Suppose there is a cube and we can colour the cube's faces

In summary, by using Burnside's theorem, we can determine that there are 64 different patterns possible when coloring a cube's faces with only black and white. However, when considering symmetry, there are only 10 distinct ways to color the cube's faces. The formula used to calculate this is derived from Polya's theorem, which may not be entirely clear.
  • #1
heman
361
0
suppose there is a cube and we can colour the cube's faces with only two colours ..i.e.
black and white ,,how many different patterns are possible...
 
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  • #2
Seems like the perfect place to apply Burnside's theorem (about orbits).
 
  • #3
Here's my take:

How many faces are on a cube? 6
How many allowable variations per face? 2
Combinations possible: 2*2*2*2*2*2 =64

Oh, but wait, this is not a strictly linear situation!
My mind is melting.
 
  • #4
all white: 1 version
1 black side: 1 version
2 black: 2 versions (adjacent/nonadjacent)
3 black: 2 versions (three in a row/three sides with common vertex)
and by symmetry...
4 black = 2 white: 2 versions
5 black: 1 version
6 black: 1 version

Total... 10 indistinguishable ways.
 
  • #5
Thanks Rach,,
i am clear with how that value 10 comes..but actually the theory related with Polya's theorem is n't much clear to me...i am actually not clear with how they have derived the formula.
 

What is a cube?

A cube is a three-dimensional shape with six congruent square faces. It is a type of geometric solid and is often used in mathematics and geometry.

What do you mean by "colour the cube's faces"?

Colouring the cube's faces means assigning a specific color to each of the six square faces of the cube. This can be done using paint, markers, or digital tools such as computer software.

Can we use any color to color the cube's faces?

Yes, you can use any color you want to color the cube's faces. There are no restrictions on the colors that can be used, as long as each face is a different color.

How many different ways can we color the cube's faces?

There are 3674160 different ways to color the cube's faces. This number can be calculated using the formula 6! / 6 = 3674160, where 6! represents the number of ways to arrange 6 different colors on the 6 faces.

What is the significance of coloring the cube's faces?

Coloring the cube's faces is a fun and creative way to explore mathematical concepts such as symmetry, patterns, and combinations. It also helps develop spatial awareness and visual perception skills. Additionally, it can be used to teach basic concepts of geometry and art.

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