coloring 3x3 chessboard with two colors… The 2019 Stack Overflow Developer Survey Results Are In Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)how many unique patterns exist for a $Ntimes N$ gridColorings of a $3times3$ chessboardCheckerboard coloring problemChessboard CombinatoricsProbability of $ 7$ white chessboard squares without neighboursCombinatorics arrangement on chessboardmaximum nonattacking black and white queens on infinite chessboardSimple chessboard exerciseThree knights on a 3x3 chess boardProblem in restricted 3 edge coloring$4times 4$ Chessboard ColoringsA computer screen shows a 98 × 98 chessboard, colored in the usual way.
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coloring 3x3 chessboard with two colors…
The 2019 Stack Overflow Developer Survey Results Are In
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)how many unique patterns exist for a $Ntimes N$ gridColorings of a $3times3$ chessboardCheckerboard coloring problemChessboard CombinatoricsProbability of $ 7$ white chessboard squares without neighboursCombinatorics arrangement on chessboardmaximum nonattacking black and white queens on infinite chessboardSimple chessboard exerciseThree knights on a 3x3 chess boardProblem in restricted 3 edge coloring$4times 4$ Chessboard ColoringsA computer screen shows a 98 × 98 chessboard, colored in the usual way.
$begingroup$
Consider a 3x3 chessboard with 9 elements. The elements are colored with black and white paints.
The task is to find the number of different chessboards of this type exist.
I don't know how to start with this problem....
combinatorics
asked Dec 13 '13 at 23:09
user108696user108696
112
$endgroup$
add a comment |
$begingroup$
Consider a 3x3 chessboard with 9 elements. The elements are colored with black and white paints.
The task is to find the number of different chessboards of this type exist.
I don't know how to start with this problem....
combinatorics
asked Dec 13 '13 at 23:09
user108696user108696
112
$endgroup$
1
$begingroup$
Does the orientation of the board matter? For instance, is the coloring that has a black square in the upper left corner and all the others squares white to be counted as different from the coloring that has a black square in the lower left corner and all the other squares white? Or are they to be counted as just one coloring, because the board can be rotated to turn one into the other?
$endgroup$
– Brian M. Scott
Dec 13 '13 at 23:12
$begingroup$
Draw a shape! And try to color it. Check if anything is ambiguous for you in the problem statement.
$endgroup$
– hhsaffar
Dec 13 '13 at 23:12
$begingroup$
This MSE link shows a solution that takes symmetries into account.
$endgroup$
– Marko Riedel
Dec 13 '13 at 23:15
$begingroup$
HINT: If the orientation of the board does matter, you’re really just counting the number of ways of choosing a subset of the $9$ elements to be colored black. How many subsets are there?
$endgroup$
– Brian M. Scott
Dec 13 '13 at 23:20
add a comment |
$begingroup$
Consider a 3x3 chessboard with 9 elements. The elements are colored with black and white paints.
The task is to find the number of different chessboards of this type exist.
I don't know how to start with this problem....
combinatorics
asked Dec 13 '13 at 23:09
user108696user108696
112
$endgroup$
Consider a 3x3 chessboard with 9 elements. The elements are colored with black and white paints.
The task is to find the number of different chessboards of this type exist.
I don't know how to start with this problem....
combinatorics
combinatorics
asked Dec 13 '13 at 23:09
user108696user108696
112
asked Dec 13 '13 at 23:09
user108696user108696
112
asked Dec 13 '13 at 23:09
user108696user108696
112
asked Dec 13 '13 at 23:09
user108696user108696
112
asked Dec 13 '13 at 23:09
user108696user108696
112
112
1
$begingroup$
Does the orientation of the board matter? For instance, is the coloring that has a black square in the upper left corner and all the others squares white to be counted as different from the coloring that has a black square in the lower left corner and all the other squares white? Or are they to be counted as just one coloring, because the board can be rotated to turn one into the other?
$endgroup$
– Brian M. Scott
Dec 13 '13 at 23:12
$begingroup$
Draw a shape! And try to color it. Check if anything is ambiguous for you in the problem statement.
