Doubt about $n$ queens problem when $gcd(n,6)=pm 1$ Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)two questions about Diophantine EquationMinimum queens to reach $8 times 8$ squares as a graph problemmaximum nonattacking black and white queens on infinite chessboardHow would someone go about proving the Frobenius Coin Problem (for m = 3)???Greatest number of non-attacking moves that queens can make on an $n times n$ chess board.the 8 queens conjecture: the next level (or dimension)Show that only finitely many positive integers $n$ have the property that for all $m$ for which $1<m<n$ and $textgcd(m,n) = 1$, $m$ is prime.Reference on properties of the queen graph $Q_n$?find $G=gcd(a^m+1,a^n+1)$ from a problemextending 8 queen chess puzzle to n*n chessboard
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Doubt about $n$ queens problem when $gcd(n,6)=pm 1$
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)two questions about Diophantine EquationMinimum queens to reach $8 times 8$ squares as a graph problemmaximum nonattacking black and white queens on infinite chessboardHow would someone go about proving the Frobenius Coin Problem (for m = 3)???Greatest number of non-attacking moves that queens can make on an $n times n$ chess board.the 8 queens conjecture: the next level (or dimension)Show that only finitely many positive integers $n$ have the property that for all $m$ for which $1<m<n$ and $textgcd(m,n) = 1$, $m$ is prime.Reference on properties of the queen graph $Q_n$?find $G=gcd(a^m+1,a^n+1)$ from a problemextending 8 queen chess puzzle to n*n chessboard
$begingroup$
So I know that the $n$-queens problem only has modular solutions if $gcd(n,6) = pm 1$
My doubt is: if $gcd(n,6) = pm 1$ then are all solutions modular? Or can there be normal solutions which are not preserved by translation?
number-theory graph-theory chessboard
asked Apr 1 at 18:37
karlaboskarlabos
406414
$endgroup$
add a comment |
$begingroup$
So I know that the $n$-queens problem only has modular solutions if $gcd(n,6) = pm 1$
My doubt is: if $gcd(n,6) = pm 1$ then are all solutions modular? Or can there be normal solutions which are not preserved by translation?
number-theory graph-theory chessboard
asked Apr 1 at 18:37
karlaboskarlabos
406414
$endgroup$
2
$begingroup$
Please explain what you mean by a modular solution.
$endgroup$
– saulspatz
Apr 1 at 18:42
$begingroup$
I mean solutions which remain solution after a (modular) translation of the board is applied (say, every queen shifts i to the right and j down, identifying the 1st row as the one that comes next the last and likewise for column) A modular solution has the property that even after a shift like that it remains a solution
$endgroup$
– karlabos
Apr 1 at 18:46
$begingroup$
How can a gcd be negative?
$endgroup$
– Servaes
Apr 1 at 21:29
add a comment |
$begingroup$
So I know that the $n$-queens problem only has modular solutions if $gcd(n,6) = pm 1$
My doubt is: if $gcd(n,6) = pm 1$ then are all solutions modular? Or can there be normal solutions which are not preserved by translation?
number-theory graph-theory chessboard
asked Apr 1 at 18:37
karlaboskarlabos
406414
$endgroup$
So I know that the $n$-queens problem only has modular solutions if $gcd(n,6) = pm 1$
My doubt is: if $gcd(n,6) = pm 1$ then are all solutions modular? Or can there be normal solutions which are not preserved by translation?
number-theory graph-theory chessboard
number-theory graph-theory chessboard
asked Apr 1 at 18:37
karlaboskarlabos
406414
asked Apr 1 at 18:37
karlaboskarlabos
406414
asked Apr 1 at 18:37
karlaboskarlabos
406414
asked Apr 1 at 18:37
karlaboskarlabos
406414
asked Apr 1 at 18:37
karlaboskarlabos
406414
406414
2
$begingroup$
Please explain what you mean by a modular solution.
$endgroup$
– saulspatz
Apr 1 at 18:42
$begingroup$
I mean solutions which remain solution after a (modular) translation of the board is applied (say, every queen shifts i to the right and j down, identifying the 1st row as the one that comes next the last and likewise for column) A modular solution has the property that even after a shift like that it remains a solution
$endgroup$
– karlabos
Apr 1 at 18:46
$begingroup$
How can a gcd be negative?
$endgroup$
– Servaes
Apr 1 at 21:29
add a comment |
2
$begingroup$
Please explain what you mean by a modular solution.
$endgroup$
– saulspatz
Apr 1 at 18:42
$begingroup$
I mean solutions which remain solution after a (modular) translation of the board is applied (say, every queen shifts i to the right and j down, identifying the 1st row as the one that comes next the last and likewise for column) A modular solution has the property that even after a shift like that it remains a solution
$endgroup$
– karlabos
Apr 1 at 18:46
$begingroup$
How can a gcd be negative?
$endgroup$
– Servaes
Apr 1 at 21:29
2
2
$begingroup$
Please explain what you mean by a modular solution.
$endgroup$
– saulspatz
Apr 1 at 18:42
$begingroup$
Please explain what you mean by a modular solution.
