Discrete Peaceful Encampments: 9 queens on a chessboard
$begingroup$
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:
- 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].
- A queen must not be threatened by other queens of a different color.
- 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
asked Jan 22 at 21:23
QuuxplusoneQuuxplusone
227110
$endgroup$
add a comment |
$begingroup$
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:
- 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].
- A queen must not be threatened by other queens of a different color.
- 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
asked Jan 22 at 21:23
QuuxplusoneQuuxplusone
227110
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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:
- 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].
- A queen must not be threatened by other queens of a different color.
- 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
asked Jan 22 at 21:23
QuuxplusoneQuuxplusone
227110
$endgroup$
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:
- 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].
- A queen must not be threatened by other queens of a different color.
- 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
asked Jan 22 at 21:23
QuuxplusoneQuuxplusone
227110
asked Jan 22 at 21:23
QuuxplusoneQuuxplusone
227110
edited Jan 22 at 21:37
Quuxplusone
asked Jan 22 at 21:23
QuuxplusoneQuuxplusone
227110
asked Jan 22 at 21:23
QuuxplusoneQuuxplusone
227110
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
add a comment |
$begingroup$
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
add a comment |
4 Answers
4
active
oldest
votes
$begingroup$
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.
answered Jan 23 at 16:43
BassBass
29k470177
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
Nine queens of each color. Some variation is possible.
answered Jan 22 at 23:07
Daniel MathiasDaniel Mathias
66318
$endgroup$
$begingroup$
Nice. Far more asymmetric than my "intended" solution!
$endgroup$
– Quuxplusone
Jan 22 at 23:10
add a comment |
$begingroup$
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.
answered Jan 22 at 21:31
Excited RaichuExcited Raichu
6,43521166
$endgroup$
$begingroup$
It's possible ;)
$endgroup$
– Brandon_J
Jan 25 at 19:16
add a comment |
$begingroup$
I got 8 Black Queens and 10 White Queens:
Also 9 and 9:
answered Jan 23 at 22:19
Brandon_JBrandon_J
1,22927
$endgroup$
add a comment |
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4 Answers
4
active
oldest
votes
4 Answers
4
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
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.
answered Jan 23 at 16:43
BassBass
29k470177
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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.
answered Jan 23 at 16:43
BassBass
29k470177
$endgroup$
$begingroup$
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
add a comment |
$begingroup$
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.
answered Jan 23 at 16:43
BassBass
29k470177
$endgroup$
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.
answered Jan 23 at 16:43
BassBass
29k470177
answered Jan 23 at 16:43
BassBass
29k470177
answered Jan 23 at 16:43
BassBass
29k470177
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
add a comment |
$begingroup$
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
add a comment |
$begingroup$
Nine queens of each color. Some variation is possible.
answered Jan 22 at 23:07
Daniel MathiasDaniel Mathias
66318
$endgroup$
$begingroup$
Nice. Far more asymmetric than my "intended" solution!
$endgroup$
– Quuxplusone
Jan 22 at 23:10
add a comment |
$begingroup$
Nine queens of each color. Some variation is possible.
answered Jan 22 at 23:07
Daniel MathiasDaniel Mathias
66318
$endgroup$
$begingroup$
Nice. Far more asymmetric than my "intended" solution!
$endgroup$
– Quuxplusone
Jan 22 at 23:10
add a comment |
$begingroup$
Nine queens of each color. Some variation is possible.
answered Jan 22 at 23:07
Daniel MathiasDaniel Mathias
66318
$endgroup$
Nine queens of each color. Some variation is possible.
answered Jan 22 at 23:07
Daniel MathiasDaniel Mathias
66318
answered Jan 22 at 23:07
Daniel MathiasDaniel Mathias
66318
answered Jan 22 at 23:07
Daniel MathiasDaniel Mathias
66318
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
add a comment |
$begingroup$
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
add a comment |
$begingroup$
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.
answered Jan 22 at 21:31
Excited RaichuExcited Raichu
6,43521166
$endgroup$
$begingroup$
It's possible ;)
$endgroup$
– Brandon_J
Jan 25 at 19:16
add a comment |
$begingroup$
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.
answered Jan 22 at 21:31
Excited RaichuExcited Raichu
6,43521166
$endgroup$
$begingroup$
It's possible ;)
$endgroup$
– Brandon_J
Jan 25 at 19:16
add a comment |
$begingroup$
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.
answered Jan 22 at 21:31
Excited RaichuExcited Raichu
6,43521166
$endgroup$
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.
answered Jan 22 at 21:31
Excited RaichuExcited Raichu
6,43521166
edited Jan 22 at 22:06
answered Jan 22 at 21:31
Excited RaichuExcited Raichu
6,43521166
answered Jan 22 at 21:31
Excited RaichuExcited Raichu
6,43521166
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
add a comment |
$begingroup$
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
add a comment |
$begingroup$
I got 8 Black Queens and 10 White Queens:
Also 9 and 9:
answered Jan 23 at 22:19
Brandon_JBrandon_J
1,22927
$endgroup$
add a comment |
$begingroup$
I got 8 Black Queens and 10 White Queens:
Also 9 and 9:
answered Jan 23 at 22:19
Brandon_JBrandon_J
1,22927
$endgroup$
add a comment |
$begingroup$
I got 8 Black Queens and 10 White Queens:
Also 9 and 9:
answered Jan 23 at 22:19
Brandon_JBrandon_J
1,22927
$endgroup$
I got 8 Black Queens and 10 White Queens:
Also 9 and 9:
answered Jan 23 at 22:19
Brandon_JBrandon_J
1,22927
edited Jan 23 at 22:47
answered Jan 23 at 22:19
Brandon_JBrandon_J
1,22927
answered Jan 23 at 22:19
Brandon_JBrandon_J
1,22927
answered Jan 23 at 22:19
Brandon_JBrandon_J
1,22927
1,22927
add a comment |
add a comment |
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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
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