Discrete Peaceful Encampments: 9 queens on a chessboard












12












$begingroup$


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?






geometry chess checkerboard







share|improve this question











$endgroup$












  • $begingroup$
    Based on the rules, why couldn't one place 64 white queens or 64 black queens?
    $endgroup$
    – Jiminion
    2 days ago






  • 1




    $begingroup$
    @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
    2 days ago








  • 2




    $begingroup$
    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
    2 days ago
















12












$begingroup$


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?






geometry chess checkerboard







share|improve this question











$endgroup$












  • $begingroup$
    Based on the rules, why couldn't one place 64 white queens or 64 black queens?
    $endgroup$
    – Jiminion
    2 days ago






  • 1




    $begingroup$
    @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
    2 days ago








  • 2




    $begingroup$
    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
    2 days ago














12












12








12





$begingroup$


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?






geometry chess checkerboard







share|improve this question











$endgroup$




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?






geometry chess checkerboard




geometry chess checkerboard






share|improve this question















share|improve this question













share|improve this question




share|improve this question








edited 2 days ago







Quuxplusone

















asked 2 days ago









QuuxplusoneQuuxplusone

26317




26317












  • $begingroup$
    Based on the rules, why couldn't one place 64 white queens or 64 black queens?
    $endgroup$
    – Jiminion
    2 days ago






  • 1




    $begingroup$
    @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
    2 days ago








  • 2




    $begingroup$
    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
    2 days ago


















  • $begingroup$
    Based on the rules, why couldn't one place 64 white queens or 64 black queens?
    $endgroup$
    – Jiminion
    2 days ago






  • 1




    $begingroup$
    @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
    2 days ago








  • 2




    $begingroup$
    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
    2 days ago
















$begingroup$
Based on the rules, why couldn't one place 64 white queens or 64 black queens?
$endgroup$
– Jiminion
2 days ago




$begingroup$
Based on the rules, why couldn't one place 64 white queens or 64 black queens?
$endgroup$
– Jiminion
2 days ago




1




1




$begingroup$
@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
2 days ago






$begingroup$
@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
2 days ago






2




2




$begingroup$
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
2 days ago




$begingroup$
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
2 days ago










4 Answers
4






active

oldest

votes


















14












$begingroup$

Nine queens of each color. Some variation is possible.




enter image description here







share|improve this answer









$endgroup$













  • $begingroup$
    Nice. Far more asymmetric than my "intended" solution!
    $endgroup$
    – Quuxplusone
    2 days ago



















6












$begingroup$

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.






share|improve this answer











$endgroup$





















    6












    $begingroup$

    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.






    share|improve this answer









    $endgroup$













    • $begingroup$
      Solution deserves upvote despite bishops attacks each other, better use knight or rook.
      $endgroup$
      – z100
      yesterday








    • 2




      $begingroup$
      @z100 The intention with the bishops was 'one or the other'
      $endgroup$
      – Daniel Mathias
      yesterday



















    3












    $begingroup$

    I got 8 Black Queens and 10 White Queens:




    Peaceful Queens




    Also 9 and 9:




    enter image description here







    share|improve this answer











    $endgroup$













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      4 Answers
      4






      active

      oldest

      votes








      4 Answers
      4






      active

      oldest

      votes









      active

      oldest

      votes






      active

      oldest

      votes









      14












      $begingroup$

      Nine queens of each color. Some variation is possible.




      enter image description here







      share|improve this answer









      $endgroup$













      • $begingroup$
        Nice. Far more asymmetric than my "intended" solution!
        $endgroup$
        – Quuxplusone
        2 days ago
















      14












      $begingroup$

      Nine queens of each color. Some variation is possible.




      enter image description here







      share|improve this answer









      $endgroup$













      • $begingroup$
        Nice. Far more asymmetric than my "intended" solution!
        $endgroup$
        – Quuxplusone
        2 days ago














      14












      14








      14





      $begingroup$

      Nine queens of each color. Some variation is possible.




      enter image description here







      share|improve this answer









      $endgroup$



      Nine queens of each color. Some variation is possible.




