Abstract
The n-queens puzzle is a well-known combinatorial problem that requires to place n queens on an \(n\times n\) chessboard so that no two queens can attack each other. Since the 19th century, this problem was studied by many mathematicians and computer scientists. While finding any solution to the n-queens puzzle is rather straightforward, it is very challenging to find the lexicographically first (or smallest) feasible solution. Solutions for this type are known in the literature for \(n\le 55\), while for some larger chessboards only partial solutions are known. The present paper was motivated by the question of whether Integer Linear Programming (ILP) can be used to compute solutions for some open instances. We describe alternative ILP-based solution approaches, and show that they are indeed able to compute (sometimes in unexpectedly-short computing times) many new lexicographically optimal solutions for n ranging from 56 to 115.
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Notes
- 1.
Alternatively, one could fix \(x_{ij}=1\), check the resulting model for feasibility, and then move to the next position.
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Acknowledgements
This research was partially supported by MiUR, Italy, through project PRIN2015 “Nonlinear and Combinatorial Aspects of Complex Networks”. We thank Donald E. Knuth for having pointed out the problem to us, and for inspiring discussions on the role of Integer Linear Programming in solving combinatorial problems arising in digital tomography.
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A New Solutions
A New Solutions
Here are the solutions we found for some open problems from the literature:
n | Solution |
|---|---|
56 | 1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 33 42 44 46 43 51 53 55 45 54 50 47 56 48 52 49 12 14 23 21 32 34 26 16 30 17 24 18 37 28 40 20 39 41 35 38 36 |
57 | 1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 12 34 43 45 47 50 52 54 44 57 49 46 56 51 48 55 53 14 28 17 33 23 16 18 30 24 37 20 32 21 26 40 35 41 39 42 36 38 |
58 | 1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 12 42 45 48 52 54 43 53 55 49 44 46 50 57 47 51 58 56 28 26 20 34 30 18 14 17 24 21 16 35 23 40 33 36 38 32 41 39 37 |
59 | 1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 12 34 36 45 47 49 52 56 53 46 57 59 48 51 54 50 55 58 16 14 17 32 23 26 20 18 33 35 28 21 43 41 37 24 40 44 30 39 42 38 |
60 | 1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 12 34 44 46 48 45 51 54 58 50 59 57 60 47 49 52 55 53 56 18 33 23 32 28 16 20 17 21 37 35 26 24 30 14 42 38 43 41 39 36 40 |
61 | 1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 12 14 35 45 47 49 52 54 56 50 60 46 61 58 48 51 53 55 57 59 23 32 16 33 21 17 26 36 18 20 38 24 28 34 40 30 41 44 42 37 39 43 |
63 | 1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 12 14 35 37 47 49 51 53 59 57 52 60 62 48 50 54 63 55 58 56 61 32 16 33 17 21 26 36 20 18 38 28 23 40 24 30 34 41 39 44 46 43 45 42 |
65 | 1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 12 14 35 37 39 49 51 53 50 56 59 63 55 64 62 65 52 54 57 60 58 61 16 30 17 21 26 36 33 20 18 41 38 23 32 24 28 48 46 34 43 40 44 47 45 42 |
67 | 1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 12 14 35 37 39 41 51 53 55 52 58 61 65 57 66 64 67 54 56 59 62 60 63 16 18 34 30 38 20 24 17 21 23 43 32 40 33 36 26 28 46 48 50 44 47 45 42 49 |
69 | 1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 12 14 35 37 39 41 43 53 55 57 54 60 63 67 59 68 66 69 56 58 61 64 62 65 17 20 16 30 24 33 40 38 18 21 34 26 23 42 49 28 32 50 36 51 46 44 52 48 45 47 |
