[1,2,3,4] means color 1 is on the bottom.
Color 2 is opposite on the top.
Color 3 is on the left.
Color 4 is opposite on the right.
[1,4,2,3] is different and can not be rotated to [1,2,3,4].
Jim FitzSimons
--- In Math4u@yahoogroups.com, "Brian Edward Jensen"
<brianejensen@...> wrote:
>
> Jim, you went to a lot of work!
> 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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