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*15,582 = 34,077,834 which is small enough to allow using a single pruning table.
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*70=51,482,970. This is a bit too much for a single coordinate, so we split the centers into fcenterF and centerB coordinates, each one having C^24_4 = 10626 positions. With symmetry reduction (16 symmetries), both coordinates are reduced to 716. 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.
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 * C^8_4 = 12,870 * 70 = 900,900. Moreover, we don't need to keep track of which center is the U center and which one is the D center. This allows to cut in half the number of cases, so 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 total size of this stage is 56,980 * 12,870 * 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 3, we only need to keep 35 cases. The edges are as the corners, except that there are two groups of edges so 420420=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 42035*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 969696 = 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 (see the appropriate section). 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 - *** not updated ***
Slice turns
------------------------
distance positions unique
0 24 14 1 48 11 2 684 99 3 7,338 997 4 68,276 8,824 5 614,616 78,097 6 5,372,580 675,305 7 41,587,696 5,206,350 8 264,525,432 33,076,413 9 1,173,434,250 146,693,452 10 2,891,653,248 361,482,039 11 4,023,107,440 502,932,549 12 4,610,360,196 576,354,995 13 4,818,898,672 602,411,843 14 2,904,398,972 363,077,183 15 804,769,384 100,607,241 16 82,031,496 10,256,713 17 2,007,656 251,493 18 9,392 1,192 ------------ ------------- 21,622,847,400 2,703,114,810
Stage 3 - *** not updated ***
Slice turns
------------------------
distance positions unique
0 12 7 1 24 6 2 300 47 3 3,112 427 4 32,620 4,241 5 338,480 42,806 6 3,434,920 430,920 7 33,776,210 4,227,153 8 311,683,476 38,977,409 9 2,439,504,410 304,981,049 10 10,729,223,804 1,341,243,036 11 9,375,305,144 1,171,989,581 12 295,853,444 36,991,377 13 10,042 1,360 14 2 1 -------------- ------------- 23,189,166,000 2,898,889,420
Stage 4 - 16 symmetries - problem with positions, not correct.
Slice turns
------------------------
distance positions unique
0 12 4 1 24 3 2 204 12 3 1,280 40 4 7,548 171 5 40,964 899 6 227,816 4,528 7 1,259,844 21,918 8 6,912,088 108,036 9 35,259,020 534,374 10 152,072,296 2,417,720 11 466,530,500 8,958,735 12 759,591,796 23,141,105 13 738,648,672 30,826,779 14 387,337,472 14,255,009 15 45,079,256 1,014,899 16 111,144 1,988 ------------- ----------- 2,593,080,000 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