I'm an eighth grader working with a graduate student on some math over the summer. He presented me with the following question: We have a cube formed of 27 small, congruent cubes. A worm sits in the central cube, and has the goal of going through every small cube exactly once. It can get from one cube to an adjacent cube only through the midpoint of the face; it cannot travel diagonally. Can this be accomplished?
After several attempts at this problem, I have been unable to find a method of solving it, and am currently trying to restrict the possibilities by figuring out which cube or class of cube (corner, center of a face, etc.) the worm has to finish its task in.
Before writing this question, I looked into the wood worm in a cube (Wood worm in a cube) problem that has already been asked here, but that problem had not in fact been given a clear answer and was closed for lack of important details.
I have been introduced to Hamiltonian Paths and Circuits, but these concern paths starting at a vertex of a graph; not the center.
Here is a Hamiltonian path starting at one corner and ending at another corner (and, by definition, passing through the center), where the vertexes represent the center of each cubelet:
But there is no Hamiltonian path that starts at the center.