Problem:
Akash's birthday cake is in the form of a inch cube. The cake has icing on the top and the four side faces, and no icing on the bottom. Suppose the cake is cut into smaller cubes, each measuring inch, as shown below. How many small pieces will have icing on exactly two sides?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
On the top side, there are cubes that are only on the top. Also, there are cubes that have icing on sides. Therefore, we have cubes on the top that have icing on exactly sides.
For the other sides, we need to find the number of cubes that have icing on exactly two sides, excluding the cubes we counted on the sides. Each face has cubes on the edge that have icing on exactly two sides. However, this would double count when taking into account the other faces, as each cube would be counted for two faces. Therefore, we need to add cubes.
The final answer is .
Thus, the correct answer is .
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions