# Some "F2L" computer analyses

I have done some "restricted" God's algorithm calculations for solving two layers of Rubik's Cube, given that the four edges of the face layer are already solved. I say it's a "restricted" calculation because it only considers certain moves or sequences of moves. For convenience, I will consider the two layers to be solved are the bottom layer (generally referred to as the D layer), and the layer above that (typically referred to by speedcubers as the E layer).

Given that four edges of the D layer are taken to be solved,
the remaining four edges being considered (those that belong in the E layer)
must be distributed among 8 possible edge locations.
So the number of configurations for those four edges is
8*7*6*5*(2^{4}) = 26880 (including orientations).
The four corner cubies being considered (those that belong in the D layer)
have 8*7*6*5*(3^{4}) = 136080 configurations (including orientations).
This makes for a total of 26880*136080 = 3,657,830,400 configurations.

By restricting the moves (or move sequences) that are considered so that the four D layer edges (these will be referred to as the cross edges from now on) remain in their solved positions, only these 3,657,830,400 configurations need to be considered. So my first calculation limited the moves considered to those listed below.

In this analysis, a position from which the solved state (i.e. the two layers of interest solved) can be reached via one of the specified 3-move sequences is considered to be a distance of 3 from solved. A position that can be solved with a U layer turn followed by one of the three move sequences is considered to be a distance of 3 + 1 = 4 from solved, and so on.

Allowed moves/sequences: counted as one move: U U' U2 counted as three moves: (L U L') (L U' L') (L U2 L') (L' U L) (L' U' L) (L' U2 L) (R U R') (R U' R') (R U2 R') (R' U R) (R' U' R) (R' U2 R) (F U F') (F U' F') (F U2 F') (F' U F) (F' U' F) (F' U2 F) (B U B') (B U' B') (B U2 B') (B' U B) (B' U' B) (B' U2 B) distance positions -------- --------- 0 1 1 0 2 0 3 24 4 40 5 0 6 444 7 2,032 8 1,732 9 8,236 10 60,212 11 116,090 12 203,792 13 1,376,580 14 4,033,014 15 6,680,438 16 25,759,944 17 80,363,704 18 148,794,392 19 400,179,802 20 1,084,980,493 21 1,124,562,280 22 624,375,623 23 139,420,738 24 11,830,455 25 5,080,152 26 182 ------------- total 3,657,830,400

Arguably, there is a flaw in this calculation in that a sequence of moves such as: (R U' R') (R' U2 R) is counted as 6 moves, but could be optimized to R U' R2 U2 R (only 5 face turns).

I have performed another calculation where 32 four-move sequences that preserve the cross are also considered. This allows all but 9 positions to be solved using 25 or less moves. See summary below.

Allowed moves/sequences: counted as one move: U U' U2 counted as three moves: (L U L') (L U' L') (L U2 L') (L' U L) (L' U' L) (L' U2 L) (R U R') (R U' R') (R U2 R') (R' U R) (R' U' R) (R' U2 R) (F U F') (F U' F') (F U2 F') (F' U F) (F' U' F) (F' U2 F) (B U B') (B U' B') (B U2 B') (B' U B) (B' U' B) (B' U2 B) counted as four moves: (L F' L' F) (L F2 L' F2) (L2 F' L2 F) (L2 F2 L2 F2) (L' B L B') (L' B2 L B2) (L2 B L2 B') (L2 B2 L2 B2) (R B' R' B) (R B2 R' B2) (R2 B' R2 B) (R2 B2 R2 B2) (R' F R F') (R' F2 R F2) (R2 F R2 F') (R2 F2 R2 F2) (F R' F' R) (F R2 F' R2) (F2 R' F2 R) (F2 R2 F2 R2) (F' L F L') (F' L2 F L2) (F2 L F2 L') (F2 L2 F2 L2) (B L' B' L) (B L2 B' L2) (B2 L' B2 L) (B2 L2 B2 L2) (B' R B R') (B' R2 B R2) (B2 R B2 R') (B2 R2 B2 R2) distance positions -------- --------- 0 1 1 0 2 0 3 24 4 64 5 72 6 444 7 3,144 8 7,951 9 18,622 10 95,670 11 384,650 12 1,033,982 13 3,175,760 14 12,081,182 15 36,424,316 16 92,555,582 17 255,605,628 18 653,124,460 19 1,096,446,464 20 1,084,487,378 21 393,409,884 22 28,509,722 23 463,652 24 1,739 25 9 ------------- total 3,657,830,400

