Probability Question (1 Viewer)

[lilainjel]

Just a random question =) Each cube of a set of equal cubes has its six faces painted red, orange, yellow, green, blue and white respectively so that no two cubes are alike and every arrangement of colours is used. How many cubes are there in the set?Thanks in advance

 

[luv2luvurmama]

Number of Ways = 4 / penis

 

[lilainjel]

Oh, my bad. Don't know why I called it probability o.o More, perms and combs I think.

 

[lolokay]

if you look at, say, the red square, then there are 5 colours that could go on the opposite face of the cube. you now have 4 remaining faces, which form the equivalent of a circle, so there are 3! = 6 ways to paint these, so 5*6=30 ways

 

[kwabon]

couldnt really grasp the question. so each of side of the cube is colored in a different color? :-S, if thats the case, how would the cubes be different?

 

[Top Secret]

you now have 4 remaining faces, which form the equivalent of a circle, so there are 3! = 6 ways

Hmm, are you sure about this?

 

[lolokay]

Hmm, are you sure about this?

maybe not, I didn't really give too much thought to the question. can you something wrong with it?

 

[biopia]

May I add my ideas to the table lol?
There is obviously going to be more than one cube in the set.
I like the idea of establishing one colour first, e.g. red.
So on a cube, the front facing side is red, this means for the opposite face there are 5 colours than can go there. As for the other four sides, they can be any arrangement of colours, so wouldn't it be 4! ways for the other sides? Dunno if that is 100% correct, but let's say it is for a minute That means there is 5x24 (5!)=120 ways of constructing the a cube where red is the front facing side. Wouldn't that means there are 120 cubes in the set?

Interesting question.

 

[Top Secret]

maybe not, I didn't really give too much thought to the question. can you something wrong with it?

I thought this would've been correct:

As for the other four sides, they can be any arrangement of colours, so wouldn't it be 4! ways for the other sides? Dunno if that is 100% correct

 

[lolokay]

have you done questions with people sitting in a circle before? consider the remaining 4 colours as being in a circle; so, by rotating the cube, even though the colours are now in different positions, it is still the same arrangement

I like the idea of establishing one colour first, e.g. red.

do a similar thing with the remaining 4 colours
if you look at it by keeping one of those colours in a fixed position, then there are only 3 colours left to arrage, i.e. 3! ways

 

[Top Secret]

have you done questions with people sitting in a circle before? consider the remaining 4 colours as being in a circle; so, by rotating the cube, even though the colours are now in different positions, it is still the same arrangement

Yes I was wondering about this.
When you talk about arrangement in a circle, the order does remain the same, but only if we allow alternating perceptions.
ABCD read clockwise is not the same as ABCD read anticlockwise
I believe the same principle applies here.

do a similar thing with the remaining 4 colours
if you look at it by keeping one of those colours in a fixed position, then there are only 3 colours left to arrage, i.e. 3! ways

But you have already fixed a colour. Cannot do that again.

 

[biopia]

That circle theory, where it is one less than the number of positions, doesn't that only count when order is important? Different combinations of colours have no order because they are different colours anyway.
If you have three men and three women in a circle, however, order is important because nothing differentiates between the three women or the three men?

And I also agree with what has just been said, since you have fixed one colour, you can't fix another.

Theoretically however, can't you unfold a cube in such a way that it is a circle anyway? It may be a weird circle, but I am sure you could. Wouldn't that make it 5! anyway? That just came into my mind though, so who knows lol.

 

[lolokay]

ok.. suppose you had a cube with a blue top, and a red bottom. you then have four other colours to colour the other 4 sides with. in how many ways can this be done, so that no two cubes are alike?

 

[Top Secret]

ok.. suppose you had a cube with no top, and a red bottom. you then have four other colours to colour the other 4 sides with. in how many ways can this be done, so that no two cubes are alike?

4! because of the fixed perception.

Remember, I'm not denying you. From what I've heard, you're the maths king around here, so I'm likely to be wrong here.

 

[biopia]

I've just thought of something else.
If you fix red, and then fix the colour opposite, there are four colours left over. Keeping them in the same order and 'rotating' them around the cube makes four different cubes. If you reverse the order of just those four cubes, it will still create four different cubes. This may be further explanation why the circle theory doesn't apply here?

I discovered this by drawing the nets of some cubes, keeping red and its opposing colour the same, and reversing the order. If you mentally construct the cubes after this, you'll see it creates two different cubes. So even though the order in the same, it creates different cubes.

Man, I don't even know if I believe any of that anymore haha. I keep confusing myself!

 

[lolokay]

4! because of the fixed perception.

it's not fixed though ? how is this any different to having four people arranged in a circle? instead, it's four colours arranged in a circle

 

[gurmies]

Can't see a flaw with lolokay's solution.

 

[Top Secret]

Can't see a flaw with lolokay's solution.

I've had a deeper look, and it seems lolokay's solution is correct.

Guess I was just over-complicating things.

 

[eldore44]

too many people should not be doing ext math. fix a colour on one side. the opposite side then can be 5 diff ones. the 4 remaining colours form a square, which is round, just like a circle. being a circle you need to fix one of them (because a cube is not a unique cube just because you rotate it 90 degrees) so this leaves us with 3! ways. = 6 5 x 6 =30 ie, as was said by lolokay or you can imagine it as a topless cube. with 5! ways of making it. however the 4 that go around the sides may be viewed from 4 directions, so divide by 4. 5!/4=30

 

[biopia]

Lol, these type of questions always start the most amazing discussions
I went to seminar for ext 1 recently, and no joke, out of 3 and a half hour seminar, we spend half an hour on one of these counting technique question. There were like 5 different answer - luckily for me, that time I was correct
Thanks for the input hehe.