$endgroup$
– hhsaffar
Dec 13 '13 at 23:12
$begingroup$
This MSE link shows a solution that takes symmetries into account.
$endgroup$
– Marko Riedel
Dec 13 '13 at 23:15
$begingroup$
HINT: If the orientation of the board does matter, you’re really just counting the number of ways of choosing a subset of the $9$ elements to be colored black. How many subsets are there?
$endgroup$
– Brian M. Scott
Dec 13 '13 at 23:20
add a comment |
1
$begingroup$
Does the orientation of the board matter? For instance, is the coloring that has a black square in the upper left corner and all the others squares white to be counted as different from the coloring that has a black square in the lower left corner and all the other squares white? Or are they to be counted as just one coloring, because the board can be rotated to turn one into the other?
$endgroup$
– Brian M. Scott
Dec 13 '13 at 23:12
$begingroup$
Draw a shape! And try to color it. Check if anything is ambiguous for you in the problem statement.
$endgroup$
– hhsaffar
Dec 13 '13 at 23:12
$begingroup$
This MSE link shows a solution that takes symmetries into account.
$endgroup$
– Marko Riedel
Dec 13 '13 at 23:15
$begingroup$
HINT: If the orientation of the board does matter, you’re really just counting the number of ways of choosing a subset of the $9$ elements to be colored black. How many subsets are there?
$endgroup$
– Brian M. Scott
Dec 13 '13 at 23:20
1
1
$begingroup$
Does the orientation of the board matter? For instance, is the coloring that has a black square in the upper left corner and all the others squares white to be counted as different from the coloring that has a black square in the lower left corner and all the other squares white? Or are they to be counted as just one coloring, because the board can be rotated to turn one into the other?
$endgroup$
– Brian M. Scott
Dec 13 '13 at 23:12
$begingroup$
Does the orientation of the board matter? For instance, is the coloring that has a black square in the upper left corner and all the others squares white to be counted as different from the coloring that has a black square in the lower left corner and all the other squares white? Or are they to be counted as just one coloring, because the board can be rotated to turn one into the other?
$endgroup$
– Brian M. Scott
Dec 13 '13 at 23:12
$begingroup$
Draw a shape! And try to color it. Check if anything is ambiguous for you in the problem statement.
$endgroup$
– hhsaffar
Dec 13 '13 at 23:12
$begingroup$
Draw a shape! And try to color it. Check if anything is ambiguous for you in the problem statement.
$endgroup$
– hhsaffar
Dec 13 '13 at 23:12
$begingroup$
This MSE link shows a solution that takes symmetries into account.
$endgroup$
– Marko Riedel
Dec 13 '13 at 23:15
$begingroup$
This MSE link shows a solution that takes symmetries into account.
$endgroup$
– Marko Riedel
Dec 13 '13 at 23:15
$begingroup$
HINT: If the orientation of the board does matter, you’re really just counting the number of ways of choosing a subset of the $9$ elements to be colored black. How many subsets are there?
$endgroup$
– Brian M. Scott
Dec 13 '13 at 23:20
$begingroup$
HINT: If the orientation of the board does matter, you’re really just counting the number of ways of choosing a subset of the $9$ elements to be colored black. How many subsets are there?
$endgroup$
– Brian M. Scott
Dec 13 '13 at 23:20
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Suppose that rotating them does not matter, then we have two colors for each square and
so a total of $2^9$ possibillities.
For the other case, (possibillities that can be made by rotation of another possibillity are considered the same) we can use group theory. I will asume here that the bottom of the board is just black. Because we have $4$ possible roations and no mirroring (because the bottom of the board is black) we get as symmetry group $C_4$, the cyclic group of order $4$. No we will check the orbis of the squares after rotations. This will give us the following:
So for the total possibillities we get (by The Counting Theorem):
$$
frac1sum_gin G |X^g| = \
frac1C_4 (1cdot 2^9 + 4 cdot 2^3 + 2 cdot 2^5+ 4 cdot 2^3)= \
frac14(1cdot 2^9 + 4 cdot 2^3 + 2 cdot 2^5+ 4 cdot 2^3)= 160.
$$
Where $X$ is the set of colourings of the chessboard and $X^g$ is the subset of $X$ consisting of those points which are left fixed by the element $g in G$.