71 | 1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 12 14 35 37 39 41 16 53 55 57 54 56 62 68 66 69 59 70 67 58 71 61 64 60 65 63 21 30 17 40 18 24 36 20 42 44 26 34 23 33 38 32 28 49 51 45 47 52 50 48 46 43 |
73 | 1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 12 14 35 37 39 41 16 44 55 57 59 56 58 63 67 69 71 73 61 70 72 65 60 62 64 66 68 20 34 21 18 42 17 38 24 43 23 28 45 33 40 36 26 32 30 54 47 50 52 46 48 53 51 49 |
77 | 1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 12 14 35 37 39 41 16 18 45 57 59 61 58 60 65 68 72 74 76 73 75 63 67 64 62 77 70 66 71 69 38 40 28 17 21 24 26 20 43 46 42 23 36 34 32 30 44 33 52 55 47 50 53 56 54 48 51 49 |
79 | 1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 12 14 35 37 39 41 16 18 45 47 59 61 63 60 62 67 70 74 71 77 79 76 78 64 68 65 69 66 73 75 72 20 38 17 21 44 24 30 23 46 48 36 42 40 34 26 28 33 50 32 53 43 57 52 58 56 54 51 49 55 |
85 | 1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 12 14 35 37 39 41 16 18 45 17 48 50 63 65 67 64 66 71 73 75 80 82 84 81 83 72 70 68 85 69 78 74 77 79 76 20 23 43 24 21 49 44 42 34 46 28 30 52 26 38 51 32 40 33 61 47 60 36 53 58 54 57 59 56 62 55 |
91 | 1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 12 14 35 37 39 41 16 18 45 17 48 20 51 53 67 69 71 68 70 75 77 79 81 85 87 90 86 91 89 72 74 76 73 80 82 84 78 83 88 21 34 26 49 46 24 47 52 43 23 30 33 55 28 42 32 54 40 36 44 64 50 38 59 61 65 57 66 60 63 56 58 62 |
93 | 1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 12 14 35 37 39 41 16 18 45 17 48 20 51 53 55 69 71 73 70 72 77 79 81 83 87 89 92 88 93 91 74 76 78 75 82 84 86 80 85 90 24 21 23 46 49 47 52 38 30 56 33 26 28 43 32 54 57 42 44 36 34 40 50 61 68 65 62 59 63 58 67 64 66 60 |
97 | 1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 12 14 35 37 39 41 16 18 45 17 48 20 51 53 21 56 71 73 75 72 74 79 81 83 85 87 89 93 95 97 94 96 76 80 77 86 78 82 84 91 88 90 92 46 24 28 52 23 49 47 34 30 26 57 50 33 61 42 44 36 32 55 43 38 54 60 66 40 70 68 63 58 69 62 65 67 64 59 |
101 | 1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 12 14 35 37 39 41 16 18 45 17 48 20 51 53 21 56 58 60 75 77 79 76 78 83 85 87 89 91 93 97 99 101 98 100 80 84 81 90 82 86 88 95 92 94 96 23 26 28 40 43 54 57 24 32 47 50 42 59 33 30 34 52 62 68 46 38 36 44 55 66 71 74 70 49 73 63 72 67 61 64 69 65 |
103 | 1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 12 14 35 37 39 41 16 18 45 17 48 20 51 53 21 56 58 60 62 77 79 81 78 80 85 87 89 91 93 95 99 101 103 100 102 82 86 83 92 84 88 90 97 94 96 98 23 26 24 30 28 36 46 55 59 52 54 44 61 34 66 33 42 32 47 49 40 38 57 73 71 63 72 43 64 70 75 50 69 67 76 74 68 65 |
109 | 1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 12 14 35 37 39 41 16 18 45 17 48 20 51 53 21 56 58 60 23 63 65 81 83 85 82 84 89 91 93 95 86 100 104 106 101 109 107 105 108 88 92 87 96 90 97 102 94 98 103 99 26 24 32 28 36 55 57 40 64 61 54 50 30 66 34 42 38 33 49 43 67 59 62 77 52 44 47 75 71 46 76 80 73 70 79 69 78 72 74 68 |
115 | 1 3 5 2 4 9 11 13 15 6 8 19 7 22 10 25 27 29 31 12 14 35 37 39 41 16 18 45 17 48 20 51 53 21 56 58 60 23 63 24 66 68 85 87 89 86 88 93 95 97 99 90 102 108 111 113 107 109 112 115 91 114 98 101 92 94 96 100 105 103 110 106 104 26 28 30 32 36 50 59 62 64 55 43 34 72 67 52 33 40 65 57 44 42 38 74 54 61 46 83 47 77 69 49 82 79 75 84 71 80 78 81 73 70 76 |
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Fischetti, M., Salvagnin, D. (2018). Chasing First Queens by Integer Programming. In: van Hoeve, WJ. (eds) Integration of Constraint Programming, Artificial Intelligence, and Operations Research. CPAIOR 2018. Lecture Notes in Computer Science(), vol 10848. Springer, Cham. https://doi.org/10.1007/978-3-319-93031-2_16
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