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There are 64 cube-shaped small one side blue-blocks. If you make new c

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There are 64 cube-shaped small one side blue-blocks. If you make new c  [#permalink]

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New post 03 Feb 2016, 21:49
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Question Stats:

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There are 64 cube-shaped small one side blue-blocks. If you make new cube such that length, width, and height consist of 4 one side blue-blocks, respectively, then what fraction of total surface area of the cube is the greatest area of one side blue-blocks to make each blue color of the blocks face outside?

A. 7/12
B. 1/2
C. 5/12
D. 1/3
E. 1/4

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Re: There are 64 cube-shaped small one side blue-blocks. If you make new c  [#permalink]

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New post 04 Feb 2016, 00:42
MathRevolution wrote:
There are 64 cube-shaped small one side blue-blocks. If you make new cube such that length, width, and height consist of 4 one side blue-blocks, respectively, then what fraction of total surface area of the cube is the greatest area of one side blue-blocks to make each blue color of the blocks face outside?

A. 7/12
B. 1/2
C. 5/12
D. 1/3
E. 1/4


* A solution will be posted in two days.


Hi MathRevolution,
Before I solve this, just a point on the wording..
Quote:
what fraction of total surface area of the cube is the greatest area of one side blue-blocks to make each blue color of the blocks face outside?
..
If you ask me, the answer is 1 as the total surface area is completely made up of one side blue blocks, so the fraction will be 1..
PLease change the wordings to:-
what is the greatest possible fraction of total surface area of the cube that can be blue coloured?

lets solve it..
I)For solving it we have to visualize how many faces of cubes will face outside..
1) 3 faces:-
all the cubes corners will have 3 faces open..
only one out of this will be blue colour- so 8..
2) 2 faces:-
all the cubes on edges less the corner ones which are already accounted for above..
so 2 in each edge. total 2*12 edges=24..
3) 1 face -
all remaining cubes = 4 on each face= 4*6=24..

total = 8+24+24=56..
surface area= 4*4*6=96..
fraction= 56/96= 7/12..
A

II) second method could be finding how many cubes are open to outside..
its a cube of 4*4*4 and its center portion of 2*2*2 will be covered from all sides..
so 64-8=56..
each of these 56 has one face blue coloured..
so for greatest blue outside, we will ensure the blue face outside, so total area=56..
total surface area= 96..
fraction=56/96=7/12
A

III) Visualization:-
If you try to visualize the cube, you will have two opposite face 16 each are blue ..
remaining 4 faces, two opposite faces will have half of 16, 8 sides, and the other pair 4..
one can see that the fraction is more than half..
only A is more than half
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Re: There are 64 cube-shaped small one side blue-blocks. If you make new c  [#permalink]

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New post 06 Feb 2016, 17:54
Attachment:
Cubes.jpg
Cubes.jpg [ 10.83 KiB | Viewed 1258 times ]


The above picture is a spread out cube as we see it. Since there is one side with blue blocks, that one side appears to be blue. At the same time, there is a side, which doesn't appear to be the same lay out. Then, the outside should be blue as much as possible, and we can come up with an equation (16*2+6*2)/16*6=7/12.
Therefore, the answer is A.
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Re: There are 64 cube-shaped small one side blue-blocks. If you make new c  [#permalink]

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New post 25 Dec 2018, 10:35
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Re: There are 64 cube-shaped small one side blue-blocks. If you make new c &nbs [#permalink] 25 Dec 2018, 10:35
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