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

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

You preceded me Sir.

Thank you.

Great explanation

Stunning

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

Thank you!

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

Regards

I think it's within the score of the GRE

If the lead quant mod says this. Turns out I am wrong.

However, I think it is slightly over. But maybe, that is me

It's in the 169.7 to 170.1 range

what prep company is mgre?

Hey jantje

It's Manhattan GRE

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

I think for this question, its better to visualize the cake, so the side pieces will have 2 sides frosted like a rubik cube but except that we don't do frosting for the bottom. fewer than 2 frosted side are the pieces in the middle and in the core of the cake. so just count it in your head and you will come up with the number 12 for both case