Question 1206044
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Grab a Rubik's cube to help visualize what's going on. 
Although the cake won't be split into 27 smaller cubes, it's a similar idea. 
The 8 corner pieces represent the pieces that have 3 sides frosted.


Since the "<font color=blue>number of pieces without frosting is equal to eight times the number of pieces that have frosting on three sides</font>", this means there will be 8*8 = 64 pieces without frosting.


This inner unfrosted block must be 4 units in side length because 4*4*4 = 64
Or you could say {{{root(3,64) = 64^(1/3)^"" = 4}}} units 


Let's look at a birds-eye-view of the cake.
*[illustration UploadedScreenshot_22.png]
The pieces marked in yellow are the 4 corners for this top face (each corner gets 3 faces frosted).
There are 8 of these pieces as mentioned earlier. 4 for the top and 4 for the bottom.


The pieces shaded gray will get 1 face painted. There are 4*4 = 16 pieces shown in the diagram. With 6 faces to the original larger cube, that's 6*16 = 96 pieces that get 1 face painted.


The unshaded parts of the diagram represent pieces of cake that get 2 faces painted.
There are 4 white little squares along any edge and 12 edges in any cube, so 12*4 = 48 pieces of cake that get 2 faces painted.


Notice the overall cube has side length 6 units, so there are 6^3 = 6*6*6 = 216 pieces total


Summary:
64 pieces with no frosting
96 pieces with one face painted
48 pieces with two faces painted
8 pieces with three faces painted
64+96+48+8 = 216 pieces total. This checksum helps verify the answer.



Edit:
Here's what the 3D view would look like
*[illustration UploadedScreenshot_24.png]
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