The New Sudoku Players' Forum

[blue]

Hi, Blue!
Very impressive and fast work!
Are you sure your lists of maximal invalid anticorner patterns and restricted minimal valid anticorner patterns are full?

Thanks !

I'm sure that the "minimal valid" shapes have puzzles, and that every shape is a superset of a minimal valid shape or a subset of maximal invalid shape.
I'll build a "pruned shape list" again, filtering out proper supersets/subsets, to be doubly sure.
It will take ~17h, once I get it going.

I'll be sure that the lists are "correct" too, if you can confirm that these shapes (29) don't have puzzles.

Hidden Text: Show

Code: Select all

...........1........1111111......111......111......111..1111111111111111111111111 (15)
........1.1......1......111......111......111.....1111..1111111..1111111111111111 (12) - diagonal symmetry
........1.....1..1..1.....1......111......111..1...111...111111111111111111111111 (12)
...........1..1...........1......111......111111...111...111111111111111111111111 (12)
..........1......11......1.......111......111..1...111..1111111..1111111111111111 (10)
..1..............1.....111.......111......111..1...111...111111...111111111111111 ( 9)
.........11.............111......111......111..1...111..1111111..1111111..1111111 ( 9)
....11.....1..............1......111......111.....1111...111111..1111111111111111 ( 9)
..1.......1..........1....1......111......111....11111...111111...111111111111111 ( 9)
..........1........1...1.....1...111..1...111..1...111..1111111..1111111..1111111 ( 9)
1........1..............111......111......111.....1111...111111..1111111.1.111111 ( 8)
..1......1..............111......111......111.....1111...111111..1111111.1.111111 ( 8)
..1.......1.............111......111......111.....1111...111111..1111111.1.111111 ( 8)
..1.......1.............111......111......1111.....111...111111..1111111.1.111111 ( 8)
.........1....1...1......11......111......111......111..1111111..1111111.1.111111 ( 8)
.........1....1....1.....11......111......111......111..1111111..1111111.1.111111 ( 8)
..........1.......1......11......111......111.....1111..1111111..1111111..1111111 ( 8)
.1.......1....1..........11......111......111......111..1111111..1111111..1111111 ( 8)
..........11........1.....1......111......111.....1111..1111111..1111111.1.111111 ( 8)
..........1..11........1..1......111......111.1...1111...111111...111111..1111111 ( 8)
.1.......1........11...1..1......111......111......111...111111...111111..1111111 ( 7)
...........1..1...1..............111......111..1...111..1111111..1111111.1.111111 ( 7)
..1.......1........11..1.........111......111......111...111111..1111111.1.111111 ( 7)
..........1.......1....1.........111......111.1....111..1111111..1111111..1111111 ( 7)
...........1.......1...1.........111......1111.....111..1111111..1111111..1111111 ( 7)
.....1.....1.......1.............111......111.1....111..1111111..1111111..1111111 ( 7)
...........1..1....1.............111......111.1....111..1111111..1111111..1111111 ( 7)
...........1.......1...1.........111......111.1....111..1111111..1111111..1111111 ( 7)
..............1....1...1...1.....1111.....1111.....111...111111...111111..1111111 ( 7)

Cheers,
Blue.

[coloin]

Indeed ... that is promising that you have found all of the patterns !!!

I haave done a much more rigerous search of many instances of this pattern ...

Code: Select all

+---+---+---+
|...|...|...|
|...|...|..1|
|..2|...|..3|
+---+---+---+
|...|123|586|
|...|456|792|
|.6.|789|134|
+---+---+---+
|...|867|345|
|7..|942|618|
|4..|531|279|
+---+---+---+   

.................X.....X..XXXX...XXXXXX...XXXXXX.X.XXXXXX...XXXXXXX..XXXXXXX..XXX   
.................1..2.....3...123586...456792.6.789134...8673457..9426184..531279   


And found these patterns with valid puzzles, which i dont think are supersets of lesser patterns

Code: Select all

Box5 removed clues
There were no C8 patterns with valid puzzles ##  6 C8 invalid patterns
Valid puzzles                                                               
.................2..4.....9.....3856....7.943.5..8.721...4123952..7381641..659287 C9
.................2..4.....9...1..856.....9347.5..7.291...5149232..7361841..982765 C9
.................1..4.....2.....5243.....1587.3..8.169...9528748..4369159..718326 C9
.................2..4.....9....2.385....3.746.5..8.921...9132542..4781631..652897 C9
                                                                                   
Box6 removed clues   
Valid puzzles                                                               
.................2..4.....9...1238.....456....5.789......5329412..6143871..978265 C7  [rarish]
.................7..3.....6...123......456....5.789.13...3712897..2956419..648735 C8
.................7..3.....6...123......659..2.5.478.3....3812797..2964159..745683 C8
.................7..4.....2...239......451..6.2.786..3...1472383..9287657..563914 C8
.................7..4.....2...127..3...546..1.2.938......7832143..2597687..461935 C8
.................3..2.....6...418......296....4.57398....6857497..9418328..732165 C8
                                                                                   
Box8 removed clues 
There were no C8 patterns with valid puzzle s##  6 C8 invalid patterns
Valid puzzles                                                               
.................7..3.....6...213568...546279.5.879413.....19827...2.6419...6.735 C9
.................7..3.....6...213568...546279.5.879413.....19827...2.6419..4..735 C9
.................7..3.....6...213568...546279.5.879413.....19829....87357...2.641 C9
.................7..3.....6...321568...546972.5.897413.....32817....26499....4735 C9
                                                                                   
Box9 removed clues
Valid puzzles                                                                   
.................8..4.....6...615243...497685.5.823179...1369..3..259...1..784... C7 all quite rare
.................8..4.....6...614253...857964.5.239187...796..53..125...1..483... C7
.................8..4.....6...615342...784965.5.293187...137...3..8592..1..426... C7
.................8..4.....6...623154...457689.5.819273...195...3..284..51..736... C7


There were numerous instances of all these patterns, maybe If there is a rarer puzzle/pattern it might be missed ...

[blue]

Hi Colin,

Nice work !
They're all minimal, and the list is almost complete.

For "Box5 removals", your 1st and 3rd puzzles have the same shape, and this one is missing:

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.................1..4.....2.....5243.....1587.3..8.169...9528748..4369159..718326 C9

For Box 6 removals, this one is missing:

Code: Select all

.................3..2.....6...418......296....4.57398....6857497..9418328..732165 C8

[coloin]

Thanks ... indeed i had the puzzles , just decided to complete it manually at the end....
It is increasingly likely that you have sorted these patterns out !!! A massive book keeping and programming exercise !!!
I presume the invalid patterns were included too.

The pattern [ one of 444 with with the 6 clues in B12347 and full B589 ] chosen I thought would have few grids !!
I thought I had generated most of them with that pattern with the full B5689 , and it certainly was hard to generate de novo ...
However having only found 32 crossing band gangsters [ from 300 originals] these were extrapolated to over 84000 grids !! ... but on check back half were missed ... possible due to multible instances of the solve/ or rotation.

There were 36 patterns with 5 clues [5plus36] and 444 patterns with 6 clues [6plus36]. It seems that most 90% ? of the 444 are supersets of the 36. And there will be fewer at higher levels ..
For some reason alll my programs stall at adding 1 clue to the puzzle ! [not sure why]

I will have a go to see if I can find a rare puzzle in another pattern ... and also invalid patterns.

[blue]

Hi, Blue!
Very impressive and fast work!
Are you sure your lists of maximal invalid anticorner patterns and restricted minimal valid anticorner patterns are full?

Thanks !

I'm sure that the "minimal valid" shapes have puzzles, and that every shape is a superset of a minimal valid shape or a subset of maximal invalid shape.
I'll build a "pruned shape list" again, filtering out proper supersets/subsets, to be doubly sure.
It will take ~17h, once I get it going.

Everything came out as expected.
The pruned shape list, had the 12793 (restricted) minimal valid shapes, and 1536 maximal invalid shapes.

The shapes are ED with respect to transformations that include diagonal reflections, but no band or stack permutations: transformations that preserve the B689 shape.
With respect to general Sudoku transformations, the 12793 are ED and the 1536 reduce to the 1428 cases mentioned here.

