QUANTITY A: The number of pieces of cake that have exactly two sides frosted
NOTE: The first time I read this question I didn't notice the part about the cake having no frosting on the bottom side.
Now that I've recognized this, the diagram below indicates (in RED) all of the pieces that have exactly TWO frosted sides:

If you can imagine the same pattern on the sides not shown, you will see that there are 12 such pieces of cake


QUANTITY B: One more than the number of pieces of cake that have fewer than two sides frosted
First, let's determine the number of pieces that have exactly ONE side frosted.
The diagram below indicates (in BLUE) all of the pieces that have exactly ONE frosted side

In total, there are 9 pieces that have exactly ONE frosted side.

Now let's determine the number of pieces that have exactly ZERO frosted sides.
- the middle piece that lies on the lowest layer has has zero frosted sides
- there's also a piece of cake at the very center that has zero frosting slides
So, there are 2 pieces that have ZERO frosted sides.

Quantity B tells us to ADD 1 to the number of pieces of cake that have fewer than two sides frosted
So, Quantity B = 9 + 2 + 1 = 12

So the two quantities are equal

Answer: C

Cheers,
Brent
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is it normal to have questions like this? I find this very time consuming

This is the real difference between an official question and one from a prep company: even if an official question is time-consuming there will be ALWAYS at least ine shortcut to solve the problem.

Hope this helps.

Regards
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Can someone please explain this? @carcass?

Question:
A cube shaped cake is frosted on all sides except the bottom and then cut into 27 equally sized cube shaped pieces
Quantity A: The number of pieces of cake that have exactly two sides frosted
Quantity B: One more than the number of pieces of cake that have fewer (than) two sides frosted


Solution:

There are 27 pieces = 3 x 3 x 3
Thus, each edge has 3 small cubes and each face has 9 small cubes

Quantity A: Pieces of cake that have exactly two sides frosted

Small cubes with 2 faces frosted will be found along the edges of the cube
There are 12 edges with 1 center small cube having 2 faces frosted (the corners have 3 faces frosted)
Thus, there are 12 small cubes.

However, if we look at the base:
There are 4 edges along the base face. The center small cube on each such edge has only 1 face frosted (not 2 - since the bottom face is not frosted). Thus, these 4 small cubes are ignored.
However, in the 4 corners of the bottom face, the 4 small cubes have 2 faces frosted (not 3 - since the bottom face is not frosted). Thus, these 4 will be added.

Hence, the total small cubes with 2 faces frosted remain 12.
Quantity A = 12

Quantity B: One more than the number of pieces of cake that have fewer (than) two sides frosted

Number of pieces of cake that have one side frosted:
Along the 5 faces, there is one center small cube with 1 face frosted i.e. 5 small cubes
Along the bottom face (base) edges, the cube in the center of each edge has 1 face frosted i.e. 4 more small cubes

Number of pieces of cake that have no side frosted:
Along the bottom face, the center small cube has no face frosted i.e. 1 small cube
Also, in the 3 x 3 x 3 cube, the center piece (obtained by stripping off the top, bottom, left, right, front and back layers) will also have no face frosted i.e. 1 more small cube

Thus, total such small cubes = 5 + 4 + 1 + 1 = 11
Thus, 1 more than the above = 12
Quantity B = 12


Hence, the quantities are equal

Answer C
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You preceded me Sir.

Thank you.

Great explanation
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Stunning


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is this a GRE level question? I find it really hard.

Thank you!
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For me it goes too far even for a GRE 5 level question.

Regards
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I think it's within the score of the GRE
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If the lead quant mod says this. Turns out I am wrong.

However, I think it is slightly over. But maybe, that is me
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It's in the 169.7 to 170.1 range
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what prep company is mgre?

Hey jantje

It's Manhattan GRE


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A total of 5*9 = 45 sides are shared by 27 cubes.
Recognize that the 4 top corners have 3 sides each, 2 cubes in the middle has 0 side.
Rest are cubes with 2 or 1 sides. Therefore, solve the system:
2a + b = 45 - 3*4
a + b = 27 - 4 - 2
cubes with 2 sides: a = 12
cubes with 1 side: b = 9
cubes with sides < 2 will be 9 + 2 = 11

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