Twenty pieces. Two permutations.
Orientation makes the difference.
Bundle stickers into the pieces they belong to. Then watch corners and edges move in synchronized cycles—carrying a twist or a flip with them.
The physical pieces
Drag to rotateTap a piece to follow it in all three views. The face letters define a fixed reference frame.
8 corners
twist ∈ {0, 1, 2}12 edges
flip ∈ {0, 1}A move acts on both sets at once
Start with a single turn, then try a sequence. The diagrams show the actual permutations; their circular layout is just a readable arrangement of the slots.
Move history
Solved
Why the pieces are not independent
A corner has three orientations, but the last corner’s twist is forced by the other seven. Likewise, the last edge’s flip is forced by the other eleven.
A quarter turn makes a 4-cycle of corners and a 4-cycle of edges. Each cycle is odd, so the two permutation parities change together.
Σ flip ≡ 0 (mod 2)
sgn(corners) = sgn(edges)
Try an impossible reassembly
These buttons start from solved and change the pieces by hand. Watch which invariant fails. Ordinary face turns cannot repair the violated invariant.
The paired swap passes all three tests and is reachable, even though neither isolated swap is.
For an ordinary 3×3 cube with a fixed center frame, these conditions are both necessary and sufficient. They reduce all type-preserving reassemblies to 1/12 of the possibilities.
There are two different kinds of “reduction”
Bundle stickers into oriented pieces
This is a change of coordinates with no loss. A corner’s three stickers become one identity and one value in ℤ/3; an edge’s two stickers become one identity and one value in ℤ/2. The 48-sticker representation is valid, but it hides these rigid bundles.
Then forget the orientations
This really does lose information. Let P be the group of possible corner/edge position permutations. The projection G → P identifies states that differ only in orientation.
Choose , then uncheck “Keep orientation information.” Every piece is home, so the positional view becomes identical to solved.
The information inside each fiber
Each position arrangement has 37 × 211 = 4,478,976 reachable orientation assignments above it. Above the identity permutation these form the normal subgroup N of orientation-only operations.
1 → N → G → P → 1
Permuting pieces also permutes where their orientation values live. That interaction gives a semidirect product:
Not an ordinary direct product: a twist attached to one slot is carried to another by a position permutation.
| Information | Choices |
|---|---|
| Corner + edge positions, with matching parity | 8! × 12! / 2 |
| Corner twists | 37 |
| Edge flips | 211 |
Where graph theory actually enters
The diagrams above visualize an action on piece positions. A chosen face turn acts as two directed cycles plus fixed points. Adding all generators gives a labeled action graph; a token also needs orientation data. This is different from a Cayley graph, whose vertices represent entire group elements (identified here with reachable cube states), and whose edges represent face turns. The latter has about 4.3 × 1019 vertices. Merely drawing 20 dots does not turn the full cube state space into a 20-vertex Cayley graph.