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

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

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

Why stop at eight?

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

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

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

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

Doubly centrosymmetric Quarter-turn symmetric Reflections produce distinct solutions

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.

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

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

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

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

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

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

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

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

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

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

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

References J. Bell, B. Stevens, A survey of known results and research areas for n-queens, Discrete Mathematics (2008), doi:10.1016/j.disc.2007.12.043. G. Kuperberg, Symmetry classes of alternating-sign matrices under one roof, Annals of Mathematics (2) 156 (2002), no. 3, 835–866. J.J. Watkins, Across the Board: The Mathematics of Chessboard Problems, Princeton University Press (2004). The N+k Queens Problem Page, http://npluskqueens.info.

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