Gear Cube

The Gear Cube was invented by Oskar van Deventer, based on an idea by Bram Cohen. It is like a 3×3×3 Rubik's Cube where all the edge pieces are cogs that turn when the outer layers are moved. If you give any face a half turn, the middle layer moves a quarter turn due to the edge cog wheels.

The puzzle is manufactured and sold by Uwe Meffert. It is quite an easy puzzle. On the standard version the edge pieces only have stickers on the rotating cog, but it is possible to put stickers on the non-rotating base part of the edge pieces to make it a little harder.

This puzzle should not be confused with the much harder Gear Cube Extreme, or Anisotropic Gear Cube.

The number of positions:

The pieces of the gear cube are very restricted in their movements, because you can only do anti-slice moves. Instead of the centres, lets use a corner piece as a fixed reference point. The corners split into two tetrads, which do not mix. The locations of the three moving corners in the tetrad with the fixed reference corner are fully determined by the locations of the four corners in the other orbit. Those four other corners have 4! = 24 possible permutations.
The four edges in any slice will always remain in the same slice. Given a corner permutation, it turns out that the edges in a slice can only be permuted in 4 ways. They also all have the same twist, for which there are three possibilities. The three slices therefore contribute a factor (4·3)³ to the number of positions.
The centres can permute but this is fully determined by the edge permutation.
This gives a total number of positions of 4!(4·3)³ = 41,472.

If the edge bases also have stickers on them, then their orientation becomes visible. There are only 4 possible orientation arrangements these can have, making the total number of positions 4!(4·3)³·4 = 165,888. Note that this is 27 times the number of positions in the cube anti-slice group because of the 3³ possible twists of the three edge slices.

I used my computer to calculate God's Algorithm for both variations of the gear cube.

	Total	1	33	579	5,921	18,072	13,977	2,889	41,472
		Multiple turns
S i n g l e t u r n s		0	1	2	3	4	5	6	Total
	0	1							1
	1		6						6
	2		6	24					30
	3		6	48	84				138
	4		6	60	276	264			606
	5		6	72	540	1,218	264		2,100
	6		3	96	1,118	3,048	1,680	96	6,041
	7			120	1,992	5,796	4,842	702	13,452
	8			108	1,610	5,721	4,650	1,189	13,278
	9			48	252	1,716	2,220	756	4,992
	10			3	36	306	285	144	774
	11				12		36		48
	12				1	3		2	6

This shows that if any number of turns of one face is considered as one move, then at most 6 moves are necessary. If each half turn of a face counts as one move, then 12 moves are sometimes needed.

The antipodes are (see below for notation)
U6 R6 F6 (a 6X pattern)
R3 U6 F6 R3 (a 2X+4H pattern, occurs in 3 orientations)
U3 F3 R6 U6 F3 U9 (a 6H pattern, occurs in 2 orientations)

	Total	1	33	579	6,029	21,960	49,941	55,377	28,080	3,888	165,888
		Multiple turns
S i n g l e t u r n s		0	1	2	3	4	5	6	7	8	Total
	0	1									1
	1		6								6
	2		6	24							30
	3		6	48	96						150
	4		6	60	276	384					726
	5		6	72	492	1,218	1,128				2,916
	6		3	96	878	3,048	3,168	2,715			9,908
	7			120	1,632	4,872	11,382	6,024	2,724		26,754
	8			108	1,850	7,101	14,562	17,987	7,629		49,237
	9			48	756	4,152	14,556	18,870	7,920	2,832	49,134
	10			3	36	1,182	4,869	8,159	8,529		22,778
	11				12		276	1,620	1,020	1,044	3,972
	12				1	3		2	258		264
	13									12	12

This shows that if any number of turns of one face is considered as one move, then 8 moves are necessary to solve the gear cube with extra stickers in the worst case. If each half turn of a face counts as one move, then 13 moves are sometimes needed.

Notation:

Let F denote a clockwise half turn of the front face, keeping the rear face stationary. Similarly, let R and U denote clockwise half turns of the Right and Upper faces respectively.

Solution:

Phase 1: Solve the corners.

Find two corner pieces that should be adjacent (they have two colours in common). Concentrating on only those two corners, do any moves needed to make them match. This is easy and takes at most two moves.
The corners now form four matching pairs. Hold the puzzle so that the matching pairs form vertical columns. Doing R or U keeps those columns intact, and by alternating those moves the columns will eventually all match up so that all the corners are correct.

Phase 2: Solve the edges.

Look at the edge piece at the UF location, shared between the top and front faces. By comparing its two colours to those of the corners, find out where the edge piece belongs.
Do one of the following, depending on where the UF edge piece belongs:
UF: Do nothing.
UB: Do F R R F
DB: Do R R
DF: Do U R R U
All four edge pieces in the vertical slice should now be correctly positioned.
If any edge is incorrectly positioned, then hold the puzzle with that edge piece at the UF location, and do steps a-b to solve it. Repeat until all edges are correct

Phase 3: oriented the edges.

If any edge needs to be twisted, then hold the puzzle with that edge piece at the UF location, and do R R R R. Repeat if necessary until all edges are twisted correctly.
If the edge base parts have visible orientation and some need to be flipped, then hold the puzzle so that any unflipped edges lie in the horizontal middle layer, and do F R F R F R.

Jaap's Puzzle Page

The Gear Cube

The number of positions:

Links to other useful pages:

Notation:

Solution: