Starts-with and Ends-With

On the old Cube-Lovers list, the terms Starts-with and Ends-with were defined as follows.  For a cube position x, StartsWith(x)=S(x) is the set of all moves with which a minimal maneuver can start and EndsWith(x)=E(x) is the set of all moves with which a minimal maneuver can end.

The concept is much older than Cube-Lovers, of course.  It's obvious that from any position except for Start itself, there must be at least one move which takes the Cube closer to Start.  The set of all such moves is simply the set of inverses of E(x).

A couple of very simple examples might be in order.  If x=URL-1, then S(x)={U} and E(x)={R,L-1}.  From position x, the moves R-1 and L will take the Cube closer to Start.  We might write this as E-1(x)={R-1,L}.  If y=URR (in the quarter turn metric), then it is also the case that y=UR-1R-1.  Hence, we have S(y)={U}, E(y)={R,R-1}, and E-1(y)={R,R-1}.

S(x) and E(x) are not very easy things to know.  Indeed, knowledge of S(x) and E(x) for every Cube position would be tantamount to full knowledge of God's Algorithm.  My current God's Algorithm program begins with the Start position and progressively determines the number of positions that are 1, 2, 3, etc. moves from Start.  I have posted results out to 13q and out to 11f.  As an artifact of the basic algorithm, the program also determines both S(x) and E(x) for every position.  The program summarizes some of the S(x) and E(x) results.  I have not posted these results before, and I will do so later in this message.

The definition of S(x) and E(x) applies to both the quarter turn metric and to the face turn metric.  In the quarter turn metric, E-1(x) partitions the set of quarter turns into those moves that take the Cube closer to Start and those moves that take the Cube further from Start.  Therefore, in the quarter turn metric it is that case that E(x) uniquely determines local maxima.  If E(x)=Q where Q is the set of quarter turns, then x is a local maximum.  Equivalently, if |E(x)|=12, then x is a local maximum.

The same is not true in the face turn metric.  A move can take the Cube closer to Start, can take the Cube further from Start, or can leave the Cube the same distance from Start.  So a position can be a local maximum in the face turn metric even if E(x) does not contain all the face turns.  For example, the Pons Asinorum position is a local maximum in the face turn metric even though E(x) only contains 6 moves.  Hence, my current God's Algorithm program cannot in general identify local maxima in the face turn metric even though it can in the quarter turn metric.

A position in the face turn metric is a strong local maximum if every move takes the position closer to Start.  A local maximum in the face turn metric that is not a strong local maximum is a weak local maximum.  That raises the interesting question of whether there are any strong local maxima.  It turns out that there are, but there's no theoretical reason that I know of why that must be so.  A strong local maximum can be identified as a position for which |E(x)|=18.  Because of that, my program can uniquely identify strong local maxima.

The tables below are antisymmetric.  That must be so because |S(x)|=|E(x-1)| etc.  The shortest local maxima in the quarter turn metric are of length 10q.  These have been reported previously on Cube-Lovers.  The shortest strong local maxima in the face turn metric are of length 9f.  I suspect that the shortest local maximum in the face turn metric is the aforementioned Pons Asinorum of length 6f, but I don't know that the possibility of a shorter local maximum in the face turn metric has ever been ruled out.

  Quarter Turn Metric
  Summary of |S(x)| and |E(x)|

|x|
   |S(x)| |E(x)|    patterns      positions
                    (positions
                     unique up
                     to symmetry)

  0   0   0               1               1
  1   1   1               1              12
  2   1   1               2              96
  2   2   2               3              18
  3   1   1              16             768
  3   1   2               4             144
  3   2   1               4             144
  3   3   3               1              12
  4   1   1             156            7296
  4   1   2              24            1152
  4   1   3               2              96
  4   2   1              24            1152
  4   2   2              10             216
  4   3   1               2              96
  4   4   4               1               3
  5   1   1            1424           68352
  5   1   2             236           10944
  5   1   3              16             768
  5   1   4               1              24
  5   2   1             236           10944
  5   2   2              40            1728
  5   2   3               4             144
  5   3   1              16             768
  5   3   2               4             144
  5   4   1               1              24
  6   1   1           13332          639456
  6   1   2            2150          102432
  6   1   3             156            7296
  6   1   4               4             192
  6   2   1            2150          102432
  6   2   2             384           16992
  6   2   3              24            1152
  6   2   4               3              36
  6   3   1             156            7296
  6   3   2              24            1152
  6   3   3               4             192
  6   4   1               4             192
  6   4   2               3              36
  6   4   4               1              24
  7   1   1          124482         5972256
  7   1   2           20022          958992
  7   1   3            1428           68544
  7   1   4              42            1920
  7   2   1           20022          958992
  7   2   2            3489          164016
  7   2   3             254           11808
  7   2   4               8             288
  7   3   1            1428           68544
  7   3   2             254           11808
  7   3   3              36            1680
  7   3   4               3             120
  7   4   1              42            1920
  7   4   2               8             288
  7   4   3               3             120
  7   4   4               8             336
  8   1   1         1161021        55725744
  8   1   2          187646         8998752
  8   1   3           13491          647184
  8   1   4             421           19824
  8   2   1          187646         8998752
  8   2   2           32110         1529640
  8   2   3            2390          113760
  8   2   4              88            3624
  8   2   5               1              48
  8   3   1           13491          647184
  8   3   2            2390          113760
  8   3   3             320           15168
  8   3   4              40            1824
  8   4   1             421           19824
  8   4   2              88            3624
  8   4   3              40            1824
  8   4   4             113            2904
  8   4   6               1              24
  8   5   2               1              48
  8   6   4               1              24
  8   6   6               2              32
  8   8   8               3              27
  9   1   1        10829774       519810912
  9   1   2         1757942        84360720
  9   1   3          127532         6119040
  9   1   4            4392          208872
  9   1   5               1              48
  9   1   6               1              48
  9   1   8               1              12
  9   2   1         1757942        84360720
  9   2   2          296216        14186208
  9   2   3           22484         1077360
  9   2   4             885           41184
  9   2   5              22            1056
  9   3   1          127532         6119040
  9   3   2           22484         1077360
  9   3   3            2675          127104
  9   3   4             301           13968
  9   3   5              27             912
  9   3   6               6             288
  9   3   8               1              48
  9   4   1            4392          208872
  9   4   2             885           41184
  9   4   3             301           13968
  9   4   4             370           16488
  9   4   5               6             288
  9   5   1               1              48
  9   5   2              22            1056
  9   5   3              27             912
  9   5   4               6             288
  9   5   5              17             696
  9   5   8               1              48
  9   6   1               1              48
  9   6   3               6             288
  9   6   6               8             288
  9   7   7               2              96
  9   8   1               1              12
  9   8   3               1              48
  9   8   5               1              48
 10   1   1       101016272      4848717648
 10   1   2        16448933       789486000
 10   1   3         1201921        57689232
 10   1   4           44187         2116176
 10   1   5             306           14688
 10   1   6              70            2976
 10   1   8              12             576
 10   2   1        16448933       789486000
 10   2   2         2747597       131784432
 10   2   3          211867        10159824
 10   2   4            9778          459432
 10   2   5             240           11520
 10   2   6              77            3528
 10   2   7               3             144
 10   2   8               8             300
 10   3   1         1201921        57689232
 10   3   2          211867        10159824
 10   3   3           22790         1093408
 10   3   4            2090           99264
 10   3   5             133            6384
 10   3   6              21            1008
 10   3   7               1              48
 10   3   8               2              96
 10   4   1           44187         2116176
 10   4   2            9778          459432
 10   4   3            2090           99264
 10   4   4            2657          115386
 10   4   5             108            5088
 10   4   6              41            1584
 10   4   7               1              48
 10   5   1             306           14688
 10   5   2             240           11520
 10   5   3             133            6384
 10   5   4             108            5088
 10   5   5             132            5832
 10   5   6               6             288
 10   6   1              70            2976
 10   6   2              77            3528
 10   6   3              21            1008
 10   6   4              41            1584
 10   6   5               6             288
 10   6   6             103            2952
 10   7   2               3             144
 10   7   3               1              48
 10   7   4               1              48
 10   7   7               6             288
 10   8   1              12             576
 10   8   2               8             300
 10   8   3               2              96
 10   8   8              15             324
 10  10  10               8             138
 10  12  12               4              42  (shortest local maxima in quarter turn metric)
 11   1   1       942084406     45219930720
 11   1   2       153748573      7379729496
 11   1   3        11315607       543131472
 11   1   4          439721        21076032
 11   1   5            4813          231024
 11   1   6            1397           64704
 11   1   7              26            1248
 11   1   8             147            6936
 11   2   1       153748573      7379729496
 11   2   2        25606674      1228857096
 11   2   3         1977557        94898112
 11   2   4           96845         4625112
 11   2   5            3065          144888
 11   2   6             771           36240
 11   2   7              41            1968
 11   2   8              76            3456
 11   2  10               2              48
 11   3   1        11315607       543131472
 11   3   2         1977557        94898112
 11   3   3          200207         9603136
 11   3   4           21623         1032600
 11   3   5            1803           86304
 11   3   6             476           21696
 11   3   7              50            2400
 11   3   8              38            1824
 11   3   9               2              96
 11   3  10               5             156
 11   3  11               2              72
 11   4   1          439721        21076032
 11   4   2           96845         4625112
 11   4   3           21623         1032600
 11   4   4           17062          807624
 11   4   5            1216           57264
 11   4   6             350           16464
 11   4   7              44            2064
 11   4   8              25             960
 11   4  10               5             240
 11   5   1            4813          231024
 11   5   2            3065          144888
 11   5   3            1803           86304
 11   5   4            1216           57264
 11   5   5             753           35568
 11   5   6             168            7632
 11   5   7              15             720
 11   5   8               9             432
 11   5  10               1              48
 11   6   1            1397           64704
 11   6   2             771           36240
 11   6   3             476           21696
 11   6   4             350           16464
 11   6   5             168            7632
 11   6   6             250           10896
 11   6   7              23            1056
 11   6   8               6             288
 11   6  10               1              48
 11   7   1              26            1248
 11   7   2              41            1968
 11   7   3              50            2400
 11   7   4              44            2064
 11   7   5              15             720
 11   7   6              23            1056
 11   7   7              39            1824
 11   7   8               6             288
 11   7   9               6             288
 11   7  10               1              48
 11   8   1             147            6936
 11   8   2              76            3456
 11   8   3              38            1824
 11   8   4              25             960
 11   8   5               9             432
 11   8   6               6             288
 11   8   7               6             288
 11   8   8              13             624
 11   8  10               1              48
 11   9   3               2              96
 11   9   7               6             288
 11   9   9               2              24
 11  10   2               2              48
 11  10   3               5             156
 11  10   4               5             240
 11  10   5               1              48
 11  10   6               1              48
 11  10   7               1              48
 11  10   8               1              48
 11  10  10               5             192
 11  11   3               2              72
 12   1   1      8783724399    421618316448
 12   1   2      1435883951     68921873016
 12   1   3       106292624      5101979448
 12   1   4         4342903       208398528
 12   1   5           65297         3129600
 12   1   6           18768          899616
 12   1   7             905           43440
 12   1   8            1840           88128
 12   1   9              16             768
 12   1  10              46            2208
 12   1  11               4             192
 12   2   1      1435883951     68921873016
 12   2   2       239147526     11478163980
 12   2   3        18587534       892133088
 12   2   4          961693        46061880
 12   2   5           38238         1832688
 12   2   6            9012          427488
 12   2   7             717           33744
 12   2   8             798           38208
 12   2   9              24            1152
 12   2  10              29            1392
 12   2  11               8             384
 12   2  12               6             240
 12   3   1       106292624      5101979448
 12   3   2        18587534       892133088
 12   3   3         1840021        88311512
 12   3   4          205576         9853488
 12   3   5           19062          913512
 12   3   6            4152          197760
 12   3   7             528           25200
 12   3   8             215           10128
 12   3   9              10             480
 12   3  10              24            1152
 12   3  11               2              96
 12   4   1         4342903       208398528
 12   4   2          961693        46061880
 12   4   3          205576         9853488
 12   4   4          148355         7027632
 12   4   5           12455          597000
 12   4   6            4580          213552
 12   4   7             480           22920
 12   4   8             538           24948
 12   4   9              31            1488
 12   4  10              99            4704
 12   4  12               2              48
 12   5   1           65297         3129600
 12   5   2           38238         1832688
 12   5   3           19062          913512
 12   5   4           12455          597000
 12   5   5            5827          279000
 12   5   6            1248           59808
 12   5   7             242           11592
 12   5   8              91            4152
 12   5   9              11             528
 12   5  10               3             144
 12   6   1           18768          899616
 12   6   2            9012          427488
 12   6   3            4152          197760
 12   6   4            4580          213552
 12   6   5            1248           59808
 12   6   6            3192          136308
 12   6   7             192            9168
 12   6   8             239           10392
 12   6   9              23            1056
 12   6  10              31            1320
 12   6  12               6             288
 12   7   1             905           43440
 12   7   2             717           33744
 12   7   3             528           25200
 12   7   4             480           22920
 12   7   5             242           11592
 12   7   6             192            9168
 12   7   7             263           12384
 12   7   8              45            2160
 12   7   9               5             240
 12   7  10               5             240
 12   8   1            1840           88128
 12   8   2             798           38208
 12   8   3             215           10128
 12   8   4             538           24948
 12   8   5              91            4152
 12   8   6             239           10392
 12   8   7              45            2160
 12   8   8             598           18684
 12   8   9               9             432
 12   8  10              22             912
 12   8  12              11             408
 12   9   1              16             768
 12   9   2              24            1152
 12   9   3              10             480
 12   9   4              31            1488
 12   9   5              11             528
 12   9   6              23            1056
 12   9   7               5             240
 12   9   8               9             432
 12   9   9              27            1264
 12   9  10               1              48
 12   9  11               1              48
 12  10   1              46            2208
 12  10   2              29            1392
 12  10   3              24            1152
 12  10   4              99            4704
 12  10   5               3             144
 12  10   6              31            1320
 12  10   7               5             240
 12  10   8              22             912
 12  10   9               1              48
 12  10  10              62            2088
 12  10  12               7             240
 12  11   1               4             192
 12  11   2               8             384
 12  11   3               2              96
 12  11   9               1              48
 12  11  11               7             240
 12  11  12               1              48
 12  12   2               6             240
 12  12   4               2              48
 12  12   6               6             288
 12  12   8              11             408
 12  12  10               7             240
 12  12  11               1              48
 12  12  12              70            1641
 13   1   1     81865248532   3929530575552
 13   1   2     13402102133    643299207408
 13   1   3       997279934     47869325232
 13   1   4        42665046      2047739232
 13   1   5          818454        39284064
 13   1   6          229038        10972032
 13   1   7           14457          693936
 13   1   8           21816         1044144
 13   1   9             383           18288
 13   1  10            1075           51312
 13   1  11              88            4224
 13   1  12              21            1008
 13   2   1     13402102133    643299207408
 13   2   2      2235171013    107285861712
 13   2   3       175593325      8428224144
 13   2   4         9707685       465665208
 13   2   5          445661        21377976
 13   2   6          111099         5311488
 13   2   7           10174          487608
 13   2   8            9274          442080
 13   2   9             607           29136
 13   2  10             671           31632
 13   2  11              46            2184
 13   2  12              35            1680
 13   3   1       997279934     47869325232
 13   3   2       175593325      8428224144
 13   3   3        17554912       842573480
 13   3   4         2068402        99220728
 13   3   5          218204        10467024
 13   3   6           52180         2492040
 13   3   7            7765          369576
 13   3   8            3982          189600
 13   3   9             622           29256
 13   3  10             410           19440
 13   3  11              81            3600
 13   3  12              25            1200
 13   4   1        42665046      2047739232
 13   4   2         9707685       465665208
 13   4   3         2068402        99220728
 13   4   4         1312252        62812668
 13   4   5          133264         6388272
 13   4   6           46023         2192544
 13   4   7            6695          319368
 13   4   8            4704          222960
 13   4   9             696           33408
 13   4  10             553           26280
 13   4  11              84            3912
 13   4  12             104            4848
 13   5   1          818454        39284064
 13   5   2          445661        21377976
 13   5   3          218204        10467024
 13   5   4          133264         6388272
 13   5   5           52878         2531688
 13   5   6           15359          734424
 13   5   7            3278          156888
 13   5   8            1378           65424
 13   5   9             224           10728
 13   5  10             139            6336
 13   5  11              19             912
 13   5  12              14             624
 13   6   1          229038        10972032
 13   6   2          111099         5311488
 13   6   3           52180         2492040
 13   6   4           46023         2192544
 13   6   5           15359          734424
 13   6   6           17433          820272
 13   6   7            2503          119184
 13   6   8            1747           81984
 13   6   9             327           15648
 13   6  10             262           12216
 13   6  11              42            1872
 13   6  12              23            1104
 13   7   1           14457          693936
 13   7   2           10174          487608
 13   7   3            7765          369576
 13   7   4            6695          319368
 13   7   5            3278          156888
 13   7   6            2503          119184
 13   7   7            1618           75984
 13   7   8             490           23472
 13   7   9             130            6024
 13   7  10              59            2832
 13   7  11              14             672
 13   7  12              11             528
 13   8   1           21816         1044144
 13   8   2            9274          442080
 13   8   3            3982          189600
 13   8   4            4704          222960
 13   8   5            1378           65424
 13   8   6            1747           81984
 13   8   7             490           23472
 13   8   8            1217           56136
 13   8   9             109            5112
 13   8  10             120            5616
 13   8  11              17             720
 13   8  12              18             864
 13   9   1             383           18288
 13   9   2             607           29136
 13   9   3             622           29256
 13   9   4             696           33408
 13   9   5             224           10728
 13   9   6             327           15648
 13   9   7             130            6024
 13   9   8             109            5112
 13   9   9              80            3264
 13   9  10              16             744
 13   9  11               1              48
 13   9  12               1              48
 13  10   1            1075           51312
 13  10   2             671           31632
 13  10   3             410           19440
 13  10   4             553           26280
 13  10   5             139            6336
 13  10   6             262           12216
 13  10   7              59            2832
 13  10   8             120            5616
 13  10   9              16             744
 13  10  10             141            6720
 13  10  11               8             384
 13  10  12              11             528
 13  11   1              88            4224
 13  11   2              46            2184
 13  11   3              81            3600
 13  11   4              84            3912
 13  11   5              19             912
 13  11   6              42            1872
 13  11   7              14             672
 13  11   8              17             720
 13  11   9               1              48
 13  11  10               8             384
 13  11  11              19             816
 13  11  12               1              48
 13  12   1              21            1008
 13  12   2              35            1680
 13  12   3              25            1200
 13  12   4             104            4848
 13  12   5              14             624
 13  12   6              23            1104
 13  12   7              11             528
 13  12   8              18             864
 13  12   9               1              48
 13  12  10              11             528
 13  12  11               1              48
 13  12  12              13             624
  Face Turn Metric
  Summary of |S(x)| and |E(x)|

|x|
   |S(x)| |E(x)|    patterns      positions
                    (positions
                     unique up
                     to symmetry)

  0   0   0               1               1
  1   1   1               1              18
  2   1   1               5             216
  2   2   2               4              27
  3   1   1              55            2592
  3   1   2              10             324
  3   2   1              10             324
  4   1   1             746           34968
  4   1   2              83            3888
  4   2   1              83            3888
  4   2   2              21             492
  4   4   4               1               3
  5   1   1            9660          463296
  5   1   2            1108           51888
  5   1   4               2              18
  5   2   1            1108           51888
  5   2   2             174            7176
  5   2   4               2              24
  5   3   3              19             576
  5   4   1               2              18
  5   4   2               2              24
  6   1   1          127718         6127560
  6   1   2           14513          692064
  6   1   3              85            3888
  6   1   4              15             504
  6   2   1           14513          692064
  6   2   2            1947           88668
  6   2   3              27            1200
  6   2   4              17             594
  6   3   1              85            3888
  6   3   2              27            1200
  6   3   3             113            4872
  6   4   1              15             504
  6   4   2              17             594
  6   4   4              24             255
  6   6   6              15             583
  7   1   1         1685940        80903592
  7   1   2          192243         9217728
  7   1   3            2077           99552
  7   1   4             282           12264
  7   1   6             132            6336
  7   2   1          192243         9217728
  7   2   2           23829         1126692
  7   2   3             599           26988
  7   2   4             108            4290
  7   2   5              21            1008
  7   2   6               7             312
  7   2   7               3             144
  7   3   1            2077           99552
  7   3   2             599           26988
  7   3   3             503           22356
  7   3   4              50            2280
  7   3   5              15             648
  7   3   6               1              12
  7   4   1             282           12264
  7   4   2             108            4290
  7   4   3              50            2280
  7   4   4             101            3072
  7   4   5               3             120
  7   4   6               1              48
  7   4   8               1              48
  7   5   2              21            1008
  7   5   3              15             648
  7   5   4               3             120
  7   5   5              26             714
  7   5   7               1              12
  7   6   1             132            6336
  7   6   2               7             312
  7   6   3               1              12
  7   6   4               1              48
  7   6   6              75            2568
  7   6   7               2              96
  7   6   8               1              48
  7   7   2               3             144
  7   7   5               1              12
  7   7   6               2              96
  7   7   7               6             168
  7   8   4               1              48
  7   8   6               1              48
  7  10  10               1               6
  8   1   1        22251034      1068019080
  8   1   2         2549011       122303784
  8   1   3           33400         1599528
  8   1   4            4198          197760
  8   1   5             287           13776
  8   1   6            2047           97560
  8   1   7              39            1872
  8   1   8               8             384
  8   2   1         2549011       122303784
  8   2   2          306251        14640522
  8   2   3            6866          327504
  8   2   4            1457           67188
  8   2   5             328           15600
  8   2   6             350           16536
  8   2   7              47            2232
  8   2   8               1              24
  8   3   1           33400         1599528
  8   3   2            6866          327504
  8   3   3            5456          255768
  8   3   4             608           28008
  8   3   5             159            7320
  8   3   6              73            3336
  8   3   7              10             480
  8   3   8               4             192
  8   4   1            4198          197760
  8   4   2            1457           67188
  8   4   3             608           28008
  8   4   4             688           25413
  8   4   5              87            4128
  8   4   6              42            1632
  8   4   7               9             432
  8   4   8               3             108
  8   5   1             287           13776
  8   5   2             328           15600
  8   5   3             159            7320
  8   5   4              87            4128
  8   5   5             124            5304
  8   5   6              22            1008
  8   5   7              11             528
  8   6   1            2047           97560
  8   6   2             350           16536
  8   6   3              73            3336
  8   6   4              42            1632
  8   6   5              22            1008
  8   6   6             338           12889
  8   6   7               3             120
  8   6   8               5             192
  8   7   1              39            1872
  8   7   2              47            2232
  8   7   3              10             480
  8   7   4               9             432
  8   7   5              11             528
  8   7   6               3             120
  8   7   7              20             648
  8   7   8               2              96
  8   7   9               1              48
  8   8   1               8             384
  8   8   2               1              24
  8   8   3               4             192
  8   8   4               3             108
  8   8   6               5             192
  8   8   7               2              96
  8   8   8              19             438
  8   8   9               2              96
  8   9   7               1              48
  8   9   8               2              96
  8   9   9               5             120
  8  10  10               5             144
  8  12  12               3              18
  9   1   1       293625442     14093870064
  9   1   2        33711631      1617992064
  9   1   3          498571        23926224
  9   1   4           68094         3253920
  9   1   5            8263          396552
  9   1   6           29839         1431000
  9   1   7            1104           52992
  9   1   8             179            8592
  9   2   1        33711631      1617992064
  9   2   2         4017656       192649008
  9   2   3          100112         4792344
  9   2   4           17262          812520
  9   2   5            2711          129480
  9   2   6            4491          213516
  9   2   7             390           18624
  9   2   8              77            3408
  9   2   9              19             912
  9   2  10               5             240
  9   2  12               1              24
  9   3   1          498571        23926224
  9   3   2          100112         4792344
  9   3   3           48218         2301096
  9   3   4            7307          347088
  9   3   5            1822           86520
  9   3   6            1093           51960
  9   3   7             223           10536
  9   3   8             128            5952
  9   3   9              13             576
  9   3  10               7             288
  9   3  12               3             144
  9   3  14               1              48
  9   4   1           68094         3253920
  9   4   2           17262          812520
  9   4   3            7307          347088
  9   4   4            5827          261840
  9   4   5             822           38520
  9   4   6             421           18558
  9   4   7              97            4512
  9   4   8              57            2400
  9   4   9               5             144
  9   4  10               2              48
  9   4  12               1              48
  9   5   1            8263          396552
  9   5   2            2711          129480
  9   5   3            1822           86520
  9   5   4             822           38520
  9   5   5            1012           44400
  9   5   6             223            9948
  9   5   7              56            2568
  9   5   8              18             768
  9   5   9              13             624
  9   5  10               6             288
  9   6   1           29839         1431000
  9   6   2            4491          213516
  9   6   3            1093           51960
  9   6   4             421           18558
  9   6   5             223            9948
  9   6   6            2279          100885
  9   6   7              51            2256
  9   6   8              60            2256
  9   6   9               4             156
  9   6  10               6             240
  9   6  12               3              56
  9   6  14               1               6
  9   7   1            1104           52992
  9   7   2             390           18624
  9   7   3             223           10536
  9   7   4              97            4512
  9   7   5              56            2568
  9   7   6              51            2256
  9   7   7             149            6432
  9   7   8              18             768
  9   7   9               3              96
  9   7  11               1              24
  9   7  12               2              48
  9   8   1             179            8592
  9   8   2              77            3408
  9   8   3             128            5952
  9   8   4              57            2400
  9   8   5              18             768
  9   8   6              60            2256
  9   8   7              18             768
  9   8   8              97            3216
  9   8   9              10             360
  9   8  10               1              24
  9   8  12               1              24
  9   9   2              19             912
  9   9   3              13             576
  9   9   4               5             144
  9   9   5              13             624
  9   9   6               4             156
  9   9   7               3              96
  9   9   8              10             360
  9   9   9              33            1232
  9   9  10               1              12
  9  10   2               5             240
  9  10   3               7             288
  9  10   4               2              48
  9  10   5               6             288
  9  10   6               6             240
  9  10   8               1              24
  9  10   9               1              12
  9  10  10              26             696
  9  11   7               1              24
  9  12   2               1              24
  9  12   3               3             144
  9  12   4               1              48
  9  12   6               3              56
  9  12   7               2              48
  9  12   8               1              24
  9  12  12               6             168
  9  12  17               1              24
  9  13  13               1              48
  9  14   3               1              48
  9  14   6               1               6
  9  14  14               4              78
  9  17  12               1              24
  9  18  18               2              32  (shortest strong local maxima in face turn metric)
 10   1   1      3872815251    185894472840
 10   1   2       445589172     21387834528
 10   1   3         7171651       344216592
 10   1   4         1016422        48752448
 10   1   5          134187         6437352
 10   1   6          413236        19823328
 10   1   7           18451          885456
 10   1   8            3743          179568
 10   1   9             277           13296
 10   1  10              94            4512
 10   1  12              27            1296
 10   1  14               4             192
 10   2   1       445589172     21387834528
 10   2   2        52606796      2524319964
 10   2   3         1400237        67156800
 10   2   4          235083        11222844
 10   2   5           39571         1893672
 10   2   6           63216         3024144
 10   2   7            5426          259560
 10   2   8            1795           85548
 10   2   9             278           13296
 10   2  10             138            6288
 10   2  11              10             384
 10   2  12              21            1008
 10   2  13               8             384
 10   2  14               8             384
 10   3   1         7171651       344216592
 10   3   2         1400237        67156800
 10   3   3          540322        25896384
 10   3   4          100168         4793472
 10   3   5           23424         1120608
 10   3   6           13014          619800
 10   3   7            3110          148896
 10   3   8            1242           58872
 10   3   9             261           12264
 10   3  10             153            7128
 10   3  11              21             984
 10   3  12              15             720
 10   3  13               8             384
 10   3  16               2              96
 10   4   1         1016422        48752448
 10   4   2          235083        11222844
 10   4   3          100168         4793472
 10   4   4           54504         2553462
 10   4   5           10843          517896
 10   4   6            6317          296772
 10   4   7            1650           78480
 10   4   8             803           37278
 10   4   9             161            7536
 10   4  10              92            4272
 10   4  11              22            1032
 10   4  12              26            1224
 10   4  13               1              24
 10   4  14               3             144
 10   5   1          134187         6437352
 10   5   2           39571         1893672
 10   5   3           23424         1120608
 10   5   4           10843          517896
 10   5   5            7258          341040
 10   5   6            2773          131496
 10   5   7             785           36984
 10   5   8             426           20352
 10   5   9             136            6288
 10   5  10              65            2880
 10   5  11              11             528
 10   5  12              12             480
 10   5  13               6             288
 10   6   1          413236        19823328
 10   6   2           63216         3024144
 10   6   3           13014          619800
 10   6   4            6317          296772
 10   6   5            2773          131496
 10   6   6           16232          748812
 10   6   7             811           38280
 10   6   8             600           26748
 10   6   9             109            5088
 10   6  10              63            2748
 10   6  11              17             744
 10   6  12              10             456
 10   6  13               1              48
 10   7   1           18451          885456
 10   7   2            5426          259560
 10   7   3            3110          148896
 10   7   4            1650           78480
 10   7   5             785           36984
 10   7   6             811           38280
 10   7   7             905           40536
 10   7   8             242           11256
 10   7   9              85            4008
 10   7  10              46            2088
 10   7  11              10             480
 10   7  12               2              96
 10   7  13               2              96
 10   8   1            3743          179568
 10   8   2            1795           85548
 10   8   3            1242           58872
 10   8   4             803           37278
 10   8   5             426           20352
 10   8   6             600           26748
 10   8   7             242           11256
 10   8   8             764           29622
 10   8   9              63            2952
 10   8  10              69            2952
 10   8  11               8             384
 10   8  12              11             528
 10   8  13               4              96
 10   8  14               2              96
 10   8  16               1               6
 10   9   1             277           13296
 10   9   2             278           13296
 10   9   3             261           12264
 10   9   4             161            7536
 10   9   5             136            6288
 10   9   6             109            5088
 10   9   7              85            4008
 10   9   8              63            2952
 10   9   9             182            7980
 10   9  10              21            1008
 10   9  11               4             192
 10   9  12               1              48
 10   9  13               3             120
 10   9  14               1              24
 10  10   1              94            4512
 10  10   2             138            6288
 10  10   3             153            7128
 10  10   4              92            4272
 10  10   5              65            2880
 10  10   6              63            2748
 10  10   7              46            2088
 10  10   8              69            2952
 10  10   9              21            1008
 10  10  10             188            6525
 10  10  11               6             288
 10  10  12               5             120
 10  10  13               2              96
 10  10  14               1              24
 10  10  16               1              48
 10  11   2              10             384
 10  11   3              21             984
 10  11   4              22            1032
 10  11   5              11             528
 10  11   6              17             744
 10  11   7              10             480
 10  11   8               8             384
 10  11   9               4             192
 10  11  10               6             288
 10  11  11              28             912
 10  11  12               2              72
 10  11  13               1              24
 10  11  15               1              24
 10  12   1              27            1296
 10  12   2              21            1008
 10  12   3              15             720
 10  12   4              26            1224
 10  12   5              12             480
 10  12   6              10             456
 10  12   7               2              96
 10  12   8              11             528
 10  12   9               1              48
 10  12  10               5             120
 10  12  11               2              72
 10  12  12              45            1016
 10  12  13               1              48
 10  12  14               2              60
 10  12  17               1              48
 10  13   2               8             384
 10  13   3               8             384
 10  13   4               1              24
 10  13   5               6             288
 10  13   6               1              48
 10  13   7               2              96
 10  13   8               4              96
 10  13   9               3             120
 10  13  10               2              96
 10  13  11               1              24
 10  13  12               1              48
 10  13  13              14             576
 10  13  14               1              48
 10  13  15               1              24
 10  14   1               4             192
 10  14   2               8             384
 10  14   4               3             144
 10  14   8               2              96
 10  14   9               1              24
 10  14  10               1              24
 10  14  12               2              60
 10  14  13               1              48
 10  14  14               8             252
 10  15  11               1              24
 10  15  13               1              24
 10  15  16               1              48
 10  16   3               2              96
 10  16   8               1               6
 10  16  10               1              48
 10  16  15               1              48
 10  16  16               9             120
 10  17  12               1              48
 10  17  17               1              24
 10  18  18               6             107
 11   1   1     51053195698   2450552215440
 11   1   2      5880076220    282241951632
 11   1   3       101690227      4881029784
 11   1   4        14806460       710565192
 11   1   5         2061080        98928720
 11   1   6         5632845       270337440
 11   1   7          283489        13605552
 11   1   8           67229         3224904
 11   1   9            7322          351456
 11   1  10            3149          151032
 11   1  11             281           13296
 11   1  12             517           24768
 11   1  13             161            7728
 11   1  14             134            6432
 11   1  16              10             480
 11   2   1      5880076220    282241951632
 11   2   2       694472589     33332098788
 11   2   3        19732343       946981272
 11   2   4         3388193       162347760
 11   2   5          584361        28034304
 11   2   6          848455        40674900
 11   2   7           84073         4033512
 11   2   8           23853         1140600
 11   2   9            5082          243432
 11   2  10            2332          110640
 11   2  11             390           18672
 11   2  12             330           15732
 11   2  13             132            6336
 11   2  14              29            1392
 11   2  16               8             384
 11   3   1       101690227      4881029784
 11   3   2        19732343       946981272
 11   3   3         6804797       326444436
 11   3   4         1457352        69882912
 11   3   5          330661        15854064
 11   3   6          182676         8750920
 11   3   7           48694         2333184
 11   3   8           18518          885648
 11   3   9            4662          223248
 11   3  10            2303          109704
 11   3  11             462           22128
 11   3  12             377           17760
 11   3  13              64            3024
 11   3  14              45            2160
 11   3  15               1              48
 11   3  16               4             192
 11   4   1        14806460       710565192
 11   4   2         3388193       162347760
 11   4   3         1457352        69882912
 11   4   4          653014        31123020
 11   4   5          153438         7353432
 11   4   6           85268         4064160
 11   4   7           22932         1098264
 11   4   8           11182          530976
 11   4   9            3198          153000
 11   4  10            1884           89280
 11   4  11             457           21864
 11   4  12             317           14568
 11   4  13             109            5160
 11   4  14              84            3864
 11   4  15               7             336
 11   4  16              10             480
 11   4  17               1              48
 11   5   1         2061080        98928720
 11   5   2          584361        28034304
 11   5   3          330661        15854064
 11   5   4          153438         7353432
 11   5   5           80491         3843576
 11   5   6           35205         1681896
 11   5   7           11518          550728
 11   5   8            5390          257208
 11   5   9            1826           87000
 11   5  10            1010           48336
 11   5  11             252           12096
 11   5  12             214           10104
 11   5  13              68            3216
 11   5  14              49            2352
 11   5  15               5             240
 11   5  16               9             432
 11   6   1         5632845       270337440
 11   6   2          848455        40674900
 11   6   3          182676         8750920
 11   6   4           85268         4064160
 11   6   5           35205         1681896
 11   6   6           95469         4496958
 11   6   7            8936          426456
 11   6   8            4830          225084
 11   6   9            1435           68496
 11   6  10            1051           48588
 11   6  11             208            9888
 11   6  12             182            7740
 11   6  13              44            2016
 11   6  14              62            2904
 11   6  15               1              48
 11   6  16               1              48
 11   6  18               4              96
 11   7   1          283489        13605552
 11   7   2           84073         4033512
 11   7   3           48694         2333184
 11   7   4           22932         1098264
 11   7   5           11518          550728
 11   7   6            8936          426456
 11   7   7            7508          348828
 11   7   8            2391          113496
 11   7   9             791           37704
 11   7  10             486           22512
 11   7  11             126            5904
 11   7  12             136            6336
 11   7  13              23             936
 11   7  14              32            1536
 11   7  15               6             288
 11   7  16               2              96
 11   8   1           67229         3224904
 11   8   2           23853         1140600
 11   8   3           18518          885648
 11   8   4           11182          530976
 11   8   5            5390          257208
 11   8   6            4830          225084
 11   8   7            2391          113496
 11   8   8            4199          178602
 11   8   9             664           31368
 11   8  10             622           28680
 11   8  11             149            7032
 11   8  12             185            8352
 11   8  13              25            1128
 11   8  14              21             912
 11   8  15               2              96
 11   8  16               4             192
 11   9   1            7322          351456
 11   9   2            5082          243432
 11   9   3            4662          223248
 11   9   4            3198          153000
 11   9   5            1826           87000
 11   9   6            1435           68496
 11   9   7             791           37704
 11   9   8             664           31368
 11   9   9             814           36744
 11   9  10             238           11208
 11   9  11              78            3720
 11   9  12              59            2736
 11   9  13              25            1104
 11   9  14              18             816
 11   9  15               3             144
 11  10   1            3149          151032
 11  10   2            2332          110640
 11  10   3            2303          109704
 11  10   4            1884           89280
 11  10   5            1010           48336
 11  10   6            1051           48588
 11  10   7             486           22512
 11  10   8             622           28680
 11  10   9             238           11208
 11  10  10            1003           41232
 11  10  11              80            3768
 11  10  12             130            5772
 11  10  13              26            1032
 11  10  14              51            2028
 11  10  16               4             144
 11  10  18               1              12
 11  11   1             281           13296
 11  11   2             390           18672
 11  11   3             462           22128
 11  11   4             457           21864
 11  11   5             252           12096
 11  11   6             208            9888
 11  11   7             126            5904
 11  11   8             149            7032
 11  11   9              78            3720
 11  11  10              80            3768
 11  11  11             129            5688
 11  11  12              52            2472
 11  11  13              16             768
 11  11  14              16             744
 11  11  15               5             216
 11  11  16               1              24
 11  11  17               4             192
 11  12   1             517           24768
 11  12   2             330           15732
 11  12   3             377           17760
 11  12   4             317           14568
 11  12   5             214           10104
 11  12   6             182            7740
 11  12   7             136            6336
 11  12   8             185            8352
 11  12   9              59            2736
 11  12  10             130            5772
 11  12  11              52            2472
 11  12  12             311           11634
 11  12  13              17             816
 11  12  14              10             432
 11  12  15               1              24
 11  12  16               9             264
 11  12  17               1              48
 11  13   1             161            7728
 11  13   2             132            6336
 11  13   3              64            3024
 11  13   4             109            5160
 11  13   5              68            3216
 11  13   6              44            2016
 11  13   7              23             936
 11  13   8              25            1128
 11  13   9              25            1104
 11  13  10              26            1032
 11  13  11              16             768
 11  13  12              17             816
 11  13  13              35            1164
 11  13  14               6             288
 11  13  15               1              48
 11  13  16               3             144
 11  13  17               4             192
 11  14   1             134            6432
 11  14   2              29            1392
 11  14   3              45            2160
 11  14   4              84            3864
 11  14   5              49            2352
 11  14   6              62            2904
 11  14   7              32            1536
 11  14   8              21             912
 11  14   9              18             816
 11  14  10              51            2028
 11  14  11              16             744
 11  14  12              10             432
 11  14  13               6             288
 11  14  14              94            3054
 11  14  16               4              96
 11  14  17               1              48
 11  14  18               3              96
 11  15   3               1              48
 11  15   4               7             336
 11  15   5               5             240
 11  15   6               1              48
 11  15   7               6             288
 11  15   8               2              96
 11  15   9               3             144
 11  15  11               5             216
 11  15  12               1              24
 11  15  13               1              48
 11  15  15              11             252
 11  16   1              10             480
 11  16   2               8             384
 11  16   3               4             192
 11  16   4              10             480
 11  16   5               9             432
 11  16   6               1              48
 11  16   7               2              96
 11  16   8               4             192
 11  16  10               4             144
 11  16  11               1              24
 11  16  12               9             264
 11  16  13               3             144
 11  16  14               4              96
 11  16  16              21             600
 11  16  17               2              96
 11  16  18               5             192
 11  17   4               1              48
 11  17  11               4             192
 11  17  12               1              48
 11  17  13               4             192
 11  17  14               1              48
 11  17  16               2              96
 11  17  17               1              12
 11  18   6               4              96
 11  18  10               1              12
 11  18  14               3              96
 11  18  16               5             192
 11  18  18              32             984

