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. The eight queens puzzle is an example of the more general n queens problem of placing n non-attacking queens on an n×n chessboard, for which solutions exist for all natural numbers n with the exception of n = 2 and n = 3.[1]

abcdefgh
8
8
77
66
55
44
33
22
11
abcdefgh
The only symmetrical solution to the eight queens puzzle (up to rotation and reflection)

History

Chess composer Max Bezzel published the eight queens puzzle in 1848. Franz Nauck published the first solutions in 1850.[2] 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. Gunther proposed a method using determinants to find solutions.[2] 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.2

Constructing and counting solutions

The problem of finding all solutions to the 8-queens problem can be quite computationally expensive, as there are 4,426,165,368 (i.e., 64C8) 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 constrains each queen to a single column (or row), though still considered brute force, it is possible to reduce the number of possibilities to 16,777,216 (that is, 88) possible combinations. Generating permutations further reduces the possibilities to just 40,320 (that is, 8!), which are then checked for diagonal attacks.

Solutions

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 (including its original form) obtained by rotating 90, 180, or 270° and then reflecting each of the four rotational variants in a mirror in a fixed position. However, should a solution be equivalent to its own 90° rotation (as happens to one solution with five queens on a 5×5 board), that fundamental solution will have only two variants (itself and its reflection). Should a solution be equivalent to its own 180° rotation (but not to its 90° rotation), it will have four variants (itself and its reflection, its 90° rotation and the reflection of that). If n > 1, it is not possible for a solution to be equivalent to its own reflection because that would require two queens to be facing each other. Of the 12 fundamental solutions to the problem with eight queens on an 8×8 board, exactly one (solution 12 below) is equal to its own 180° rotation, and none is equal to its 90° rotation; thus, the 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

These brute-force algorithms to count the number of solutions are computationally manageable for n = 8, but would be intractable for problems of n ≥ 20, 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.[3] These solutions exhibit stair-stepped patterns, as in the following examples for n = 8, 9 and 10:

Counting solutions

The following tables give the number of solutions for placing n queens on an n × n board, both fundamental (sequence A002562 in the OEIS) and all (sequence A000170 in the OEIS).

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 six queens puzzle has fewer solutions than the five queens puzzle.

There is no known formula for the exact number of solutions, or even for its asymptotic behaviour. The 27×27 board is the highest-order board that has been completely enumerated.[4]

  • Higher dimensions
Find the number of non-attacking queens that can be placed in a d-dimensional chess 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 nk queens do not suffice to attack all spaces.[5][6]
  • Using pieces other than queens
On an 8×8 board one can place 32 knights, or 14 bishops, 16 kings or eight rooks, so that no two pieces attack each other. Fairy chess pieces have also been substituted for queens. 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. Eight rooks can be placed along a long diagonal (amongst thousands of other solutions), and 16 kings are placed on the board by dividing it into 2 by 2 squares and placing the kings at equivalent points on each square.
  • 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 pawns and n+k mutually nonattacking dragon kings on an n×n shogi board.[7]
In mathematics, a permutation matrix can be regarded geometrically as a set of n points lying on the squares of a n×n chessboard, such that each row or column contains only one point. Thus, an order-n permutation matrix is a solution to an n-rooks puzzle.
  • Nonstandard boards
Pólya studied the n queens problem on a 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.[8] In 2009 Pearson and Pearson algorithmically populated three-dimensional boards (n×n×n) with n2 queens, and proposed that multiples of these can yield solutions for a four-dimensional version of the puzzle.[9]
  • 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.[10][11]
  • 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[12] or placing queens and pawns so that no two queens attack each other.[13]
In 1992, Demirörs, Rafraf, and Tanik published a method for converting some magic squares into n-queens solutions, and vice versa.[14]
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.
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 4n − 6 nontrivial diagonals of the board. The matrix has n2 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
A 2017 paper[15] investigated the problem "Given an n×n chessboard on which some queens are already placed, can you place a queen in every remaining row so that no two queens attack each other?" and several related problems. The authors asserted that these problems are NP-complete[16] and #P-complete.

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 technique can be used in a way that is much more efficient than the naïve brute-force search algorithm, which considers all 648 = 248 = 281,474,976,710,656 possible blind placements of eight queens, and then filters these to remove all placements that place two queens either on the same square (leaving only 64!/56! = 178,462,987,637,760 possible placements) or in mutually attacking positions. This very poor algorithm will, among other things, produce the same results over and over again in all the different permutations of the assignments of the eight queens, as well as repeating the same computations over and over again for the different sub-sets of each solution. A better brute-force algorithm places a single queen on each row, leading to only 88 = 224 = 16,777,216 blind placements.

