You have a cube sitting on one square of a square grid. The cube may move to any empty square sharing an edge with its current square. The cube moves by performing a 90 degree rotation about the edge between the square it is currently on and the square it is moving too. For example, if the cube's d face is against the square the cube is sitting on and the f face is facing forward, than moving one square forward will put the f face against the grid and the u face facing forward.
Given this kind of movement, I have the following questions: 1. Given sufficient space in which to move around a given square of the grid, can the cube be placed onto the given square in all 24 orientations? 2. What is the minimal space needed to allow a cube to be placed onto a given square in all possible orientations? 3. Are any extra restrictions of orientation placed on the cube if the target square is along an edge of the grid or in a corner? 4. Are there any physical puzzles that use this kind of movement? 5. Would a 15 puzzle with this type of movement allow for the swapping of two cubes while having all cubes in the correct orientation? 6. How many states would a 15 puzzle with this kind of movement have?
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