2x2x2 Scramble Math Question

[Gaétan Guimond]

The heart of the cube is the 8 corners because it's number can not change.
Pure system http://pages.videotron.com/toulou/gaetan/

The cube popularity took a dive after hungary budapest championship of the world in 1982. The return in competition after 21 years was the world championship in toronto canada in 2003. Exactly had the same place at the science fair my web page photo that I placed on my web site that I took on the national championship of 1982.

The name of my domain rubikscuberecord.com and I'm the only one to have solved the cube blindfolded. If you don't believe in the one that has brought back the cube you will have to answer to the irreversables evidence. Contrary to it's return in 2003 in the store where the cube sales were influenced by the championship wich was not the case in 1982.

The cube is'nt musical (method & math) partition exchange only but it's has competitive.

The cube is a puzzle where the genius of the teenager's suffice to reach world records. I never said that it was not for children or adults because human curiosity has no age.

The emails that I kept in my reception box before 2003 from hotmail speak for themselves. Journalists outside the province of quebec never heard from me because the cuber's of my generation present on the web before 2003 never told my name with my story. People are sold by popular culture.

http://www.youtube.com/watch?v=iE1RH2S1fWQ

 

[ShadenSmith]

I love you Guimond.

 

[Stefan]

He's back! Pointless as always!

The emails that I kept in my reception box before 2003 from hotmail speak for themselves.

The emails you didn't get after 2003 also speak for themselves.

 

[reThinking the Cube]

He's back! Pointless as always!

Exactly what I was thinking at first, but then I began to wonder what he was trying to get at here...

The heart of the cube is the 8 corners because it's number can not change.
Pure system http://pages.videotron.com/toulou/gaetan/

You should take a closer look! After deciphering, this reveals itself to be an ingenius way to represent the corner pieces for all cubes, and a demonstration for solving ANY corner position with at most 3 face turns (i.e. X=F,Y=U,Z=R). It looks to me like the pyramid could be assigned as (1=UBL, 2=ULF, 3=URB, 4=DFL, 5=DRF, 6=DBR, and UFR is the picture =(0)). DLB would be the solved reference piece. Using this idea, all corner cases can be represented using only 6 positions, since 2 (UFR, DLB) are always solved by default. I have the thought now, that the 6 positions can be seen as arranged in a hexagon around UFR (instead of a pyramid). This is inspiring towards a better system for representing all corner cases.

..

 

[reThinking the Cube]

So I can probably sort out the positions in the file, particular which ones are SL/ZL and the (ULF, URB, DRF, DLB) case, and similarly sort out for the n=5 and n=6 positions.

I am very much looking forward to this. For convenience let's make DL = (ULF,URB,DRF,DLB). Good job spotting the previous oversite. If possible, you could even use up to a 2-move conjugation for reducing, since the main value of this is to get numbers for the fewest possible cases that are required to represent them all. How they are actually represented, or solved is of secondary importance. Just to have the list reduced to a reasonable minimal number of cases to get all n=>4 positions would be very sweet. Guimond's post has also given me an idea on how to better represent in a non-arbitrary way, but I am still rethinking on this one.

Bruce, could you generate for the #of moves away, all 3674160 are from any n>=4 position? That would reveal the potential for human transitions into the easier known cases.

It was rather easy to alter my code to treat all positions with 4 or more pieces solved with respect to each other as if they were "solved," and perform a breadth-first search to calculate how many moves are required to get to this (funny notion of) "solved" from each possible state.

[B]
Number of moves required to get at least 4 cubies of the 2x2x2 solved
with respect to each other, in terms of number of positions and
number of positions as reduced by symmetry and antisymmetry.

moves   positions    reduced
-----   ---------    -------             
  0         34786        529
  1        247092       2863
  2        961942      10606
  3       1884540      20389
  4        544192       5884
  5          1608         25
          -------      -----
total     3674160      40296
[/B]

That is very significant! Over 85% transition to some n=>4 case in 3 moves or less. Could you also do this calculation for some of the n=1 positions below - especially the bolded one?

moves   n=1      n=2      n=3      n=4      n=5      n=6      n=8   totals   n >= 2

