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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