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# How many combinations of colours are there on a 6 sided cube

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1. I'm really confused on how to find out how many combinations there are for 6 colours on a 6 sided cube. I've read that it's 30 but I thought it would be more along the lines of 216. Can someone help me with the formula and maybe the answer?
2. (Original post by mattyeend)
I'm really confused on how to find out how many combinations there are for 6 colours on a 6 sided cube. I've read that it's 30 but I thought it would be more along the lines of 216. Can someone help me with the formula and maybe the answer?
Does this help?
3. Not really no, but thanks
4. 6x5x4x3x2
5. (Original post by offhegoes)
6x5x4x3x2
That is one way, depends on how you look at the problem, but I think really it wants the ones that are symmetrically unique
6. I would want to know how many ways there are to colour the cube and how many unique ways are there to colour the cube?

Offhegoes: how did you get that?
7. (Original post by 13 1 20 8 42)
That is one way, depends on how you look at the problem, but I think really it wants the ones that are symmetrically unique
6 colours, unique combinations?
8. (Original post by mattyeend)
I would want to know how many ways there are to colour the cube and how many unique ways are there to colour the cube?

Offhegoes: how did you get that?
That is how many unique ways there are to colour the cube using exactly 6 colours.

Without going into the theory, imagine picking a colour to paint first and seeing how many options you have. 6 right? Then, for each of those first choices, how many possible locations for the next colour? 5 right? So 6x5 combinations so far. And so on...

The next questions is whether you consider the colouring to be unique if an identical looking cube can be made just be turning it around, if you see what I mean?
9. I understand what you mean for the first bit, so for the second bit, would that be the 30 number I've seen everywhere?
10. (Original post by mattyeend)
Not really no, but thanks
Basically, if we're talking about unique ways to colour the cube i.e. we care precisely which face each colour is on, it is what offhegoes said and that is because we have 6 choices of face for the first colour, then 5 choices of face for the second colour, and so on, so we have 6 lots of 5 lots of 4 lots of 3 lots of 2 (lots of 1).

But if we're talking about ways that just aesthetically look different, i.e. you can rotate the cube around however much you like and it still won't look the same, then that first long answer on the page Zacken linked is the gist of it but I'll try to explain it a bit more.

So for argument's sake let's say our colours are red blue yellow green black and grey. Now, we can decide the bottom face is red because it doesn't matter which particular face it is, you can always rotate the cube around so that red is on the bottom and it will look the same. Then the blue face is either adjacent to it or opposite it.

If it is opposite to it, we then can decide to make the yellow face be the face facing us, because the yellow face facing us is symmetrically identical to it being anywhere else. But now it starts to matter where things are. Looking at the cube we have a yellow face facing us, a red face on the bottom, and a blue face on top. If we put a colour on the left side as we look it looks different to putting it on the right side. So we make a choice of which colour to put on the left side, this is out of 3 choices. Then we make a choice of which of the 2 remaining to put on the right side, this is out of 2. Then the last choice is fixed. This gives 3*2 choices = 6 choices.

Now, if the blue face is adjacent, like we did before rotate the cube around so the blue face faces us. Then as we look, it indeed matters which colour we put on the "left side" as to how the cube looks. So we have 4 choices for that, then 3 choices for the right side, then 2 for the top, then the last is fixed. This gives 4*3*2 = 24 choices.

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