My answers are
Question 1 = 15, you agree
Question 2 = 45, Jim got 135
Question 3 = 30
I have a feeling I made another mistake. Seems to me that once you
have the ring of 4 colors around the cube, there are only 2 ways to
assign the remaining 2 colors so answer 3 should be twice answer 2.
I'll look at it tonight.
So Jim's answer is 3 times my answer for question 2. It could be
interpretation. I look at Jim's list of 135 possibilities. I would
interpret [1234], [1432], and [3412] as duplicates because they each
have 1 and 3 opposite. If you divide 135 by 3 you get 45. So Jim and
I are on the same track.
Looking at Jim's solution for question 3, I can't tell which sides
are opposite. If the three pairs of opposites were the same, there
would still be two unique solutions. We could call them right hand or
left hand with the thumb on one axis pointing toward increasing value
and the fingers in the direction of rotation. Don't know how we'd
decide the direction.
Regards,
Brian
--- In Math4u@yahoogroups.com, "w7anf" <cherry@...> wrote:
>
> Question:-
> Given six different colours and a cube.
> 1. In how many ways two opposite faces of the cube can
> be coloured with two different colours?
> 2. In how many ways four faces of the cube can be
> coloured with four different colours of which two
> faces are opposite and other two faces are also
> opposite?
> 3. In how many ways all six faces of the cube can be
> coloured with six different colours?
>
> Colors 1,2,3,4,5,6
> Sides 1 bottom, 2 top, 3 left, 4 right, 5 back, 6 front
>
> 1. C(6,2)=6*5/2=15
> [[1,2],[1,3],[1,4],[1,5],[1,6],[2,3],[2,4],[2,5],[2,6],
> [3,4],[3,5],[3,6],[4,5],[4,6],[5,6]]
>
> 2. 15*4*3=180
> Flip and rotate.
> Without duplicates there are
> 135 different ways.
> [[1,2,3,4],[1,2,3,5],[1,2,3,6],[1,2,4,3],[1,2,4,5],
> [1,2,4,6],[1,2,5,3],[1,2,5,4],[1,2,5,6],[1,2,6,3],
> [1,2,6,4],[1,2,6,5],[1,3,2,4],[1,3,2,5],[1,3,2,6],
> [1,3,4,2],[1,3,4,5],[1,3,4,6],[1,3,5,2],[1,3,5,4],
> [1,3,5,6],[1,3,6,2],[1,3,6,4],[1,3,6,5],[1,4,2,3],
> [1,4,2,5],[1,4,2,6],[1,4,3,2],[1,4,3,5],[1,4,3,6],
> [1,4,5,2],[1,4,5,3],[1,4,5,6],[1,4,6,2],[1,4,6,3],
> [1,4,6,5],[1,5,2,3],[1,5,2,4],[1,5,2,6],[1,5,3,2],
> [1,5,3,4],[1,5,3,6],[1,5,4,2],[1,5,4,3],[1,5,4,6],
> [1,5,6,2],[1,5,6,3],[1,5,6,4],[1,6,2,3],[1,6,2,4],
> [1,6,2,5],[1,6,3,2],[1,6,3,4],[1,6,3,5],[1,6,4,2],
> [1,6,4,3],[1,6,4,5],[1,6,5,2],[1,6,5,3],[1,6,5,4],
> [2,3,1,4],[2,3,1,5],[2,3,1,6],[2,3,4,5],[2,3,4,6],
> [2,3,5,4],[2,3,5,6],[2,3,6,4],[2,3,6,5],[2,4,1,3],
> [2,4,1,5],[2,4,1,6],[2,4,3,5],[2,4,3,6],[2,4,5,3],
> [2,4,5,6],[2,4,6,3],[2,4,6,5],[2,5,1,3],[2,5,1,4],
> [2,5,1,6],[2,5,3,4],[2,5,3,6],[2,5,4,3],[2,5,4,6],
> [2,5,6,3],[2,5,6,4],[2,6,1,3],[2,6,1,4],[2,6,1,5],
> [2,6,3,4],[2,6,3,5],[2,6,4,3],[2,6,4,5],[2,6,5,3],
> [2,6,5,4],[3,4,1,2],[3,4,1,5],[3,4,1,6],[3,4,2,5],
> [3,4,2,6],[3,4,5,6],[3,4,6,5],[3,5,1,2],[3,5,1,4],
> [3,5,1,6],[3,5,2,4],[3,5,2,6],[3,5,4,6],[3,5,6,4],
> [3,6,1,2],[3,6,1,4],[3,6,1,5],[3,6,2,4],[3,6,2,5],
> [3,6,4,5],[3,6,5,4],[4,5,1,2],[4,5,1,3],[4,5,1,6],
> [4,5,2,3],[4,5,2,6],[4,5,3,6],[4,6,1,2],[4,6,1,3],
> [4,6,1,5],[4,6,2,3],[4,6,2,5],[4,6,3,5],[5,6,1,2],
> [5,6,1,3],[5,6,1,4],[5,6,2,3],[5,6,2,4],[5,6,3,4]]
>
> 3. 135*2=270
> Flip and rotate.
