A328283 The maximum number m such that m white, m black and m red queens can coexist on an n X n chessboard without attacking each other.

0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 14

Offset: 1

Author

Dmitry Kamenetsky, Oct 11 2019

Keywords

• nonn
• more
• hard

Comments

This is the peaceable queens problem (A250000) for 3 players.

For n >= 11, it seems that a(n) is simply 2n - 14. However this turns out to be false as a(18) >= 23.

In the limit of large n, Arthur O'Dwyer (see links) showed that the optimal value is lower bounded by 0.0718*n^2. All currently known best solutions follow this formula (when rounded down). - M. A. Achterberg, Dec 01 2022

Examples

			a(8) = 4, because 4 queens of each color can co-exist without attacking queens of another color. Note that in this case both red (6) and white (5) have more than 4 queens.
+ - - - - - - - - +
| R . R . R . . . |
| R . . . . . . . |
| . . . . . W . W |
| R . R . . . . . |
| . . . . . W . W |
| . B . B . . . . |
| . . . . . . . W |
| . B . B . . . . |
+ - - - - - - - - +
		

Links

• M. A. Achterberg, Best known solutions for n <= 30, Dec 01 2022.
• Dmitry Kamenetsky, Best known solutions for n <= 24.
• Arthur O'Dwyer, Discrete Peaceful Encampments, 2019.
• Arthur O'Dwyer, Discrete Peaceful Encampments: Player 3 has entered the game!, Puzzling StackExchange, 2019.
• Arthur O'Dwyer, Peaceful Encampments, round 2, 2019.

Crossrefs

Cf. A250000.