Sudoku computation

Given two completed sudoku grids, can we quickly determine if one can be tranformed to the other by a sequence of sudoku-preserving operations? The sudoku group may be a basis for a Rubik's cube style puzzle.

If they are different (i.e., they cannot be transformed), by what algorithm can we minimize the difference between them? What is the maximum possible minimized difference (minimax)? What is the largest possible clique such that the pairwise distance between any two is that minimax distance?

Sudoku may be generalized to larger sizes. What is the largest size completed grid that has ever been produced? What is the computational complexity as the grid size gets larger? What is the computational complexity of producing two grids of maximum distance? The motivation for the previous question is "how can we produce a large number of qualitatively different grids?" -- given one grid, it is easy to perturb (or sudoku-preservingly transform) it to a different puzzle, but such constructions do not seem qualitatively very different.

We call a puzzle (not a completed grid) is minimal if no clue may be removed while preserving the uniqueness of the completed solution. Whereas producing a short proof of non-uniquess is straightforward (simply give two solutions), how can we produce a short proof of uniqueness? What is the largest puzzle that has been proved minimal?

Sudoku may generalized to more dimensions. Regular sudoku is almost four-dimensional, using only three of the six planes though each point. For a given number of dimensions, what is the minimum width greater than one of the hypercube? (Is the answer constantly 2?) Does 6D sudoku of width 3, a 3x3 collection of 9x9 puzzles with Choose(6,2)=15 constraints per point, exist? What is the complexity of producing completed and minimal grids of a given width and dimension?