 0        0        0        0        0        0        0        1        1        1
 1        0        0        0        9        0        0        0        9        9
 2        0       54        0        0        0        0        0       54       54
 3       48      186        0       87        0        0        0      321      273
 4      264      956      528       99        0        0        0     1847     1583
 5     1092     6302     1944      654        0        0        0     9992     8900
 6     7092    35976     6408      660        0        0        0    50136    43044
 7    44904   151900    26592     3948      192        0        0   227536   182632
 8   186864   577424    99688     5328      768        0        0   870072   683208
 9   [B][SIZE="2"]451728[/SIZE][/B]  1235424   186288    13524      784        0        0  1887748  1436020
10   151512   401360    62496     7932      384      116        0   623800   472288
11      240     1776      328      276        0       24        0     2644     2404
     ------  -------   ------    -----      ---      ---        -  -------  -------
     843744  2411358   384272    32517     2128      140        1  3674160  2830416

Edit: average number of moves added, per Stefan's request
     8.8199   8.7447   8.6894   8.6794   8.6391  10.1714   0.0000   8.7556   8.7364

...

 

[cuBerBruce]

Could you also do this calculation for some of the n=1 positions below - especially the bolded one?

OK, first I did this for all n=1 positions. So I considered all positions with no two pieces solved with respect to each other as "solved." So not even the solved position is considered "solved." I then did a breadth-first search to determine number of moves to reach one of these positions from every possible position.

moves  positions  cumulative
  0       843744      843744
  1      2021802     2865546
  2       800585     3666131
  3         8029     3674160

Average number of moves = 0.9926

Next I did the same for only the n=1 postions that are 9 moves from the solved position (as this seems to be what reThinking the Cube asked for). Note there is one position that requires nine moves to reach one of these positions. I hope you can figure out what position that is.

moves  positions  cumulative

  0       451728      451728
  1      1772136     2223864
  2      1349636     3573500
  3        88180     3661680
  4        10240     3671920
  5         1844     3673764
  6          329     3674093
  7           57     3674150
  8            9     3674159
  9            1     3674160

Average number of moves = 1.3033

I'll try to get the categorized list of the cases for 4 or more solved pieces later this weekend.

 

[reThinking the Cube]

Next I did the same for only the n=1 postions that are 9 moves from the solved position (as this seems to be what reThinking the Cube asked for). Note there is one position that requires nine moves to reach one of these positions. I hope you can figure out what position that is.

Good trivial question, but I should have been more specific earlier.

I want to compare the following results for (n=any, m=any) 3674160 positions -

Bruce, could you generate for the #of moves away, all 3674160 are from any n>=4 position? That would reveal the potential for human transitions into the easier known cases.

Number of moves required to get at least 4 cubies of the 2x2x2 solved with respect to each other,in terms of number of positions and
number of positions as reduced by symmetry and antisymmetry.

moves   positions    reduced
-----   ---------    -------             
  0         34786        529
  1        247092       2863
  2        961942      10606
  3       1884540      20389
  4        544192       5884
  5          1608         25
          -------      -----
total     3674160      40296

That is very significant! Over 85% transition to some n=>4 case in 3 moves or less.

- with the results that would be generated, for the (n=1, m=9) 451728 positions in this table.

moves   n=1      n=2      n=3      n=4      n=5      n=6      n=8   totals   n >= 2

 0        0        0        0        0        0        0        1        1        1
 1        0        0        0        9        0        0        0        9        9
 2        0       54        0        0        0        0        0       54       54
 3       48      186        0       87        0        0        0      321      273
 4      264      956      528       99        0        0        0     1847     1583
 5     1092     6302     1944      654        0        0        0     9992     8900
 6     7092    35976     6408      660        0        0        0    50136    43044
 7    44904   151900    26592     3948      192        0        0   227536   182632
 8   186864   577424    99688     5328      768        0        0   870072   683208
 9   [B][SIZE="2"]451728[/SIZE][/B]  1235424   186288    13524      784        0        0  1887748  1436020
10   151512   401360    62496     7932      384      116        0   623800   472288
11      240     1776      328      276        0       24        0     2644     2404
     ------  -------   ------    -----      ---      ---        -  -------  -------
     843744  2411358   384272    32517     2128      140        1  3674160  2830416

Edit: average number of moves added, per Stefan's request
     8.8199   8.7447   8.6894   8.6794   8.6391  10.1714   0.0000   8.7556   8.7364

The purpose, is to determine the relative difficulty for human transposition into the n=>4 cases, starting from scrambles that are n=1, as opposed to scrambles that are n=any.