      enter image description here








      share|improve this answer












      share|improve this answer



      share|improve this answer










      answered 2 days ago









      Daniel MathiasDaniel Mathias

      6336




      6336












      • $begingroup$
        Nice. Far more asymmetric than my "intended" solution!
        $endgroup$
        – Quuxplusone
        2 days ago


















      • $begingroup$
        Nice. Far more asymmetric than my "intended" solution!
        $endgroup$
        – Quuxplusone
        2 days ago
















      $begingroup$
      Nice. Far more asymmetric than my "intended" solution!
      $endgroup$
      – Quuxplusone
      2 days ago




      $begingroup$
      Nice. Far more asymmetric than my "intended" solution!
      $endgroup$
      – Quuxplusone
      2 days ago











      6












      $begingroup$

      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.






      share|improve this answer











      $endgroup$


















        6












        $begingroup$

        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.






        share|improve this answer











        $endgroup$
















          6












          6








          6





          $begingroup$

          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.






          share|improve this answer











          $endgroup$



          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.







          share|improve this answer














          share|improve this answer



          share|improve this answer








          edited 2 days ago

























          answered 2 days ago









          Excited RaichuExcited Raichu

          6,42021066




          6,42021066























              6












              $begingroup$

              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.






              share|improve this answer









              $endgroup$













              • $begingroup$
                Solution deserves upvote despite bishops attacks each other, better use knight or rook.
                $endgroup$
                – z100
                yesterday








              • 2




                $begingroup$
                @z100 The intention with the bishops was 'one or the other'
                $endgroup$
                – Daniel Mathias
                yesterday
















              6












              $begingroup$

              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.






              share|improve this answer









              $endgroup$













              • $begingroup$
                Solution deserves upvote despite bishops attacks each other, better use knight or rook.
                $endgroup$
                – z100
                yesterday








              • 2




                $begingroup$
                @z100 The intention with the bishops was 'one or the other'
                $endgroup$
                – Daniel Mathias
                yesterday














              6












              6








              6





              $begingroup$

              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.






              share|improve this answer









              $endgroup$



              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.







              share|improve this answer












              share|improve this answer



              share|improve this answer










              answered yesterday









              BassBass

              28.4k469175




              28.4k469175












              • $begingroup$
                Solution deserves upvote despite bishops attacks each other, better use knight or rook.
                $endgroup$
                – z100
                yesterday








              • 2




                $begingroup$
                @z100 The intention with the bishops was 'one or the other'
                $endgroup$
                – Daniel Mathias
                yesterday


















              • $begingroup$
                Solution deserves upvote despite bishops attacks each other, better use knight or rook.
                $endgroup$
                – z100
                yesterday








              • 2




                $begingroup$
                @z100 The intention with the bishops was 'one or the other'
                $endgroup$
                – Daniel Mathias
                yesterday
















              $begingroup$
              Solution deserves upvote despite bishops attacks each other, better use knight or rook.
              $endgroup$
              – z100
              yesterday






              $begingroup$
              Solution deserves upvote despite bishops attacks each other, better use knight or rook.
              $endgroup$
              – z100
              yesterday






              2




              2




              $begingroup$
              @z100 The intention with the bishops was 'one or the other'
              $endgroup$
              – Daniel Mathias
              yesterday




              $begingroup$
              @z100 The intention with the bishops was 'one or the other'
              $endgroup$
              – Daniel Mathias
              yesterday











              3












              $begingroup$

              I got 8 Black Queens and 10 White Queens:




              Peaceful Queens




              Also 9 and 9:




              enter image description here







              share|improve this answer











              $endgroup$


















                3












                $begingroup$

                I got 8 Black Queens and 10 White Queens:




                Peaceful Queens




                Also 9 and 9:




                enter image description here







                share|improve this answer











                $endgroup$
















                  3












                  3








                  3





                  $begingroup$

                  I got 8 Black Queens and 10 White Queens:




                  Peaceful Queens




                  Also 9 and 9:




                  enter image description here







                  share|improve this answer











                  $endgroup$



                  I got 8 Black Queens and 10 White Queens:




                  Peaceful Queens




                  Also 9 and 9:




                  enter image description here








                  share|improve this answer














                  share|improve this answer



                  share|improve this answer








                  edited yesterday

























                  answered yesterday









                  Brandon_JBrandon_J

                  1,05825




                  1,05825






























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