$endgroup$
– saulspatz
Apr 1 at 18:42
$begingroup$
I mean solutions which remain solution after a (modular) translation of the board is applied (say, every queen shifts i to the right and j down, identifying the 1st row as the one that comes next the last and likewise for column) A modular solution has the property that even after a shift like that it remains a solution
$endgroup$
– karlabos
Apr 1 at 18:46
$begingroup$
I mean solutions which remain solution after a (modular) translation of the board is applied (say, every queen shifts i to the right and j down, identifying the 1st row as the one that comes next the last and likewise for column) A modular solution has the property that even after a shift like that it remains a solution
$endgroup$
– karlabos
Apr 1 at 18:46
$begingroup$
How can a gcd be negative?
$endgroup$
– Servaes
Apr 1 at 21:29
$begingroup$
How can a gcd be negative?
$endgroup$
– Servaes
Apr 1 at 21:29
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
It seems that all $5$ queens solutions are modular, but here is a $7$ queens solution which is not:
$$
beginarrayc
hline
&Qrule[-0.6ex]0ex3ex\
hline
&&&Qrule[-0.6ex]0ex3ex\
hline
Qrule[-0.6ex]0ex3ex\
hline
&&&&&&Qrule[-0.6ex]0ex3ex\
hline
&&&&Qrule[-0.6ex]0ex3ex\
hline
&&Qrule[-0.6ex]0ex3ex\
hline
&&&&&Qrule[-0.6ex]0ex3ex\
hline
endarray
$$
Rotating this board to the left or right will cause a conflict between the queens on the third and fourth rows.
answered Apr 1 at 21:38
FredHFredH
3,6851024
$endgroup$
add a comment |
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1 Answer
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1 Answer
1
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oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
It seems that all $5$ queens solutions are modular, but here is a $7$ queens solution which is not:
$$
beginarrayc
hline
&Qrule[-0.6ex]0ex3ex\
hline
&&&Qrule[-0.6ex]0ex3ex\
hline
Qrule[-0.6ex]0ex3ex\
hline
&&&&&&Qrule[-0.6ex]0ex3ex\
hline
&&&&Qrule[-0.6ex]0ex3ex\
hline
&&Qrule[-0.6ex]0ex3ex\
hline
&&&&&Qrule[-0.6ex]0ex3ex\
hline
endarray
$$
Rotating this board to the left or right will cause a conflict between the queens on the third and fourth rows.
answered Apr 1 at 21:38
FredHFredH
3,6851024
$endgroup$
add a comment |
$begingroup$
It seems that all $5$ queens solutions are modular, but here is a $7$ queens solution which is not:
$$
beginarrayc
hline
&Qrule[-0.6ex]0ex3ex\
hline
&&&Qrule[-0.6ex]0ex3ex\
hline
Qrule[-0.6ex]0ex3ex\
hline
&&&&&&Qrule[-0.6ex]0ex3ex\
hline
&&&&Qrule[-0.6ex]0ex3ex\
hline
&&Qrule[-0.6ex]0ex3ex\
hline
&&&&&Qrule[-0.6ex]0ex3ex\
hline
endarray
$$
Rotating this board to the left or right will cause a conflict between the queens on the third and fourth rows.
answered Apr 1 at 21:38
FredHFredH
3,6851024
$endgroup$
add a comment |
$begingroup$
It seems that all $5$ queens solutions are modular, but here is a $7$ queens solution which is not:
$$
beginarrayc
hline
&Qrule[-0.6ex]0ex3ex\
hline
&&&Qrule[-0.6ex]0ex3ex\
hline
Qrule[-0.6ex]0ex3ex\
hline
&&&&&&Qrule[-0.6ex]0ex3ex\
hline
&&&&Qrule[-0.6ex]0ex3ex\
hline
&&Qrule[-0.6ex]0ex3ex\
hline
&&&&&Qrule[-0.6ex]0ex3ex\
hline
endarray
$$
Rotating this board to the left or right will cause a conflict between the queens on the third and fourth rows.
answered Apr 1 at 21:38
FredHFredH
3,6851024
$endgroup$
It seems that all $5$ queens solutions are modular, but here is a $7$ queens solution which is not:
$$
beginarrayc
hline
&Qrule[-0.6ex]0ex3ex\
hline
&&&Qrule[-0.6ex]0ex3ex\
hline
Qrule[-0.6ex]0ex3ex\
hline
&&&&&&Qrule[-0.6ex]0ex3ex\
hline
&&&&Qrule[-0.6ex]0ex3ex\
hline
&&Qrule[-0.6ex]0ex3ex\
hline
&&&&&Qrule[-0.6ex]0ex3ex\
hline
endarray
$$
Rotating this board to the left or right will cause a conflict between the queens on the third and fourth rows.
answered Apr 1 at 21:38
FredHFredH
3,6851024
answered Apr 1 at 21:38
FredHFredH
3,6851024
answered Apr 1 at 21:38
FredHFredH
3,6851024
answered Apr 1 at 21:38
FredHFredH
3,6851024
3,6851024
add a comment |
add a comment |
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2
$begingroup$
Please explain what you mean by a modular solution.
$endgroup$
– saulspatz
Apr 1 at 18:42
$begingroup$
I mean solutions which remain solution after a (modular) translation of the board is applied (say, every queen shifts i to the right and j down, identifying the 1st row as the one that comes next the last and likewise for column) A modular solution has the property that even after a shift like that it remains a solution
$endgroup$
– karlabos
Apr 1 at 18:46
$begingroup$
How can a gcd be negative?
$endgroup$
– Servaes
Apr 1 at 21:29