...

 

[cuBerBruce]

OK, I believe what reThinking the Cube wants is for each set of positions in the table giving how many positions at each distance from solved for each n value, to further break down these sets of positions by how many moves they are from at least 4 pieces being solved. For the specific case of n=1 and 9 moves from solved, the breakdown is

moves
from              reduced
n>=4   positions   cases
------ ---------  -------
  2       72768      793
  3      269208     2861
  4      109224     1159
  5         528        7
         ------     ----
total    451728     4820

For the complete set of n=1,2,3 values, see the spoiler below.

Spoiler

"n" is the maximum number of pieces solved with respect to each other

"distance from n >= 4" is the number of moves (HTM) required to get
at least four pieces solved with respect to each other.

"distance from solved" is the number of moves (HTM) required to reach
the solved state.

"positions" is the number of 2x2x2 positions meeting the given conditions.

"reduced cases" is the symmetry/antisymmetry-reduced number of equivalence
classes for these positions.

      distance   distance
        from       from                 reduced
 n     n >= 4     solved    positions    cases
===   ========   ========   =========   =======
 1         2         3           48         3
 1         2         4          264         7
 1         2         5          744        16
 1         2         6         1620        26
 1         2         7        11856       140
 1         2         8        31848       358
 1         2         9        72768       793
 1         2        10        33696       381
 1         2        11          192         2
 1         3         5          348         8
 1         3         6         4368        55
 1         3         7        23784       270
 1         3         8       114312      1214
 1         3         9       269208      2861
 1         3        10        82848       885
 1         3        11           48         1
 1         4         6         1104        12
 1         4         7         9264       101
 1         4         8        40512       428
 1         4         9       109224      1159
 1         4        10        34728       373
 1         5         8          192         2
 1         5         9          528         7
 1         5        10          240         3

 2         1         2           54         4
 2         1         3          186         7
 2         1         4          246         7
 2         1         5          390        13
 2         1         6         5256        68
 2         1         7         9768       132
 2         1         8        25656       300
 2         1         9        53624       628
 2         1        10        31528       379
 2         1        11          672        12
 2         2         4          686        15
 2         2         5         3432        47
 2         2         6        12360       149
 2         2         7        52332       578
 2         2         8       165152      1801
 2         2         9       366496      3948
 2         2        10       122880      1386
 2         2        11          568        13
 2         3         4           24         2
 2         3         5         2480        38
 2         3         6        15864       186
 2         3         7        74800       836
 2         3         8       310168      3324
 2         3         9       632520      6808
 2         3        10       193400      2139
 2         3        11          528        10
 2         4         6         2496        31
 2         4         7        15000       164
 2         4         8        76184       826
 2         4         9       182560      1965
 2         4        10        53448       594
 2         5         8          264         4
 2         5         9          224         4
 2         5        10          104         3
 2         5        11            8         1

 3         1         4          528         7
 3         1         5         1152        13
 3         1         6         2856        34
 3         1         7         8064        89
 3         1         8        30960       331
 3         1         9        54192       588
 3         1        10        21840       248
 3         1        11          120         3
 3         2         5          192         2
 3         2         6         1776        19
 3         2         7         5568        62
 3         2         8        19680       214
 3         2         9        43488       475
 3         2        10        14232       169
 3         2        11           64         2
 3         3         5          600         8
 3         3         6         1488        16
 3         3         7        11280       123
 3         3         8        43512       466
 3         3         9        79704       868
 3         3        10        23160       269
 3         3        11           96         2
 3         4         6          288         3
 3         4         7         1680        19
 3         4         8         5536        62
 3         4         9         8904       107
 3         4        10         3216        39
 3         4        11           48         1
 3         5        10           48         1

EDIT:
OK, I now have created a new version of the "4_or_more_solved" file that is sorted by the pattern of unsolved piece positions. The cycle notation is preceded by [letter,number] where letter indicates the pattern of unsolved pieces and the number is the distance from solved. (I have picked out letters to use for the n=5 and n=6 cases - V,P,K and I,A,C.)

