Discrete Peaceful Encampments: 9 queens on a chessboard












14












$begingroup$


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
    Jan 22 at 21:31






  • 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
    Jan 22 at 21:37








  • 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
    Jan 23 at 1:43
















14












$begingroup$


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
    Jan 22 at 21:31






  • 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
    Jan 22 at 21:37








  • 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
    Jan 23 at 1:43














14












14








14





$begingroup$


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 Jan 22 at 21:37







Quuxplusone

















asked Jan 22 at 21:23









QuuxplusoneQuuxplusone

227110




227110












  • $begingroup$
    Based on the rules, why couldn't one place 64 white queens or 64 black queens?
    $endgroup$
    – Jiminion
    Jan 22 at 21:31






  • 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
    Jan 22 at 21:37








  • 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
    Jan 23 at 1:43


















  • $begingroup$
    Based on the rules, why couldn't one place 64 white queens or 64 black queens?
    $endgroup$
    – Jiminion
    Jan 22 at 21:31






  • 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
    Jan 22 at 21:37








  • 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
    Jan 23 at 1:43
















$begingroup$
Based on the rules, why couldn't one place 64 white queens or 64 black queens?
$endgroup$
– Jiminion
Jan 22 at 21:31




$begingroup$
Based on the rules, why couldn't one place 64 white queens or 64 black queens?
$endgroup$
– Jiminion
Jan 22 at 21:31




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
Jan 22 at 21:37






$begingroup$
@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






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
Jan 23 at 1:43




$begingroup$
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










4 Answers
4






active

oldest

votes


















7












$begingroup$

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
    Jan 23 at 20:43








  • 2




    $begingroup$
    @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



















15












$begingroup$

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
    Jan 22 at 23:10



















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











$endgroup$













  • $begingroup$
    It's possible ;)
    $endgroup$
    – Brandon_J
    Jan 25 at 19:16



















4












$begingroup$

I got 8 Black Queens and 10 White Queens:




Peaceful Queens




Also 9 and 9:




enter image description here







share|improve this answer











$endgroup$













    Your Answer





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    4 Answers
    4






    active

    oldest

    votes








    4 Answers
    4






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    7












    $begingroup$

    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
      Jan 23 at 20:43








    • 2




      $begingroup$
      @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
















    7












    $begingroup$

    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
      Jan 23 at 20:43








    • 2




      $begingroup$
      @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














    7












    7








    7





    $begingroup$

    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 Jan 23 at 16:43









    BassBass

    29k470177




    29k470177












    • $begingroup$
      Solution deserves upvote despite bishops attacks each other, better use knight or rook.
      $endgroup$
      – z100
      Jan 23 at 20:43








    • 2




      $begingroup$
      @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


















    • $begingroup$
      Solution deserves upvote despite bishops attacks each other, better use knight or rook.
      $endgroup$
      – z100
      Jan 23 at 20:43








    • 2




      $begingroup$
      @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
















    $begingroup$
    Solution deserves upvote despite bishops attacks each other, better use knight or rook.
    $endgroup$
    – z100
    Jan 23 at 20:43






    $begingroup$
    Solution deserves upvote despite bishops attacks each other, better use knight or rook.
    $endgroup$
    – z100
    Jan 23 at 20:43






    2




    2




    $begingroup$
    @z100 The intention with the bishops was 'one or the other'
    $endgroup$
    – Daniel Mathias
    Jan 23 at 20:50




    $begingroup$
    @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




    $begingroup$
    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











    15












    $begingroup$

    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
      Jan 22 at 23:10
















    15












    $begingroup$

    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
      Jan 22 at 23:10














    15












    15








    15





    $begingroup$

    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 Jan 22 at 23:07









    Daniel MathiasDaniel Mathias

    66318




    66318












    • $begingroup$
      Nice. Far more asymmetric than my "intended" solution!
      $endgroup$
      – Quuxplusone
      Jan 22 at 23:10


















    • $begingroup$
      Nice. Far more asymmetric than my "intended" solution!
      $endgroup$
      – Quuxplusone
      Jan 22 at 23:10
















    $begingroup$
    Nice. Far more asymmetric than my "intended" solution!
    $endgroup$
    – Quuxplusone
    Jan 22 at 23:10




    $begingroup$
    Nice. Far more asymmetric than my "intended" solution!
    $endgroup$
    – Quuxplusone
    Jan 22 at 23:10











    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











    $endgroup$













    • $begingroup$
      It's possible ;)
      $endgroup$
      – Brandon_J
      Jan 25 at 19:16
















    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











    $endgroup$













    • $begingroup$
      It's possible ;)
      $endgroup$
      – Brandon_J
      Jan 25 at 19:16














    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 Jan 22 at 22:06

























    answered Jan 22 at 21:31









    Excited RaichuExcited Raichu

    6,43521166




    6,43521166












    • $begingroup$
      It's possible ;)
      $endgroup$
      – Brandon_J
      Jan 25 at 19:16


















    • $begingroup$
      It's possible ;)
      $endgroup$
      – Brandon_J
      Jan 25 at 19:16
















    $begingroup$
    It's possible ;)
    $endgroup$
    – Brandon_J
    Jan 25 at 19:16




    $begingroup$
    It's possible ;)
    $endgroup$
    – Brandon_J
    Jan 25 at 19:16











    4












    $begingroup$

    I got 8 Black Queens and 10 White Queens:




    Peaceful Queens




    Also 9 and 9:




    enter image description here







    share|improve this answer











    $endgroup$


















      4












      $begingroup$

      I got 8 Black Queens and 10 White Queens:




      Peaceful Queens




      Also 9 and 9:




      enter image description here







      share|improve this answer











      $endgroup$
















        4












        4








        4





        $begingroup$

        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 Jan 23 at 22:47

























        answered Jan 23 at 22:19









        Brandon_JBrandon_J

        1,22927




        1,22927






























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            Title Spacing in Bjornstrup Chapter, Removing Chapter Number From Contents

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            up vote 5 down vote favorite 2 I would like to make it so that the vertical space below the title of each chapter is the same when using the Bjornstrup package (i.e. so that the descenders do not affect the spacing - as shown by the red arrow in the picture below). I think I'm fairly happy with the spacing around "A chapter", so I'd really like titles without descenders (e.g. "Contents") to have the same vertical spacing as "A chapter" below the title. I would also like to remove the large "0" in the contents title. Any help appreciated, as always. documentclass[11pt,a4paper]{book} % Packages[![enter image description here][1]][1] usepackage[left=3.5cm, right=3.5cm, top=4.3cm, bottom=4.25cm]{geometry} usepackage[utf8]{inputenc} usepackage{libertine} usepackage[libertine]{newtx
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