SOLUTION: A cube composed of 64 1 x 1 x 1 cubes is painted and then broken apart again into the smaller cubes. How many of the 1 x 1 x 1 cubes will have exactly 2 painted faces? (A) 8 (B)

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Question 206586: A cube composed of 64 1 x 1 x 1 cubes is painted and then broken apart again into the smaller cubes. How many of the 1 x 1 x 1 cubes will have exactly 2 painted faces?
(A) 8 (B) 16 (C) 24 (D) 36 (E) 48
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
you have 64 cubes.
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cube root of 64 is 4 which means your cubic structure has a length of 4 cubes and a width of 4 cubes and a depth of 4 cubes.
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out of these 64 cubes there will be 8 cubes that will have 0 faces painted.
these cubes are in the interior of the cube.
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that leaves 64 - 8 = 56 cubes that have at least one face painted.
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if you look at each side of the 6 sides of the cube, you will see that there are 4 cubes on each side that will only have 1 face painted. Those are the interior cubes of each side.
since 6 * 4 = 24, you have 24 more cubes that will not have 2 sides painted because they only have 1 side painted.
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56 - 24 = 32 cubes are left that will have at least 2 sides painted.
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if you look at the corners of your cube you will see that each corner cube will have 3 sides painted.
since there are 8 corners to the cube, that makes 8 cubes that will have more than 2 sides painted.
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32 - 8 = 24 cubes are left that will have exactly 2 sides painted.
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that makes your answer = 24.
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another way to look at this is to look at each side of your cube.
there are 16 faces of cubes on each side.
out of those 16 only the on the edge of each side of the cube and not on the corners are the candidates for having 2 faces painted.
there are 8 of these faces on each side of the cube that will have a companion face on the other side of the cube only.
6 sides of the cube times 8 = 48 faces thaat are painted that are members of the set of 2 faces to a cube.
48 / 2 = 24 again.
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i would say 24 cubes is your answer.
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a picture of your cubes can be seen on this website.
http://theo.x10hosting.com/
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go there and click on 206586.
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if it's not there, give it 1/2 hour and try again.