# 2-phase Curvy Copter analysis (sans jumbling)



## cuBerBruce (Mar 9, 2018)

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.


Spoiler





```
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     26,127,360,000
```




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.



Spoiler





```
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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