> Without duplicates there are
> 210 different ways.
> [[1,2,3,4,5,6],[1,2,3,4,6,5],[1,2,3,5,4,6],[1,2,3,5,6,4],
> [1,2,3,6,4,5],[1,2,3,6,5,4],[1,2,4,3,5,6],[1,2,4,3,6,5],
> [1,2,4,5,3,6],[1,2,4,5,6,3],[1,2,4,6,3,5],[1,2,4,6,5,3],
> [1,2,5,3,4,6],[1,2,5,3,6,4],[1,2,5,4,3,6],[1,2,5,4,6,3],
> [1,2,5,6,3,4],[1,2,5,6,4,3],[1,2,6,3,4,5],[1,2,6,3,5,4],
> [1,2,6,4,3,5],[1,2,6,4,5,3],[1,2,6,5,3,4],[1,2,6,5,4,3],
> [1,3,2,4,5,6],[1,3,2,4,6,5],[1,3,2,5,4,6],[1,3,2,5,6,4],
> [1,3,2,6,4,5],[1,3,2,6,5,4],[1,3,4,2,5,6],[1,3,4,2,6,5],
> [1,3,4,5,2,6],[1,3,4,5,6,2],[1,3,4,6,2,5],[1,3,4,6,5,2],
> [1,3,5,2,4,6],[1,3,5,2,6,4],[1,3,5,4,2,6],[1,3,5,4,6,2],
> [1,3,5,6,2,4],[1,3,5,6,4,2],[1,3,6,2,4,5],[1,3,6,2,5,4],
> [1,3,6,4,2,5],[1,3,6,4,5,2],[1,3,6,5,2,4],[1,3,6,5,4,2],
> [1,4,2,3,5,6],[1,4,2,3,6,5],[1,4,2,5,3,6],[1,4,2,5,6,3],
> [1,4,2,6,3,5],[1,4,2,6,5,3],[1,4,3,2,5,6],[1,4,3,2,6,5],
> [1,4,3,5,2,6],[1,4,3,5,6,2],[1,4,3,6,2,5],[1,4,3,6,5,2],
> [1,4,5,2,3,6],[1,4,5,2,6,3],[1,4,5,3,2,6],[1,4,5,3,6,2],
> [1,4,5,6,2,3],[1,4,5,6,3,2],[1,4,6,2,3,5],[1,4,6,2,5,3],
> [1,4,6,3,2,5],[1,4,6,3,5,2],[1,4,6,5,2,3],[1,4,6,5,3,2],
> [1,5,2,3,4,6],[1,5,2,3,6,4],[1,5,2,4,3,6],[1,5,2,4,6,3],
> [1,5,2,6,3,4],[1,5,2,6,4,3],[1,5,3,2,4,6],[1,5,3,2,6,4],
> [1,5,3,4,2,6],[1,5,3,4,6,2],[1,5,3,6,2,4],[1,5,3,6,4,2],
> [1,5,4,2,3,6],[1,5,4,2,6,3],[1,5,4,3,2,6],[1,5,4,3,6,2],
> [1,5,4,6,2,3],[1,5,4,6,3,2],[1,5,6,2,3,4],[1,5,6,2,4,3],
> [1,5,6,3,2,4],[1,5,6,3,4,2],[1,5,6,4,2,3],[1,5,6,4,3,2],
> [1,6,2,3,4,5],[1,6,2,3,5,4],[1,6,2,4,3,5],[1,6,2,4,5,3],
> [1,6,2,5,3,4],[1,6,2,5,4,3],[1,6,3,2,4,5],[1,6,3,2,5,4],
> [1,6,3,4,2,5],[1,6,3,4,5,2],[1,6,3,5,2,4],[1,6,3,5,4,2],
> [1,6,4,2,3,5],[1,6,4,2,5,3],[1,6,4,3,2,5],[1,6,4,3,5,2],
> [1,6,4,5,2,3],[1,6,4,5,3,2],[1,6,5,2,3,4],[1,6,5,2,4,3],
> [1,6,5,3,2,4],[1,6,5,3,4,2],[1,6,5,4,2,3],[1,6,5,4,3,2],
> [2,3,1,4,5,6],[2,3,1,4,6,5],[2,3,1,5,4,6],[2,3,1,5,6,4],
> [2,3,1,6,4,5],[2,3,1,6,5,4],[2,3,4,5,1,6],[2,3,4,6,1,5],
> [2,3,5,4,1,6],[2,3,5,6,1,4],[2,3,6,4,1,5],[2,3,6,5,1,4],
> [2,4,1,3,5,6],[2,4,1,3,6,5],[2,4,1,5,3,6],[2,4,1,5,6,3],
> [2,4,1,6,3,5],[2,4,1,6,5,3],[2,4,3,5,1,6],[2,4,3,6,1,5],
> [2,4,5,3,1,6],[2,4,5,6,1,3],[2,4,6,3,1,5],[2,4,6,5,1,3],
> [2,5,1,3,4,6],[2,5,1,3,6,4],[2,5,1,4,3,6],[2,5,1,4,6,3],
> [2,5,1,6,3,4],[2,5,1,6,4,3],[2,5,3,4,1,6],[2,5,3,6,1,4],
> [2,5,4,3,1,6],[2,5,4,6,1,3],[2,5,6,3,1,4],[2,5,6,4,1,3],
> [2,6,1,3,4,5],[2,6,1,3,5,4],[2,6,1,4,3,5],[2,6,1,4,5,3],
> [2,6,1,5,3,4],[2,6,1,5,4,3],[2,6,3,4,1,5],[2,6,3,5,1,4],
> [2,6,4,3,1,5],[2,6,4,5,1,3],[2,6,5,3,1,4],[2,6,5,4,1,3],
> [3,4,1,2,5,6],[3,4,1,2,6,5],[3,4,1,5,2,6],[3,4,1,6,2,5],
> [3,4,2,5,1,6],[3,4,2,6,1,5],[3,4,5,6,1,2],[3,4,6,5,1,2],
