Can you uniquely define a cube knowing the color of the faces but not its orientation?

I'm doing an university project for one of my subjects. I have the 6 faces of the Rubik's cube but i don't know if they have been rotated or not, i just know their colors. Is that enough information to reconstruct the original cube? If that's not the case, from how many faces do i have to know the orientation?

Solution 1:

A Rubik's cube configuration does contain more information than just the colours of the squares. Specifically one can draw an arrow on each center square. If one performs a series of moves taking a completed Rubik's cube to a completed Rubik's cube, the directions of the arrows may be rotated some multiples of $90^\circ$ clockwise (facing the cube).

The directions of the center arrows and the colours of the squares do completely determine the state of the cube, as squares of the same colour cannot be permuted (whilst preserving the colours of all squares) because they are physically attached to unique combinations of other colours. Similarly they may not be rotated (other than the previously discussed center squares) because they are physically attached to distinct colours over the outer edges.

So as well the colours of all the squares, one requires an element of the group $C_4^6$ to completely describe the state of a Rubik's cube.

This begs the question: Which subgroup of $C_4^6$ is the set of elements which can be attained through a sequence of moves that fixes the colours of all the squares?

If the subgroup is trivial, then the directions of the arrows give no new information, as they are determined by the colours of the squares.

In fact this subgroup is index $2$ in $C_4^6$: Suppose we perform a clockwise $90^\circ$ turn on a face, and then another clockwise $90^\circ$ turn on an adjacent face. The resulting permutation on squares consists of cycles of length $3,15, 7,7$ so has order $105$.

That is after $105$ pairs of clockwise $90^\circ$ turns, we will have returned all squares to their starting position, but rotated a pair of adjacent arrows clockwise $90^\circ$.

These pairs of clockwise $90^\circ$ turns generate an index $2$ subgroup in $C_4^6$. To see that only elements of this subgroup can be induced by sequences of moves that preserve the colours of the squares, note:

A single $90^\circ$ clockwise turn of a face, permutes squares in five $4$-cycles hence is odd. Thus to return all squares to where they start, an even number of $90^\circ$ clockwise turns must have been been made.