The historical root of the problem is the case n=8 which was discussed
in the 19th century; it was posed in 1848 by Max Bezzel, and also the famous
mathematician Carl Friedrich Gauss dealt with it.
The most common  and most useful  generalization
is the "torus problem".
What means "torus problem"? It means that the queens are not limited
by the borders in their movement. There are two ways to think of it: first
you can think of the chess board as wrapped around a cylinder. Thats all
you need concerning the n queens torus problem, although the wrapped board
is only half the way from rectangle to a torus. The second step is that
you form a ring  just like a life belt  from the cylinder, removing the
other two borders. That's what mathematicians call a torus, and it's different
from a cylinder in other aspects  yet equivalent for the n queens.
The second way to imagine the torus problem is to think of a borderless,
endless plane and of infinite many queens on it in a periodic arrangement;
the n changes its meaning from the size of the board to the periodic distance
of these queens. We have a solution of the torus problem if there are exactly
n queens in every n by n rectangle of the endless plane, and if every queen
"attacks" only her "periodic sisters".
Main tool: group actions
To classify and to count the solutions, there is a main tool in my opinion.
That tool is the idea of groups acting on a set. That is not discussed
in the literature, as far as I know; but it should be discussed.
Dealing with that idea, it is natural to handle not only "normal" and
torus solutions, but also some other sets in which these are contained.
To come to group actions, we start with some group of motions moving
the fields of the board, and look if they induce an action on the solutions.
The most natural group acting on the board is the group consisting
of reflection and rotation by 90 ° or a multiple of it.
It induces actions on both normal and torus solutions: it is clear
that a normal solution remains a normal solution if you rotate or reflect
it. It is also clear that a torus solution remains a torus solution under
reflection and rotation. This group is called the dihedral group D4 by
the mathematicians.
The next bigger group contains also shifts on the torus. I call that
group the group of congruent motions. However, that gives a complication
for the normal solutions: a shifted solution may no longer be a solution.
Clearly, every solution is equivalent to a permutation. In the language
of chess, that is the "n rooks problem" instead of the n queens problem.
The set of all permutations is a set where the group of congruent motions
acts properly.
There are also subsets of permutations which we should study when we
search normal n queens solutions. We can assign to each permutation the
maximum number of queens which share the same torus diagonal or torus antidiagonal.
Then the set of permutations having 1 as that maximum is just the set of
torus solutions. The set of permutations having a maximum <= 2 contains
all normal solutions  and it is a set which is stable under the action
of congruent motions.
There are still two further steps for group actions; that means, there
are two bigger groups which act on the solutions or a set containing them.
One is the group of similar motions, the other the group of all affine
motions. For similar motions, the last set described above works fine;
for all affine motions, the set has to be extended. I do not know what
is most appropriate until now; it is surely a subset of "n of n²",
but maybe it may and should be confined in some way.
By the way: I have learned a lot on group actions in the book
"Applied Finite Group Actions" by A.Kerber, 2nd edition, Springer 1999,
ISBN 3540659412 (and I hope to learn still more), although the n queens
problem is not mentioned in the book.
Results for counting
For counting the solutions, some information on the problem is contained
in OnLine
Encyclopedia of Integer Sequences. In the following table, you find
links to special sequences.
The table is organized by the different sets and by the symmetries.
Tables of sequences:
Permutations as base sets (i.e. rooks instead of
queens)


Permutations

Congruencies in the square, i.e. reflect and rotate

Congruencies on the torus, i.e. shift, reflect and rotate allowed

Similarity

?? regular affine mappings ??

all affine mappings

all

A000142

A000903

A006841







central symmetry
(i.e. for rotation of 180°)

A037223











rotational symmetry (90°)

A037224











shift symmetries













other symmetries

Fixed by composition of rotation and reflection
A000085











Normal queens


Permutations

Congruencies in the square, i.e. reflect and rotate

Congruencies on the torus, i.e. shift, reflect and rotate allowed

Similarity

?? regular affine mappings ??

all affine mappings

all

 classic case 
A000170

A002562

A062164

A062165





central symmetry
(i.e. for rotation of 180°)

A032522











rotational symmetry (90°)

A033148











shift symmetries













other symmetries













Torus queens


Permutations

Congruencies in the square, i.e. reflect and rotate

Congruencies on the torus, i.e. shift, reflect and rotate allowed

Similarity

?? regular affine mappings ??

all affine mappings

all

A007705



A053994

A062166





central symmetry
(i.e. for rotation of 180°)













rotational symmetry (90°)













shift symmetries













other symmetries





A054500
 02







Permutations having at most 2 queens on a diagonal


Permutations

Congruencies in the square, i.e. reflect and rotate

Congruencies on the torus, i.e. shift, reflect and rotate allowed

Similarity

?? regular affine mappings ??

all affine mappings

all





A062167

A062168





central symmetry
(i.e. for rotation of 180°)













rotational symmetry (90°)













shift symmetries













other symmetries













As you see, there are still many unknown sequences which
are also of some interest.
Two colorful images for n=13
For more details, there are images at Torus images for n=13.
For a first glance, the next two images show some
torus solutions for a 13 × 13 field, chosen at random; different solutions
appear in different colors.
Additional information
Prof.Kosters in Leiden (Netherlands) maintains a bibliografy
containing 80 (or more) articles on the nQueens problem.
Start of page
Prepared by Matthias Engelhardt
mailto:
Matthias.R.Engelhardt@web.de
last updated: 20010812