So this will give us a total of $160$ chessboards,
If the back of the chessboard is also in account, for example see-through, please add this to the question. This will give the symmetric group $D_4$ and some extra possobillities, for if you want to work it out.
answered Dec 14 '13 at 0:12
user112167user112167
1,2621715
$endgroup$
add a comment |
$begingroup$
If you consider rotations and reflections to be the same.
Number of $ntimes n$ binary matrices under action of dihedral group of the square $D_4$.
http://oeis.org/A054247
$$a(n)=begincasesdfrac18left(2^n^2+2cdot2^n^2/4+3cdot2^n^2/2+2cdot2^(n^2+n)/2right) & n= 0pmod2 \
dfrac18left(2^n^2+2cdot2^(n^2+3)/4+2^(n^2+1)/2+4cdot2^(n^2+n)/2right)& n= 1pmod2endcases$$
beginalign*
a(3)&=,dfrac18left(2^3^2+2cdot2^(3^2+3)/4+2^(3^2+1)/2+4cdot2^(3^2+3)/2right)\
&=,dfrac18left(2^9+2cdot2^3+2^5+4cdot2^6right)=dfrac2^9+2^4+2^5+2^88=,quadlargecolorred102
endalign*
answered Mar 25 at 5:34
D.MatthewD.Matthew
414
$endgroup$
add a comment |
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2 Answers
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2 Answers
2
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active
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oldest
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$begingroup$
Suppose that rotating them does not matter, then we have two colors for each square and
so a total of $2^9$ possibillities.
For the other case, (possibillities that can be made by rotation of another possibillity are considered the same) we can use group theory. I will asume here that the bottom of the board is just black. Because we have $4$ possible roations and no mirroring (because the bottom of the board is black) we get as symmetry group $C_4$, the cyclic group of order $4$. No we will check the orbis of the squares after rotations. This will give us the following:
So for the total possibillities we get (by The Counting Theorem):
$$
frac1sum_gin G |X^g| = \
frac1C_4 (1cdot 2^9 + 4 cdot 2^3 + 2 cdot 2^5+ 4 cdot 2^3)= \
frac14(1cdot 2^9 + 4 cdot 2^3 + 2 cdot 2^5+ 4 cdot 2^3)= 160.
$$
Where $X$ is the set of colourings of the chessboard and $X^g$ is the subset of $X$ consisting of those points which are left fixed by the element $g in G$.
So this will give us a total of $160$ chessboards,
If the back of the chessboard is also in account, for example see-through, please add this to the question. This will give the symmetric group $D_4$ and some extra possobillities, for if you want to work it out.
answered Dec 14 '13 at 0:12
user112167user112167
1,2621715
$endgroup$
add a comment |
$begingroup$
Suppose that rotating them does not matter, then we have two colors for each square and
so a total of $2^9$ possibillities.
For the other case, (possibillities that can be made by rotation of another possibillity are considered the same) we can use group theory. I will asume here that the bottom of the board is just black. Because we have $4$ possible roations and no mirroring (because the bottom of the board is black) we get as symmetry group $C_4$, the cyclic group of order $4$. No we will check the orbis of the squares after rotations. This will give us the following:
So for the total possibillities we get (by The Counting Theorem):
$$
frac1sum_gin G |X^g| = \
frac1C_4 (1cdot 2^9 + 4 cdot 2^3 + 2 cdot 2^5+ 4 cdot 2^3)= \
frac14(1cdot 2^9 + 4 cdot 2^3 + 2 cdot 2^5+ 4 cdot 2^3)= 160.
$$
Where $X$ is the set of colourings of the chessboard and $X^g$ is the subset of $X$ consisting of those points which are left fixed by the element $g in G$.
So this will give us a total of $160$ chessboards,
If the back of the chessboard is also in account, for example see-through, please add this to the question. This will give the symmetric group $D_4$ and some extra possobillities, for if you want to work it out.
answered Dec 14 '13 at 0:12
user112167user112167
1,2621715
$endgroup$
add a comment |
$begingroup$
Suppose that rotating them does not matter, then we have two colors for each square and
so a total of $2^9$ possibillities.
