . . W . . . W . . . W |

%e | . . B . . . B . . . B . . |

%e | W . . . W . . . W . . . W |

%e | . . B . . . B . . . B . . |

%e | W . . . W . . . W . . . W |

%e | . . B . . . B . . . B . . |

%e | W . . . W . . . W . . . W |

%e | . . B . . . B . . . B . . |

%e | W . . . W . . . W . . . W |

%e | . . B . . . B . . . B . . |

%e | W . . . W . . . W . . . W |

%e +---------------------------+

%e n=15, a fully symmetrical optimal configuration from _Paul Tabatabai_, Oct 16 2018:

%e +-------------------------------+

%e | B . B . B . . . . . B . B . B |

%e | . . . . . . W W W . . . . . . |

%e | B . B . B . . . . . B . B . B |

%e | . . . . . . W . W . . . . . . |

%e | B . B . . . . . . . . . B . B |

%e | . . . . . . W . W . . . . . . |

%e | . W . W . W . W . W . W . W . |

%e | . W . . . . W . W . . . . W . |

%e | . W . W . W . W . W . W . W . |

%e | . . . . . . W . W . . . . . . |

%e | B . B . . . . . . . . . B . B |

%e | . . . . . . W . W . . . . . . |

%e | B . B . B . . . . . B . B . B |

%e | . . . . . . W W W . . . . . . |

%e | B . B . B . . . . . B . B . B |

%e +-------------------------------+

%e n=17: A 42-queen arrangement (the best presently known) for n=17, from _Rob Pratt_, Feb 07 2014:

%e +-----------------------------------+

%e | . . . . W W W W W . . . . . . . . |

%e | . . . . W W W W W . . . . . . . . |

%e | . . . . W W W W W . . . . . . . W |

%e | . . . . W W W W . . . . . . . W W |

%e | . . . . W W W . . . . . . . W W W |

%e | . . . . . W . . . . . . . W W W W |

%e | . . . . . . . . . . . . . W W W W |

%e | . . . . . . . . . . . . . W W W . |

%e | . . . . . . . . . . . . . W W . . |

%e | . . B B . . . . . . . . . . . . . |

%e | . B B B . . . . . . . . . . . . . |

%e | B B B B . . . . . . . . . . . . . |

%e | B B B B . . . . . . . B . . . . . |

%e | B B B B . . . . . . B B B . . . . |

%e | B B B B . . . . . B B B B . . . . |

%e | B B B . . . . . . B B B B . . . . |

%e | B B . . . . . . . B B B B . . . . |

%e +-----------------------------------+

%e From _Bob Selcoe_, Feb 09 2015 (Start):

%e Two alternative 42-queen arrangements for n=17 (inspired by _Rob Pratt_). Other arrangements exist.

%e Alternative 1:

%e +-----------------------------------+

%e | . . . . . W W W W W . . . . . . . |

%e | . . . . . W W W W W . . . . . . . |

%e | . . . . . W W W W W . . . . . . W |

%e | . . . . . W W W W . . . . . . W W |

%e | . . . . . W W W . . . . . . W W W |

%e | . . . . . . W . . . . . . W W W W |

%e | . . . . . . . . . . . . . W W W W |

%e | . . . . . . . . . . . . . W W W . |

%e | . . . . . . . . . . . . . W W . . |

%e | . . . B B . . . . . . . . . . . . |

%e | . . B B B . . . . . . . . . . . . |

%e | . B B B B . . . . . . . . . . . . |

%e | B B B B B . . . . . . B B . . . . |

%e | B B B B B . . . . . B B B . . . . |

%e | B B B B . . . . . . B B B . . . . |

%e | B B B . . . . . . . B B B . . . . |

%e | B B . . . . . . . . B B B . . . . |

%e +-----------------------------------+

%e Alternative 2:

%e +-----------------------------------+

%e | . . . . W W W W . . . . . . . . W |

%e | . . . . W W W W . . . . . . . W W |

%e | . . . . W W W W . . . . . . W W W |

%e | . . . . W W W W . . . . . W W W W |

%e | . . . . . W W . . . . . . W W W W |

%e | . . . . . . . . . . . . . W W W W |

%e | . . . . . . . . . . . . . W W W . |

%e | . . . . . . . . . . . . . W W . . |

%e | . . . . . . . . . . . . . W . . . |

%e | . . B B . . . . . . . . . . . . . |

%e | . B B B . . . . . . . . . . . . . |

%e | B B B B . . . . . . . B . . . . . |

%e | B B B B . . . . . . B B B . . . . |

%e | B B B . . . . . . B B B B . . . . |

%e | B B . . . . . . B B B B B . . . . |

%e | B . . . . . . . B B B B B . . . . |

%e | . . . . . . . . B B B B B . . . . |

%e +-----------------------------------+

%e Example of an alternative n=20, 58-queen arrangement with "cracked" blocks from _Bob Selcoe_, May 23 2017:

%e +-----------------------------------------+

%e | . . . . . W W W W W . . . . . . . . W . |

%e | . . . . . W W W W W . . . . . . . W . W |

%e | . . . . . W W W W W . . . . . . W . W W |

%e | . . . . . W W W W W . . . . . W . W W W |

%e | . . . . . W W W W . . . . . . . W W W W |

%e | . . . . . W W W . . . . . . . W W W W W |

%e | . . . . . . W . . . . . . . . W W W W . |

%e | . . . . . . . . . . . . . . . W W W . . |

%e | . . . . . . . . . . . . . . . W W . . . |

%e | . . . . . . . . . W . . . . . W . . . . |

%e | . . . B B . . . . . . . . . . . . . . . |

%e | . . B B B . . . . . . . . . . . . . . . |

%e | . B B B B . . . . . . . . . . . . . . . |

%e | B B B B B . . . . . . . B . . . . . . . |

%e | B B B B . . . . . . . B B B . . . . . . |

%e | B B B . B . . . . . B B B B B . . . . . |

%e | B B . B . . . . . . B B B B B . . . . . |

%e | B . B . . . . . . . B B B B B . . . . . |

%e | . B . . . . . . . . B B B B B . . . . . |

%e | B . . . . . . . . . B B B B B . . . . . |

%e +-----------------------------------------+

%e Pattern for n = 4m; four chessboards total.

%e Board 1: n=12, a(12)=21:

%e +-------------------------+

%e | . . . W W W . . . . . . |

%e | . . . W W W . . . . . W |

%e | . . . W W W . . . . W W |

%e | . . . . W . . . . W W W |

%e | . . . . . . . . . W W W |

%e | . . . . . . . . . W W . |

%e | . . B . . . . . . . . . |

%e | . B B . . . . . . . . . |

%e | B B B . . . . . B . . . |

%e | B B B . . . . B B . . . |

%e | B B . . . . B B B . . . |

%e | B . . . . . B B B . . . |

%e +-------------------------+

%e Board 2: n=16, 37-queen arrangement:

%e +---------------------------------+

%e | . . . . W W W W . . . . . . . . |

%e | . . . . W W W W . . . . . . . W |

%e | . . . . W W W W . . . . . . W W |

%e | . . . . W W W W . . . . . W W W |

%e | . . . . . W W . . . . . W W W W |

%e | . . . . . . . . . . . . W W W W |

%e | . . . . . . . . . . . . W W W . |

%e | . . . . . . . . . . . . W W . . |

%e | . . . B . . . . . . . . . . . . |

%e | . . B B . . . . . . . . . . . . |

%e | . B B B . . . . . . . . . . . . |

%e | B B B B . . . . . . B B . . . . |

%e | B B B B . . . . . B B B . . . . |

%e | B B B . . . . . B B B B . . . . |

%e | B B . . . . . . B B B B . . . . |

%e | B . . . . . . . B B B B . . . . |

%e +---------------------------------+

%e Board 3: n=20, 58-queen arrangement:

%e +-----------------------------------------+

%e | . . . . . W W W W W . . . . . . . . . . |

%e | . . . . . W W W W W . . . . . . . . . W |

%e | . . . . . W W W W W . . . . . . . . W W |

%e | . . . . . W W W W W . . . . . . . W W W |

%e | . . . . . W W W W W . . . . . . W W W W |

%e | . . . . . . W W W . . . . . . W W W W W |

%e | . . . . . . . W . . . . . . . W W W W W |

%e | . . . . . . . . . . . . . . . W W W W . |

%e | . . . . . . . . . . . . . . . W W W . . |

%e | . . . . . . . . . . . . . . . W W . . . |

%e | . . . . B . . . . . . . . . . . . . . . |

%e | . . . B B . . . . . . . . . . . . . . . |

%e | . . B B B . . . . . . . . . . . . . . . |

%e | . B B B B . . . . . . . . B . . . . . . |

%e | B B B B B . . . . . . . B B B . . . . . |

%e | B B B B B . . . . . . B B B B . . . . . |

%e | B B B B . . . . . . B B B B B . . . . . |

%e | B B B . . . . . . . B B B B B . . . . . |

%e | B B . . . . . . . . B B B B B . . . . . |

%e | B . . . . . . . . . B B B B B . . . . . |

%e +-----------------------------------------+

%e Board 4: n=24, 83-queen arrangement:

%e +-------------------------------------------------+

%e | . . . . . . W W W W W W . . . . . . . . . . . . |

%e | . . . . . . W W W W W W . . . . . . . . . . . W |

%e | . . . . . . W W W W W W . . . . . . . . . . W W |

%e | . . . . . . W W W W W W . . . . . . . . . W W W |

%e | . . . . . . W W W W W W . . . . . . . . W W W W |

%e | . . . . . . W W W W W W . . . . . . . W W W W W |

%e | . . . . . . . W W W W . . . . . . . W W W W W W |

%e | . . . . . . . . W W . . . . . . . . W W W W W W |

%e | . . . . . . . . . . . . . . . . . . W W W W W . |

%e | . . . . . . . . . . . . . . . . . . W W W W . . |

%e | . . . . . . . . . . . . . . . . . . W W W . . . |

%e | . . . . . . . . . . . . . . . . . . W W . . . . |

%e | . . . . . B . . . . . . . . . . . . . . . . . . |

%e | . . . . B B . . . . . . . . . . . . . . . . . . |

%e | . . . B B B . . . . . . . . . . . . . . . . . . |

%e | . . B B B B . . . . . . . . . . . . . . . . . . |

%e | . B B B B B . . . . . . . . . B B . . . . . . . |

%e | B B B B B B . . . . . . . . B B B B . . . . . . |

%e | B B B B B B . . . . . . . B B B B B . . . . . . |

%e | B B B B B . . . . . . . B B B B B B . . . . . . |

%e | B B B B . . . . . . . . B B B B B B . . . . . . |

%e | B B B . . . . . . . . . B B B B B B . . . . . . |

%e | B B . . . . . . . . . . B B B B B B . . . . . . |

%e | B . . . . . . . . . . . B B B B B B . . . . . . |

%e +-------------------------------------------------+

%e (End)

%e Example of an alternative n=20, 58-queen arrangement with "cracked" blocks from _Bob Selcoe_, May 23 2017:

%e +-----------------------------------------+

%e | . . . . . W W W W W . . . . . . . . W . |

%e | . . . . . W W W W W . . . . . . . W . W |

%e | . . . . . W W W W W . . . . . . W . W W |

%e | . . . . . W W W W W . . . . . W . W W W |

%e | . . . . . W W W W . . . . . . . W W W W |

%e | . . . . . W W W . . . . . . . W W W W W |

%e | . . . . . . W . . . . . . . . W W W W . |

%e | . . . . . . . . . . . . . . . W W W . . |

%e | . . . . . . . . . . . . . . . W W . . . |

%e | . . . . . . . . . W . . . . . W . . . . |

%e | . . . B B . . . . . . . . . . . . . . . |

%e | . . B B B . . . . . . . . . . . . . . . |

%e | . B B B B . . . . . . . . . . . . . . . |

%e | B B B B B . . . . . . . B . . . . . . . |

%e | B B B B . . . . . . . B B B . . . . . . |

%e | B B B . B . . . . . B B B B B . . . . . |

%e | B B . B . . . . . . B B B B B . . . . . |

%e | B . B . . . . . . . B B B B B . . . . . |

%e | . B . . . . . . . . B B B B B . . . . . |

%e | B . . . . . . . . . B B B B B . . . . . |

%e +-----------------------------------------+

%e ------------------------

%e n=24, 84-queen arrangement (variation on n=4m pattern above; for m>=6, the new pattern is generalizable as an improvement over the previously shown pattern and can be used to achieve arrangements showing that a(n) >= floor(7*n^2/48), as in _Peter Karpov_'s Apr 03 2016 observation). - _Bob Selcoe_, Apr 14 2016

%e +-------------------------------------------------+

%e | . . . . . . W W W W W W . . . . . . . . . . . . |

%e | . . . . . . W W W W W W . . . . . . . . . . . W |

%e | . . . . . . W W W W W W . . . . . . . . . . W W |

%e | . . . . . . W W W W W W . . . . . . . . . W W W |

%e | . . . . . . W W W W W W . . . . . . . . W W W W |

%e | . . . . . . W W W W W . . . . . . . . W W W W W |

%e | . . . . . . . W W W . . . . . . . . W W W W W W |

%e | . . . . . . . . W . . . . . . . . . W W W W W W |

%e | . . . . . . . . . . . . . . . . . . W W W W W W |

%e | . . . . . . . . . . . . . . . . . . W W W W W . |

%e | . . . . . . . . . . . . . . . . . . W W W W . . |

%e | . . . . . . . . . . . . . . . . . . W W W . . . |

%e | . . . . B B . . . . . . . . . . . . . . . . . . |

%e | . . . B B B . . . . . . . . . . . . . . . . . . |

%e | . . B B B B . . . . . . . . . . . . . . . . . . |

%e | . B B B B B . . . . . . . . . . . . . . . . . . |

%e | B B B B B B . . . . . . . . . . B . . . . . . . |

%e | B B B B B B . . . . . . . . . B B B . . . . . . |

%e | B B B B B B . . . . . . . . B B B B . . . . . . |

%e | B B B B B . . . . . . . . B B B B B . . . . . . |

%e | B B B B . . . . . . . . B B B B B B . . . . . . |

%e | B B B . . . . . . . . . B B B B B B . . . . . . |

%e | B B . . . . . . . . . . B B B B B B . . . . . . |

%e | B . . . . . . . . . . . B B B B B B . . . . . . |

%e +-------------------------------------------------+

%Y A260680 gives number of solutions.

%Y Cf. A002620, A274947, A274948, A286283 (lower bound).

%Y See A000170, A002562, A319284, etc., for the classic non-attacking queens problem.

%K nonn,nice,more

%O 1,4

%A _Don Knuth_, Aug 01 2014

%E Uniqueness of n = 5 example corrected by _Rob Pratt_, Nov 30 2014

%E a(12)-a(13) obtained from Prestwich/Beck paper by _Rob Pratt_, Nov 30 2014

%E More examples from _Rob Pratt_, Dec 01 2014

%E a(1)-a(13) confirmed and bounds added for n = 14 to 20 obtained via integer linear programming by _Rob Pratt_, Dec 01 2014

%E 28 <= a(14) <= 43, 32 <= a(15) <= 53, 37 <= a(16) <= 64, 42 <= a(17) <= 72, 47 <= a(18) <= 81, 52 <= a(19) <= 90, 58 <= a(20) <= 100. - _Rob Pratt_, Dec 01 2014

%E Bounds obtained by simulated annealing: a(21) >= 64, a(22) >= 70, a(23) >= 77, a(24) >= 84. - _Peter Karpov_, Apr 03 2016

%E a(14)-a(15) from _Paul Tabatabai_ using integer programming, Oct 16 2018

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Last modified November 17 00:14 EST 2018. Contains 317275 sequences. (Running on oeis4.)