In case JPF and/or Serg are interested in confirming it by other means, the shapes that are ED with respect to the reduced set of transformations, have this breakdown by (B123457) clue count:

Code: Select all

clues |     patterns
------+-------------
    0 |            1
    1 |            4
    2 |           23
    3 |          125
    4 |          630
    5 |         2970
    6 |        13089
    7 |        53801
    8 |       206531
    9 |       739539
   10 |      2468234
   11 |      7674378
   12 |     22221573
   13 |     59917001
   14 |    150468286
   15 |    352048728
   16 |    767736056
   17 |   1561293914
   18 |   2962363840
   19 |   5246685692
   20 |   8678086628
   21 |  13410257969
   22 |  19368071661
   23 |  26152415035
   24 |  33024031396
   25 |  39006180915
   26 |  43101117092
   27 |  44559029938
   28 |  43101117092
   29 |  39006180915
   30 |  33024031396
   31 |  26152415035
   32 |  19368071661
   33 |  13410257969
   34 |   8678086628
   35 |   5246685692
   36 |   2962363840
   37 |   1561293914
   38 |    767736056
   39 |    352048728
   40 |    150468286
   41 |     59917001
   42 |     22221573
   43 |      7674378
   44 |      2468234
   45 |       739539
   46 |       206531
   47 |        53801
   48 |        13089
   49 |         2970
   50 |          630
   51 |          125
   52 |           23
   53 |            4
   54 |            1
------+-------------
      | 432307140160

[Serg]

Hi!
I cofirm invalidity of 1428 ED maximal invalid anticorner patterns, presented by Blue. (Including 34 additional to my list maximal invalid patterns.)
I confirm that those 1428 patterns are not only invalid, but maximal invalid patterns, i.e. adding any 1 clue makes every pattern valid.

Serg

[blue]

Hi Serg,

That was quick ... the last 29 patterns.
Thanks.

I tried to understand your "birthing" method, but nothing really "clicked".

There exists equivalent procedure of giving birth to a (restricted) valid anticorner pattern by 2 maximal invalid anticorner patterns.

Would you outline that procedure too, please ?

[coloin]

Well it does seem that you have cracked this code...
I get your method now ... and it is a massive undertaking ...
You do it by generating valid puzzles .. and adding clues which rule out superset patterns.
Rule out other patterns with known invalid patterns
Leaving you with a small list of patterns which Serg checked !!!!

Of course you now have the data for valid puzzles with only the crossing pattern [B12347]

Fwiw i have now been able to ? reliably find invalid patterns on this pattern, but would be interested to see how he does his method too

Code: Select all

........5
........2
..8......
...123748
...456139
...789256
..4935871
.7.812564
5..674923

i was able to generate 45416 ED instances of the pattern , and easily resolving the individual B5689 boxes.
Fairly certain that all puzzles were found using expansion from all found B12347, and no new random ones could be generated.

For this pattern there were many instances of puzzles with added clues

Code: Select all

Box5 -  53632 puzzles with 4 clues , 4 patterns , none with 3 clues
Box6 -   3736 puzzles with 2 clues , 4 patterns , none with 1 clue
Box8 - 121129 puzzles with 3 clues , 4 patterns , none with 2 clues
Box9 -      9 puzzles with 0 clues added all same pattern !


Code: Select all

........4........6..5.........126593...458762...379418..9513....4.782...2..694...
........4........6..5.........316592...572468...489713..9153....4.728...2..694...
........4........6..5.........612593...584762...379418..9153....4.728...2..496...
........4........6..5.........629513...578462...314798..9153....4.782...2..496...
........4........8..6.........345261...769485...812379..3126....8.957...5..483...
........4........8..6.........345261...869475...712389..3126....8.957...5..483...
........4........8..6.........345261...967485...812379..3126....8.759...5..483...
........4........8..6.........412365...765489...389271..3126....8.957...5..843...
........4........8..6.........425361...986475...317289..3162....8.759...5..843...

Code: Select all

+---+---+---+
|...|...|..4|
|...|...|..6|
|..5|...|...|
+---+---+---+
|...|126|593|
|...|458|762|
|...|379|418|
+---+---+---+
|..9|513|...|
|.4.|782|...|
|2..|694|...|
+---+---+---+  Valid puzzle


One of these puzzles with this pattern is already in my list of 37plus6 puzzlers however published as

Code: Select all

.....8........3.....9........7...295.4....7388.....641...314986...795423...286157

That said, these puzzles wont be easily found - there are no others within { -3+3] !

the Box6 pattern with the least puzzles found was this with only 172 ED puzzles, but almost certainly its in your list !

Code: Select all

+---+---+---+
|...|...|..3|
|...|...|..9|
|..8|...|...|
+---+---+---+
|...|752|...|
|...|643|...|
|...|918|7.6|
+---+---+---+
|..7|126|458|
|.6.|489|237|
|4..|375|691|
+---+---+---+


[Serg]

Hi, Blue!

I tried to understand your "birthing" method, but nothing really "clicked".

There exists equivalent procedure of giving birth to a (restricted) valid anticorner pattern by 2 maximal invalid anticorner patterns.

Would you outline that procedure too, please ?

OK. I'll describe "birthing" procedure in more detail.

Case 1 - two restricted minimal patterns give rise third pattern - candidate to invalid pattern.

Let's consider 2 restricted minimal valid patterns - A and B.

Code: Select all

          A                      B

+-----+-----+-----+     +-----+-----+-----+
|. . .|. . .|. . .|     |. . .|. . .|. . .|
|. . .|. . .|. . x|     |. . .|. . .|. . x|
|. . .|. . x|. . .|     |. . .|. . x|. . .|
+-----+-----+-----+     +-----+-----+-----+
|. . x|. . .|x x x|     |. . x|. . .|x x x|
|. . x|. . x|x x x|     |. . x|. . x|x x x|
|. . x|. . x|x x x|     |. . x|. x .|x x x|
+-----+-----+-----+     +-----+-----+-----+
|. x .|x x x|x x x|     |. x .|x x x|x x x|
|. x .|x x x|x x x|     |. x .|x x x|x x x|
|x . .|x x x|x x x|     |. x .|x x x|x x x|
+-----+-----+-----+     +-----+-----+-----+

Merging (union) A and B patterns gives "starting set":

Code: Select all

+-----+-----+-----+
|. . .|. . .|. . .|
|. . .|. . .|. . x|
|. . .|. . x|. . .|
+-----+-----+-----+
|. . x|. . .|x x x|
|. . x|. . x|x x x|
|. . x|. x x|x x x|
+-----+-----+-----+
|. x .|x x x|x x x|
|. x .|x x x|x x x|
|x x .|x x x|x x x|
+-----+-----+-----+

Intersecting A and B patterns gives "removal set":

Code: Select all

+-----+-----+-----+
|. . .|. . .|. . .|
|. . .|. . .|. . x|
|. . .|. . x|. . .|
+-----+-----+-----+
|. . x|. . .|x x x|
|. . x|. . x|x x x|
|. . x|. . .|x x x|
+-----+-----+-----+
|. x .|x x x|x x x|
|. x .|x x x|x x x|
|. . .|x x x|x x x|
+-----+-----+-----+

To produce family of candidates to invalid patterns we must remove 1 clue from starting set (located in B123457 area) presented in "removal set". So, we'll get 8 candidate patterns:

Code: Select all

         P1                      P2                      P3                      P4

+-----+-----+-----+     +-----+-----+-----+     +-----+-----+-----+     +-----+-----+-----+
|. . .|. . .|. . .|     |. . .|. . .|. . .|     |. . .|. . .|. . .|     |. . .|. . .|. . .|
|. . .|. . .|. . .|     |. . .|. . .|. . x|     |. . .|. . .|. . x|     |. . .|. . .|. . x|
|. . .|. . x|. . .|     |. . .|. . .|. . .|     |. . .|. . x|. . .|     |. . .|. . x|. . .|
+-----+-----+-----+     +-----+-----+-----+     +-----+-----+-----+     +-----+-----+-----+
|. . x|. . .|x x x|     |. . x|. . .|x x x|     |. . .|. . .|x x x|     |. . x|. . .|x x x|
|. . x|. . x|x x x|     |. . x|. . x|x x x|     |. . x|. . x|x x x|     |. . .|. . x|x x x|
|. . x|. x x|x x x|     |. . x|. x x|x x x|     |. . x|. x x|x x x|     |. . x|. x x|x x x|
+-----+-----+-----+     +-----+-----+-----+     +-----+-----+-----+     +-----+-----+-----+
|. x .|x x x|x x x|     |. x .|x x x|x x x|     |. x .|x x x|x x x|     |. x .|x x x|x x x|
|. x .|x x x|x x x|     |. x .|x x x|x x x|     |. x .|x x x|x x x|     |. x .|x x x|x x x|
|x x .|x x x|x x x|     |x x .|x x x|x x x|     |x x .|x x x|x x x|     |x x .|x x x|x x x|
+-----+-----+-----+     +-----+-----+-----+     +-----+-----+-----+     +-----+-----+-----+

