Great effort. My answers are:-
Answer 1: 15.
Answer 2: 45.
Answer 3: 30.
Sanjiva Dayal
--- Brian Edward Jensen <brianejensen@prodigy.net>
wrote:
> OK,
> 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],
>
=== message truncated ===
Sanjiva Dayal, B.Tech.(I.I.T. Kanpur)
Address:A-602, Twin Towers, Lakhanpur, Kanpur-208024, INDIA.
Phones:+91-512-2581532,2581426.
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Business email:sanjivadayal@yahoo.com
Personal email:sanjivadayal@hotmail.com
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