Editing Eight queens puzzle (section)

==Related problems==
;Higher dimensions
:Find the number of non-attacking queens that can be placed in a ''d''-dimensional chess {{boardgloss|gamespace|space}} of size ''n''.  More than ''n'' queens can be placed in some higher dimensions (the smallest example is four non-attacking queens in a 3×3×3 chess space), and it is in fact known that for any ''k'', there are higher dimensions where ''n''<sup>''k''</sup> queens do not suffice to attack all spaces.<ref>J. Barr and S. Rao (2006), The n-Queens Problem in Higher Dimensions, [[Elemente der Mathematik]], vol 61 (4), pp. 133–137.</ref><ref>{{cite web | url= http://queens.lyndenlea.info/beyond2d.php | title= Queens On A Chessboard – Beyond The 2nd Dimension | access-date=2020-01-27 | author= Martin S. Pearson | format= php | language= en}}</ref>
;Using pieces other than queens
:On an 8×8 board one can place 32 [[knight (chess)|knight]]s, or 14 [[bishop (chess)|bishop]]s, 16 [[king (chess)|king]]s or 8 [[rook (chess)|rook]]s, so that no two pieces attack each other. In the case of knights, an easy solution is to place one on each square of a given color, since they move only to the opposite color. The solution is also easy for rooks and kings. Sixteen kings can be placed on the board by dividing it into 2-by-2 squares and placing the kings at equivalent points on each square.  Placements of ''n'' rooks on an ''n''×''n'' board are in direct correspondence with order-''n'' [[permutation matrices]].
;Chess variations
:Related problems can be asked for [[chess variations]] such as [[shogi]]. For instance, the ''n''+''k'' dragon kings problem asks to place ''k'' [[Shogi#equipment|shogi pawns]] and ''n''+''k'' mutually nonattacking [[Shogi#equipment|dragon kings]] on an ''n''×''n'' shogi board.<ref>{{Cite journal|last=Chatham|first=Doug|date=1 December 2018|title=Reflections on the n +k dragon kings problem|journal=Recreational Mathematics Magazine|language=en|volume=5|issue=10|pages=39–55|doi=10.2478/rmm-2018-0007|doi-access=free}}</ref>
;Nonstandard boards
:[[George Pólya|Pólya]] studied the ''n'' queens problem on a [[torus|toroidal]] ("donut-shaped") board and showed that there is a solution on an ''n''×''n'' board if and only if ''n'' is not divisible by 2 or 3.<ref>G. Pólya, Uber die "doppelt-periodischen" Losungen des n-Damen-Problems, George Pólya:    Collected papers Vol. IV, G-C. Rota, ed., MIT Press, Cambridge, London, 1984, pp.&nbsp;237–247
</ref>
;Domination
:Given an ''n''×''n'' board, the '''domination number''' is the minimum number of queens (or other pieces) needed to attack or occupy every square. For ''n'' = 8 the queen's domination number is 5.<ref>{{citation|last1=Burger|first1=A. P.|last2=Cockayne|first2=E. J.|last3=Mynhardt|first3=C. M.|author3-link=Kieka Mynhardt|doi=10.1016/0012-365X(95)00327-S|issue=1–3|journal=Discrete Mathematics|mr=1428557|pages=47–66|title=Domination and irredundance in the queens' graph|volume=163|year=1997|hdl=1828/2670|hdl-access=free}}</ref><ref>{{citation|last=Weakley|first=William D.|editor1-last=Gera|editor1-first=Ralucca|editor1-link=Ralucca Gera|editor2-last=Haynes|editor2-first=Teresa W.|editor2-link=Teresa W. Haynes|editor3-last=Hedetniemi|editor3-first=Stephen T.|contribution=Queens around the world in twenty-five years|doi=10.1007/978-3-319-97686-0_5|location=Cham|mr=3889146|pages=43–54|publisher=Springer|series=Problem Books in Mathematics|title=Graph Theory: Favorite Conjectures and Open Problems – 2|year=2018|isbn=978-3-319-97684-6 }}</ref>
;Queens and other pieces
:Variants include mixing queens with other pieces; for example, placing ''m'' queens and ''m'' knights on an ''n''×''n'' board so that no piece attacks another<ref>{{Cite web |url=http://www.vector.org.uk/archive/v213/hui213.htm |title=Queens and knights problem |access-date=20 September 2005 |archive-url=https://web.archive.org/web/20051016004909/http://www.vector.org.uk/archive/v213/hui213.htm |archive-date=16 October 2005 |url-status=dead }}</ref> or placing queens and pawns so that no two queens attack each other.<ref>{{cite journal|last1=Bell |first1=Jordan |last2=Stevens |first2=Brett |title=A survey of known results and research areas for '''n'''-queens  |journal=Discrete Mathematics|volume=309 |number=1|year=2009|pages=1–31|doi=10.1016/j.disc.2007.12.043|doi-access=free}}</ref>
;[[Magic square]]s
:In 1992, Demirörs, Rafraf, and Tanik published a method for converting some magic squares into ''n''-queens solutions, and vice versa.<ref>O. Demirörs, N. Rafraf, and M.M. Tanik. Obtaining n-queens solutions from magic squares and constructing magic squares from n-queens solutions. Journal of Recreational Mathematics, 24:272–280, 1992</ref>
;[[Latin square]]s
:In an ''n''×''n'' matrix, place each digit 1 through ''n'' in ''n'' locations in the matrix so that no two instances of the same digit are in the same row or column.
;[[Exact cover]]
:Consider a matrix with one primary column for each of the ''n'' ranks of the board, one primary column for each of the ''n'' files, and one secondary column for each of the 4''n'' − 6 nontrivial diagonals of the board.  The matrix has ''n''<sup>2</sup> rows: one for each possible queen placement, and each row has a 1 in the columns corresponding to that square's rank, file, and diagonals and a 0 in all the other columns.  Then the ''n'' queens problem is equivalent to choosing a subset of the rows of this matrix such that every primary column has a 1 in precisely one of the chosen rows and every secondary column has a 1 in at most one of the chosen rows; this is an example of a generalized [[exact cover]] problem, of which [[sudoku]] is another example.
; ''n''-queens completion
:The completion problem asks whether, given an ''n''×''n'' chessboard on which some queens are already placed, it is possible to place a queen in every remaining row so that no two queens attack each other.  This and related questions are [[NP-complete]] and [[Sharp-P-complete|#P-complete]].<ref>{{cite journal
 | last1 = Gent
 | first1 = Ian P.
 | last2 = Jefferson
 | first2 = Christopher
 | last3 = Nightingale
 | first3 = Peter
 | title = Complexity of ''n''-Queens Completion
 | journal = [[Journal of Artificial Intelligence Research]]
 | volume = 59
 | pages = 815–848
 | date = August 2017
 | language = en
 | url = https://jair.org/index.php/jair/article/view/11079/26262
 | issn = 1076-9757
 | doi = 10.1613/jair.5512
 | access-date = 7 September 2017 | doi-access = free
 | hdl = 10023/11627
 | hdl-access = free
 }}</ref> Any placement of at most ''n''/60 queens can be completed, while there are partial configurations of roughly ''n''/4 queens that cannot be completed.<ref>{{cite journal |last1=Glock |first1=Stefan |last2=Correia |first2=David Munhá |last3=Sudakov |first3=Benny |title=The ''n''-queens completion problem |journal=Research in the Mathematical Sciences |date=6 July 2022 |volume=9 |issue=41 |page=41 |doi=10.1007/s40687-022-00335-1 |pmid=35815227 |pmc=9259550 |s2cid=244478527 |doi-access=free }}</ref>