         P5                      P6                      P7                      P8

+-----+-----+-----+     +-----+-----+-----+     +-----+-----+-----+     +-----+-----+-----+
|. . .|. . .|. . .|     |. . .|. . .|. . .|     |. . .|. . .|. . .|     |. . .|. . .|. . .|
|. . .|. . .|. . x|     |. . .|. . .|. . x|     |. . .|. . .|. . x|     |. . .|. . .|. . x|
|. . .|. . x|. . .|     |. . .|. . x|. . .|     |. . .|. . x|. . .|     |. . .|. . x|. . .|
+-----+-----+-----+     +-----+-----+-----+     +-----+-----+-----+     +-----+-----+-----+
|. . x|. . .|x x x|     |. . x|. . .|x x x|     |. . x|. . .|x x x|     |. . x|. . .|x x x|
|. . x|. . .|x x x|     |. . x|. . x|x x x|     |. . x|. . x|x x x|     |. . .|. . x|x x x|
|. . x|. x x|x x x|     |. . .|. x x|x x x|     |. . x|. x x|x x x|     |. . x|. x x|x x x|
+-----+-----+-----+     +-----+-----+-----+     +-----+-----+-----+     +-----+-----+-----+
|. x .|x x x|x x x|     |. x .|x x x|x x x|     |. . .|x x x|x x x|     |. x .|x x x|x x x|
|. x .|x x x|x x x|     |. x .|x x x|x x x|     |. x .|x x x|x x x|     |. . .|x x x|x x x|
|x x .|x x x|x x x|     |x x .|x x x|x x x|     |x x .|x x x|x x x|     |x x .|x x x|x x x|
+-----+-----+-----+     +-----+-----+-----+     +-----+-----+-----+     +-----+-----+-----+

Union of 2 valid patterns obviously gives valid pattern. One clue removal from this union can potentially gives invalid pattern. It's no sense to remove clues presented in A pattern, but not presented in B pattern and vice versa (resulting patterns will be always valid). Removing clues presented in both patterns gives chance to get invalid pattern.

This is outline of "birth" procedure, when 2 known maximal invalid anticorner patterns give birth to a valid anticorner pattern.
1. Intersect 2 maximal patterns clue sets to get starting pattern.
2. Form "addition" clue set by adding clues not presented both in the first and in the second patterns.
3. For all clues in "addition" clue set add in turn 1 next clue to starting pattern to get candidate pattern. Typically candidate pattern will be invalid, but sometimes it will be valid (or will have unknown status).

It's curious but usage of 2 the same patterns in "birth" procedure gives a reasonable result. 2 equal maximal invalid patterns give birth definitely valid pattern (adding 1 clue to maximal pattern) and 2 equal restricted minimal valid patterns give birth invalid pattern (removal 1 clue from minimal pattern).

Serg

[Edited. I changed term ""removal" clue set" to ""addition" clue set" in the description of "birth" procedure.]

[JPF]

In case JPF and/or Serg are interested in confirming it by other means, the shapes that are ED with respect to the reduced set of transformations, have this breakdown by (B123457) clue count:
------+-------------
432307140160

I confirm the total number and ...I need to get my program back that provides the breakdown by number of clues.

JPF

[JPF]

I finally managed to find my old programs.
If the question is still relevant, I confirm the numbers provided by Blue.
Some elements for their calculation according to the method explained here:
Number of permutations: 2 x 6^6 = 93312
Number of conjugacy classes : 405
Class number, size, cycle decomposition:

