What is the maximum number of queens that can be placed on an 
chessboard such that
no two attack one another? The answer is 
queens for 
or 
and 
queens otherwise, which gives eight queens for the usual 
board (Madachy 1979; Steinhaus
1999, p. 29). The number of different ways the 
queens can be placed on an 
chessboard so that no two queens may attack each other
for the first few 
are 1, 0, 0, 2, 10, 4, 40, 92, ... (OEIS A000170;
Madachy 1979; Steinhaus 1999, p. 29). The number of rotationally and reflectively
distinct solutions of these are 1, 0, 0, 1, 2, 1, 6, 12, 46, 92, ... (OEIS A002562;
Dudeney 1970; p. 96). The 12 distinct solutions for 
are illustrated above, and the remaining 80 are generated
by rotation and reflection
(Madachy 1979, Steinhaus 1999).
The minimum number of queens needed to occupy or attack all squares of an 
chessboard (i.e., domination numbers for the 
queen graphs) are given
for 
,
2, ... by 1, 1, 1, 2, 3, 3, 4, 5, 5, 5, 5, 6, 7, 8, 9, 9, 9, 9, 10, ... (OEIS A075458), where Steinhaus 1999 (p. 29) notes
.
Dudeney (1970, pp. 95-96) gave the following results for the number of distinct arrangements 
of 
queens attacking or occupying every square of an 
board for which every queen is attacked ("protected")
by at least one other, with the 
value given by Steinhaus (1999, p. 29). The 4860 solutions
in the 
case may be obtained from 638 fundamental arrangements by rotation
and reflection.
 queens |  |  |
| 2 | 4 | 3 |
| 3 | 5 | 37 |
| 3 | 6 | 1 |
| 4 | 7 | 5 |
| 5 | 8 | 4860 |
Dudeney (1970, pp. 95-96) also gave the following results for the number of distinct arrangements 
of 
queens attacking or occupying every square of an 
board for which no two queens attack one another (they
are "not protected").
 queens |  |  |
| 1 | 2 | 1 |
| 1 | 3 | 1 |
| 3 | 4 | 2 |
| 3 | 5 | 2 |
| 4 | 6 | 17 |
| 4 | 7 | 1 |
| 5 | 8 | 91 |
Vardi (1991) generalizes the problem from a square chessboard to one with the topology of the torus. The number of solutions for 
queens with 
odd are 1, 0, 10, 28, 0, 88,
... (OEIS A007705). Vardi (1991) also considers
the toroidal "semiqueens" problem, in which a semiqueen can move like a
rook or bishop, but only on positive broken diagonals.
The number of solutions to this problem for 
queens with 
odd are 1, 3, 15, 133, 2025,
37851, ... (OEIS A006717), and 0 for even 
.
Velucchi gives the solution to the question, "How many different arrangements of 
queens are possible on an order 
chessboard?" as 1/8th of the coefficient
of 
in the polynomial
 |
(1)
|
Velucchi also considers the nondominating queens problem, which consists of placing 
queens on an order 
chessboard to leave a maximum number 
of unattacked vacant cells. The first few values are 0,
0, 0, 1, 3, 5, 7, 11, 18, 22, 30, 36, 47, 56, 72, 82, ... (OEIS A001366).
The results can be generalized to 
queens on an 
board.
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