N k Queens Reflections

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Nk Queens Reflections

• Doug Chatham
• Morehead State University
• August 1, 2008

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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.

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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.

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Why stop at eight?
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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.

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Symmetries of N-queens solutions

• Kraitchik (1954) classified solutions to the
  n-queens problem into 3 types
• Ordinary
• Centrosymmetric
• Doubly centrosymmetric

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Ordinary

• No symmetries
• Reflections and rotations produce distinct
  solutions
• Most solutions are ordinary

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Centrosymmetric

• Half-turn symmetric
• Other rotations and reflections produce distinct
  solutions

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Doubly centrosymmetric

• Quarter-turn symmetric
• Reflections produce distinct solutions

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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.

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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

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Reflections across vertical line (by
contradiction, N even)

• Columns in center must have queens
• Those queens must be adjacent

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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.

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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!

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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

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Reflections across diagonal Conclusion

• At most one queen on main diagonal
• At least one other queen
• Attack by symmetric duplicate

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Number of centrosymmetric solutions to the Nk
Queens problem
 
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  Number of solutions to the Nk Queens problem.
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Doubly centrosymmetric is rare

• No doubly centrosymmetric solutions with 1,2, or
  3 pawns.

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Doubly centrosymmetric example
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Doubly centrosymmetric example
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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.

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Your move!

• Any questions?