In that case (where for [1,2,3,4] 1 and 2 are opposite, 3 and 4 are
opposite) I believe
[1234] and [1243] and [3412] are duplicates
[1235] and [1253] and [3512] are duplicates
[1236] and [1263] and [3612] are duplicates
For example we can rotate [1,2,3,4] about the 1,2 axis and get
[1,2,4,3]
135 printed out ways divided by 3 duplicates in each group equals 45,
ANSWER.
Another way of looking at the problems:
Question 1, 6 different colors; find the number of ways of picking
unique colors for 1 pair of opposite sides.
Question 2, 6 different colors; find the number of ways of picking
unique colors for 2 pairs of opposite sides.
Question 3, 6 different colors; find the number of ways of picking
unique colors for 3 pairs of opposite sides.
I am feeling more confident in my answers
Question 1 we agree the answer is 6*5/2=15
Question 2
Ways of picking 4 colors, any order = 6*5*4*3/(4*3*2*1)=15
Ways of excluding 2 colors, any order = 6*5/(2*1)=15
Pick any color. It has 3 choices for the opposite color. So the
answer is
15*3=45
Question 3
Method A
Pick any color.
You have 5 choices for the opposite side.
Pick any of the remaining 4 colors.
You have 3 choices for the opposite side
You have 2 choices for the next 2 colors.
5*3*2=30 Answer
Method B
Assign a color to a side.
You have 5*4*3*2*1 or 120 ways of assigning the other colors.
Keeping our chosen color to the front, we can rotate the cube into 4
positions.
120/4=30 Answer
Method C
Don't move the cube
Ways of assigning the colors = 6*5*4*3*2*1=720
There are 6 different faces that can be put in front and for each
face the cube can be rotated into 4 positions so there are 24
orientations of the cube.
720/24=30 answer.
Method D
We take our answer of 45 from question 2. There are 2 ways to add
the last two colors. But we will have three duplicates for each
combination.
45*2/3=30 answer.
Regards,
Brian
--- In Math4u@yahoogroups.com, "w7anf" <cherry@...> wrote:
>
> Sides are numbered 1 bottom, 2 top, 3 left, 4 right, 5 back, 6
front.
> [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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