Code: Select all

1,1,()
2,4,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)
3,4,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,30,29)(34,36,35)(40,42,41)(46,48,47)(49,51,50)(52,54,53)
4,6,(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)
5,12,(1,3,2)(10,21,11,19,12,20)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(28,30,29)(34,36,35)(40,42,41)(46,48,47)(49,51,50)(52,54,53)
6,9,(2,3)(10,19)(11,21)(12,20)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(29,30)(35,36)(41,42)(47,48)(50,51)(53,54)
7,4,(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,44,38)(33,45,39)
8,8,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,44,38)(33,45,39)
9,8,(1,3,2)(10,12,11)(19,21,20)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,44,38)(33,45,39)(46,48,47)(49,51,50)(52,54,53)
10,16,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,44,38)(33,45,39)(46,48,47)(49,51,50)(52,54,53)
11,12,(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,44,38)(33,45,39)
12,24,(1,3,2)(10,21,11,19,12,20)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,44,38)(33,45,39)(46,48,47)(49,51,50)(52,54,53)
13,12,(2,3)(11,12)(20,21)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,44,38)(33,45,39)(47,48)(50,51)(53,54)
14,24,(1,19,10)(2,21,11,3,20,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,44,38)(33,45,39)(47,48)(50,51)(53,54)
15,36,(2,3)(10,19)(11,21)(12,20)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,44,38)(33,45,39)(47,48)(50,51)(53,54)
16,4,(4,6,5)(13,15,14)(22,24,23)(28,40,34)(29,41,35)(30,42,36)(31,45,38)(32,43,39)(33,44,37)
17,16,(1,19,10)(2,20,11)(3,21,12)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,26,17)(9,27,18)(28,40,34)(29,41,35)(30,42,36)(31,45,38)(32,43,39)(33,44,37)
18,16,(1,21,11)(2,19,12)(3,20,10)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,26,17)(9,27,18)(28,42,35)(29,40,36)(30,41,34)(31,45,38)(32,43,39)(33,44,37)(46,48,47)(49,51,50)(52,54,53)
19,24,(4,6,5)(10,19)(11,20)(12,21)(13,24,14,22,15,23)(16,25)(17,26)(18,27)(28,40,34)(29,41,35)(30,42,36)(31,45,38)(32,43,39)(33,44,37)
20,48,(1,3,2)(4,6,5)(10,21,11,19,12,20)(13,24,14,22,15,23)(16,25)(17,26)(18,27)(28,42,35)(29,40,36)(30,41,34)(31,45,38)(32,43,39)(33,44,37)(46,48,47)(49,51,50)(52,54,53)
21,36,(2,3)(4,6,5)(10,19)(11,21)(12,20)(13,24,14,22,15,23)(16,25)(17,26)(18,27)(28,40,34)(29,42,35,30,41,36)(31,45,38)(32,43,39)(33,44,37)(47,48)(50,51)(53,54)
22,6,(34,40)(35,41)(36,42)(37,43)(38,44)(39,45)
23,12,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(34,40)(35,41)(36,42)(37,43)(38,44)(39,45)
24,12,(1,3,2)(10,12,11)(19,21,20)(28,30,29)(34,42,35,40,36,41)(37,43)(38,44)(39,45)(46,48,47)(49,51,50)(52,54,53)
25,24,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,30,29)(34,42,35,40,36,41)(37,43)(38,44)(39,45)(46,48,47)(49,51,50)(52,54,53)
26,18,(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(34,40)(35,41)(36,42)(37,43)(38,44)(39,45)
27,36,(1,3,2)(10,21,11,19,12,20)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(28,30,29)(34,42,35,40,36,41)(37,43)(38,44)(39,45)(46,48,47)(49,51,50)(52,54,53)
28,18,(2,3)(11,12)(20,21)(29,30)(34,40)(35,42)(36,41)(37,43)(38,44)(39,45)(47,48)(50,51)(53,54)
29,36,(1,19,10)(2,21,11,3,20,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(29,30)(34,40)(35,42)(36,41)(37,43)(38,44)(39,45)(47,48)(50,51)(53,54)
30,54,(2,3)(10,19)(11,21)(12,20)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(29,30)(34,40)(35,42)(36,41)(37,43)(38,44)(39,45)(47,48)(50,51)(53,54)
31,12,(4,6,5)(13,15,14)(22,24,23)(31,33,32)(34,40)(35,41)(36,42)(37,45,38,43,39,44)
32,24,(1,19,10)(2,20,11)(3,21,12)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,26,17)(9,27,18)(31,33,32)(34,40)(35,41)(36,42)(37,45,38,43,39,44)
33,24,(1,3,2)(4,6,5)(10,12,11)(13,15,14)(19,21,20)(22,24,23)(28,30,29)(31,33,32)(34,42,35,40,36,41)(37,45,38,43,39,44)(46,48,47)(49,51,50)(52,54,53)
34,48,(1,21,11)(2,19,12)(3,20,10)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,26,17)(9,27,18)(28,30,29)(31,33,32)(34,42,35,40,36,41)(37,45,38,43,39,44)(46,48,47)(49,51,50)(52,54,53)
35,36,(4,6,5)(10,19)(11,20)(12,21)(13,24,14,22,15,23)(16,25)(17,26)(18,27)(31,33,32)(34,40)(35,41)(36,42)(37,45,38,43,39,44)
36,72,(1,3,2)(4,6,5)(10,21,11,19,12,20)(13,24,14,22,15,23)(16,25)(17,26)(18,27)(28,30,29)(31,33,32)(34,42,35,40,36,41)(37,45,38,43,39,44)(46,48,47)(49,51,50)(52,54,53)
37,36,(2,3)(4,6,5)(11,12)(13,15,14)(20,21)(22,24,23)(29,30)(31,33,32)(34,40)(35,42)(36,41)(37,45,38,43,39,44)(47,48)(50,51)(53,54)
38,72,(1,19,10)(2,21,11,3,20,12)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,26,17)(9,27,18)(29,30)(31,33,32)(34,40)(35,42)(36,41)(37,45,38,43,39,44)(47,48)(50,51)(53,54)
39,108,(2,3)(4,6,5)(10,19)(11,21)(12,20)(13,24,14,22,15,23)(16,25)(17,26)(18,27)(29,30)(31,33,32)(34,40)(35,42)(36,41)(37,45,38,43,39,44)(47,48)(50,51)(53,54)
40,9,(5,6)(14,15)(23,24)(32,33)(34,40)(35,41)(36,42)(37,43)(38,45)(39,44)
41,36,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,24,14,6,23,15)(7,25,16)(8,26,17)(9,27,18)(32,33)(34,40)(35,41)(36,42)(37,43)(38,45)(39,44)
42,36,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,24,14,6,23,15)(7,25,16)(8,26,17)(9,27,18)(28,30,29)(32,33)(34,42,35,40,36,41)(37,43)(38,45)(39,44)(46,48,47)(49,51,50)(52,54,53)
43,54,(5,6)(10,19)(11,20)(12,21)(13,22)(14,24)(15,23)(16,25)(17,26)(18,27)(32,33)(34,40)(35,41)(36,42)(37,43)(38,45)(39,44)
44,108,(1,3,2)(5,6)(10,21,11,19,12,20)(13,22)(14,24)(15,23)(16,25)(17,26)(18,27)(28,30,29)(32,33)(34,42,35,40,36,41)(37,43)(38,45)(39,44)(46,48,47)(49,51,50)(52,54,53)
45,81,(2,3)(5,6)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,25)(17,26)(18,27)(29,30)(32,33)(34,40)(35,42)(36,41)(37,43)(38,45)(39,44)(47,48)(50,51)(53,54)
46,4,(46,52,49)(47,53,50)(48,54,51)
47,8,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(46,52,49)(47,53,50)(48,54,51)
48,8,(1,3,2)(10,12,11)(19,21,20)(28,30,29)(34,36,35)(40,42,41)(46,54,50)(47,52,51)(48,53,49)
49,16,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,30,29)(34,36,35)(40,42,41)(46,54,50)(47,52,51)(48,53,49)
50,12,(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(46,52,49)(47,53,50)(48,54,51)
51,24,(1,3,2)(10,21,11,19,12,20)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(28,30,29)(34,36,35)(40,42,41)(46,54,50)(47,52,51)(48,53,49)
52,12,(2,3)(11,12)(20,21)(29,30)(35,36)(41,42)(46,52,49)(47,54,50,48,53,51)
53,24,(1,19,10)(2,21,11,3,20,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(29,30)(35,36)(41,42)(46,52,49)(47,54,50,48,53,51)
54,36,(2,3)(10,19)(11,21)(12,20)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(29,30)(35,36)(41,42)(46,52,49)(47,54,50,48,53,51)
