Using the Ponting Square Packing, squares of size 1-49 can be packed in a 7x7 array so that the 25 interior squares are completely surrounded.
Another way to look at the above squares is with the following grid:
Is there some similar method that allows cubes of size 1 to 125 to be packed so that 27 interior cubes are completely surrounded? If so, would the central section of cubes have a cross section similar to the following?
If such a 3D packing exists and is extendible, what does the shape look like for a lot of cubes? As an example, here are squares of size 1 to 2025.