# Last Layer Optimal Solving

I have run all 8020 symmetrically distinct "last layer" positions of Rubik's Cube with the optimal solver of the Cube Explorer program. All these positions could be solved in 16 face turns or less. I also used the number of positions associated with each of these 8020 symmetry class representatives to determine the precise distribution of distances of all 62208 "last layer" (abbreviated LL) positions. Note that this analysis considers solving the last layer with respect to the already solved first two layers.

I note that Helmstetter (see here) has done a similar analysis previously, but his analysis basically only considers solving the last layer pieces relative to themselves, and does not consider what cases may need an additional move to align the last layer properly with the first two layers. My analysis includes all moves needed to solve the last layer with respect to the first two layers. So Helmstetter considered only 1212 cases (15552 "relative" positions reduced by symmetry and antisymmetry), while I considered 8020 cases (62208 "absolute" positions reduced by symmetry).

First, I give a summary for the entire set. (I note that with Cube Explorer, the optimal solver reports the solved cube as 8f*, but here I report it as a 0f* position.)

Last Layer optimal FTM analysis Moves Patterns Positions Product ----- -------- --------- ------- 0f* 1 1 0 1f* 2 3 3 6f* 2 16 96 7f* 11 88 616 8f* 25 200 1600 9f* 50 394 3546 10f* 143 1102 11020 11f* 443 3470 38170 12f* 1249 9807 117684 13f* 2611 20505 266565 14f* 2683 20768 290752 15f* 762 5630 84450 16f* 38 224 3584 ---- ----- ------ Total 8020 62208 818086 Average distance is approximately 13.15 face turns. Patterns = number of positions as reduced by the use of symmetry Product = Moves * Positions (used to compute average distance)

I note that Helmstetter arrived at an average of 12.58 face turns for solving the last layer pieces with respect to themselves.

I also split up the positions with respect to corner orientation as well as with respect to edge orientation. The following table gives the summary of these results.

Last Layer optimal FTM analysis - categorized by corner orientation corners 2 adj corners 2 opp corners 3 corners 4 corners 4 corners oriented twisted twisted twisted twisted ++-- twisted +-+- ---------- ------------- ------------- ------------- ------------- ------------- Moves pat pos pat pos pat pos pat pos pat pos pat pos ----- --- --- --- --- --- --- --- --- --- --- --- --- 0f* 1 1 1f* 2 3 6f* 2 16 7f* 6 48 5 40 8f* 5 40 2 16 18 144 9f* 6 42 20 160 13 104 9 72 2 16 10f* 24 174 55 436 30 224 23 184 10 76 1 8 11f* 17 120 155 1236 75 576 141 1128 37 284 18 126 12f* 45 305 343 2708 195 1536 431 3448 164 1292 71 518 13f* 118 833 795 6240 393 3100 764 6112 380 3000 161 1220 14f* 102 678 776 6016 377 2948 721 5768 441 3416 266 1942 15f* 21 134 198 1468 95 692 182 1456 147 1108 119 772 16f* 4 14 13 64 4 20 10 80 3 24 4 22 ---- ---- ---- ----- --- ---- ---- ----- ---- ---- --- ---- Total 340 2304 2368 18432 1184 9216 2304 18432 1184 9216 640 4608 Avg. distance 12.87 13.08 13.06 13.08 13.39 13.60 Explanations ------------ pat = number of symmetrically distinct patterns. pos = number of positions. corners oriented = all LL corners correctly oriented. 2 adj corners twisted = two adjacent LL corners misoriented, the other two correctly oriented. 2 opp corners twisted = two diagonally opposite LL corners misoriented, the other two correctly oriented. 4 corners twisted ++-- = two adjacent LL corners misoriented clockwise, the other two misoriented counterclockwise. 4 corners twisted +-+- = two diagonally opposite LL corners misoriented clockwise, the other two misoriented counterclockwise. Last Layer optimal FTM analysis - categorized by edge orientation edges adj edges opp edges 4 edges oriented misoriented misoriented misoriented ---------- ----------- ----------- ----------- Moves pat pos pat pos pat pos pat pos ----- --- --- --- --- --- --- --- --- 0f* 1 1 1f* 2 3 6f* 1 8 1 8 7f* 3 24 5 40 3 24 8f* 11 88 12 96 2 16 9f* 18 138 25 200 7 56 10f* 55 422 65 500 23 180 11f* 84 628 230 1832 104 814 25 196 12f* 181 1372 647 5164 303 2378 118 893 13f* 386 2978 1318 10492 654 5082 253 1953 14f* 271 1998 1299 10328 702 5304 411 3138 15f* 21 122 306 2412 224 1590 211 1506 16f* 1 2 4 32 17 100 16 90 ---- ---- ---- ----- --- ---- ---- ----- Total 1034 7776 3912 31104 2040 15552 1034 7776 Avg. distance 12.64 13.11 13.24 13.66 Explanations ------------ edges oriented = all LL edges correctly oriented. 2 adj edges misoriented = two adjacent LL edges misoriented, the other two correctly oriented. 2 opp corners twisted = two diagonally opposite LL corners twisted, the other two correctly oriented. 4 corners twisted ++-- = two adjacent LL corners twisted clockwise, the other two oriented counterclockwise. 4 corners twisted +-+- = two diagonally opposite LL corners twisted clockwise, the other two oriented counterclockwise.

I have also tabulated the results for the 288 positions where the last layer pieces are all correctly oriented. This is often referred to as permutations of the last layer or PLL. The results are shown below. I note that Helmstetter computed an average of 11.21 face turns for solving these cases without regard to correct positioning with respect to the first two layers. (I confirm that value.) My value for the average number of moves required for solving the last layer including correct positioning with respect to the first two layers is 11.64 face turns. I've also shown the distances for each of the 14 categories (including the "PLL skip" category) that speedcubers use to identify the various configurations.

Optimal PLL (FTM) - All 288 positions Moves All Skip A E F G H J N R T U V Y Z ----- --- ---- - - - - - - - - - - - - - 0f* 1 1 1f* 3 3 9f* 18 8 2 8 10f* 82 24 2 24 8 24 11f* 16 8 8 12f* 40 32 8 13f* 84 12 32 4 24 12 14f* 40 4 4 4 8 16 4 15f* 4 4 --- -- -- -- -- -- -- -- -- -- -- -- -- -- -- Total 288 4 32 8 16 64 4 32 8 32 16 32 16 16 8