# Essays/4 Queens Problem

A problem was posed on the Wikipedia Eight Queens Problem discussion page: Can 4 queens cover (attack all squares of) the (8,8) chessboard?

Since there are only` 4!8*8 `or 635376 different placements
of 4 queens on an (8,8) chessboard, it is feasible to
do an exhaustive analysis of all possible placements. Thus:

comb=: 4 : 0 k=. i.>:d=.y-x z=. (d$<i.0 0),<i.1 0 for. i.x do. z=. k ,.&.> ,&.>/\. >:&.> z end. ; z ) cov=: 3 : 0 " 2 r=. ,(|:y){"_1 M d=. N#. (*./"1@(e.&(i.N)) # ]) ((#D)#y)+(y*&#D)$D I *./@e. r,d ) qcover=: 4 : 0 N=: y I=: i.N*N M=: (,: |:) i.N,N D=: 0 0 -.~ (,.~ , ] ,. |.)i: N-1 (cov@((N,N)&#:) # ]) x comb N*N )

`x comb y `finds all size-`x `combinations of` i.y ` and is from the` for. `page of the
dictionary.

`cov y `is 1 if and only if` y `covers the` N,N `chessboard,
where` y `is a 2-column matrix of the row and column indices of the placement
of the queens. Within` cov` ,` r `are the squares covered by rectangular moves
and` d `are those covered by diagonal moves.

`x qcover y `finds all solutions of covering the` (y,y) `chessboard using` x `queens.

The question is answered in the negative:

$ 4 qcover 8 0 4

That is, it is not possible to cover the (8,8) chessboard with 4 queens.

We now exhibit all solutions of covering the (7,7) chessboard with 4 queens.

$ t= 4 qcover 7 86 4

`var `and` uvar `from the Queens and Knights page
are used to suppress rotational and reflectional variants.

var =: ~.@(] , |:&.> , |.&.> , |."1&.>)^:3 uvar=: ~. @: ({.@(/:~)@var&>) u=: uvar <"0 ((i.7 7) e. ])&.> <"1 t $ u 13 load 'viewmat' rgb=: 192,0,:255 0 0 rgb&viewmat&> (2 <. (2|i.7 7) + 2 * ])&.> u

**See also**

Contributed by Roger Hui.