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A cube marked 1, 2, 3, 4, 5, and 6 on its six faces. Three [#permalink]

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14 Jan 2010, 14:21

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A cube marked 1, 2, 3, 4, 5, and 6 on its six faces. Three colors, red, blue, and green are used to paint the six faces of the cube. If the adjacent faces are painted with the different colors, in how many ways can the cube be painted?

A cube marked 1, 2, 3, 4, 5, and 6 on its six faces. Three colors, red, blue, and green are used to paint the six faces of the cube. If the adjacent faces are painted with the different colors, in how many ways can the cube be painted? (A) 3 (B) 6 (C) 8 (D) 12 (E) 27

Any one side of the cube will be having 4 sides which can be termed as adjacent.. Isn't it ? Say for e.g side A it will be having 4 sides adjacent to it, one on left, one on right, one above and one below.

nitishmahajan, I agree with your assessment. There are 4 adjacent sides for every face of the cube.

Let's say side 1 is painted red, then the 4 adjacent sides can be either green or blue alternating. This can be done in 2 ways. GBGB BGBG Sixth side should be the same color as side 1.

For each color chosen for side 1(and side6) there are 2 ways of painting side 2,3,4 and 5. No. of colors that can be chosen for side 1(and side6) is 3. So 3*2 = 6..

Re: A cube marked 1, 2, 3, 4, 5, and 6 on its six faces. Three [#permalink]

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20 Dec 2012, 03:59

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apoorvasrivastva wrote:

A cube marked 1, 2, 3, 4, 5, and 6 on its six faces. Three colors, red, blue, and green are used to paint the six faces of the cube. If the adjacent faces are painted with the different colors, in how many ways can the cube be painted?

(A) 3 (B) 6 (C) 8 (D) 12 (E) 27

If the base of the cube is red, then in order the adjacent faces to be painted with the different colors, the top must also be red. 4 side faces can be painted in Green-Blue-Green-Blue OR Blue-Green-Blue-Green (2 options).

But we can have the base painted in either of the three colors, thus the total number of ways to paint the cube is 3*2=6.

Re: A cube marked 1, 2, 3, 4, 5, and 6 on its six faces. Three [#permalink]

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28 Dec 2012, 01:48

So, do they asked this on test day? This drove me nuts.

Imagine a flattened cube... The three colored region will establish the other colors of the remaining faces of the cube.

For example: We assumed the sequence of color in the given image as RED on face#1 and BLUE on face#2 and GREEN on face#3. Since face#1 is RED then we know #4 and #5 cannot be RED. Since face #2 is BLUE, we know that #5 and #6 cannot be BLUE. Since face#3 is GREEN, we know #4 and #6 (the bottom) cannot be GREEN.

So, all we need is to count the possible number of arrangements of 3 colors.

Re: A cube marked 1, 2, 3, 4, 5, and 6 on its six faces. Three [#permalink]

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28 Dec 2012, 19:30

mbaiseasy wrote:

So, do they asked this on test day? This drove me nuts.

Imagine a flattened cube... The three colored region will establish the other colors of the remaining faces of the cube.

For example: We assumed the sequence of color in the given image as RED on face#1 and BLUE on face#2 and GREEN on face#3. Since face#1 is RED then we know #4 and #5 cannot be RED. Since face #2 is BLUE, we know that #5 and #6 cannot be BLUE. Since face#3 is GREEN, we know #4 and #6 (the bottom) cannot be GREEN.

So, all we need is to count the possible number of arrangements of 3 colors.

3! = 6

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Re: A cube marked 1, 2, 3, 4, 5, and 6 on its six faces. Three [#permalink]

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12 Sep 2013, 03:26

How about this approach? There are 3 colors and 6 sides. Same color can't be next to each other so put them on opposite sides. There are 3 opposite sides. So 3 colors 3 sides- no. of ways 3*2*1 = 6

Re: A cube marked 1, 2, 3, 4, 5, and 6 on its six faces. Three [#permalink]

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13 Feb 2014, 09:00

Actually this one works out like this. Cube has 6 faces and we are told adjacent are different therefore base different from side different from front. Three sides to choose the colors to paint them with. Well since we have three colors then 3! =6 (B) is the right answer

Re: A cube marked 1, 2, 3, 4, 5, and 6 on its six faces. Three [#permalink]

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02 Mar 2015, 18:45

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Re: A cube marked 1, 2, 3, 4, 5, and 6 on its six faces. Three [#permalink]

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15 May 2016, 10:07

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