Based on geometric observations of the cube cake, we are able to not only generate the table, but also generate the formula for an n x n x n cake:

Total Number of Pieces = 8 + 12(n - 2) + 6(n - 2)² + (n - 2)³

This formula now allows us to use algebra to verify our answer to the problem. When substituting the value of 6 for n, we can justify our answer of 216 pieces of cake (and 216 math teachers):

8 + 12(6 - 2) + 6(6 - 2)² + (6 - 2)³ =

8 + (12 • 4) + (6 • 16) + 64 =

8 + 48 + 96 + 64 =

216

Remember that the Reasoning and Proof Standard calls for you to make conjectures and then test and refine your ideas. In both the Frosted Cube Cake and the Building Rafts with Rods problem, the visual model will help you form conjectures. From those conjectures, you can begin to generalize for any number of Cuisenaire rods or for any size frosted cube cake. In addition to making, testing, and refining conjectures, the visual model enables you to develop arguments to support your conclusions. For example, the number of pieces of cake with one side frosted will always be the number of pieces in the center of one face of the cube (n - 2)² times the six faces of the original cube. You can justify the pattern for each arrangement of frosting using the original model.

  Reflect on the Frosted Cube Cake problem