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I did the same thing for the

I did the same thing for the cube restricted to the corners permutations only. Here are the QTM results:
|x|     |S(x)|  |E(x)|  positions

0       0       0          1
0                          1
 
1       1       1         12
1                         12

2       1       1         96
2       2       2         18
2                        114

3       1       1        480
3       1       2         96
3       2       1         96
3       2       2        144
3       3       3         60
3                        876

4       1       1       1824
4       1       2        480
4       2       1        480
4       2       2        864
4       2       3        240
4       3       2        240
4       3       3        336
4       4       4        315
4       5       5         96
4       6       6         56
4                       4931

5       1       1       1536
5       1       2        336
5       2       1        336
5       2       2       1224
5       2       3        624
5       2       4        216
5       3       2        624
5       3       3       1152
5       3       4        336
5       3       5         96
5       3       6         48
5       4       2        216
5       4       3        336
5       4       4       1056
5       4       5        384
5       4       6         48
5       4       8         24
5       5       3         96
5       5       4        384
5       5       5        576
5       5       6        192
5       5       7        192
5       5       8         48
5       6       3         48
5       6       4         48
5       6       5        192
5       6       6        672
5       6       7         96
5       6       8         48
5       7       5        192
5       7       6         96
5       7       7        312
5       7       8         48
5       7       9         48
5       8       4         24
5       8       5         48
5       8       6         48
5       8       7         48
5       8       8        468
5       8       10        48
5       9       7         48
5       9       9        256
5       10      8         48
5       10      10        48
5       12      12         8
5                      12972

6       1       1        192
6       1       2         96
6       1       3         96
6       2       1         96
6       2       2        504
6       2       3        144
6       2       4        192
6       2       6         48
6       3       1         96
6       3       2        144
6       3       3        192
6       3       5         96
6       3       6         96
6       3       7         48
6       3       8         72
6       4       2        192
6       4       4        456
6       4       5         96
6       4       6        168
6       4       8        216
6       4       9         48
6       4       10        72
6       5       3         96
6       5       4         96
6       5       5        288
6       5       6        240
6       5       7         48
6       5       8        240
6       5       9        192
6       5       10        48
6       5       11        48
6       6       2         48
6       6       3         96
6       6       4        168
6       6       5        240
6       6       6        648
6       6       7        144
6       6       8        408
6       6       9        192
6       6       10       432
6       6       11        48
6       7       3         48
6       7       5         48
6       7       6        144
6       7       7        384
6       7       8        168
6       7       10        48
6       7       11        48
6       8       3         72
6       8       4        216
6       8       5        240
6       8       6        408
6       8       7        168
6       8       8        864
6       8       9         96
6       8       10        48
6       8       11       192
6       8       12        96
6       9       4         48
6       9       5        192
6       9       6        192
6       9       8         96
6       9       9        192
6       9       10        96
6       9       12        96
6       10      4         72
6       10      5         48
6       10      6        432
6       10      7         48
6       10      8         48
6       10      9         96
6       10      10       360
6       10      11       192
6       10      12        96
6       11      5         48
6       11      6         48
6       11      7         48
6       11      8        192
6       11      10       192
6       11      11       384
6       11      12       192
6       12      8         96
6       12      9         96
6       12      10        96
6       12      11       192
6       12      12       810
6                      15066

7       10      10        24
7       10      12        96
7       11      11       144
7       11      12       192
7       12      10        96
7       12      11       192
7       12      12      5556
7                       6300

8       12      12        48
8                         48

The Two Shortest Strong Local Maxima in the Face Turn Metric

Here are the two positions that are the shortest strong local maxima in the face turn metric:
D2 F2 L2 D' U L2 F2 D' U'  (9f*) 
U2 B2 L2 D U' R2 B2 D' U'  (9f*) 
The positions look extremely symmetric, but the first one only has two symmetries and the second one only has six symmetries. For the first position, the two symmtries are the identity and the central inversion. For the second position, the six symmetries are the identity, a 120 degree rotation about a major diagonal axis, a 240 degree rotation about the same major diagonal axis, and the central inversion composed with each of the three rotations. The two positions look almost identical, and differ in subtle ways. They both have the following property. Consider a cube that is colored according to the "opposite faces differ by yellow" convention. Namely, white is opposite yellow, blue is opposite green, and red is opposite orange. Call the white and yellow faces "whitish". Call the blue and green faces "bluish". Call the red and orange faces "redish". Then, both of these positions have the property that they preserve the whitishness of the whitesh faces, they preserve the bluishness of the bluish faces, and they preserve the redishness of the redish faces.

Preservation of bluishness etc.

The two positions are in (U2,D2,R2,L2,F2,B2), where all positions have this property.

L2 U2 B2 D2 R2 D2 R2 B2 L2 U2 (10f)
L2 D2 F2 U2 R2 D2 L2 F2 R2 U2 (10f)

I did not look if these positions also are strong local maxima in this subgroup.