Goal:
- Orient the corner cubies.
- Put the u- and d-layer edges into those two layers. (A d-layer edge may be in u layer, and a u-layer edge may be in the d layer.)
Turns allowed:
U, U', U2, u, u', u2, D, D', D2, d, d', d2,
L, L', L2, l, l', l2, R, R', R2, r, r', r2,
F, F', F2, f, f', f2, B, B', B2, b, b', b2
One-time whole cube rotations allowed: 120-degree turns (either direction) about the UFL-DBR axis.
There are 3 possibilities for each corner orientation and 8 corners so 3^8 possibilities. However, when setting the orientation of 7 corners, the 8th is fixed so only 3^7 = 2,187 cases.
For the edges, we have to record the position of 8 edges among 24 slots, so C^24_8 = 735,471 cases. This coordinate is reduced by symmetry. There are 48 symmetries, which let to only 15582 cases.
The overall space of this stage is 2,187 x 15,582 = 34,077,834.
Goal:
- Put front and back centers onto the front and back faces into one of the twelve configurations that can be solved using only half-turn moves.
- Arrange u- and d-layer edges within the u- and d-layers so that they will be in one of the 96 configurations that can be solved using only half-turn moves.
Turns allowed:
U, U', U2, u, u', u2, D, D', D2, d, d', d2,
L2, l, l', l2, R2, r, r', r2,
F2, f, f', f2, B2, b, b', b2
One-time whole cube rotations allowed: 90-degree turn about U-D axis.
As for edges of stage 1, there are C^24_8 = 735,471 cases for storing the position of the 8 F/B centers, and we also need to keep track of which center are F and which are B, requiring another C^8_4 = 70, so a total of 735,471 x 70 = 51,482,970. However, we don't need to track the exactly colour of those centers, but only if two centers are from the same or different colours. This reduce by a factor of two, giving 735,471 x 35 = 25,741,485. This (large) coordinate is reduced by symmetry (16 symmetries), giving 1,612,515 unique coordinates.
For the edges, the total number of permutations is 8! = 40,320 and there are 96 configurations that are considered solved (square group), so the coordinate is 40,320/96 = 420.
The overall space of this stage is 1,612,515 x 420 = 677,256,300
Goal:
- Put centers for left and right faces into the left and right faces so that they are in one of the 12 configurations that can be solved using only half-turn moves. This leaves the centers for the U and D faces arbitrarily arranged in the U and D faces.
- Put top and bottom layer edges into positions such that the U or D facelet is facing either up or down. Also, put these edges into an even permutation.
Turns allowed:
U, U', U2, u2, D, D', D2, d2,
L2, l2, R2, r2,
F2, f, f', f2, B2, b, b', b2
For centers, this is the same as for stage 2, except that now there are only 16 remaining slots for one center, as F and B faces are filled during stage 2. Using the same principle, the center coordinate is C^16_8 x C^8_4 = 12,870 x 35 = 450,450. Using symmetry reduction (8 in this stage), we only need to store 56,980 positions.
For edges, we need to put half of the edges in half of the positions, so C^16_8 = 12,870 cases. The even permutation required gives a extra factor of 2.
The overall space of this stage is 56,980 x 12,870 x 2 = 1,466,665,200.
Goal:
- Put corners into one of the 96 configurations that can be solved using only half-turn moves.
- Put U and D centers into one of the 12 configurations that can be solved using only half-turn moves.
- Put all U- and D-layer edges into a configuration that can be solved using only half-turn moves. This consists of 96 possible configurations for the l- and r-layer edges, and 96 for the f- and b-layer edges.
Turns allowed:
U, U',U2, u2, D, D', D2, d2,
L2, l2, R2, r2,
F2, f2, B2, b2
The corner coordinate is exactly like the edge coordinate from stage 2, which gives 420 cases.
The remaining centers are already in the right faces, so we only need to put them in the right order: C^8_4 = 70. Using the same trick as for stage 2 and 3, we only need to keep 35 cases.
The edges are as the corners, except that there are two groups of edges so 420 x 420 = 176,400. In fact, only half of the cases happen, because of the parity condition from stage 3, so 88,200 real cases. We are doing the symmetry reduction on this coordinate (16 symmetries), which leave only 5,968 cases.
The overall size is 420 x 35 x 5,968 = 87,729,600
Goal:
- Put all cubies into their solved position.
Turns allowed:
U2, u2, D2, d2,
L2, l2, R2, r2,
F2, f2, B2, b2
There are 96 positions for corners.
Edges are like 3 independent groups of corners, so 96 x 96 x 96 = 884,736 positions. We are doing a symmetry reduction, and we use a trick to get as much as 192 different symmetries, so that this coordinate has only 7,444 positions. In addition to the usual 48 symmetries of the cube, we add 4 cube rotations (generated by x2 and y2) because the cube can be in four different solved positions. This allows us to reduce this coordinate by a factor of 48 x 4 = 192.
