%I #35 Feb 06 2022 14:59:23

%S 0,0,2,3,4,6,7,10,12,15,19

%N Largest aggressor for the maximum number of peaceable coexisting queens as given in A250000.

%C Sequence A250000 is the maximum number m such that m white queens and m black queens can coexist on an n X n chessboard without attacking each other. However, one of the players can have more than m queens, being a bigger 'aggressor' in peaceful times. The current sequence lists the largest aggressors with k queens when the opponent has m queens for an n X n chessboard (from A250000).

%C The idea and name of the sequence was first mentioned by _Bob Selcoe_ on May 29 2019 in the comment section of A250000.

%C The sequence was initially generated by _Roy van Rijn_ using a SAT solver and is optimal for n=1 to n=11 (as of June 12 2019).

%C _Bob Selcoe_ has shown it is possible to construct a 15 X 15 board with 32 queens of one color and 34 of another but this hasn't yet been proved to be optimal.

%C Many of these values have already been obtained by Stephen Ainley in 1977 (see links).

%C Conjecture: a(n) - A250000(n) <= 2 for all n. - _Dmitry Kamenetsky_, Oct 14 2019

%H Stephen Ainley, <a href="/A250000/a250000_6.png">Mathematical Puzzles</a>, London: G Bell & Sons, 1977. [Annotated scan of a portion of page 32]

%H Dmitry Kamenetsky, <a href="/A308632/a308632_1.txt">Best known solutions for 12 <= n <= 30</a>.

%F a(n) >= A250000(n).

%e Examples (omitted cases where the largest aggressor is equal to A250000):

%e n=1: white queens 0, black queens 0

%e n=2: white queens 0, black queens 0

%e n=3: white queens 1, black queens 2

%e n=4: white queens 2, black queens 3

%e +---------+

%e | . W . W |

%e | . . . . |

%e | B . B . |

%e | . . B . |

%e +---------+

%e n=5: white queens 4, black queens 4

%e n=6: white queens 5, black queens 6

%e +-------------+

%e | . W . . . . |

%e | W . W . . . |

%e | . . . . B B |

%e | . . . B . B |

%e | W W . . . . |

%e | . . . B . B |

%e +-------------+

%e n=7: white queens 7, black queens 7

%e n=8: white queens 9, black queens 10

%e +-----------------+

%e | . . . B B B . . |

%e | W W . . . . . . |

%e | . . . B . . . B |

%e | . . . . . . B B |

%e | . . . . . B B B |

%e | . W W . . . . . |

%e | W W W . . . . . |

%e | W W . . . . . . |

%e +-----------------+

%e n=9: white queens 12, black queens 12

%e n=10: white queens 14, black queens 15

%e +---------------------+

%e | . . B B . . . . B B |

%e | . . B B . . . B B B |

%e | . . B . . . . B B B |

%e | . . . . . . . B B . |

%e | . W . . . . . . . . |

%e | W W . . . . . . . . |

%e | W W . . . . . . . . |

%e | W . . . . W W . . . |

%e | . . . . W W W . . . |

%e | . . . . W W W . . . |

%e +---------------------+

%e n=11: white queens 17, black queens 19

%e +-----------------------+

%e | W . W . . . . . W . W |

%e | . . . . B B B . . . . |

%e | W . W . . . . . W . W |

%e | . . . . B . B . . . . |

%e | . B . . . B . B . B . |

%e | . B . . B . B . . B . |

%e | . B . . . B . B . B . |

%e | . . . . B . B . . . . |

%e | W . W . . . . . W . W |

%e | . . . W . . . . . . . |

%e | W . W . . . . . W . W |

%e +-----------------------+

%Y Cf. A250000.

%K nonn,hard,more

%O 1,3

%A _Roy van Rijn_, Jun 12 2019