55,8,(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,44,38)(33,45,39)(46,52,49)(47,53,50)(48,54,51)
56,16,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,44,38)(33,45,39)(46,52,49)(47,53,50)(48,54,51)
57,16,(1,3,2)(10,12,11)(19,21,20)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,44,38)(33,45,39)(46,54,50)(47,52,51)(48,53,49)
58,32,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,44,38)(33,45,39)(46,54,50)(47,52,51)(48,53,49)
59,24,(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,44,38)(33,45,39)(46,52,49)(47,53,50)(48,54,51)
60,48,(1,3,2)(10,21,11,19,12,20)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,44,38)(33,45,39)(46,54,50)(47,52,51)(48,53,49)
61,24,(2,3)(11,12)(20,21)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,44,38)(33,45,39)(46,52,49)(47,54,50,48,53,51)
62,48,(1,19,10)(2,21,11,3,20,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,44,38)(33,45,39)(46,52,49)(47,54,50,48,53,51)
63,72,(2,3)(10,19)(11,21)(12,20)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,44,38)(33,45,39)(46,52,49)(47,54,50,48,53,51)
64,8,(4,6,5)(13,15,14)(22,24,23)(31,33,32)(37,39,38)(43,45,44)(46,52,49)(47,53,50)(48,54,51)
65,16,(1,19,10)(2,20,11)(3,21,12)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,26,17)(9,27,18)(31,33,32)(37,39,38)(43,45,44)(46,52,49)(47,53,50)(48,54,51)
66,16,(1,3,2)(4,6,5)(10,12,11)(13,15,14)(19,21,20)(22,24,23)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,42,41)(43,45,44)(46,54,50)(47,52,51)(48,53,49)
67,32,(1,21,11)(2,19,12)(3,20,10)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,26,17)(9,27,18)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,42,41)(43,45,44)(46,54,50)(47,52,51)(48,53,49)
68,24,(4,6,5)(10,19)(11,20)(12,21)(13,24,14,22,15,23)(16,25)(17,26)(18,27)(31,33,32)(37,39,38)(43,45,44)(46,52,49)(47,53,50)(48,54,51)
69,48,(1,3,2)(4,6,5)(10,21,11,19,12,20)(13,24,14,22,15,23)(16,25)(17,26)(18,27)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,42,41)(43,45,44)(46,54,50)(47,52,51)(48,53,49)
70,24,(2,3)(4,6,5)(11,12)(13,15,14)(20,21)(22,24,23)(29,30)(31,33,32)(35,36)(37,39,38)(41,42)(43,45,44)(46,52,49)(47,54,50,48,53,51)
71,48,(1,19,10)(2,21,11,3,20,12)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,26,17)(9,27,18)(29,30)(31,33,32)(35,36)(37,39,38)(41,42)(43,45,44)(46,52,49)(47,54,50,48,53,51)
72,72,(2,3)(4,6,5)(10,19)(11,21)(12,20)(13,24,14,22,15,23)(16,25)(17,26)(18,27)(29,30)(31,33,32)(35,36)(37,39,38)(41,42)(43,45,44)(46,52,49)(47,54,50,48,53,51)
73,16,(4,6,5)(13,15,14)(22,24,23)(28,40,34)(29,41,35)(30,42,36)(31,45,38)(32,43,39)(33,44,37)(46,52,49)(47,53,50)(48,54,51)
74,32,(1,19,10)(2,20,11)(3,21,12)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,26,17)(9,27,18)(28,40,34)(29,41,35)(30,42,36)(31,45,38)(32,43,39)(33,44,37)(46,52,49)(47,53,50)(48,54,51)
75,32,(1,3,2)(4,6,5)(10,12,11)(13,15,14)(19,21,20)(22,24,23)(28,42,35)(29,40,36)(30,41,34)(31,45,38)(32,43,39)(33,44,37)(46,54,50)(47,52,51)(48,53,49)
76,64,(1,21,11)(2,19,12)(3,20,10)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,26,17)(9,27,18)(28,42,35)(29,40,36)(30,41,34)(31,45,38)(32,43,39)(33,44,37)(46,54,50)(47,52,51)(48,53,49)
77,48,(4,6,5)(10,19)(11,20)(12,21)(13,24,14,22,15,23)(16,25)(17,26)(18,27)(28,40,34)(29,41,35)(30,42,36)(31,45,38)(32,43,39)(33,44,37)(46,52,49)(47,53,50)(48,54,51)
78,96,(1,3,2)(4,6,5)(10,21,11,19,12,20)(13,24,14,22,15,23)(16,25)(17,26)(18,27)(28,42,35)(29,40,36)(30,41,34)(31,45,38)(32,43,39)(33,44,37)(46,54,50)(47,52,51)(48,53,49)
79,48,(2,3)(4,6,5)(11,12)(13,15,14)(20,21)(22,24,23)(28,40,34)(29,42,35,30,41,36)(31,45,38)(32,43,39)(33,44,37)(46,52,49)(47,54,50,48,53,51)
80,96,(1,19,10)(2,21,11,3,20,12)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,26,17)(9,27,18)(28,40,34)(29,42,35,30,41,36)(31,45,38)(32,43,39)(33,44,37)(46,52,49)(47,54,50,48,53,51)
81,144,(2,3)(4,6,5)(10,19)(11,21)(12,20)(13,24,14,22,15,23)(16,25)(17,26)(18,27)(28,40,34)(29,42,35,30,41,36)(31,45,38)(32,43,39)(33,44,37)(46,52,49)(47,54,50,48,53,51)
82,12,(34,40)(35,41)(36,42)(37,43)(38,44)(39,45)(46,52,49)(47,53,50)(48,54,51)
83,24,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(34,40)(35,41)(36,42)(37,43)(38,44)(39,45)(46,52,49)(47,53,50)(48,54,51)
84,24,(1,3,2)(10,12,11)(19,21,20)(28,30,29)(34,42,35,40,36,41)(37,43)(38,44)(39,45)(46,54,50)(47,52,51)(48,53,49)
85,48,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,30,29)(34,42,35,40,36,41)(37,43)(38,44)(39,45)(46,54,50)(47,52,51)(48,53,49)
86,36,(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(34,40)(35,41)(36,42)(37,43)(38,44)(39,45)(46,52,49)(47,53,50)(48,54,51)
87,72,(1,3,2)(10,21,11,19,12,20)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(28,30,29)(34,42,35,40,36,41)(37,43)(38,44)(39,45)(46,54,50)(47,52,51)(48,53,49)
88,36,(2,3)(11,12)(20,21)(29,30)(34,40)(35,42)(36,41)(37,43)(38,44)(39,45)(46,52,49)(47,54,50,48,53,51)
89,72,(1,19,10)(2,21,11,3,20,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(29,30)(34,40)(35,42)(36,41)(37,43)(38,44)(39,45)(46,52,49)(47,54,50,48,53,51)
90,108,(2,3)(10,19)(11,21)(12,20)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(29,30)(34,40)(35,42)(36,41)(37,43)(38,44)(39,45)(46,52,49)(47,54,50,48,53,51)
91,24,(4,6,5)(13,15,14)(22,24,23)(31,33,32)(34,40)(35,41)(36,42)(37,45,38,43,39,44)(46,52,49)(47,53,50)(48,54,51)
92,48,(1,19,10)(2,20,11)(3,21,12)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,26,17)(9,27,18)(31,33,32)(34,40)(35,41)(36,42)(37,45,38,43,39,44)(46,52,49)(47,53,50)(48,54,51)
93,48,(1,3,2)(4,6,5)(10,12,11)(13,15,14)(19,21,20)(22,24,23)(28,30,29)(31,33,32)(34,42,35,40,36,41)(37,45,38,43,39,44)(46,54,50)(47,52,51)(48,53,49)
94,96,(1,21,11)(2,19,12)(3,20,10)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,26,17)(9,27,18)(28,30,29)(31,33,32)(34,42,35,40,36,41)(37,45,38,43,39,44)(46,54,50)(47,52,51)(48,53,49)
95,72,(4,6,5)(10,19)(11,20)(12,21)(13,24,14,22,15,23)(16,25)(17,26)(18,27)(31,33,32)(34,40)(35,41)(36,42)(37,45,38,43,39,44)(46,52,49)(47,53,50)(48,54,51)
96,144,(1,3,2)(4,6,5)(10,21,11,19,12,20)(13,24,14,22,15,23)(16,25)(17,26)(18,27)(28,30,29)(31,33,32)(34,42,35,40,36,41)(37,45,38,43,39,44)(46,54,50)(47,52,51)(48,53,49)
97,72,(2,3)(4,6,5)(11,12)(13,15,14)(20,21)(22,24,23)(29,30)(31,33,32)(34,40)(35,42)(36,41)(37,45,38,43,39,44)(46,52,49)(47,54,50,48,53,51)
98,144,(1,19,10)(2,21,11,3,20,12)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,26,17)(9,27,18)(29,30)(31,33,32)(34,40)(35,42)(36,41)(37,45,38,43,39,44)(46,52,49)(47,54,50,48,53,51)
99,216,(2,3)(4,6,5)(10,19)(11,21)(12,20)(13,24,14,22,15,23)(16,25)(17,26)(18,27)(29,30)(31,33,32)(34,40)(35,42)(36,41)(37,45,38,43,39,44)(46,52,49)(47,54,50,48,53,51)
100,12,(5,6)(14,15)(23,24)(32,33)(38,39)(44,45)(46,52,49)(47,53,50)(48,54,51)
