I am currently researching a problem that involves mathematics and computer science, and I am hoping to get advice or guidance from experts in the field. My question pertains to finding a mapping that can project the state space of a Rubik's cube onto a Euclidean space of a certain dimension.
Specifically, I am looking for a proof that there exists a high-dimensional Euclidean space that can meet the following description.
Given that most Rubik's cube states require at least 18 moves for resolution, if we suppose that the solved state is at the center, then the shape is approximated as a sphere, where most points in the space are located on the surface of a sphere with a radius of 18. In this space, if we choose any point from the set, the distance to most other points in the set should be 18. This seems rather contradictory.
One conjecture is that this space could be an ($n-1$) dimensional surface in $R^n$. For example, in a two-dimensional sphere, if we form a regular tetrahedron within the sphere, then within this spherical space, the distances between the vertices are all equidistant (thus, the solved state would no longer be the center point, but one of the vertices).
My goal is to learn the states of the Rubik's cube through this mapping using a neural network, and to optimize the state space search algorithm using the properties of this space. I have conducted some preliminary experiments, but the results were not satisfactory. I suspect that this may be related to the design of the loss function. Therefore, I hope to confirm whether such a space does indeed exist, and to design a more effective loss function based on the structure of this space.
More precisely, I just want to know whether there is such a space, without actually finding it, and design the loss function by understanding the structure. (I plan to use AutoEncoder to train this mapping function. The bottleneck output uses the distance formula to calculate the distance and calculate the loss with the real distance. Therefore, I need to understand this space to design this formula)
I very much look forward to your thoughts, suggestions, or guidance. Thank you for your time and assistance!