Eight queens puzzle

The eight queens puzzle is the problem of placing eight chess queens on an 8×8 chessboard so that no two queens threaten each other; thus, a solution requires that no two queens share the same row, column, or diagonal. There are 92 solutions. The problem was first posed in the mid-19th century. In the modern era, it is often used as an example problem for various computer programming techniques.
The eight queens puzzle is a special case of the more general n queens problem of placing n non-attacking queens on an n×''n chessboard. Solutions exist for all natural numbers n'' with the exception of n = 2 and n = 3. Although the exact number of solutions is only known for n ≤ 27, the asymptotic growth rate of the number of solutions is approximately ⁿ.

History

published the eight queens puzzle in 1848. Franz Nauck published the first solutions in 1850. Nauck also extended the puzzle to the n queens problem, with n queens on a chessboard of n×''n squares.
Since then, many mathematicians, including Carl Friedrich Gauss, have worked on both the eight queens puzzle and its generalized n''-queens version. In 1874, S. Günther proposed a method using determinants to find solutions. J.W.L. Glaisher refined Gunther's approach.
In 1972, Edsger Dijkstra used this problem to illustrate the power of what he called structured programming. He published a highly detailed description of a depth-first backtracking algorithm.

Constructing and counting solutions when ''n'' = 8

The problem of finding all solutions to the 8-queens problem can be quite computationally expensive, as there are 4,426,165,368 possible arrangements of eight queens on an 8×8 board, but only 92 solutions. It is possible to use shortcuts that reduce computational requirements or rules of thumb that avoids brute-force computational techniques. For example, by applying a simple rule that chooses one queen from each column, it is possible to reduce the number of possibilities to 16,777,216 possible combinations. Generating permutations further reduces the possibilities to just 40,320, which can then be checked for diagonal attacks.
The eight queens puzzle has 92 distinct solutions. If solutions that differ only by the symmetry operations of rotation and reflection of the board are counted as one, the puzzle has 12 solutions. These are called fundamental solutions; representatives of each are shown below.
A fundamental solution usually has eight variants obtained by rotating 90, 180, or 270° and then reflecting each of the four rotational variants in a mirror in a fixed position. However, one of the 12 fundamental solutions is identical to its own 180° rotation, so has only four variants. Thus, the total number of distinct solutions is 11×8 + 1×4 = 92.
All fundamental solutions are presented below:
Solution 10 has the additional property that no three queens are in a straight line.

Existence of solutions

Brute-force algorithms to count the number of solutions are computationally manageable for, but would be intractable for problems of, as 20! = 2.433 × 10¹⁸. If the goal is to find a single solution, one can show solutions exist for all n ≥ 4 with no search whatsoever.
These solutions exhibit stair-stepped patterns, as in the following examples for n = 8, 9 and 10:
The examples above can be obtained with the following formulas. Let be the square in column i and row j on the n × n chessboard, k an integer.
One approach is

If the remainder from dividing n by 6 is not 2 or 3 then the list is simply all even numbers followed by all odd numbers not greater than n.
Otherwise, write separate lists of even and odd numbers.
If the remainder is 2, swap 1 and 3 in odd list and move 5 to the end.
If the remainder is 3, move 2 to the end of even list and 1,3 to the end of odd list.
Append odd list to the even list and place queens in the rows given by these numbers, from left to right.

For this results in fundamental solution 1 above. A few more examples follow.

14 queens : 2, 4, 6, 8, 10, 12, 14, 3, 1, 7, 9, 11, 13, 5.
15 queens : 4, 6, 8, 10, 12, 14, 2, 5, 7, 9, 11, 13, 15, 1, 3.
20 queens : 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 3, 1, 7, 9, 11, 13, 15, 17, 19, 5.
Counting solutions for other sizes ''n''

Exact enumeration

There is no known formula for the exact number of solutions for placing n queens on an board i.e. the number of independent sets of size n in an queen's graph. The 27×27 board is the highest-order board that has been completely enumerated. The following tables give the number of solutions to the n queens problem, both fundamental and all, for all known cases.