101,24,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,24,14,6,23,15)(7,25,16)(8,26,17)(9,27,18)(32,33)(38,39)(44,45)(46,52,49)(47,53,50)(48,54,51)
102,24,(1,3,2)(5,6)(10,12,11)(14,15)(19,21,20)(23,24)(28,30,29)(32,33)(34,36,35)(38,39)(40,42,41)(44,45)(46,54,50)(47,52,51)(48,53,49)
103,48,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,24,14,6,23,15)(7,25,16)(8,26,17)(9,27,18)(28,30,29)(32,33)(34,36,35)(38,39)(40,42,41)(44,45)(46,54,50)(47,52,51)(48,53,49)
104,36,(5,6)(10,19)(11,20)(12,21)(13,22)(14,24)(15,23)(16,25)(17,26)(18,27)(32,33)(38,39)(44,45)(46,52,49)(47,53,50)(48,54,51)
105,72,(1,3,2)(5,6)(10,21,11,19,12,20)(13,22)(14,24)(15,23)(16,25)(17,26)(18,27)(28,30,29)(32,33)(34,36,35)(38,39)(40,42,41)(44,45)(46,54,50)(47,52,51)(48,53,49)
106,36,(2,3)(5,6)(11,12)(14,15)(20,21)(23,24)(29,30)(32,33)(35,36)(38,39)(41,42)(44,45)(46,52,49)(47,54,50,48,53,51)
107,72,(1,19,10)(2,21,11,3,20,12)(4,22,13)(5,24,14,6,23,15)(7,25,16)(8,26,17)(9,27,18)(29,30)(32,33)(35,36)(38,39)(41,42)(44,45)(46,52,49)(47,54,50,48,53,51)
108,108,(2,3)(5,6)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,25)(17,26)(18,27)(29,30)(32,33)(35,36)(38,39)(41,42)(44,45)(46,52,49)(47,54,50,48,53,51)
109,24,(5,6)(14,15)(23,24)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,45,38,33,44,39)(46,52,49)(47,53,50)(48,54,51)
110,48,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,24,14,6,23,15)(7,25,16)(8,26,17)(9,27,18)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,45,38,33,44,39)(46,52,49)(47,53,50)(48,54,51)
111,48,(1,3,2)(5,6)(10,12,11)(14,15)(19,21,20)(23,24)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,45,38,33,44,39)(46,54,50)(47,52,51)(48,53,49)
112,96,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,24,14,6,23,15)(7,25,16)(8,26,17)(9,27,18)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,45,38,33,44,39)(46,54,50)(47,52,51)(48,53,49)
113,72,(5,6)(10,19)(11,20)(12,21)(13,22)(14,24)(15,23)(16,25)(17,26)(18,27)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,45,38,33,44,39)(46,52,49)(47,53,50)(48,54,51)
114,144,(1,3,2)(5,6)(10,21,11,19,12,20)(13,22)(14,24)(15,23)(16,25)(17,26)(18,27)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,45,38,33,44,39)(46,54,50)(47,52,51)(48,53,49)
115,72,(2,3)(5,6)(11,12)(14,15)(20,21)(23,24)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,45,38,33,44,39)(46,52,49)(47,54,50,48,53,51)
116,144,(1,19,10)(2,21,11,3,20,12)(4,22,13)(5,24,14,6,23,15)(7,25,16)(8,26,17)(9,27,18)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,45,38,33,44,39)(46,52,49)(47,54,50,48,53,51)
117,216,(2,3)(5,6)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,25)(17,26)(18,27)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,45,38,33,44,39)(46,52,49)(47,54,50,48,53,51)
118,36,(5,6)(14,15)(23,24)(32,33)(34,40)(35,41)(36,42)(37,43)(38,45)(39,44)(46,52,49)(47,53,50)(48,54,51)
119,72,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,24,14,6,23,15)(7,25,16)(8,26,17)(9,27,18)(32,33)(34,40)(35,41)(36,42)(37,43)(38,45)(39,44)(46,52,49)(47,53,50)(48,54,51)
120,72,(1,3,2)(5,6)(10,12,11)(14,15)(19,21,20)(23,24)(28,30,29)(32,33)(34,42,35,40,36,41)(37,43)(38,45)(39,44)(46,54,50)(47,52,51)(48,53,49)
121,144,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,24,14,6,23,15)(7,25,16)(8,26,17)(9,27,18)(28,30,29)(32,33)(34,42,35,40,36,41)(37,43)(38,45)(39,44)(46,54,50)(47,52,51)(48,53,49)
122,108,(5,6)(10,19)(11,20)(12,21)(13,22)(14,24)(15,23)(16,25)(17,26)(18,27)(32,33)(34,40)(35,41)(36,42)(37,43)(38,45)(39,44)(46,52,49)(47,53,50)(48,54,51)
123,216,(1,3,2)(5,6)(10,21,11,19,12,20)(13,22)(14,24)(15,23)(16,25)(17,26)(18,27)(28,30,29)(32,33)(34,42,35,40,36,41)(37,43)(38,45)(39,44)(46,54,50)(47,52,51)(48,53,49)
124,108,(2,3)(5,6)(11,12)(14,15)(20,21)(23,24)(29,30)(32,33)(34,40)(35,42)(36,41)(37,43)(38,45)(39,44)(46,52,49)(47,54,50,48,53,51)
125,216,(1,19,10)(2,21,11,3,20,12)(4,22,13)(5,24,14,6,23,15)(7,25,16)(8,26,17)(9,27,18)(29,30)(32,33)(34,40)(35,42)(36,41)(37,43)(38,45)(39,44)(46,52,49)(47,54,50,48,53,51)
126,324,(2,3)(5,6)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,25)(17,26)(18,27)(29,30)(32,33)(34,40)(35,42)(36,41)(37,43)(38,45)(39,44)(46,52,49)(47,54,50,48,53,51)
127,4,(7,9,8)(16,18,17)(25,27,26)(46,52,49)(47,53,50)(48,54,51)
128,16,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,27,17)(8,25,18)(9,26,16)(46,52,49)(47,53,50)(48,54,51)
129,16,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,23,14)(6,24,15)(7,27,17)(8,25,18)(9,26,16)(28,30,29)(34,36,35)(40,42,41)(46,54,50)(47,52,51)(48,53,49)
130,24,(7,9,8)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,27,17,25,18,26)(46,52,49)(47,53,50)(48,54,51)
131,48,(1,3,2)(7,9,8)(10,21,11,19,12,20)(13,22)(14,23)(15,24)(16,27,17,25,18,26)(28,30,29)(34,36,35)(40,42,41)(46,54,50)(47,52,51)(48,53,49)
132,36,(2,3)(7,9,8)(10,19)(11,21)(12,20)(13,22)(14,23)(15,24)(16,27,17,25,18,26)(29,30)(35,36)(41,42)(46,52,49)(47,54,50,48,53,51)
133,16,(7,9,8)(16,18,17)(25,27,26)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,44,38)(33,45,39)(46,52,49)(47,53,50)(48,54,51)
134,32,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,27,17)(8,25,18)(9,26,16)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,44,38)(33,45,39)(46,52,49)(47,53,50)(48,54,51)
135,32,(1,3,2)(7,9,8)(10,12,11)(16,18,17)(19,21,20)(25,27,26)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,44,38)(33,45,39)(46,54,50)(47,52,51)(48,53,49)
136,64,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,23,14)(6,24,15)(7,27,17)(8,25,18)(9,26,16)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,44,38)(33,45,39)(46,54,50)(47,52,51)(48,53,49)
137,48,(7,9,8)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,27,17,25,18,26)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,44,38)(33,45,39)(46,52,49)(47,53,50)(48,54,51)
138,96,(1,3,2)(7,9,8)(10,21,11,19,12,20)(13,22)(14,23)(15,24)(16,27,17,25,18,26)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,44,38)(33,45,39)(46,54,50)(47,52,51)(48,53,49)
139,48,(2,3)(7,9,8)(11,12)(16,18,17)(20,21)(25,27,26)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,44,38)(33,45,39)(46,52,49)(47,54,50,48,53,51)
140,96,(1,19,10)(2,21,11,3,20,12)(4,22,13)(5,23,14)(6,24,15)(7,27,17)(8,25,18)(9,26,16)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,44,38)(33,45,39)(46,52,49)(47,54,50,48,53,51)
141,144,(2,3)(7,9,8)(10,19)(11,21)(12,20)(13,22)(14,23)(15,24)(16,27,17,25,18,26)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,44,38)(33,45,39)(46,52,49)(47,54,50,48,53,51)
142,16,(4,6,5)(7,9,8)(13,15,14)(16,18,17)(22,24,23)(25,27,26)(28,40,34)(29,41,35)(30,42,36)(31,45,38)(32,43,39)(33,44,37)(46,52,49)(47,53,50)(48,54,51)