For centers, each pairs of opposite centers have 12 different configurations, so 121212 = 1,728 positions.
The overall size is 967,4441,728 = 1,234,870,272
There are 24 "edge" cubies, numbered 0 to 23. The home positions of these cubies are labeled in the diagram below. Each edge cubie has two exposed faces, so there are two faces labelled with each number.
-------------
| 5 1 |
|12 U 10|
| 8 14|
| 0 4 |
-------------------------------------------------
| 12 8 | 0 4 | 14 10 | 1 5 |
|22 L 16|16 F 21|21 R 19|19 B 22|
|18 20|20 17|17 23|23 18|
| 9 13 | 6 2 | 11 15 | 7 3 |
-------------------------------------------------
| 6 2 |
|13 D 11|
| 9 15|
| 3 7 |
-------------
There are 8 "corner" cubies, numbered 0 to 7. The home positions of these cubies are labeled in the diagram below. Each corner cubie has three exposed faces, so there are three faces labelled with each number. Asterisks mark the primary facelet position. Orientation will be the number of clockwise rotations the primary facelet is from the primary facelet position where it is located.
+----------+
|*5* *1*|
| U |
|*0* *4*|
+----------+----------+----------+----------+
| 5 0 | 0 4 | 4 1 | 1 5 |
| L | F | R | B |
| 3 6 | 6 2 | 2 7 | 7 3 |
+----------+----------+----------+----------+
|*6* *2*|
| D |
|*3* *7*|
+----------+
There are 24 "center" cubies. They are numbered 0 to 23 as shown.
-------------
| |
| 3 1 |
| 0 2 |
| |
-------------------------------------------------
| | | | |
| 10 8 | 16 19 | 14 12 | 21 22 |
| 9 11 | 18 17 | 13 15 | 23 20 |
| | | | |
-------------------------------------------------
| |
| 6 4 |
| 5 7 |
| |
-------------
Stage 1 - 48 symmetries
Slice turns
------------------------
distance positions unique
-------- --------- ------
0 3 1
1 6 1
2 144 4
3 2,796 66
4 48,324 1,033
5 745,302 15,620
6 10,030,470 209,273
7 103,416,912 2,155,397
8 575,138,592 11,984,424
9 826,559,202 17,222,730
10 92,489,544 1,927,399
11 43,782 916
------------- -----------
1,608,475,077 33,516,864
Stage 2 - 16 symmetries
Slice turns
------------------------
distance positions unique
-------- --------- ------
0 12 6
1 36 8
2 684 54
3 9,254 661
4 103,998 6,785
5 1,149,674 73,297
6 11,929,486 750,382
7 92,729,838 5,803,099
8 447,778,202 27,991,967
9 1,247,722,776 77,990,037
10 1,930,825,644 120,695,743
11 2,215,400,576 138,498,874
12 2,607,462,418 163,000,022
13 1,828,141,454 114,282,664
14 426,682,056 26,675,281
15 1,487,536 93,585
16 56 5
------------ -------------
10,811,423,700 675,862,470
Stage 3 - 8 symmetries
Slice turns
------------------------
distance positions unique
-------- --------- ------
0 6 6
1 12 4
2 150 28
3 1,556 230
4 16,310 2,185
5 169,240 21,630
6 1,717,460 216,142
7 16,888,105 2,115,779
8 155,841,738 19,496,147
9 1,219,752,205 152,510,075
10 5,364,611,902 670,664,810
11 4,687,652,572 586,031,875
12 147,926,722 18,500,776
13 5,021 732
14 1 1
-------------- -------------
11,594,583,000 1,466,665,200
Stage 4 - 16 symmetries
Slice turns
------------------------
distance positions unique
-------- --------- ------
0 6 4
1 12 3
2 102 12
3 640 56
4 3,774 285
5 20,482 1,475
6 113,908 7,655
7 629,922 40,711
8 3,456,044 219,404
9 17,629,510 1,110,842
10 76,036,148 4,774,517
11 233,265,250 14,621,516
12 379,795,898 23,799,062
13 369,324,336 23,143,536
14 193,668,736 12,142,383
15 22,539,628 1,420,768
16 55,572 3,983
17 32 8
------------- -----------
1,296,540,000 81,286,220
Stage 5 ("Squares Coset") - 192 symmetries
Slice turns
--------------------------
distance positions unique
-------- --------- ------
0 4 1
1 48 2
2 420 7
3 3,456 36
4 27,168 228
5 203,752 1,429
6 1,451,996 9,127
7 9,527,856 55,967
8 56,528,036 320,517
9 295,097,696 1,636,219
10 1,306,291,304 7,145,262
11 4,761,203,264 25,797,686
12 13,820,728,272 74,257,367
13 29,956,341,744 159,930,965
14 43,427,866,752 231,079,243
15 36,297,535,208 193,022,572
16 14,711,566,720 78,368,608
17 2,063,584,704 11,055,492
18 59,082,112 320,252
19 45,056 244
--------------- -------------
146,767,085,568 783,001,224