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# A cube of side 5cm is painted on all of its sides. If it is

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A cube of side 5cm is painted on all of its sides. If it is [#permalink]

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01 Aug 2011, 08:23
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A cube of side 5cm is painted on all of its sides. If it is sliced into 1 cubic cm cubes, how many 1 cubic cm cubes will have exactly one of their sides painted?

A. 9
B. 61
C. 98
D. 54
E. 64

[Reveal] Spoiler:
D. This came straight from the GMAT club math review, but I'm a bit tripped up on the explanation
[Reveal] Spoiler: OA
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01 Aug 2011, 08:30
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A 5 cm^3 cube divided in 1cm^3 cubes = 125 cubes in total. A cube has 6 faces.
On each face, all the cubes surrounding the boundary will have more than 1 side painted, which makes total of 5+5+3+3 = 16 cubes.

So we are left with 25-16 = 9 cubes on each face (which has only 1 side painted) totaling to 9*6=54 cubes.
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01 Aug 2011, 08:52
Is the quickest way to solve the problem to draw a diagram of the cube?
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01 Aug 2011, 09:12
I would definitely avoid drawing a diagram of the complete cube. One face would be sufficient.
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01 Aug 2011, 09:16
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I think the quickest way is to imagine in your mind 1 side of the cube. Understand that the border cubes will have more than one side painted, so the ones with only one side are the ones in the middle, in this case 3x3=9. Then multiply that by 6 sides. If you will make a diagram, make only one side and then multiply by 6. Don't waste the little precious time that you have.
See the picture below to understand it better. The green squares have adjacent surfaces painted. Only the blue ones have 1 side only painted.
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5cm cube.jpg [ 3.96 KiB | Viewed 3610 times ]

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01 Aug 2011, 11:46
all the boundary cubes will have more than one face painted. only the center ones on each side of the larger cube, will have paint on just one side.

if we observer one face, there are 9 such cubes which have only one face painted.

so this times 6 faces we will have 9*6 = 54 such small cubes .

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01 Aug 2011, 13:50
the smaller cubes which will share an edge will have more than 2 faces painted.. so only the center cubes will have to be considered.. so .. 3*3 on each face.. 6 faces = 54
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02 Nov 2011, 11:34
Nice explanations.
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23 Sep 2012, 03:04
puneet478 wrote:

A 5 cm^3 cube divided in 1cm^3 cubes = 125 cubes in total. A cube has 6 faces.
On each face, all the cubes surrounding the boundary will have more than 1 side painted, which makes total of 5+5+3+3 = 16 cubes.

So we are left with 25-16 = 9 cubes on each face (which has only 1 side painted) totaling to 9*6=54 cubes.

Hi.. is there any formula to calculate total number of cubes after division.
Let say a 6cm cube is divided into 2cm cubes ?
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Re: A cube of side 5cm is painted on all of its sides. If it is [#permalink]

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21 Feb 2013, 01:00
Yes there is
it will be 6^3 / 2 ^3 = 27 cubes made
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21 Feb 2013, 05:41
cano wrote:
I think the quickest way is to imagine in your mind 1 side of the cube. Understand that the border cubes will have more than one side painted, so the ones with only one side are the ones in the middle, in this case 3x3=9. Then multiply that by 6 sides. If you will make a diagram, make only one side and then multiply by 6. Don't waste the little precious time that you have.
See the picture below to understand it better. The green squares have adjacent surfaces painted. Only the blue ones have 1 side only painted.

For those who have a problem in visualising they can atleast come to a conlusion that such cubes will be present on all the six sides. Now, whatever be this number(9 in this case), the answer will be a multiple of 6. Thus, just choose an option which is a multiple of 6.
However, GMAT can give 2 options where in more than 1 option is a multiple of 6. In this question, luckily we have only one such option.

D.
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Re: 3-D Cube Question   [#permalink] 21 Feb 2013, 05:41
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