1. N+k Queens Reflections
Doug Chatham
Morehead State University
August 1, 2008
Doug Chatham, August 1, 2008

2. N+k Queens Reflections
Acknowledgments
This work is part of research partially supported by an NSF KY-EPSCoR Research Enhancement Grant.
This is part of joint work with Blankenship, Doyle, Fricke, Hufford, Miller, Skaggs, Ward, Wahle, et. al.
Doug Chatham, August 1, 2008

3. The Eight Queens Problem
N+k Queens Reflections
The Eight Queens Problem
Place eight queens on a standard chessboard so that no two attack each other.
First posed, 1848.
Generalized to N queens on an N x N board, 1869.
Mention applications: training example for many CS techniques such as backtracking. Also used as benchmarking problem. Also parallel memory storage
Doug Chatham, August 1, 2008

4. Why stop at eight?

5. N+k Queens Reflections
Theorem: For each k, for large enough N we can place k pawns and N+k queens and an N-by-N board so that no two queens attack each other.
Doug Chatham, August 1, 2008

6. Symmetries of N-queens solutions
Kraitchik (1954) classified solutions to the n-queens problem into 3 types:
Ordinary
Centrosymmetric
Doubly centrosymmetric

7. Ordinary No symmetries
Reflections and rotations produce distinct solutions
Most solutions are ordinary

8. Centrosymmetric Half-turn symmetric
Other rotations and reflections produce distinct solutions

9. Doubly centrosymmetric
Quarter-turn symmetric
Reflections produce distinct solutions

10. What about N+k queens solutions?
Reflections always produce distinct solutions (except for the trivial solution: N=1).
All (nontrivial) solutions are either ordinary, centrosymmetric, or doubly centrosymmetric.

11. Alternating Lemma (AL)
N+k Queens Reflections
Alternating Lemma (AL)
Given a solution to the N+k queens problem:
The first and last piece in each row and column is a queen
Two pawns in the same row (column) have at least one queen between them
Two queens in the same row, column, or diagonal have at least one pawn between them
Mention ASM here
Doug Chatham, August 1, 2008

12. Reflections across vertical line (by contradiction, N even)
N+k Queens Reflections
Reflections across vertical line (by contradiction, N even)
Columns in center must have queens
Those queens must be adjacent
Doug Chatham, August 1, 2008

13. Reflections across vertical line (by contradiction, N odd)
N+k Queens Reflections
Reflections across vertical line (by contradiction, N odd)
Lemma: No square in the central column is empty.
1st row: Q must be in central column
Other row: Consider closest piece to central column. It has a duplicate on other side. AL gives contradiction.
Note: For vertical flips we would use a similar argument. (Switch “columns” and “rows”)
Doug Chatham, August 1, 2008

14. Reflections across vertical line (by contradiction, N odd)
N+k Queens Reflections
Reflections across vertical line (by contradiction, N odd)
Every other square in central column has a queen.
Those queens attack every square in the adjacent columns!
Note: For vertical flips we would use a similar argument. (Switch “columns” and “rows”)
Doug Chatham, August 1, 2008

15. Reflections across diagonal: There are no pawns
N+k Queens Reflections
Reflections across diagonal: There are no pawns
S(c): no pawns in upper left c-by-c square corner
S(1) true by AL
Induction step
So, no pawns
Note: For the other kind of diagonal we would use a similar argument.
Doug Chatham, August 1, 2008

16. Reflections across diagonal: Conclusion
At most one queen on main diagonal
At least one other queen
Attack by symmetric duplicate

17. N+k Queens Reflections
N+k Queens Reflections N
k=1
k=2
k=3
k=4
k=5
5 or less 6 7 4 8 9 20 16 10 11 72
124 32 36 12 52
20(4) 13
200
568
492
564
260 14
1,008
804 15
2,608
6,284
6,164
12,932 17
17,040
For a C solution, k odd  N odd.
Number of centrosymmetric solutions
to the N+k Queens problem
Doug Chatham, August 1, 2008

18. N+k Queens Reflections
5 or less 6 16 7 20 4 8
128 44 9
396
280 10
2,288
1,304
528 88 11
11,152
12,452
5,976
1,688
196 12
65,172
105,012
77,896
30,936
7,032 13
437,848
977,664
1,052,884
627,916
225,884 14
3,118,664
9,239,816
13,666,360
11,546,884 15
23,387,448
90,776,620
179,787,988
183,463,680
897,446,092 17
1,474,699,536
Number of solutions to the
N+k Queens problem.
Doug Chatham, August 1, 2008

19. Doubly centrosymmetric is rare
No doubly centrosymmetric solutions with 1,2, or 3 pawns.

20. Doubly centrosymmetric example
N+k Queens Reflections
Doubly centrosymmetric example
Smallest one with pawns
Doug Chatham, August 1, 2008

21. Doubly centrosymmetric example
N+k Queens Reflections
Doubly centrosymmetric example
Doug Chatham, August 1, 2008

22. References
J. Bell, B. Stevens, A survey of known results and research areas for n-queens, Discrete Mathematics (2008), doi: /j.disc
G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, Annals of Mathematics (2) 156 (2002), no. 3, 835–866.
J.J. Watkins, Across the Board: The Mathematics of Chessboard Problems, Princeton University Press (2004).
The N+k Queens Problem Page,

23. N+k Queens Reflections
Your move!
Any questions?
Doug Chatham, August 1, 2008