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N k Queens Reflections

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Title: N k Queens Reflections


1
Nk Queens Reflections
  • Doug Chatham
  • Morehead State University
  • August 1, 2008

2
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.

3
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.

4
Why stop at eight?
5
Nk 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.

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 Nk queens solutions?
  • Reflections always produce distinct solutions
    (except for the trivial solution N1).
  • All (nontrivial) solutions are either ordinary,
    centrosymmetric, or doubly centrosymmetric.

11
Alternating 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

12
Reflections across vertical line (by
contradiction, N even)
  • Columns in center must have queens
  • Those queens must be adjacent

13
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.

14
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!

15
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

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

17
 
Number of centrosymmetric solutions to the Nk
Queens problem
 
18
  Number of solutions to the Nk Queens problem.
19
Doubly centrosymmetric is rare
  • No doubly centrosymmetric solutions with 1,2, or
    3 pawns.

20
Doubly centrosymmetric example
21
Doubly centrosymmetric example
22
References
  • 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.

23
Your move!
  • Any questions?
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