16
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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?

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  • $\begingroup$ Based on the rules, why couldn't one place 64 white queens or 64 black queens? $\endgroup$ – Jiminion Jan 22 at 21:31
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    $\begingroup$ @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". $\endgroup$ – Quuxplusone Jan 22 at 21:37
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    $\begingroup$ 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! :) $\endgroup$ – Quuxplusone Jan 23 at 1:43
9
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Can I claim Nine-and-a-half? :-)

enter image description here

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

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  • $\begingroup$ Solution deserves upvote despite bishops attacks each other, better use knight or rook. $\endgroup$ – z100 Jan 23 at 20:43
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    $\begingroup$ @z100 The intention with the bishops was 'one or the other' $\endgroup$ – Daniel Mathias Jan 23 at 20:50
  • $\begingroup$ Solver confirms that this and one other 10+9 solution is optimal, giving no solutions for 10+10. $\endgroup$ – Daniel Mathias Jan 26 at 0:36
16
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Nine queens of each color. Some variation is possible.

enter image description here

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  • $\begingroup$ Nice. Far more asymmetric than my "intended" solution! $\endgroup$ – Quuxplusone Jan 22 at 23:10
6
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Here's 8 peaceful queens of each color:

enter image description here

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

enter image description here

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

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  • $\begingroup$ It's possible ;) $\endgroup$ – Brandon_J Jan 25 at 19:16
6
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I got 8 Black Queens and 10 White Queens:

Peaceful Queens

Also 9 and 9:

enter image description here

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