# Corner-turning Gear Octahedron The Corner-turning Gear Octahedron like a geared version of Trajber's Octahedron. The six vertex pieces and the twelve edge pieces are all eight-toothed gears. The triangular face centres are not geared. When you twist a vertex, the gears mesh so that the edges twist as you do a move. When the pieces have travelled around by a quarter turn, the edges have all twisted 5/8ths of a turn, and the vertex has twisted 7/8ths of a turn.

## The number of positions:

There are 6 unmoving vertex pieces with 8 orientations, 12 moving edge pieces also with 8 orientations, and 8 moving face centres, giving a maximum of 86·12!·812·8! positions. This limit is not reached because:

• The total twist of the vertices has the same parity as the corner permutation (2)
• Every move is an even permutation so only even permutations of pieces are possible (2)
• The total twist of the edges is determined by the twist of the vertices (8)

This leaves 86·12!·812·8! / (2·2·8) = 10,872,446,393,440,151,739,039,744,000 or 1.09·1028 positions.

## Notation:

The vertices are denoted by the letters L, R, F, B, U, D (Left, Right, Front, Back, Up and Down). Clockwise quarter turns are denoted by the appropriate letter, anti-clockwise turns by the letter followed by an apostrophe (e.g. L' means turn the Left vertex 90 degrees anti-clockwise). To indicate moves of more than a quarter turn, a number is added to show ho many quarter turns are done. For example F2 means a clockwise half turn of the Front vertex, and F'2 an anti-clockwise half turn.

## Solution:

Phase 1: Solve vertex pieces.

1. Rotate the F, B, L, R and U vertices so that the colours of the vertex pieces match each other. Remember to do complete moves, so that the face centres still lie at the centre of each face. The D vertex will be done later.

Phase 2: Place the U edges correctly.

1. First find any edge that belongs next to the U vertex but which is incorrectly positioned.
2. If the edge is not at the D vertex, then hold the puzzle so that the side with the edge piece is at the front and use one of the following sequences to bring it to the bottom:
Move FR to the D vertex: Do F D F'.
Move UF to the D vertex: Do F2 D F'2.
Move FL to the D vertex: Do F' D F.
3. Hold the puzzle so that the edge belongs at the front, i.e. at FU.
4. Rotate D to place the piece at the right hand side, i.e. at DR.
5. Put it in place by using the move sequence F2 D' F2.
6. If the UF edge gear needs to be twisted, then do one or more of the following sequences:
Eight of a turn: R F R' F D3 F'2
Quarter turn: F2 D F'2 R U' R U R'2
Half turn: F2 D4 F'2
7. Repeat the above until all U edges are solved.

Phase 3: Place middle edges correctly.

1. Find an edge that belongs in the middle layer, but which is not in the correct position.
2. If it lies in the middle layer (at the wrong location) then hold the puzzle so that it is at the FR location and do F D F' to put it next to the D vertex.
3. Hold the puzzle so that the edge belongs at FR.
4. Rotate D to bring the edge to the DR position.
5. Insert the edge at FR by the sequence F D' F'.
6. If the FR edge gear needs to be twisted, then do one or more of the following sequences
Eight of a turn: F D'2 F' R' D' R
Quarter turn: F D F'2 R F R'2 D R
Half turn: F D4 F'
7. Repeat the above until all middle layer edges are solved.

Phase 4: Solve last vertex piece.

1. Turn the puzzle upside down so that the last vertex is at U.
2. Rotate U until the vertex piece matches the others.

Phase 5: Solve the four U edges.

1. Find the edge piece that belongs at the UF location. Depending on where it is, do one of the following sequences to put it in position:
The UF location: do nothing.
The UR location: do F U2 F' U' F U' F' (UR->UF->UB->UR)
The UB location: do F U F' U F U'2 F' (UB->UF->UR->UB)
The UL location: do F U F' U'2 F U F' (UL->UF->UR->UL)
2. Now that the UF edge is in location, we can orient it using the following sequences one or more times:
Clock wise 1/8th turn: R' F R F' U'3 F' U3 F
Clock wise 1/4th turn: R2 U D' F2 U' F'2 U' D R'2 U
Half turn: F4 U F'4 U'
3. Turn the whole puzzle to the left so that the solved UF piece now lies at the UL location.
4. Repeat steps a-c for each of the edge pieces in turn so that they are all solved. The last edge piece should be automatically correct, because otherwise the puzzle has been disassembled or the gears have slipped past each other.

Phase 6: Solve the face centres

1. Hold the puzzle so that there is an incorrect face centre at the DFR face, and so that the face where it belongs lies in the top half, and so that there is at least one other incorrect face centre in the top half as well. This is always possible if there are incorrect centres. If there are only two incorrect centres, then the puzzle has been taken apart and reassembled incorrectly, or has been incorrectly re-stickered.
2. Move the centres using the following steps:
1. Find the face that the DFR centre belongs to.
2. Turn the U vertex to bring part of that face to the UFR location.
3. Do R' D' R to replace the face centre piece.
4. Turn the U vertex so that the other incorrect face centre lies at the UFR location.
5. Do R' D R to replace the face centre piece.
6. Turn U back to its original state.
3. Repeat a-b until all face centres are correct.