cube files
Indexed Cube Lovers Archive
Starts-with and Ends-With
Submitted by Jerry Bryan on Thu, 12/21/2006 - 16:12.
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
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