Each of 27 white 1-centimeter cubes will have exactly one face painted red. If these 27 cubes are joined together to form one large cube, as shown above, what is the greatest possible fraction of the surface area that could be red?

A. 11/27
B. 13/27
C. 1/2
D. 5/9
E. 19/27

PS93602.01
Quantitative Review 2020 NEW QUESTION



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Number of cubes having 3 faces outside- 8 (corner ones)

Number of cubes having 2 faces outside- 12*(n-2)= 12*(3-2)= 12

Number of cubes having 1 face outside- \(6*(n-2)^2\) =\( 6*{(3-2)^2}\) = 6

Number of cubes having 0 face outside- \((n-2)^2\) = \({(3-2)^2}\)=1

Maximum number of colored faces outside=8+12+6=26

greatest possible fraction of the surface area that could be red= \(\frac{26}{54} \)= \(\frac{13}{27}\)

Solution

Given
In this question, we are given

• Each of 27 white 1-centimeter cubes will have exactly one face painted red.
• These 27 cubes are joined together to form one large cube, as shown in the given diagram.

To Find
We need to determine

• The greatest possible fraction of the surface area that could be red.

Approach & Working
To maximise the surface area, count as red, except the cube which is at the centre, all the other cubes must have their red surface area exposed.

• Number of cubes with red surface area exposed = 27 – 1 = 26

However, in each surface of the bigger cube, there will be 9 smaller cube surfaces, and there will be total 6 surfaces for the bigger cube.

• Hence, total number of cube surfaces exposed = 9 * 6 = 54
• Therefore, the greatest fraction = 26/54 = 13/27

Hence, the correct answer is option B.


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EgmatQuantExpert ScottTargetTestPrep Bhanupriya05

I'm sorry, I don't understand the solution(s) provided, especially when you say all the 'exposed' sides, why do we consider the exposed sides (as per the diagram) when even the ones on the bottom could also have been painted red? [Bunuel could you please clarify on this?]

As per the 'exposed' sides, the most bottom right cube would be considered as two red faces, but the question stem clearly says, "Each of 27 white 1-centimeter cubes will have exactly one face painted red"

I'm not sure where am I getting confused in here, and would sincerely appreciate any help on this.

Thanks in advance.

Let’s rephrase the question to make sure we clearly understand what the question is asking: We have 27 small cubes with a side length of 1. We paint exactly one face of each cube red and then stack these smaller cubes to form the shape depicted on the diagram. Though that’s not exactly what the question is asking, we actually need to figure out how many red faces we will see at most.

Having that said, I think you are getting confused about the number of visible faces (which is 56) versus the number of small cubes (which is 27). Let me provide an alternative explanation to overcome that. Let’s classify the smaller cubes into four groups: a) The cubes with 3 exposed faces (the cubes which are on the corner of the big cube, there are 8 of them) b) The cubes with 2 exposed faces (the cubes which are on the edge but not on the corner of the big cube, there are 12 of them) c) The cubes with 1 exposed face (the cubes which are at the center of each face, there are 6 of them) d) The cube with no exposed face (the only cube which is at the core of the larger cube)

Let’s pick one face from each group besides the no visible face group and paint it red. Then, we will have 8 + 12 + 6 = 26 of them. That’s why you can have at most 26 faces painted red.
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This must be super simple but I am not getting the part where we subtract 1 from 27. ie: 26 (all except the one at the center). why is that one not exposed?
Why? Pls explain:=)))