A "Sudoku cube" is a 3x3x3 uncoloured Rubik's cube. In the solved state, each face has the digits 1 through 9 arranged in ascending rows from top to bottom, and all of the digits on a given face have the same orientation. When designing such a cube, each of the 6 faces has 4 possible orientations in a fixed frame of reference. Assume that cube designs that can be transformed into one another by rotation of the entire cube are equivalent.
The orientations of the faces affects the number of possible solutions to the cube, if each cuboid is taken as distinct. Take the following cube, where the orientation of a face is given as the direction of the top of the digits on that face:
There are four possible positions that the left and right faces can each take while preserving the solved state, yielding 16 obvious solutions and probably many more.
What set of orientations for all of the cube's faces yields the lowest number of valid solutions, and what is that lower bound?