For the other case, (possibillities that can be made by rotation of another possibillity are considered the same) we can use group theory. I will asume here that the bottom of the board is just black. Because we have $4$ possible roations and no mirroring (because the bottom of the board is black) we get as symmetry group $C_4$, the cyclic group of order $4$. No we will check the orbis of the squares after rotations. This will give us the following:
So for the total possibillities we get (by The Counting Theorem):
$$
frac1sum_gin G |X^g| = \
frac1C_4 (1cdot 2^9 + 4 cdot 2^3 + 2 cdot 2^5+ 4 cdot 2^3)= \
frac14(1cdot 2^9 + 4 cdot 2^3 + 2 cdot 2^5+ 4 cdot 2^3)= 160.
$$
Where $X$ is the set of colourings of the chessboard and $X^g$ is the subset of $X$ consisting of those points which are left fixed by the element $g in G$.
So this will give us a total of $160$ chessboards,
If the back of the chessboard is also in account, for example see-through, please add this to the question. This will give the symmetric group $D_4$ and some extra possobillities, for if you want to work it out.
answered Dec 14 '13 at 0:12
user112167user112167
1,2621715
$endgroup$
Suppose that rotating them does not matter, then we have two colors for each square and
so a total of $2^9$ possibillities.
For the other case, (possibillities that can be made by rotation of another possibillity are considered the same) we can use group theory. I will asume here that the bottom of the board is just black. Because we have $4$ possible roations and no mirroring (because the bottom of the board is black) we get as symmetry group $C_4$, the cyclic group of order $4$. No we will check the orbis of the squares after rotations. This will give us the following:
So for the total possibillities we get (by The Counting Theorem):
$$
frac1sum_gin G |X^g| = \
frac1C_4 (1cdot 2^9 + 4 cdot 2^3 + 2 cdot 2^5+ 4 cdot 2^3)= \
frac14(1cdot 2^9 + 4 cdot 2^3 + 2 cdot 2^5+ 4 cdot 2^3)= 160.
$$
Where $X$ is the set of colourings of the chessboard and $X^g$ is the subset of $X$ consisting of those points which are left fixed by the element $g in G$.
So this will give us a total of $160$ chessboards,
If the back of the chessboard is also in account, for example see-through, please add this to the question. This will give the symmetric group $D_4$ and some extra possobillities, for if you want to work it out.
answered Dec 14 '13 at 0:12
user112167user112167
1,2621715
edited Dec 14 '13 at 0:54
answered Dec 14 '13 at 0:12
user112167user112167
1,2621715
answered Dec 14 '13 at 0:12
user112167user112167
1,2621715
answered Dec 14 '13 at 0:12
user112167user112167
1,2621715
1,2621715
add a comment |
add a comment |
$begingroup$
If you consider rotations and reflections to be the same.
Number of $ntimes n$ binary matrices under action of dihedral group of the square $D_4$.
http://oeis.org/A054247
$$a(n)=begincasesdfrac18left(2^n^2+2cdot2^n^2/4+3cdot2^n^2/2+2cdot2^(n^2+n)/2right) & n= 0pmod2 \
dfrac18left(2^n^2+2cdot2^(n^2+3)/4+2^(n^2+1)/2+4cdot2^(n^2+n)/2right)& n= 1pmod2endcases$$
beginalign*
a(3)&=,dfrac18left(2^3^2+2cdot2^(3^2+3)/4+2^(3^2+1)/2+4cdot2^(3^2+3)/2right)\
&=,dfrac18left(2^9+2cdot2^3+2^5+4cdot2^6right)=dfrac2^9+2^4+2^5+2^88=,quadlargecolorred102
endalign*
answered Mar 25 at 5:34
D.MatthewD.Matthew
414
$endgroup$
add a comment |
$begingroup$
If you consider rotations and reflections to be the same.
