# Discrete Peaceful Encampments: 9 queens on a chessboard

Here's a discrete variation of yesterday's puzzle Peaceful Encampments.

You have 8 white queens and 8 black queens. Place all these pieces onto a normal 8x8 chessboard in such a way that no white queen threatens a black queen (nor vice versa).

Or, phrasing the puzzle in a way parallel to Black and white queens on an 8x8 chessboard — changing only one word from that puzzle — I would say:

What is the largest number of queens that can be placed on a regular 8×8 chessboard, if the following rules are met:

1. A queen can be either black or white, and there can be unequal numbers of each type [but if so, we count the smaller population].
2. A queen must not be threatened by other queens of a different color.
3. Queens threaten all squares in the same row, column, or diagonal (as in chess). Also, threats are blocked by other queens [not that this matters].

Can you find a way to place more than 8 queens of each color "peacefully" on an 8x8 chessboard?

• Based on the rules, why couldn't one place 64 white queens or 64 black queens? Jan 22, 2019 at 21:31
• @Jiminion: Someone commented the same thing on puzzling.stackexchange.com/questions/28926/… ! :) I've edited that part of the question to reflect that if you place, e.g., 9 white queens and 7 black queens, your score is "7", not "9". And if you place 64 white queens and 0 black queens, your score is "0", not "64". Jan 22, 2019 at 21:37
• Or, phrasing the puzzle another way, what is the continuation of this sequence? 0, 0, 1, 2, 4, 5, 7, 9, 12, 14, 17, 21, ... It turns out that this is OEIS sequence A250000 and fairly well studied! :) Jan 23, 2019 at 1:43

Can I claim Nine-and-a-half? :-)

You can replace either bishop with a tenth queen, but then the other bishop's square must remain empty.

• Solution deserves upvote despite bishops attacks each other, better use knight or rook.
– z100
Jan 23, 2019 at 20:43
• @z100 The intention with the bishops was 'one or the other' Jan 23, 2019 at 20:50
• Solver confirms that this and one other 10+9 solution is optimal, giving no solutions for 10+10. Jan 26, 2019 at 0:36

Nine queens of each color. Some variation is possible.

• Nice. Far more asymmetric than my "intended" solution! Jan 22, 2019 at 23:10

I got 8 Black Queens and 10 White Queens:

Also 9 and 9:

Here's 8 peaceful queens of each color:

After a lot of messing around, I snuck in a 9th white queen (black still at 8)

I'll keep looking for a way to do 9 for each side, but it may not be possible.

• It's possible ;) Jan 25, 2019 at 19:16