n	fundamental	all
1	1	1
2	0	0
3	0	0
4	1	2
5	2	10
6	1	4
7	6	40
8	12	92
9	46	352
10	92	724
11	341	2,680
12	1,787	14,200
13	9,233	73,712
14	45,752	365,596
15	285,053	2,279,184
16	1,846,955	14,772,512
17	11,977,939	95,815,104
18	83,263,591	666,090,624
19	621,012,754	4,968,057,848
20	4,878,666,808	39,029,188,884
21	39,333,324,973	314,666,222,712
22	336,376,244,042	2,691,008,701,644
23	3,029,242,658,210	24,233,937,684,440
24	28,439,272,956,934	227,514,171,973,736
25	275,986,683,743,434	2,207,893,435,808,352
26	2,789,712,466,510,289	22,317,699,616,364,044
27	29,363,495,934,315,694	234,907,967,154,122,528

The number of placements in which furthermore no three queens lie on any straight line is known for .

Asymptotic enumeration

In 2021, Michael Simkin proved that for large numbers n, the number of solutions of the n queens problem is approximately. More precisely, the number of solutions has asymptotic growth
where is a constant that lies between 1.939 and 1.945.
If one instead considers a toroidal chessboard, it is only possible to place n queens on an board if In this case, the asymptotic number of solutions is

Exercise in algorithm design

Finding all solutions to the eight queens puzzle is a good example of a simple but nontrivial problem. For this reason, it is often used as an example problem for various programming techniques, including nontraditional approaches such as constraint programming, logic programming or genetic algorithms. Most often, it is used as an example of a problem that can be solved with a recursive algorithm, by phrasing the n queens problem inductively in terms of adding a single queen to any solution to the problem of placing n−1 queens on an n×''n chessboard. The induction bottoms out with the solution to the 'problem' of placing 0 queens on the chessboard, which is the empty chessboard. This process can be made more efficient by applying the observation that each column of the chessboard must contain exactly one queen, and restricting the positions at which the algorithm tries adding the i''th queen to the ith row of the chessboard, in columns that are not already attacked by other queens. These choices significantly reduce the number of placements that need to be tried, allowing the algorithm to find all solutions on an chessboard in a time so short as to appear instantaneous.
An alternative to exhaustive search is an 'iterative repair' algorithm, which typically starts with all queens on the board, for example with one queen per column. It then counts the number of conflicts, and uses a heuristic to determine how to improve the placement of the queens. The 'minimum-conflicts' heuristic – moving the piece with the largest number of conflicts to the square in the same column where the number of conflicts is smallest – is particularly effective: it easily finds a solution to even the 1,000,000 queens problem.
Unlike the backtracking search outlined above, iterative repair does not guarantee a solution: like all greedy procedures, it may get stuck on a local optimum. On the other hand, it can solve problem sizes that are several orders of magnitude beyond the scope of a depth-first search.
As an alternative to backtracking, solutions can be counted by recursively enumerating valid partial solutions, one row at a time. Rather than constructing entire board positions, blocked diagonals and columns are tracked with bitwise operations. This does not allow the recovery of individual solutions.

Sample program

The following program is a translation of Niklaus Wirth's solution into the Python programming language, but does without the index arithmetic found in the original and instead uses lists to keep the program code as simple as possible. By using a coroutine in the form of a generator function, both versions of the original can be unified to compute either one or all of the solutions. Only 15,720 possible queen placements are examined.

def queens:
if i < n:
for j in range:
if j not in a and i + j not in b and i - j not in c:
yield from queens
else:
yield a
for solution in queens:
print
The following program is an implementation of Donald Knuth's informal description of the solution on Page 31, Section 7.2.2 Backtrack Programming from The Art of Computer Programming, Volume 4B into the Python programming language.

def property -> bool:
for k in range:
for j in range:
if j < k:
if perm perm:
return False
elif abs k - j:
return False
return True
def extend:
new_perm =
for p in perm:
for i in range:
new_perm.append
return new_perm
def n_queens -> int:
domain = list
perm =
for i in range:
new_perm = list
perm = new_perm
return len