§3

State vector: (cp, co, ep, eo)

A 3×3×3 cube has 8 corners and 12 edges. Centres are fixed (they define the colour scheme). The full state is captured by four arrays:

c_{p} \in S_{8}

positions of the 8 corners (permutation)

c_{o} \in (Z /3)^{8}

orientation of each corner (twist)

e_{p} \in S_{12}

positions of the 12 edges

e_{o} \in (Z /2)^{12}

orientation of each edge (flip)

These are the standard coordinates Kociemba chose in the 1990s for his two-phase solver [7]. All of cube algebra is built on this 4-tuple: each generator is a fixed "permute cp + add offsets to co/ep/eo" combination.

3.1 Unfolded layout & cubie indexing

The 3×3×3 unfolded: 54 stickers but only 26 cubies (8 corners × 3 stickers each, 12 edges × 2, 6 centres × 1). Centres are immobile, so the state needs only describe the 8 + 12 = 20 movable cubies.

The cubie indexing used throughout (following Kociemba):

corners: 0:URF 1:UFL 2:ULB 3:UBR 4:DFR 5:DLF 6:DBL 7:DRB edges: 0:UR 1:UF 2:UL 3:UB 4:DR 5:DF 6:DL 7:DB 8:FR 9:FL 10:BL 11:BR

3.2 Orientation convention

Each corner at a fixed position has 3 possible orientations (its "U/D-coloured sticker" can face one of three axes). Each edge has 2 orientations ("good" or "flipped" relative to the F/B-axis convention). So:

corner orientation ∈ ℤ/3 = {0, 1, 2} (0 = aligned, +1 = CW, +2 = CCW)
edge orientation ∈ ℤ/2 = {0, 1} (0 = good, 1 = flipped)

The exact definition depends on convention. Here we use Kociemba's: corner orientation tracked by the U/D sticker; edge orientation by the F/B sticker. Singmaster and others use slightly different bases — the algebraic structure is unchanged.

3.3 Tensor representation of generators

Each face turn corresponds to a (permutation, orientation offset) pair. For example, R:

R_{c_{p}} = (0473)

4-cycle on corners

R_{c_{o}} = (+ 2, 0, 0, + 1, + 1, 0, 0, + 2)

corner twist deltas

R_{e_{p}} = (08114)

4-cycle on edges

R_{e_{o}} = 0

R does not affect EO (since R is on the RL-axis)

Similarly: F flips the EO of 4 edges (UF, DF, FR, FL each +1 mod 2). U/D turns change neither CO nor EO — only positions. This clear axis-specific structure makes both state compression and solver search highly efficient:

state size \approx corners (8 perm + 8 ori bits) + edges (12 perm + 12 ori bits) \approx 10 bytes

Compare: naively storing colours for all 54 stickers takes 162 bits ≈ 21 bytes, with no compression. The structured encoding halves memory and intrinsically excludes illegal states.

Interactive § State tensor

A cube state is the 4-tuple (cp, co, ep, eo). Type any alg, watch the four arrays mutate.

alg

cp ∈ S₈ — corner permutation

co ∈ (ℤ/3)⁸ — corner twist

ep ∈ S₁₂ — edge permutation

1010

1111

eo ∈ (ℤ/2)¹² — edge flip

100

110

corner cycle type

2-cycle × 2-cycle

edge cycle type

3-cycle

3.4 Counting degrees of freedom

Treat cp / co / ep / eo as four independent coordinates. Their raw cardinalities:

Coord	Codomain	Size
$c_{p}$	$S_{8}$	8! = 40,320
$c_{o}$	$(Z /3)^{8}$	3⁸ = 6,561
$e_{p}$	$S_{12}$	12! = 479,001,600
$e_{o}$	$(Z /2)^{12}$	2¹² = 4,096
product (free space F)	$∣ F ∣ = 5.19 \times 1 0^{20}$
cube group G	$∣ G ∣ = ∣ F ∣/12 \approx 4.33 \times 1 0^{19}$

So "position + orientation" carries $lo g_{2} ∣ F ∣ \approx 69.1$ bits of information, but G only sits in $lo g_{2} ∣ G ∣ \approx 65.2$ of those. The missing $lo g_{2} 12 \approx 3.58$ bits are precisely the three invariants of §5 (ℤ/3 × ℤ/2 × ℤ/2).

3.5 ℓ₁ and ℓ∞ distances to the origin

Embed state vectors into $R^{40}$ (20 position indices + 8 mod-3 + 12 mod-2) and ask "distance to origin." Two natural norms:

$ℓ_{1}$ mismatch count = #(cubies off-position) + #(orientation deltas). Max ≈ 36 (8 + 12 + 8 + 12 minus a few combinatorial constraints) — the seed of Korf's simple admissible heuristic in §22.
$ℓ_{\infty}$ worst-cubie offset = max over all cubies. For a random state this is almost always 1 (at least one cubie misplaced). A very loose lower bound on HTM distance.

Crucial: these norms are not equivalent to the HTM metric $∣ g ∣_{S}$ (§2.2). They are weak lower bounds on d_S(e, g), used as heuristics by §22's solvers. The true d_S(e, g) is the Cayley-graph distance — no closed form; computed only by Korf IDA* or Kociemba two-phase.

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