143,64,(1,19,10)(2,20,11)(3,21,12)(4,24,14)(5,22,15)(6,23,13)(7,27,17)(8,25,18)(9,26,16)(28,40,34)(29,41,35)(30,42,36)(31,45,38)(32,43,39)(33,44,37)(46,52,49)(47,53,50)(48,54,51)
144,64,(1,21,11)(2,19,12)(3,20,10)(4,24,14)(5,22,15)(6,23,13)(7,27,17)(8,25,18)(9,26,16)(28,42,35)(29,40,36)(30,41,34)(31,45,38)(32,43,39)(33,44,37)(46,54,50)(47,52,51)(48,53,49)
145,96,(4,6,5)(7,9,8)(10,19)(11,20)(12,21)(13,24,14,22,15,23)(16,27,17,25,18,26)(28,40,34)(29,41,35)(30,42,36)(31,45,38)(32,43,39)(33,44,37)(46,52,49)(47,53,50)(48,54,51)
146,192,(1,3,2)(4,6,5)(7,9,8)(10,21,11,19,12,20)(13,24,14,22,15,23)(16,27,17,25,18,26)(28,42,35)(29,40,36)(30,41,34)(31,45,38)(32,43,39)(33,44,37)(46,54,50)(47,52,51)(48,53,49)
147,144,(2,3)(4,6,5)(7,9,8)(10,19)(11,21)(12,20)(13,24,14,22,15,23)(16,27,17,25,18,26)(28,40,34)(29,42,35,30,41,36)(31,45,38)(32,43,39)(33,44,37)(46,52,49)(47,54,50,48,53,51)
148,24,(7,9,8)(16,18,17)(25,27,26)(34,40)(35,41)(36,42)(37,43)(38,44)(39,45)(46,52,49)(47,53,50)(48,54,51)
149,48,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,27,17)(8,25,18)(9,26,16)(34,40)(35,41)(36,42)(37,43)(38,44)(39,45)(46,52,49)(47,53,50)(48,54,51)
150,48,(1,3,2)(7,9,8)(10,12,11)(16,18,17)(19,21,20)(25,27,26)(28,30,29)(34,42,35,40,36,41)(37,43)(38,44)(39,45)(46,54,50)(47,52,51)(48,53,49)
151,96,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,23,14)(6,24,15)(7,27,17)(8,25,18)(9,26,16)(28,30,29)(34,42,35,40,36,41)(37,43)(38,44)(39,45)(46,54,50)(47,52,51)(48,53,49)
152,72,(7,9,8)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,27,17,25,18,26)(34,40)(35,41)(36,42)(37,43)(38,44)(39,45)(46,52,49)(47,53,50)(48,54,51)
153,144,(1,3,2)(7,9,8)(10,21,11,19,12,20)(13,22)(14,23)(15,24)(16,27,17,25,18,26)(28,30,29)(34,42,35,40,36,41)(37,43)(38,44)(39,45)(46,54,50)(47,52,51)(48,53,49)
154,72,(2,3)(7,9,8)(11,12)(16,18,17)(20,21)(25,27,26)(29,30)(34,40)(35,42)(36,41)(37,43)(38,44)(39,45)(46,52,49)(47,54,50,48,53,51)
155,144,(1,19,10)(2,21,11,3,20,12)(4,22,13)(5,23,14)(6,24,15)(7,27,17)(8,25,18)(9,26,16)(29,30)(34,40)(35,42)(36,41)(37,43)(38,44)(39,45)(46,52,49)(47,54,50,48,53,51)
156,216,(2,3)(7,9,8)(10,19)(11,21)(12,20)(13,22)(14,23)(15,24)(16,27,17,25,18,26)(29,30)(34,40)(35,42)(36,41)(37,43)(38,44)(39,45)(46,52,49)(47,54,50,48,53,51)
157,48,(4,6,5)(7,9,8)(13,15,14)(16,18,17)(22,24,23)(25,27,26)(31,33,32)(34,40)(35,41)(36,42)(37,45,38,43,39,44)(46,52,49)(47,53,50)(48,54,51)
158,96,(1,19,10)(2,20,11)(3,21,12)(4,24,14)(5,22,15)(6,23,13)(7,27,17)(8,25,18)(9,26,16)(31,33,32)(34,40)(35,41)(36,42)(37,45,38,43,39,44)(46,52,49)(47,53,50)(48,54,51)
159,96,(1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,30,29)(31,33,32)(34,42,35,40,36,41)(37,45,38,43,39,44)(46,54,50)(47,52,51)(48,53,49)
160,192,(1,21,11)(2,19,12)(3,20,10)(4,24,14)(5,22,15)(6,23,13)(7,27,17)(8,25,18)(9,26,16)(28,30,29)(31,33,32)(34,42,35,40,36,41)(37,45,38,43,39,44)(46,54,50)(47,52,51)(48,53,49)
161,144,(4,6,5)(7,9,8)(10,19)(11,20)(12,21)(13,24,14,22,15,23)(16,27,17,25,18,26)(31,33,32)(34,40)(35,41)(36,42)(37,45,38,43,39,44)(46,52,49)(47,53,50)(48,54,51)
162,288,(1,3,2)(4,6,5)(7,9,8)(10,21,11,19,12,20)(13,24,14,22,15,23)(16,27,17,25,18,26)(28,30,29)(31,33,32)(34,42,35,40,36,41)(37,45,38,43,39,44)(46,54,50)(47,52,51)(48,53,49)
163,144,(2,3)(4,6,5)(7,9,8)(11,12)(13,15,14)(16,18,17)(20,21)(22,24,23)(25,27,26)(29,30)(31,33,32)(34,40)(35,42)(36,41)(37,45,38,43,39,44)(46,52,49)(47,54,50,48,53,51)
164,288,(1,19,10)(2,21,11,3,20,12)(4,24,14)(5,22,15)(6,23,13)(7,27,17)(8,25,18)(9,26,16)(29,30)(31,33,32)(34,40)(35,42)(36,41)(37,45,38,43,39,44)(46,52,49)(47,54,50,48,53,51)
165,432,(2,3)(4,6,5)(7,9,8)(10,19)(11,21)(12,20)(13,24,14,22,15,23)(16,27,17,25,18,26)(29,30)(31,33,32)(34,40)(35,42)(36,41)(37,45,38,43,39,44)(46,52,49)(47,54,50,48,53,51)
166,36,(5,6)(7,9,8)(14,15)(16,18,17)(23,24)(25,27,26)(32,33)(34,40)(35,41)(36,42)(37,43)(38,45)(39,44)(46,52,49)(47,53,50)(48,54,51)
167,144,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,24,14,6,23,15)(7,27,17)(8,25,18)(9,26,16)(32,33)(34,40)(35,41)(36,42)(37,43)(38,45)(39,44)(46,52,49)(47,53,50)(48,54,51)
168,144,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,24,14,6,23,15)(7,27,17)(8,25,18)(9,26,16)(28,30,29)(32,33)(34,42,35,40,36,41)(37,43)(38,45)(39,44)(46,54,50)(47,52,51)(48,53,49)
169,216,(5,6)(7,9,8)(10,19)(11,20)(12,21)(13,22)(14,24)(15,23)(16,27,17,25,18,26)(32,33)(34,40)(35,41)(36,42)(37,43)(38,45)(39,44)(46,52,49)(47,53,50)(48,54,51)
170,432,(1,3,2)(5,6)(7,9,8)(10,21,11,19,12,20)(13,22)(14,24)(15,23)(16,27,17,25,18,26)(28,30,29)(32,33)(34,42,35,40,36,41)(37,43)(38,45)(39,44)(46,54,50)(47,52,51)(48,53,49)
171,324,(2,3)(5,6)(7,9,8)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,27,17,25,18,26)(29,30)(32,33)(34,40)(35,42)(36,41)(37,43)(38,45)(39,44)(46,52,49)(47,54,50,48,53,51)
172,6,(49,52)(50,53)(51,54)
173,12,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(49,52)(50,53)(51,54)
174,12,(1,3,2)(10,12,11)(19,21,20)(28,30,29)(34,36,35)(40,42,41)(46,48,47)(49,54,50,52,51,53)
175,24,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,30,29)(34,36,35)(40,42,41)(46,48,47)(49,54,50,52,51,53)
176,18,(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(49,52)(50,53)(51,54)
177,36,(1,3,2)(10,21,11,19,12,20)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(28,30,29)(34,36,35)(40,42,41)(46,48,47)(49,54,50,52,51,53)
178,18,(2,3)(11,12)(20,21)(29,30)(35,36)(41,42)(47,48)(49,52)(50,54)(51,53)
179,36,(1,19,10)(2,21,11,3,20,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(29,30)(35,36)(41,42)(47,48)(49,52)(50,54)(51,53)
180,54,(2,3)(10,19)(11,21)(12,20)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(29,30)(35,36)(41,42)(47,48)(49,52)(50,54)(51,53)
181,12,(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,44,38)(33,45,39)(49,52)(50,53)(51,54)
182,24,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,44,38)(33,45,39)(49,52)(50,53)(51,54)
183,24,(1,3,2)(10,12,11)(19,21,20)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,44,38)(33,45,39)(46,48,47)(49,54,50,52,51,53)
184,48,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,44,38)(33,45,39)(46,48,47)(49,54,50,52,51,53)
185,36,(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,44,38)(33,45,39)(49,52)(50,53)(51,54)
186,72,(1,3,2)(10,21,11,19,12,20)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,44,38)(33,45,39)(46,48,47)(49,54,50,52,51,53)
187,36,(2,3)(11,12)(20,21)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,44,38)(33,45,39)(47,48)(49,52)(50,54)(51,53)
188,72,(1,19,10)(2,21,11,3,20,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,44,38)(33,45,39)(47,48)(49,52)(50,54)(51,53)
189,108,(2,3)(10,19)(11,21)(12,20)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,44,38)(33,45,39)(47,48)(49,52)(50,54)(51,53)
