Results 1 to 6 of 6

Math Help - Coloring of a Cube

  1. #1
    Newbie
    Joined
    Mar 2008
    Posts
    11

    Angry Coloring of a Cube

    Hey.. needed some help in the analysis of this problem. Here it goes:

    "A cube has six distinct colours on its faces. How many such unique cubes are possible? "

    PS: The initial looks are lethally deceptive.Please don't get carried away by the simplicity of the statement :P
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by p vs np View Post
    Hey.. needed some help in the analysis of this problem. Here it goes:

    "A cube has six distinct colours on its faces. How many such unique cubes are possible? "

    PS: The initial looks are lethally deceptive.Please don't get carried away by the simplicity of the statement :P
    I think it is 6! for such number of colorings divided by number of cube arrangment which is 6\cdot 5 = 30. Thus we get 720/30=24.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by ThePerfectHacker View Post
    I think it is 6! for such number of colorings divided by number of cube arrangment which is 6\cdot 5 = 30. Thus we get 720/30=24.
    Haven't you got the 24 and the 30 the wrong way round? There are 24 (orientation-preserving) symmetries of the cube, therefore 6!/24 = 30 possible colourings.

    To see this in a totally elementary way, call the six colours A, B, C, D, E and F. We can always place the cube on a table so that the face with colour A is facing downwards. There are 5 possibilities for the colour of the top face. Say this face has colour B. Then we can rotate the cube so that the face with colour C is pointing south (say). There are then 3!=6 distinct ways of colouring the remaining three faces. So the total number of distinct colourings is 56=30.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by Opalg View Post
    Haven't you got the 24 and the 30 the wrong way round? There are 24 (orientation-preserving) symmetries of the cube, therefore 6!/24 = 30 possible colourings.
    Yes! That is exactly what I wanted to say.

    Another way to solve this problem is using Burnside's formula from group theory. Let G be the set of all rotations of the group under the binary operation of composition. There are six faces to choose and for each one there are four more positions (to preserve the order of the cube) thus there are 24 rotations. Thus, |G| = 24 and note that G is a group. Let X be the set of all possible arrangements of the cube (even counting similar ones under rotations) thus |X| = 6! = 720. Let G act on X in an obvious way i.e. it changes the cube under the rotation. Now two cube representations x_1,x_2 are similar if and only if there is a g so that x_2 = gx_1 i.e. x_1,x_2 are in the same orbit. It remains to count the number of orbits. By the formula this is \frac{1}{|G|} \sum_{g\in G}|X_g| . Now |X_g| = \{x\in X | gx = x \}. Note if g is not the identity then |X_g| = 0 and if g is the identity then |X_g| = |X| = 720. Thus, we get \tfrac{720}{24} = 30.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Mar 2008
    Posts
    11

    Neat solution..

    Thank you Mr Opalg.. That was a real cool way to look at the problem.
    and of course, thank you Mr PerfectHacker as well
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member
    Joined
    Mar 2008
    Posts
    934
    Thanks
    33
    Awards
    1
    Quote Originally Posted by p vs np View Post
    Hey.. needed some help in the analysis of this problem. Here it goes:

    "A cube has six distinct colours on its faces. How many such unique cubes are possible? "

    PS: The initial looks are lethally deceptive.Please don't get carried away by the simplicity of the statement :P
    Place the cube on a tabletop so that color #1 is on top. There are then 5 possibilities for the color of the bottom face and 4! / 4 possible ways to color the side faces (dividing by 4 because of the 4 rotations of the cube, keeping the original face on top).

    So there are 5 * 4! / 4 = 30 unique cubes in all.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. graph coloring
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: March 2nd 2011, 02:30 AM
  2. Graph coloring
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: November 2nd 2010, 11:49 AM
  3. Coloring K10
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: December 15th 2009, 10:42 AM
  4. The Hilbert cube homeomorphic to the cube?
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: June 12th 2009, 09:35 AM
  5. Coloring problem
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: December 30th 2008, 06:49 AM

Search Tags


/mathhelpforum @mathhelpforum