Number of $ntimes n$ binary matrices under action of dihedral group of the square $D_4$.
http://oeis.org/A054247
$$a(n)=begincasesdfrac18left(2^n^2+2cdot2^n^2/4+3cdot2^n^2/2+2cdot2^(n^2+n)/2right) & n= 0pmod2 \
dfrac18left(2^n^2+2cdot2^(n^2+3)/4+2^(n^2+1)/2+4cdot2^(n^2+n)/2right)& n= 1pmod2endcases$$
beginalign*
a(3)&=,dfrac18left(2^3^2+2cdot2^(3^2+3)/4+2^(3^2+1)/2+4cdot2^(3^2+3)/2right)\
&=,dfrac18left(2^9+2cdot2^3+2^5+4cdot2^6right)=dfrac2^9+2^4+2^5+2^88=,quadlargecolorred102
endalign*
answered Mar 25 at 5:34
D.MatthewD.Matthew
414
$endgroup$
add a comment |
$begingroup$
If you consider rotations and reflections to be the same.
Number of $ntimes n$ binary matrices under action of dihedral group of the square $D_4$.
http://oeis.org/A054247
$$a(n)=begincasesdfrac18left(2^n^2+2cdot2^n^2/4+3cdot2^n^2/2+2cdot2^(n^2+n)/2right) & n= 0pmod2 \
dfrac18left(2^n^2+2cdot2^(n^2+3)/4+2^(n^2+1)/2+4cdot2^(n^2+n)/2right)& n= 1pmod2endcases$$
beginalign*
a(3)&=,dfrac18left(2^3^2+2cdot2^(3^2+3)/4+2^(3^2+1)/2+4cdot2^(3^2+3)/2right)\
&=,dfrac18left(2^9+2cdot2^3+2^5+4cdot2^6right)=dfrac2^9+2^4+2^5+2^88=,quadlargecolorred102
endalign*
answered Mar 25 at 5:34
D.MatthewD.Matthew
414
$endgroup$
If you consider rotations and reflections to be the same.
Number of $ntimes n$ binary matrices under action of dihedral group of the square $D_4$.
http://oeis.org/A054247
$$a(n)=begincasesdfrac18left(2^n^2+2cdot2^n^2/4+3cdot2^n^2/2+2cdot2^(n^2+n)/2right) & n= 0pmod2 \
dfrac18left(2^n^2+2cdot2^(n^2+3)/4+2^(n^2+1)/2+4cdot2^(n^2+n)/2right)& n= 1pmod2endcases$$
beginalign*
a(3)&=,dfrac18left(2^3^2+2cdot2^(3^2+3)/4+2^(3^2+1)/2+4cdot2^(3^2+3)/2right)\
&=,dfrac18left(2^9+2cdot2^3+2^5+4cdot2^6right)=dfrac2^9+2^4+2^5+2^88=,quadlargecolorred102
endalign*
answered Mar 25 at 5:34
D.MatthewD.Matthew
414
answered Mar 25 at 5:34
D.MatthewD.Matthew
414
answered Mar 25 at 5:34
D.MatthewD.Matthew
414
answered Mar 25 at 5:34
D.MatthewD.Matthew
414
414
add a comment |
add a comment |
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$begingroup$
Does the orientation of the board matter? For instance, is the coloring that has a black square in the upper left corner and all the others squares white to be counted as different from the coloring that has a black square in the lower left corner and all the other squares white? Or are they to be counted as just one coloring, because the board can be rotated to turn one into the other?
$endgroup$
– Brian M. Scott
Dec 13 '13 at 23:12
$begingroup$
Draw a shape! And try to color it. Check if anything is ambiguous for you in the problem statement.
$endgroup$
– hhsaffar
Dec 13 '13 at 23:12
$begingroup$
This MSE link shows a solution that takes symmetries into account.
$endgroup$
– Marko Riedel
Dec 13 '13 at 23:15
$begingroup$
HINT: If the orientation of the board does matter, you’re really just counting the number of ways of choosing a subset of the $9$ elements to be colored black. How many subsets are there?
$endgroup$
– Brian M. Scott
Dec 13 '13 at 23:20