Given a three dimensional n×n×n cube containing n^3 cubes of 1×1×1,
Define a cube tube to be a sequence of 1×1×1 cubes from the n×n×n cube such that:
1) No cube appears in the sequence more than once.
2) Each cube in the sequence shares a face with the cube directly after it in the sequence.
Define a cube loop to be a cube tube where the first and last cubes in the sequence share a face.
What is the shortest cube tube such that its projection in any of the three planes is a full n×n square? What about the shortest cube loop?
Example: For n=2 the answer is 6 for both the cube tube and the cube loop.