 

[reThinking the Cube]

OK, I believe what reThinking the Cube wants is for each set of positions in the table giving how many positions at each distance from solved for each n value, to further break down these sets of positions by how many moves they are from at least 4 pieces being solved. For the specific case of n=1 and 9 moves from solved, the breakdown is

moves
from              reduced
n>=4   positions   cases
------ ---------  -------
  2       72768      793
  3      269208     2861
  4      109224     1159
  5         528        7
         ------     ----
total    451728     4820

For the complete set of n=1,2,3 values, see the spoiler below.

Spoiler

"n" is the maximum number of pieces solved with respect to each other

"distance from n >= 4" is the number of moves (HTM) required to get
at least four pieces solved with respect to each other.

"distance from solved" is the number of moves (HTM) required to reach
the solved state.

"positions" is the number of 2x2x2 positions meeting the given conditions.

"reduced cases" is the symmetry/antisymmetry-reduced number of equivalence
classes for these positions.

      distance   distance
        from       from                 reduced
 n     n >= 4     solved    positions    cases
===   ========   ========   =========   =======
 1         2         3           48         3
 1         2         4          264         7
 1         2         5          744        16
 1         2         6         1620        26
 1         2         7        11856       140
 1         2         8        31848       358
 1         2         9        72768       793
 1         2        10        33696       381
 1         2        11          192         2
 1         3         5          348         8
 1         3         6         4368        55
 1         3         7        23784       270
 1         3         8       114312      1214
 1         3         9       269208      2861
 1         3        10        82848       885
 1         3        11           48         1
 1         4         6         1104        12
 1         4         7         9264       101
 1         4         8        40512       428
 1         4         9       109224      1159
 1         4        10        34728       373
 1         5         8          192         2
 1         5         9          528         7
 1         5        10          240         3

 2         1         2           54         4
 2         1         3          186         7
 2         1         4          246         7
 2         1         5          390        13
 2         1         6         5256        68
 2         1         7         9768       132
 2         1         8        25656       300
 2         1         9        53624       628
 2         1        10        31528       379
 2         1        11          672        12
 2         2         4          686        15
 2         2         5         3432        47
 2         2         6        12360       149
 2         2         7        52332       578
 2         2         8       165152      1801
 2         2         9       366496      3948
 2         2        10       122880      1386
 2         2        11          568        13
 2         3         4           24         2
 2         3         5         2480        38
 2         3         6        15864       186
 2         3         7        74800       836
 2         3         8       310168      3324
 2         3         9       632520      6808
 2         3        10       193400      2139
 2         3        11          528        10
 2         4         6         2496        31
 2         4         7        15000       164
 2         4         8        76184       826
 2         4         9       182560      1965
 2         4        10        53448       594
 2         5         8          264         4
 2         5         9          224         4
 2         5        10          104         3
 2         5        11            8         1

 3         1         4          528         7
 3         1         5         1152        13
 3         1         6         2856        34
 3         1         7         8064        89
 3         1         8        30960       331
 3         1         9        54192       588
 3         1        10        21840       248
 3         1        11          120         3
 3         2         5          192         2
 3         2         6         1776        19
 3         2         7         5568        62
 3         2         8        19680       214
 3         2         9        43488       475
 3         2        10        14232       169
 3         2        11           64         2
 3         3         5          600         8
 3         3         6         1488        16
 3         3         7        11280       123
 3         3         8        43512       466
 3         3         9        79704       868
 3         3        10        23160       269
 3         3        11           96         2
 3         4         6          288         3
 3         4         7         1680        19
 3         4         8         5536        62
 3         4         9         8904       107
 3         4        10         3216        39
 3         4        11           48         1
 3         5        10           48         1

Thanks Bruce, that is perfect. Your tables have shown that scrambles resulting in n=1, m=9 cases have considerably less easy solve potential than scrambles that are uniformly random.

(In the spoiler table above) I was now wondering how many cases have transposition to n=>4 end up being equivalent to minimum distance from solved. In other words, how many of these minimum move solves actually go directly to some n=>4 that ends up to be the shortest distance from solved anyway? Taking the first case as example - if it takes 2 moves to get to n=>4, and 3 moves to optimal solve, then the last move is the n=4, m=1 case, and therefore the transistion to n=>4 for this case must have been optimal. Another one that pops out at me is (n=1, m=11). So how many of these n=>4 transposition cases are optimal then? And if they are not optimal, how many moves are being added, by first transposing to a n=>4 case?

