Challenge:Classify the Unit Cubes (Math)?

Question

Consider a cube composed of (n^2) unit cubes. If we paint the surface of the composed cube, some of the unit cubes may have no sides colored and some may have more than one side colored. Using (n), classify the cube breaking down how many of each type of unit cubes there are.


NOTE: You have two jobs. The first is to consider different sized cubes, such as for n=1,2, etc. Next for each of these sizes give the categories determined by the unit cubes and determine the number of unit cubes in each category.

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Show the solution...

Answer 1

the classification depends on the size of the cubes

Answer 2

n = 1 and n = 2 are special cases. For n = 1, there is only one unit cube, and it has six sides painted. For n = 2, all eight unit cubes have three sides painted. For n = 3 and anything greater, the eight unit cubes at the corners have three sides painted. 12*(n-2) unit cubes on the edges (but not at the corners) have two adjacent sides painted. The remaining 6*(n-2)^2 unit cubes on the exterior have a single face painted. There is a smaller cube on the interior with side length n - 2, consisting of unit cubes without any faces painted, so there are (n - 2)^3 totally unpainted cubes.