It is possible to do much better than this. One algorithm solves the eight rooks puzzle by generating the permutations of the numbers 1 through 8 (of which there are 8! = 40,320), and uses the elements of each permutation as indices to place a queen on each row. Then it rejects those boards with diagonal attacking positions. The backtracking depth-first search program, a slight improvement on the permutation method, constructs the search tree by considering one row of the board at a time, eliminating most nonsolution board positions at a very early stage in their construction. Because it rejects rook and diagonal attacks even on incomplete boards, it examines only 15,720 possible queen placements. A further improvement, which examines only 5,508 possible queen placements, is to combine the permutation based method with the early pruning method: the permutations are generated depth-first, and the search space is pruned if the partial permutation produces a diagonal attack. Constraint programming can also be very effective on this problem.

min-conflicts solution to 8 queens

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.[17] It then counts the number of conflicts (attacks), 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 finds a solution to the 1,000,000 queen problem in less than 50 steps on average. This assumes that the initial configuration is 'reasonably good' – if a million queens all start in the same row, it will take at least 999,999 steps to fix it. A 'reasonably good' starting point can for instance be found by putting each queen in its own row and column so that it conflicts with the smallest number of queens already on the board.

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. (In such a case, the algorithm may be restarted with a different initial configuration.) On the other hand, it can solve problem sizes that are several orders of magnitude beyond the scope of a depth-first search.

This animation illustrates backtracking to solve the problem. A queen is placed in a column that is known not to cause conflict. If a column is not found the program returns to the last good state and then tries a different column.

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.[18][19]

Sample program

The following is a Pascal program by Niklaus Wirth in 1976.[20] It finds one solution to the eight queens problem.

program eightqueen1(output);
 
var i : integer; q : boolean;
    a : array[ 1 .. 8] of boolean;
    b : array[ 2 .. 16] of boolean;
    c : array[ −7 .. 7] of boolean;
    x : array[ 1 .. 8] of integer;
 
procedure try( i : integer; var q : boolean);
    var j : integer;
    begin 
    j := 0;
    repeat 
        j := j + 1; 
        q := false;
        if a[ j] and b[ i + j] and c[ i − j] then
            begin 
            x[ i    ] := j;
            a[ j    ] := false; 
            b[ i + j] := false; 
            c[ i − j] := false;
            if i < 8 then
                begin
                try( i + 1, q);
                if not q then
                    begin 
                    a[ j] := true; 
                    b[ i + j] := true; 
                    c[ i − j] := true;
                    end
                end 
            else 
                q := true
            end
    until q or (j = 8);
    end;
 
begin
for i := 1 to  8 do a[ i] := true;
for i := 2 to 16 do b[ i] := true;
for i := −7 to  7 do c[ i] := true;
try( 1, q);
if q then
    for i := 1 to 8 do write( x[ i]:4);
writeln
end.

See also

References
  1. E. J. Hoffman et al., "Construction for the Solutions of the m Queens Problem". Mathematics Magazine, Vol. XX (1969), pp. 66–72.
  2. W. W. Rouse Ball (1960) "The Eight Queens Problem", in Mathematical Recreations and Essays, Macmillan, New York, pp. 165–171.
  3. Bo Bernhardsson (1991). "Explicit Solutions to the N-Queens Problem for All N". SIGART Bull. 2 (2): 7. doi:10.1145/122319.122322.
  4. The Q27 Project
  5. J. Barr and S. Rao (2006), The n-Queens Problem in Higher Dimensions, Elemente der Mathematik, vol 61 (4), pp. 133–137.
  6. Martin S. Pearson. "Queens On A Chessboard – Beyond The 2nd Dimension" (php). Retrieved 27 January 2020.
  7. Chatham, Doug (1 December 2018). "Reflections on the n +k dragon kings problem". Recreational Mathematics Magazine. 5 (10): 39–55. doi:10.2478/rmm-2018-0007.
  8. 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. 237–247
  10. Burger, A. P.; Cockayne, E. J.; Mynhardt, C. M. (1997), "Domination and irredundance in the queens' graph", Discrete Mathematics, 163 (1–3): 47–66, doi:10.1016/0012-365X(95)00327-S, hdl:1828/2670, MR 1428557
  11. Weakley, William D. (2018), "Queens around the world in twenty-five years", in Gera, Ralucca; Haynes, Teresa W.; Hedetniemi, Stephen T. (eds.), Graph Theory: Favorite Conjectures and Open Problems – 2, Problem Books in Mathematics, Cham: Springer, pp. 43–54, doi:10.1007/978-3-319-97686-0_5, MR 3889146
  12. "Queens and knights problem". Archived from the original on 16 October 2005. Retrieved 20 September 2005.
  13. Nine queens problem
  14. 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
  15. Gent, Ian P.; Jefferson, Christopher; Nightingale, Peter (August 2017). "Complexity of n-Queens Completion". Journal of Artificial Intelligence Research. 59: 815–848. doi:10.1613/jair.5512. ISSN 1076-9757. Retrieved 7 September 2017.
  16. "The 8-Queens Puzzle". www.claymath.org. Clay Mathematics Institute. 2 September 2017.
  17. A Polynomial Time Algorithm for the N-Queen Problem by Rok Sosic and Jun Gu, 1990. Describes run time for up to 500,000 Queens which was the max they could run due to memory constraints.
  18. Qiu, Zongyan (February 2002). "Bit-vector encoding of n-queen problem". ACM SIGPLAN Notices. 37 (2): 68–70. doi:10.1145/568600.568613.
  19. Richards, Martin (1997). Backtracking Algorithms in MCPL using Bit Patterns and Recursion (PDF) (Technical report). University of Cambridge Computer Laboratory. UCAM-CL-TR-433.
  20. Wirth, 1976, p. 145

Further reading
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