Consider a cube centered at the origin in $\mathbb{R}^3$, with each face perpendicular to one of the coordinate axes. We label the faces as follows:
\[ \begin{aligned} U &: z = +1, \quad& D &: z = -1, \\ R &: x = +1, \quad& L &: x = -1, \\ F &: y = -1, \quad& B &: y = +1. \end{aligned} \]
The 8 corner positions correspond to the points \[ (x,y,z) \in \{\pm1\}^3, \] each determining three faces. Example: $UFR$ corresponds to $(1,-1,1)$.
Fix the corner indexing: \[ \begin{aligned} 0&=UFR, & 1&=URB, & 2&=UBL, & 3&=ULF, \\ 4&=DFR, & 5&=DRB, & 6&=DBL, & 7&=DLF. \end{aligned} \]
A cube state is given by:
Valid states satisfy: \[ \sum_{i=0}^7 o_i \equiv 0 \pmod{3}. \]
Thus the state space is \[ \mathcal{C} = \{ (p,o) \in S_8 \times (\mathbb{Z}_3)^8 : \sum o_i = 0 \pmod 3 \}. \]
Each move $M$ acts by a permutation $\pi_M$ and orientation increments $\Delta_M$: \[ \begin{aligned} p'_i &= p_{\pi_M^{-1}(i)}, \\ o'_i &= o_{\pi_M^{-1}(i)} + \Delta_{M,i} \pmod{3}. \end{aligned} \]
For example, the $R$ turn cycles: \[ (0\,1\,5\,4) \] and its orientation vector has entries \[ \Delta_R = (\delta_0,\delta_1,0,0,\delta_4,\delta_5,0,0). \]
If $M_1, M_2$ are moves, \[ \pi_{M_2\circ M_1} = \pi_{M_2}\circ \pi_{M_1}, \] \[ \Delta_{M_2\circ M_1}(i) = \Delta_{M_2}(i) + \Delta_{M_1}(\pi_{M_2}^{-1}(i)). \]
The moves generate a subgroup of the semidirect product \[ G = S_8 \ltimes (\mathbb{Z}_3)^8. \]
Number of reachable states: \[ \frac{8!\cdot 3^7}{24} = 3,674,160. \]
If a scramble is \[ g = M_1 M_2 \cdots M_k, \] then a solution is \[ g^{-1} = M_k^{-1}\cdots M_2^{-1}M_1^{-1}. \]
Shortest solutions can be searched by BFS or IDA* in the state graph.
For moves $X,Y$: \[ [X,Y] = XYX^{-1}Y^{-1} \] produces localized permutations.
Conjugation \[ C[X,Y]C^{-1} \] relocates the effect.
The $2\times 2$ cube is modeled by corner permutations and $\mathbb{Z}_3$ orientations. Moves form a subgroup of a semidirect product, and solving corresponds to inverting a group element. Commutators and conjugations generate practical solving algorithms.