CFOP PLL Tutorial

With F2L solved and all U-layer pieces oriented, the last layer has 288 legal permutation states (4! × 4! / 2 = 288 equal-parity corner/edge pairs). These 288 states are grouped into 22 structural orbits under AUF + viewing-rotation equivalence. One orbit is the solved/AUF-only state; the remaining 21 orbits are the named PLL cases.

Denominator 72 (unconditional). Per-case and per-group probability tables use the standard denominator 72 — they include PLL-skip (1/72 of all legal PLL states). For example, EPLL covers 11/72 ≈ 15.3% of the uniform probability space. Denominator 71 (excluding skip). The learning-stage roadmap uses coverage among solves that require a PLL algorithm, dividing by 71 instead. The two denominators serve different purposes and must not be mixed.

• Four headlights distinguish EPLL from every non-skip teaching group.
• Ua and Ub have one full bar — the three-edge cycle anchor.
• H and Z have no full bar and no colour block.
• On H, the middle edge on every headlights side is the colour opposite the headlights colour; on Z it is an adjacent colour.

B/R/F/L directions: the full bar (when present) supplies the orientation anchor. For H and Z, any headlights side may serve as the reference — the opposite/adjacent edge-colour comparison holds on all four sides.

1. Is there one Full Bar? → Yes → U-perm: remaining edges cycle clockwise → Ua; cycle counterclockwise → Ub.
2. Is there one Full Bar? → No: middle edge is opposite-colour on every headlights side → H.
3. Middle edge is adjacent-colour on every headlights side → Z.

Ua ↔ Ub are both mirrors and inverses. H and Z are self-inverse and self-mirror.

Stage 0 order: Ua → Ub → H → Z. No retained cross-group confusion after the four-headlights observation.

• E has no headlights and no colour blocks — the only case in this category.
• Aa and Ab have exactly one headlight and two disconnected colour blocks.
• Using the headlights side as reference, one corner serves as the anchor around which the other three cycle.

B/R/F/L directions: mentally treat the headlights side as 12 o'clock. The two disconnected colour blocks sit on the sides adjacent to the headlights — their positions relative to the anchor corner distinguish Aa from Ab.

1. How many Headlights? → 0, no Colour Blocks → E.
2. 1 Headlight, two disconnected Colour Blocks → A-perm: three corners cycle clockwise around the anchor corner → Aa; counterclockwise → Ab.

Aa ↔ Ab are both mirrors and inverses. E is self-inverse and self-mirror.

CPLL ↔ Adjacent — shared one-headlight, no-full-bar first impression. Any connected colour block means Adjacent; none plus two disconnected blocks means CPLL/A. CPLL ↔ Diagonal — shared zero-headlights. Zero colour blocks means E; any colour block means Diagonal. CPLL ↔ G-perms — shared one-headlight, no full bar, no connected block. Count colour blocks: two means CPLL/A; one means G-perms.

Stage 0 order within CPLL: Aa → Ab → E.

• Full-bar branch (Ja, Jb, F): Ja and Jb have three additional same-direction colour blocks outside the bar; F has none outside it.
• No-full-bar branch (T, Ra, Rb): T has two connected colour blocks; Ra and Rb have one.
• Mirror direction is read relative to the headlights / full-bar side, never from an absolute face.

B/R/F/L directions: centre the headlights or full-bar side at 12 o'clock. 'Lean clockwise' means the colour block uses the corner at the clockwise end of that side; 'lean counterclockwise' means the opposite end.

1. Are the Headlights a Full Bar? → Yes → no Colour Block outside the bar → F.
2. Three outside Colour Blocks, all leaning the same way → J-perm: all lean counterclockwise → Jb; all lean clockwise → Ja.
3. Are the Headlights a Full Bar? → No → two Connected Colour Blocks → T.
4. One Connected Colour Block → R-perm: block on the headlights' counterclockwise adjacent side → Ra; clockwise adjacent side → Rb.

Ja ↔ Jb and Ra ↔ Rb are mirror pairs whose members are each self-inverse. T and F are self-inverse and self-mirror.

CPLL ↔ Adjacent — CPLL shares one headlight and no full bar, but has no connected block and has two disconnected blocks. G-perms ↔ Adjacent — G-perms also have one headlight, but have neither a full bar nor a connected block and have only one disconnected block.

Curriculum order differs from gallery order. Stage 1: T → Jb → Ja. Stage 2: Ra → Rb → F. T is established before R/F comparisons.

• Y and V have two colour blocks — in V the two blocks meet at the same corner; in Y they are disjoint.
• Na and Nb have four colour blocks, all leaning consistently around the perimeter.
• No headlights — this immediately distinguishes Diagonal from all Groups except CPLL's E case.

B/R/F/L directions: with zero headlights, use any side as the starting reference. Scan all four sides for colour blocks and trace whether they meet at a common corner (V) or remain separate (Y). For Na/Nb, all four sides display a leaning colour block — check the consistent direction.

1. How many Colour Blocks? → 2: blocks are disjoint → Y.
2. Blocks meet at the same corner piece → V.
3. 4 Colour Blocks: all lean counterclockwise → Na; all lean clockwise → Nb.

Na ↔ Nb are mirrors whose members are each self-inverse. Y and V are self-inverse and self-mirror.

At zero headlights, E can initially resemble a Diagonal case — but E has no color blocks at all, while every Diagonal case has two or four. One complete scan settles it.

Stage 1 introduces Y as the no-headlights reference. Stage 2 adds V. Stage 4 adds Na → Nb — compare both back to Y and V.

• Use the headlights side as 12 o'clock.
• The edge between the headlights is either an adjacent colour or the opposite colour relative to the headlights colour — this is the first split.
• The sole disconnected colour block sits either on the counterclockwise adjacent side, the clockwise adjacent side, or the side opposite the headlights — its position and lean complete the identification.

B/R/F/L directions: centre the headlights at 12 o'clock. The edge colour between the headlights (adjacent vs opposite) determines the row; the block position and lean determine the column in the 2×2 layout.

1. Colour between the Headlights? → Adjacent colour → sole block on counterclockwise adjacent side → Ga; on clockwise adjacent side → Gc.
2. Opposite colour → sole block on the side opposite the headlights: block leans counterclockwise → Gb; leans clockwise → Gd.

Inverse rows: Ga ↔ Gb and Gc ↔ Gd. Mirror columns: Ga ↔ Gc and Gb ↔ Gd. None of the four is self-inverse. The 2×2 layout is: top row Ga (adjacent, counterclockwise) and Gb (opposite, counterclockwise); bottom row Gc (adjacent, clockwise) and Gd (opposite, clockwise).

CPLL ↔ G-perms — Aa/Ab share one headlight, no full bar, and no connected block, but have two disconnected colour blocks rather than one. Adjacent ↔ G-perms — Adjacent cases have either a full bar or at least one connected colour block; G-perms have neither.

Stage 3 order: Ga → Gb → Gc → Gd, keeping inverse pairs consecutive.

Each PLL Group has a characteristic pair of cycle types — one describing how the four corners move, the other describing the four edges. Introduce just four structures: 3-cycle (three pieces cycle, the fourth stays — two swaps, even parity); double transposition (two disjoint pairs swap — two swaps, even); single transposition (one pair swaps — odd); and fixed (no movement for that piece type).

Every row has matching corner and edge parity — this is the readable reason a legal cube cannot swap just two corners or just two edges.

Assume F2L solved, all U-layer pieces oriented, only the four corners and four edges permuted. Index the four U-layer slots and solved labels clockwise 0, 1, 2, 3. A fixed-frame state is a pair (c, e) where c, e ∈ S₄ are the corner and edge arrangements, with equal parity (sign(c) = sign(e)). Each S₄ has 12 even and 12 odd permutations, so

|X| = 12×12 + 12×12 = 288

Two states share the same named PLL structure when they differ only by pre-AUF, target-frame/final-AUF equivalence, or whole-cube y rotation — together an action of H = C₄ × C₄ on X.

Burnside's lemma counts the orbits: identity (0,0) fixes 288; the four 4-cycle pairs (a,b ∈ {1,3}) each fix 8 states; the single (2,2) pair fixes 32; the 10 mismatched-cycle-type pairs fix 0.

(288 + 4×8 + 32) / 16 = 352 / 16 = 22 orbits

One is the solved state → 21 non-solved PLL cases.

Do not quotient inverse or mirror — Aa and Ab, Ua and Ub, Na and Nb, and all four G-perms remain distinct named cases. The frequency table (denominator 72) follows from the orbit-size distribution: 16 orbits of size 16, 2 of size 8, and 4 of size 4 (including skip).

Choose fixed-frame representatives: α = Aa (corner 3-cycle, edges fixed), ε = Ua (edge 3-cycle, corners fixed), r = U (simultaneous 4-cycle on corners and edges). Their closure ⟨α, ε, r⟩ = X, with ⟨Aa, Ua, U⟩ = 288.

Proof sketch: conjugates rᵏαr⁻ᵏ move the corner 3-cycle to different triples, generating all even corner permutations (A₄). Conjugates rᵏεr⁻ᵏ similarly generate all even edge permutations (A₄). Together they reach the even-even subgroup A₄ × A₄ (144 states). The 4-cycle r is odd, reaching the odd-odd coset (144 more states) → 288 total. This is a mathematical generating set, not a recommended solve method — never execute Aa and Ua repeatedly in a timed solve instead of learning one-step PLL algorithms. Theory relates the cases structurally but does not choose finger tricks — each Case's actual algorithm is in the quick reference at the end of this page.

Conjugate — X A X⁻¹ moves the work area, performs the core effect, moves back. e.g. UᵏAU⁻ᵏ relocates the corner 3-cycle to a different set of three corners. Commutator — [A,B] = A B A⁻¹ B⁻¹. When A and B overlap locally, the cancellation can leave a small net cycle — this supplies intuition for longer algorithms that temporarily disturb many pieces.

Inverse: Aa→Ab, Ua→Ub, Ga→Gb, Gc→Gd; E, F, H, Ja, Jb, Na, Nb, Ra, Rb, T, V, Y, Z are self-inverse. Mirror: Aa→Ab, Ua→Ub, Ja→Jb, Ra→Rb, Na→Nb; inverse(Ga) = Gb while mirror(Ga) = Gc — a decisive contrast that prevents the common mistake of assuming mirror equals inverse. Mirror relations describe case geometry, not finger-trick implementation.

These elementary structures (3-cycles, double transpositions, coordinated corner/edge transpositions) explain why algorithms cluster and why setups, inverses, and mirrors reuse ideas. They do not uniquely determine a move sequence — many algorithms realise the same permutation with different grips, rotations, and intermediate disturbance.