Title: N k Queens Reflections
1Nk Queens Reflections
- Doug Chatham
- Morehead State University
- August 1, 2008
2Acknowledgments
- 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.
3The 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.
4Why stop at eight?
5Nk queens
- Theorem For each k, for large enough N we can
place k pawns and Nk queens and an N-by-N board
so that no two queens attack each other.
6Symmetries of N-queens solutions
- Kraitchik (1954) classified solutions to the
n-queens problem into 3 types - Ordinary
- Centrosymmetric
- Doubly centrosymmetric
7Ordinary
- No symmetries
- Reflections and rotations produce distinct
solutions - Most solutions are ordinary
8Centrosymmetric
- Half-turn symmetric
- Other rotations and reflections produce distinct
solutions
9Doubly centrosymmetric
- Quarter-turn symmetric
- Reflections produce distinct solutions
10What about Nk queens solutions?
- Reflections always produce distinct solutions
(except for the trivial solution N1). - All (nontrivial) solutions are either ordinary,
centrosymmetric, or doubly centrosymmetric.
11Alternating Lemma (AL)
- Given a solution to the Nk 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
12Reflections across vertical line (by
contradiction, N even)
- Columns in center must have queens
- Those queens must be adjacent
13Reflections 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.
14Reflections 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!
15Reflections 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
16Reflections across diagonal Conclusion
- At most one queen on main diagonal
- At least one other queen
- Attack by symmetric duplicate
17Â
Number of centrosymmetric solutions to the Nk
Queens problem
Â
18Â Number of solutions to the Nk Queens problem.
19Doubly centrosymmetric is rare
- No doubly centrosymmetric solutions with 1,2, or
3 pawns.
20Doubly centrosymmetric example
21Doubly centrosymmetric example
22References
- J. Bell, B. Stevens, A survey of known results
and research areas for n-queens, Discrete
Mathematics (2008), doi10.1016/j.disc.2007.12.043
. - G. Kuperberg, Symmetry classes of
alternating-sign matrices under one roof, Annals
of Mathematics (2) 156 (2002), no. 3, 835866. - J.J. Watkins, Across the Board The Mathematics
of Chessboard Problems, Princeton University
Press (2004). - The Nk Queens Problem Page, http//npluskqueens.i
nfo.
23Your move!