3
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What is the largest number of queens that can be placed on a regular $8\times8$ 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.
  2. A queen must not be threatened by other queens of the same color.
  3. Queens threaten all squares in the same row, column, or diagonal (as in chess). Also, threats are blocked by other queens.

Bonus:

Would this number change if rule 1 was changed to enforce equal numbers of black and white queens?

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  • $\begingroup$ Related: puzzling.stackexchange.com/questions/24092/… $\endgroup$ – manshu Mar 14 '16 at 15:46
  • $\begingroup$ Threats are blocked by the other color(s). So, yes. $\endgroup$ – Bojidar Marinov Mar 14 '16 at 18:01
  • $\begingroup$ The conventional meaning of threaten is that you can only threaten a man of the other colour. It would confuse chess players less to make rule 2 "No queen may protect another queen of the same colour". $\endgroup$ – Peter Taylor Mar 15 '16 at 6:00
10
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I'll guess

32 queens.

Every other row is filled with alternating queens. Starting queens on each filled row alternate.

W B W B W B W B
- - - - - - - -
B W B W B W B W
- - - - - - - -
W B W B W B W B
- - - - - - - -
B W B W B W B W

I think this is ideal, since for a single queen, cutting off all lines of sight would require:

B - B - B
- - - - -
- B W B -
- - - - -
B - B - B

Filling white queens into the spaces and repeating the pattern gives us the 32 solution.

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  • 4
    $\begingroup$ If you divide the board into 16 2x2 blocks, no block can have more than one queen of each color. $\endgroup$ – f'' Mar 14 '16 at 16:06
  • 1
    $\begingroup$ Are you sure that opposite queens block the threat? $\endgroup$ – Etoplay Mar 14 '16 at 16:51
  • $\begingroup$ With his comment of "as in chess", I read it as "the queen needs a direct path". $\endgroup$ – charfellow Mar 14 '16 at 17:44
  • $\begingroup$ @Etoplay If opposite queens didn't block each other, the question would only be asking for two eight-queens solutions that don't share a square. $\endgroup$ – f'' Mar 14 '16 at 17:50
  • $\begingroup$ @Etoplay @f - Clarified, they do block. $\endgroup$ – Bojidar Marinov Mar 14 '16 at 18:04

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