# 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
```

## Comment viewing options

### 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.