The 9 distance-25 positions in the above analysis can be reduced
to 3 patterns by using symmetry.
These patterns can be generated by:

(L U L') (F U' F') (R' U R) (R U2 R') (B' R2 B R2) (B' U2 B) (F' U F) (L' U' L)

(L U' L') (F' U F) (R' U' R) (R U2 R') U2 (F U F') (R U' R') (L U2 L') (L' U' L)

(L U L') (R' U2 R) (R U' R') (R' F2 R F2) U (R U2 R') U (R' U' R) U (L U L')

Obviously, from the sequences given above, these positions can all be solved in 24 face turns or less using trivial optimizations of the given sequences.

I performed a third calculation where D layer moves are also considered, but not the four-move sequences from the previous calculation. With D layer turns the four cross edges can now have four different configurations, so the number of positions considered becomes 4*3,657,830,400 = 14,631,321,600. Note that the four cross edges are merely permuted amongst themselves, so the other edges are still limited to the same remaining 8 edge positions.

Allowed moves/sequences: counted as one move: U U' U2 D D' D2 counted as three moves: (L U L') (L U' L') (L U2 L') (L' U L) (L' U' L) (L' U2 L) (R U R') (R U' R') (R U2 R') (R' U R) (R' U' R) (R' U2 R) (F U F') (F U' F') (F U2 F') (F' U F) (F' U' F) (F' U2 F) (B U B') (B U' B') (B U2 B') (B' U B) (B' U' B) (B' U2 B) positions with: --------------------------------------------- A. solved B. cross rotated C. cross rotated total positions distance cross +/- 90 deg 180 deg (A+2B+C) -------- ----------- ------------- ------------- --------------- 0 1 0 0 1 1 0 1 1 3 2 0 0 0 0 3 24 0 0 24 4 40 48 48 184 5 72 128 128 456 6 564 80 80 804 7 2,032 1,464 1,460 6,420 8 6,292 9,520 9,440 34,772 9 30,020 21,908 22,040 95,876 10 86,220 61,424 60,928 269,996 11 282,834 380,052 373,172 1,416,110 12 1,442,740 1,412,144 1,420,360 5,687,388 13 4,253,940 3,550,728 3,556,892 14,912,288 14 10,775,680 12,628,394 12,522,088 48,554,556 15 45,721,886 47,613,567 47,699,038 188,648,058 16 125,038,958 114,134,678 114,145,388 467,453,702 17 248,217,526 263,563,353 261,694,326 1,037,038,558 18 713,565,520 720,915,551 720,143,790 2,875,540,412 19 1,149,313,518 1,080,898,530 1,082,829,610 4,393,940,188 20 916,944,377 956,586,249 956,485,206 3,786,602,081 21 396,869,429 410,799,168 411,375,646 1,629,843,411 22 36,579,071 32,271,898 32,645,183 133,768,050 23 8,096,077 12,405,618 12,260,201 45,167,514 24 492,748 9,753 19,231 531,485 25 110,828 566,116 566,125 1,809,185 26 3 28 19 78 ------------- ------------- ------------- -------------- total 3,657,830,400 3,657,830,400 3,657,830,400 14,631,321,600

The three positions with solved cross at distance 26 have all cubies
in the correct place, but one of them has all four non-cross edges flipped,
and the other two have all four corners twisted,
with diagonally opposite corners having the same twist,
but neighboring corners having opposite twist.
Sequences for these positions are:

(L U L') (L' U L) (F U2 F') (R U' R') D2 (B U B') (B' U B) (R' U' R) (F' U' F) D2

(L U L') (L' U L) (F' U F) (R' U' R) U' (F' U F) (L' U2 L) U (L U' L') (R' U2 R)

(L U L') (L' U' L) (B U' B') D2 (L U' L') (B' U' B) (L' U' L) (B U' B') D2 (L U2 L')

Clearly, these sequences can be trivially optimized to sequences of 25 face turns or less.