> [3,5,1,2,4,6],[3,5,1,2,6,4],[3,5,1,4,2,6],[3,5,1,6,2,4],
> [3,5,2,4,1,6],[3,5,2,6,1,4],[3,5,4,6,1,2],[3,5,6,4,1,2],
> [3,6,1,2,4,5],[3,6,1,2,5,4],[3,6,1,4,2,5],[3,6,1,5,2,4],
> [3,6,2,4,1,5],[3,6,2,5,1,4],[3,6,4,5,1,2],[3,6,5,4,1,2],
> [4,5,1,2,3,6],[4,5,1,3,2,6],[4,5,1,6,2,3],[4,5,2,3,1,6],
> [4,5,2,6,1,3],[4,5,3,6,1,2],[4,6,1,2,3,5],[4,6,1,3,2,5],
> [4,6,1,5,2,3],[4,6,2,3,1,5],[4,6,2,5,1,3],[4,6,3,5,1,2],
> [5,6,1,2,3,4],[5,6,1,3,2,4],[5,6,1,4,2,3],[5,6,2,3,1,4],
> [5,6,2,4,1,3],[5,6,3,4,1,2]]
>
> Jim FitzSimons
>
> --- In Math4u@yahoogroups.com, "Brian Edward Jensen"
> <brianejensen@> wrote:
> >
> > Question 1
> > I agree with Jim's answer. There are 6 color choices for one
> square
> > and 5 remaining color choices for the opposite square. But the
> cube
> > can be flipped over causing half the choices to be duplicates.
So
> > the answer is 6*5/2=15
> > Question 2
> > First of all we chose 4 colors out of 6 different colors.
> > I get 6*5*4*3=360.
> > Let's chose two colors to exclude:
> > I get 6*5=30
> > What a discrepancy! We should get the same answer picking 4
> colors
> > or excluding 2 colors.
> > Obviously, if we chose 6 colors out of six different colors,
there
> is
> > only one possibility. So we must chose the lower number. There
> are
> > 30 different ways to choose 4 colors out of 6.
> > Let's call these colors 1,2,3,and 4.
> > These 4 colors are to be arranged in a band around the cube. One
> of
> > the sides will have color 1. There are three remaining colors to
> > choose from to be opposite color 1.
> > So I'd say the answer is 30*3=90.
> > Question 3
> > We have a cube colored with 6 colors.
> > There are 5 choices for the color opposite color 1. Let's call
> the
> > remaining colors 3,4,5,and 6. Color 3 will be between the first
> two
> > colors.
> > There are 3 choices for the color opposite color 3.
> > There are 2 remaining choices for arranging the last 2 colors.
> > So I'd say the answer is 5*3*2=30
> > Regards,
> > Brian
> >
> >
> > --- In Math4u@yahoogroups.com, "sanjivadayal" <sanjivadayal@>
> > wrote:
> > >
> > > Question:-
> > > Given six different colours and a cube.
> > > 1. In how many ways two opposite faces of the cube can
> > > be coloured with two different colours?
> > > 2. In how many ways four faces of the cube can be
> > > coloured with four different colours of which two
> > > faces are opposite and other two faces are also
> > > opposite?
> > > 3. In how many ways all six faces of the cube can be
> > > coloured with six different colours?
> > >
> >
>
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