There have been found many Simple Perfect Squared Squares. Refer to http://www.squaring.net
for background information, please. It has been proven that a three-dimensional analogon "Simple Perfect Cubed Cube" does not exist. However, we could think about derived problems with relaxed requirements.
This model shows a Perfect Cubed Cube Frame (43x43x43) consisting of 19 cubes called the frame "elements". One edge has order 2, all others have order 3. The elements are all of different size:
18 6 19
20 10 13
3 2 16 7
21 8 14
22 4 17
It is not as easy as expected to find small examples. If you find a good (minimum) one, please let me know. Minimum criteria are frame size and number of elements.
Finding solutions of an appropriate set of linear equations is not sufficient. All the edge sums must be equal to the frame size, of course. But elements also must not collide within the bounding cube!
"Magic cube" (some kind of three-dimensional "magic square") solutions all have small cubes at the edges which makes them unusable for forming a PCCF.
Is it possible to make a similar model with all edges having order 3 using cubes of sizes 1,2,... up to 20?