This is in continuation with our earlier post on the same topic and extends into various other applications and branches in the same domain. I will be discussing today about the Cube Coloring (including the Pólya enumeration theorem) , the minimum number of cuts problem and a typical problem involving painted cubes which will make use of our last article's concepts.
Let's start !
Cube and Cuboid Coloring
1) To paint a cube on all its faces using n colors , we apply the Pólya enumeration theorem , which states that the number of ways to paint a cube using upto n colors(valid only for n <=4) is given by :
(n^6 + 3.n^4 + 12.n^3 + 8.n^2)/24
2) Another problem that is common to this area is finding the number of ways to paint the faces of cube using 6 colors. The answer to that is given by simply 6!/24 or can be seen as painting any one of the faces, choosing another out of the remaining 5 faces and finally permuting for the 3 different orientations possible with last one being automatically decided hence giving us : 5c1*3! = 30 which is same as 6!/24.
NOTE : The common occurrence here of a factor of 24 is the total number of distinct rotational orientations that a cube can assume. This can be visualized as follows :
a) There are 6 faces and so we can place it on each face in turn in 4 possible ways giving 6*4 = 24 possible orientations. How?
Let's say we have face 1 touching the surface face down. A face 2 is automatically decided for any such face 1 which would be exactly opposite to it and face up on the top for us. Now taking these fixed positions we can have any of the four faces facing us giving us 4 different orientations and this assumed face 1 could be any of the 6 faces. Hence 6*4 = 24 rotational orientations of cube.
b) There are 8 vertices and so we can place it on each vertex in a given position in 3 possible ways giving 8*3 = 24 possible orientations. How?
A similar line of thought can be followed by fixing a vertex and seeing how each produces only 3 different orientations giving us 8*3 = 24 orientations for our 8 vertices.
3) Number of ways to paint all the faces of a cuboid with 6 colors is given as : 6!/4 due to only 4 distinct orientations.
Minimum Cuts Problem
This problem asks for minimum number of cuts required for dividing a cube or cuboid of volume N into identical pieces.
Solution to the problem is simple :
Factorize N into three closest factors, say a,b,and c.[Reason : Read article on Minima and Maxima]
Then,
Minimum number of cuts required = (a - 1) + (b - 1) + (c - 1)
Typical Problem on Painted Cubes
A typical problem on painted cubes involves painting a cube on all faces, then cutting a piece from the corner, repainting the exposed surfaces and placing the piece back and notice what impact it has on the number of cubes having none,exactly 1,exactly 2 and exactly 3 faces painted.
In such problems, best approach to follow is the layered numbering that we discussed in the previous article of the same topic. Let's take an example and discuss how the layers of that cube will turn out and take us to our answer.
We have the following situation where a cube is painted , a piece is cut and exposed surfaces are repainted after which the piece is put back.
Going from top to bottom layer by layer we'll get the following layers :
[Before you go ahead, make layers L1 to L5 yourself for a general 5x5x5 cube without any cutting and repositioning. Then make it for this situation. Finally compare that with the following layer structure to confirm your understanding of concept. Make layers from top to bottom to match unlike last article where we went along the breadth in a zoom in direction ]
The red ones show the change due to cutting repainting and repositioning back into slot of small piece of cube(3x3x3) out of a larger cube(5x5x5)
Counting them up you' ll have :
No face painted : 8 [ In normal conditions, would have been (5 - 2)^3 = 27 ]
Exactly 1 : 48 [ In normal conditions, would have been 6(5 - 2)^2 = 54 ]
Exactly 2 : 51 [ In normal conditions, would have been 12(5 - 2) = 36 ]
Exactly 3 : 18 [ In normal conditions, would have been 8 ]
Total : 125
This brings us to the close of all about Painted cubes and cuboids, coloring them and various possibilities that exist in this domain. The layered approach should take you through other variants too and help you master this area of problems.
Leave comments in case of any doubts or queries. Rate, Like and Share if you like the work.
Cheers!
AS
Let's start !
Cube and Cuboid Coloring
1) To paint a cube on all its faces using n colors , we apply the Pólya enumeration theorem , which states that the number of ways to paint a cube using upto n colors(valid only for n <=4) is given by :
(n^6 + 3.n^4 + 12.n^3 + 8.n^2)/24
2) Another problem that is common to this area is finding the number of ways to paint the faces of cube using 6 colors. The answer to that is given by simply 6!/24 or can be seen as painting any one of the faces, choosing another out of the remaining 5 faces and finally permuting for the 3 different orientations possible with last one being automatically decided hence giving us : 5c1*3! = 30 which is same as 6!/24.
NOTE : The common occurrence here of a factor of 24 is the total number of distinct rotational orientations that a cube can assume. This can be visualized as follows :
a) There are 6 faces and so we can place it on each face in turn in 4 possible ways giving 6*4 = 24 possible orientations. How?
Let's say we have face 1 touching the surface face down. A face 2 is automatically decided for any such face 1 which would be exactly opposite to it and face up on the top for us. Now taking these fixed positions we can have any of the four faces facing us giving us 4 different orientations and this assumed face 1 could be any of the 6 faces. Hence 6*4 = 24 rotational orientations of cube.
b) There are 8 vertices and so we can place it on each vertex in a given position in 3 possible ways giving 8*3 = 24 possible orientations. How?
A similar line of thought can be followed by fixing a vertex and seeing how each produces only 3 different orientations giving us 8*3 = 24 orientations for our 8 vertices.
3) Number of ways to paint all the faces of a cuboid with 6 colors is given as : 6!/4 due to only 4 distinct orientations.
Minimum Cuts Problem
This problem asks for minimum number of cuts required for dividing a cube or cuboid of volume N into identical pieces.
Solution to the problem is simple :
Factorize N into three closest factors, say a,b,and c.[Reason : Read article on Minima and Maxima]
Then,
Minimum number of cuts required = (a - 1) + (b - 1) + (c - 1)
Typical Problem on Painted Cubes
A typical problem on painted cubes involves painting a cube on all faces, then cutting a piece from the corner, repainting the exposed surfaces and placing the piece back and notice what impact it has on the number of cubes having none,exactly 1,exactly 2 and exactly 3 faces painted.
In such problems, best approach to follow is the layered numbering that we discussed in the previous article of the same topic. Let's take an example and discuss how the layers of that cube will turn out and take us to our answer.
We have the following situation where a cube is painted , a piece is cut and exposed surfaces are repainted after which the piece is put back.
Going from top to bottom layer by layer we'll get the following layers :
[Before you go ahead, make layers L1 to L5 yourself for a general 5x5x5 cube without any cutting and repositioning. Then make it for this situation. Finally compare that with the following layer structure to confirm your understanding of concept. Make layers from top to bottom to match unlike last article where we went along the breadth in a zoom in direction ]
The red ones show the change due to cutting repainting and repositioning back into slot of small piece of cube(3x3x3) out of a larger cube(5x5x5)
Counting them up you' ll have :
No face painted : 8 [ In normal conditions, would have been (5 - 2)^3 = 27 ]
Exactly 1 : 48 [ In normal conditions, would have been 6(5 - 2)^2 = 54 ]
Exactly 2 : 51 [ In normal conditions, would have been 12(5 - 2) = 36 ]
Exactly 3 : 18 [ In normal conditions, would have been 8 ]
Total : 125
This brings us to the close of all about Painted cubes and cuboids, coloring them and various possibilities that exist in this domain. The layered approach should take you through other variants too and help you master this area of problems.
Leave comments in case of any doubts or queries. Rate, Like and Share if you like the work.
Cheers!
AS
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