Possibilities to decorate a cube with different patterns

Asked Nov 14 '17 at 23:49

Active Nov 15 '17 at 00:21

Viewed 38 times

I have the following problem:

You're trying to decorate a cube. How many essentially different cubes are there, if there are six different symmetric patterns to choose from for every side? Also, how many essentially different cubes are there, if it's the same situation, except for the fact that one of them can only be used once?

For the first problem I found the solution, by using the formula: $$\frac{1}{24}(1*x^6+3*2*x^3+3*x^4+6*x^3+4*2*x^2)$$ for x being the amount of patterns to choose from for every side. So for x=6 there should be 2226 possibilities. But for the second problem, I have no clue how to solve it. There are two cases:

1) This particular pattern is not on the cube. So there are according to the formula 800 possibilities.

2) The pattern is there exactly once. But how can I find the possible decorations in that case?

Can someone help me?

edited Nov 15 '17 at 00:21

asked Nov 14 '17 at 23:49

Tobi92sr

1,691

Is the pattern symmetric? That is, should you take into account rotations of the pattern by 90 or 180 or270 degrees? – Bram28 Nov 14 '17 at 23:57
Yes, the patterns are all symmetric. I'll edit it. – Tobi92sr Nov 15 '17 at 00:20
So you could think of it as having 6 different colors, right? – Bram28 Nov 15 '17 at 00:23
Yes, you are right. – Tobi92sr Nov 15 '17 at 00:29
Possible duplicate of Painting the faces of a cube with distinct colours – iamwhoiam Nov 15 '17 at 00:40
In my problem the colours could appear more than once. So there could be one side blue and the other five could be red. – Tobi92sr Nov 15 '17 at 01:25

Possibilities to decorate a cube with different patterns

0 Answers0