A308632 Largest aggressor for the maximum number of peaceable coexisting queens as given in A250000.

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

Offset: 1

Author

Roy van Rijn, Jun 12 2019

Keywords

• nonn
• hard
• more

Comments

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).

The idea and name of the sequence was first mentioned by Bob Selcoe on May 29 2019 in the comment section of A250000.

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).

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.

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

Conjecture: a(n) - A250000(n) <= 2 for all n. - Dmitry Kamenetsky, Oct 14 2019

Examples

			Examples (omitted cases where the largest aggressor is equal to A250000):
  n=1: white queens 0, black queens 0
  n=2: white queens 0, black queens 0
  n=3: white queens 1, black queens 2
  n=4: white queens 2, black queens 3
  +---------+
  | . W . W |
  | . . . . |
  | B . B . |
  | . . B . |
  +---------+
  n=5: white queens 4, black queens 4
  n=6: white queens 5, black queens 6
  +-------------+
  | . W . . . . |
  | W . W . . . |
  | . . . . B B |
  | . . . B . B |
  | W W . . . . |
  | . . . B . B |
  +-------------+
  n=7: white queens 7, black queens 7
  n=8: white queens 9, black queens 10
  +-----------------+
  | . . . B B B . . |
  | W W . . . . . . |
  | . . . B . . . B |
  | . . . . . . B B |
  | . . . . . B B B |
  | . W W . . . . . |
  | W W W . . . . . |
  | W W . . . . . . |
  +-----------------+
  n=9: white queens 12, black queens 12
  n=10: white queens 14, black queens 15
  +---------------------+
  | . . B B . . . . B B |
  | . . B B . . . B B B |
  | . . B . . . . B B B |
  | . . . . . . . B B . |
  | . W . . . . . . . . |
  | W W . . . . . . . . |
  | W W . . . . . . . . |
  | W . . . . W W . . . |
  | . . . . W W W . . . |
  | . . . . W W W . . . |
  +---------------------+
  n=11: white queens 17, black queens 19
  +-----------------------+
  | W . W . . . . . W . W |
  | . . . . B B B . . . . |
  | W . W . . . . . W . W |
  | . . . . B . B . . . . |
  | . B . . . B . B . B . |
  | . B . . B . B . . B . |
  | . B . . . B . B . B . |
  | . . . . B . B . . . . |
  | W . W . . . . . W . W |
  | . . . W . . . . . . . |
  | W . W . . . . . W . W |
  +-----------------------+
		

Links

• Stephen Ainley, Mathematical Puzzles, London: G Bell & Sons, 1977. [Annotated scan of a portion of page 32]
• Dmitry Kamenetsky, Best known solutions for 12 <= n <= 30.

Crossrefs

Cf. A250000.

Formula

a(n) >= A250000(n).