cuBerBruce
Veteran
I have done a 2-phase analysis of the Curvy Copter puzzle (sans jumbling). The first phase brings the puzzle into the <UF, UR, UB, UL, FR, BR> subgroup. The 2nd phase solves the puzzle using the 6 generators of that subgroup. This analysis gives an upper bound of 48 moves to solve the Curvy Copter (without jumbling moves).
The first phase puts six edges (DF, DR, DB, DL, FL, and BL) into the correct orientation, solves the DLF and DBL corner pieces, and also solves six face pieces (or center pieces, if you prefer to call them that) - the six adjacent to the DLF and DBL corners along with the other two on the D face. All other pieces are ignored. There are 26,127,360,000 positions (2^6 edge piece positions, (6*5)^4 face piece positions, and (8*7)*(3^2) corner piece positions). It takes 20 moves to solve phase 1 (worst case).
The distance distribution for phase 1 is in this spoiler.
Phase 2, or solving the <UF, UR, UB, UL, FR, BR> subgroup, has 2^6 edge piece configurations, and 12^4 face piece positions for a given edge configuration, and (6!/2)*(3^5) corner positions for a given edge configuration. Thus, there are 116,095,057,920 total positions. It takes 28 generator moves to solve the worst case positions.
The distance distribution is given in this spoiler.
The first phase puts six edges (DF, DR, DB, DL, FL, and BL) into the correct orientation, solves the DLF and DBL corner pieces, and also solves six face pieces (or center pieces, if you prefer to call them that) - the six adjacent to the DLF and DBL corners along with the other two on the D face. All other pieces are ignored. There are 26,127,360,000 positions (2^6 edge piece positions, (6*5)^4 face piece positions, and (8*7)*(3^2) corner piece positions). It takes 20 moves to solve phase 1 (worst case).
The distance distribution for phase 1 is in this spoiler.
Code:
distance positions cumulative
0 1 1
1 6 7
2 31 38
3 165 203
4 864 1,067
5 4,352 5,419
6 21,132 26,551
7 100,631 127,182
8 468,229 595,411
9 2,118,409 2,713,820
10 9,281,707 11,995,527
11 39,138,308 51,133,835
12 156,719,874 207,853,709
13 581,020,617 788,874,326
14 1,902,594,303 2,691,468,629
15 5,016,787,825 7,708,256,454
16 8,884,790,934 16,593,047,388
17 7,611,727,098 24,204,774,486
18 1,864,880,454 26,069,654,940
19 57,654,306 26,127,309,246
20 50,754 26,127,360,000Phase 2, or solving the <UF, UR, UB, UL, FR, BR> subgroup, has 2^6 edge piece configurations, and 12^4 face piece positions for a given edge configuration, and (6!/2)*(3^5) corner positions for a given edge configuration. Thus, there are 116,095,057,920 total positions. It takes 28 generator moves to solve the worst case positions.
The distance distribution is given in this spoiler.
Code:
distance positions cumulative
0 1 1
1 6 7
2 23 30
3 81 111
4 283 394
5 988 1,382
6 3,387 4,769
7 11,430 16,199
8 38,106 54,305
9 125,309 179,614
10 406,330 585,944
11 1,299,761 1,885,705
12 4,096,514 5,982,219
13 12,693,276 18,675,495
14 38,567,114 57,242,609
15 114,475,814 171,718,423
16 329,879,912 501,598,335
17 913,955,559 1,415,553,894
18 2,399,554,839 3,815,108,733
19 5,830,823,995 9,645,932,728
20 12,564,754,273 22,210,687,001
21 22,296,829,536 44,507,516,537
22 29,378,479,034 73,885,995,571
23 25,557,087,775 99,443,083,346
24 13,003,762,062 112,446,845,408
25 3,312,266,440 115,759,111,848
26 327,974,315 116,087,086,163
27 7,958,990 116,095,045,153
28 12,767 116,095,057,920
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