OK, I now have created a new version of the "4_or_more_solved" file that is sorted by the pattern of unsolved piece positions. The cycle notation is preceded by [letter,number] where letter indicates the pattern of unsolved pieces and the number is the distance from solved. (I have picked out letters to use for the n=5 and n=6 cases - V,P,K and I,A,C.)

Sorting the patterns this way is much easier to work with. Are you going to also attempt reducing these by conjugations?

...

 

[Gaétan Guimond]

You should write a book with the following title .....

Giving his life to the Rubik's puzzle

By Stephan pochmann world record post message

...................................................................................

http://www.youtube.com/watch?v=DAMRPcICix4

 

[rachmaninovian]

Dear Mr Guimond, you make no sense.

but i'm a fan of yours so please make your website public again.

otherwise you make no sense.

regards,
someone.

 

[cuBerBruce]

I was now wondering how many cases have transposition to n=>4 end up being equivalent to minimum distance from solved. In other words, how many of these minimum move solves actually go directly to some n=>4 that ends up to be the shortest distance from solved anyway? Taking the first case as example - if it takes 2 moves to get to n=>4, and 3 moves to optimal solve, then the last move is the n=4, m=1 case, and therefore the transistion to n=>4 for this case must have been optimal. Another one that pops out at me is (n=1, m=11). So how many of these n=>4 transposition cases are optimal then? And if they are not optimal, how many moves are being added, by first transposing to a n=>4 case?

OK, I've started looking into this. First I note that for each position, there may be several optimal solutions to solve it and several optimal solutions for reaching an n>=4 case. One optimal path to n>=4 might be along the path to solve optimally, while another for the same position might not. If I am to assume we consider all optimal paths to n>=4 and see if at least one of them is along the optimal path to solve the position, then it looks to me like the answer is somewhere in the 9% to 10% range. If you choose an arbitrary optimal path to n>=4, then the chances of it being along an optimal path to solve the position is significantly smaller (about 3% I believe).

I also note that this property may not always be the same between a position and its inverse. Therefore, I need to consider the 77802 symmetry equivalence classes instead of the 40296 symmetry/antisymmetry equivalence classes.

Sorting the patterns this way is much easier to work with. Are you going to also attempt reducing these by conjugations?

...

I note that a 1-move conjugation can always produce a type D,Z,P, or I case (or you could pick A or C in place of I). So a one-move conjugation allows you to at least limit the cases to these four sets of positions.

EDIT: My program has calculated 76994 symmetry-reduced cases for N={1,2,3}. Of these, there exists an optimal path to n>=4 that is also along the optimal path to solved for 7013 of the 76994 cases. This is approximately 9.11% of the symmetry reduced cases. This also corresponds to 328608 out of the 3639374 positions or about 9.03% of the n={1,2,3} positions.

For n=1, m=11, there are 6 symmetry-reduced cases, of which 2 have an optimal solution to n>=4 that is also along the optimal solve path. This corresponds to 96 out of 240 positions. Two symmetry-reduced cases have 1 optimal solution for n>=4 that is not along the optimal solve path. Two symmetry-reduced cases have 2 optimal solutions for n>=4 (2 moves). For one of these, both are along an optimal path, while for the other only one is along an optimal path. For the other two symmetry-reduced cases, there are 14 optimal solutions to n>=4 (3 moves), but none of them along an optimal path.

Some further detail for the n=1 m=11 cases: The two cases with one optimal solution for n>=4 (2 moves), the n>=4 positions reached are 10 moves from solved (while the initial positions were 11 moves from solved). For the two cases of 2 optimal solutions for n>=4, in one case both of these reduce the position from 11 moves from solved to 9 moves from solved, and in the other case, one of the cases reaches a position 10 moves from solved. Of the two cases with 14 optimal solutions (3 moves) for reaching n>=4, in one case 12 reach an m=9 position while 2 reach an m=10 position, while in the other case, 10 reach m=9 while 4 reach m=10.