190,12,(4,6,5)(13,15,14)(22,24,23)(31,33,32)(37,39,38)(43,45,44)(49,52)(50,53)(51,54)
191,24,(1,19,10)(2,20,11)(3,21,12)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,26,17)(9,27,18)(31,33,32)(37,39,38)(43,45,44)(49,52)(50,53)(51,54)
192,24,(1,3,2)(4,6,5)(10,12,11)(13,15,14)(19,21,20)(22,24,23)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,42,41)(43,45,44)(46,48,47)(49,54,50,52,51,53)
193,48,(1,21,11)(2,19,12)(3,20,10)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,26,17)(9,27,18)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,42,41)(43,45,44)(46,48,47)(49,54,50,52,51,53)
194,36,(4,6,5)(10,19)(11,20)(12,21)(13,24,14,22,15,23)(16,25)(17,26)(18,27)(31,33,32)(37,39,38)(43,45,44)(49,52)(50,53)(51,54)
195,72,(1,3,2)(4,6,5)(10,21,11,19,12,20)(13,24,14,22,15,23)(16,25)(17,26)(18,27)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,42,41)(43,45,44)(46,48,47)(49,54,50,52,51,53)
196,36,(2,3)(4,6,5)(11,12)(13,15,14)(20,21)(22,24,23)(29,30)(31,33,32)(35,36)(37,39,38)(41,42)(43,45,44)(47,48)(49,52)(50,54)(51,53)
197,72,(1,19,10)(2,21,11,3,20,12)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,26,17)(9,27,18)(29,30)(31,33,32)(35,36)(37,39,38)(41,42)(43,45,44)(47,48)(49,52)(50,54)(51,53)
198,108,(2,3)(4,6,5)(10,19)(11,21)(12,20)(13,24,14,22,15,23)(16,25)(17,26)(18,27)(29,30)(31,33,32)(35,36)(37,39,38)(41,42)(43,45,44)(47,48)(49,52)(50,54)(51,53)
199,24,(4,6,5)(13,15,14)(22,24,23)(28,40,34)(29,41,35)(30,42,36)(31,45,38)(32,43,39)(33,44,37)(49,52)(50,53)(51,54)
200,48,(1,19,10)(2,20,11)(3,21,12)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,26,17)(9,27,18)(28,40,34)(29,41,35)(30,42,36)(31,45,38)(32,43,39)(33,44,37)(49,52)(50,53)(51,54)
201,48,(1,3,2)(4,6,5)(10,12,11)(13,15,14)(19,21,20)(22,24,23)(28,42,35)(29,40,36)(30,41,34)(31,45,38)(32,43,39)(33,44,37)(46,48,47)(49,54,50,52,51,53)
202,96,(1,21,11)(2,19,12)(3,20,10)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,26,17)(9,27,18)(28,42,35)(29,40,36)(30,41,34)(31,45,38)(32,43,39)(33,44,37)(46,48,47)(49,54,50,52,51,53)
203,72,(4,6,5)(10,19)(11,20)(12,21)(13,24,14,22,15,23)(16,25)(17,26)(18,27)(28,40,34)(29,41,35)(30,42,36)(31,45,38)(32,43,39)(33,44,37)(49,52)(50,53)(51,54)
204,144,(1,3,2)(4,6,5)(10,21,11,19,12,20)(13,24,14,22,15,23)(16,25)(17,26)(18,27)(28,42,35)(29,40,36)(30,41,34)(31,45,38)(32,43,39)(33,44,37)(46,48,47)(49,54,50,52,51,53)
205,72,(2,3)(4,6,5)(11,12)(13,15,14)(20,21)(22,24,23)(28,40,34)(29,42,35,30,41,36)(31,45,38)(32,43,39)(33,44,37)(47,48)(49,52)(50,54)(51,53)
206,144,(1,19,10)(2,21,11,3,20,12)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,26,17)(9,27,18)(28,40,34)(29,42,35,30,41,36)(31,45,38)(32,43,39)(33,44,37)(47,48)(49,52)(50,54)(51,53)
207,216,(2,3)(4,6,5)(10,19)(11,21)(12,20)(13,24,14,22,15,23)(16,25)(17,26)(18,27)(28,40,34)(29,42,35,30,41,36)(31,45,38)(32,43,39)(33,44,37)(47,48)(49,52)(50,54)(51,53)
208,18,(34,40)(35,41)(36,42)(37,43)(38,44)(39,45)(49,52)(50,53)(51,54)
209,36,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(34,40)(35,41)(36,42)(37,43)(38,44)(39,45)(49,52)(50,53)(51,54)
210,36,(1,3,2)(10,12,11)(19,21,20)(28,30,29)(34,42,35,40,36,41)(37,43)(38,44)(39,45)(46,48,47)(49,54,50,52,51,53)
211,72,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,30,29)(34,42,35,40,36,41)(37,43)(38,44)(39,45)(46,48,47)(49,54,50,52,51,53)
212,54,(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(34,40)(35,41)(36,42)(37,43)(38,44)(39,45)(49,52)(50,53)(51,54)
213,108,(1,3,2)(10,21,11,19,12,20)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(28,30,29)(34,42,35,40,36,41)(37,43)(38,44)(39,45)(46,48,47)(49,54,50,52,51,53)
214,54,(2,3)(11,12)(20,21)(29,30)(34,40)(35,42)(36,41)(37,43)(38,44)(39,45)(47,48)(49,52)(50,54)(51,53)
215,108,(1,19,10)(2,21,11,3,20,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(29,30)(34,40)(35,42)(36,41)(37,43)(38,44)(39,45)(47,48)(49,52)(50,54)(51,53)
216,162,(2,3)(10,19)(11,21)(12,20)(13,22)(14,23)(15,24)(16,25)(17,26)(18,27)(29,30)(34,40)(35,42)(36,41)(37,43)(38,44)(39,45)(47,48)(49,52)(50,54)(51,53)
217,36,(4,6,5)(13,15,14)(22,24,23)(31,33,32)(34,40)(35,41)(36,42)(37,45,38,43,39,44)(49,52)(50,53)(51,54)
218,72,(1,19,10)(2,20,11)(3,21,12)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,26,17)(9,27,18)(31,33,32)(34,40)(35,41)(36,42)(37,45,38,43,39,44)(49,52)(50,53)(51,54)
219,72,(1,3,2)(4,6,5)(10,12,11)(13,15,14)(19,21,20)(22,24,23)(28,30,29)(31,33,32)(34,42,35,40,36,41)(37,45,38,43,39,44)(46,48,47)(49,54,50,52,51,53)
220,144,(1,21,11)(2,19,12)(3,20,10)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,26,17)(9,27,18)(28,30,29)(31,33,32)(34,42,35,40,36,41)(37,45,38,43,39,44)(46,48,47)(49,54,50,52,51,53)
221,108,(4,6,5)(10,19)(11,20)(12,21)(13,24,14,22,15,23)(16,25)(17,26)(18,27)(31,33,32)(34,40)(35,41)(36,42)(37,45,38,43,39,44)(49,52)(50,53)(51,54)
222,216,(1,3,2)(4,6,5)(10,21,11,19,12,20)(13,24,14,22,15,23)(16,25)(17,26)(18,27)(28,30,29)(31,33,32)(34,42,35,40,36,41)(37,45,38,43,39,44)(46,48,47)(49,54,50,52,51,53)
223,108,(2,3)(4,6,5)(11,12)(13,15,14)(20,21)(22,24,23)(29,30)(31,33,32)(34,40)(35,42)(36,41)(37,45,38,43,39,44)(47,48)(49,52)(50,54)(51,53)
224,216,(1,19,10)(2,21,11,3,20,12)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,26,17)(9,27,18)(29,30)(31,33,32)(34,40)(35,42)(36,41)(37,45,38,43,39,44)(47,48)(49,52)(50,54)(51,53)
225,324,(2,3)(4,6,5)(10,19)(11,21)(12,20)(13,24,14,22,15,23)(16,25)(17,26)(18,27)(29,30)(31,33,32)(34,40)(35,42)(36,41)(37,45,38,43,39,44)(47,48)(49,52)(50,54)(51,53)
226,18,(5,6)(14,15)(23,24)(32,33)(38,39)(44,45)(49,52)(50,53)(51,54)
227,36,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,24,14,6,23,15)(7,25,16)(8,26,17)(9,27,18)(32,33)(38,39)(44,45)(49,52)(50,53)(51,54)
228,36,(1,3,2)(5,6)(10,12,11)(14,15)(19,21,20)(23,24)(28,30,29)(32,33)(34,36,35)(38,39)(40,42,41)(44,45)(46,48,47)(49,54,50,52,51,53)
229,72,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,24,14,6,23,15)(7,25,16)(8,26,17)(9,27,18)(28,30,29)(32,33)(34,36,35)(38,39)(40,42,41)(44,45)(46,48,47)(49,54,50,52,51,53)
230,54,(5,6)(10,19)(11,20)(12,21)(13,22)(14,24)(15,23)(16,25)(17,26)(18,27)(32,33)(38,39)(44,45)(49,52)(50,53)(51,54)
231,108,(1,3,2)(5,6)(10,21,11,19,12,20)(13,22)(14,24)(15,23)(16,25)(17,26)(18,27)(28,30,29)(32,33)(34,36,35)(38,39)(40,42,41)(44,45)(46,48,47)(49,54,50,52,51,53)
232,54,(2,3)(5,6)(11,12)(14,15)(20,21)(23,24)(29,30)(32,33)(35,36)(38,39)(41,42)(44,45)(47,48)(49,52)(50,54)(51,53)
233,108,(1,19,10)(2,21,11,3,20,12)(4,22,13)(5,24,14,6,23,15)(7,25,16)(8,26,17)(9,27,18)(29,30)(32,33)(35,36)(38,39)(41,42)(44,45)(47,48)(49,52)(50,54)(51,53)
234,162,(2,3)(5,6)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,25)(17,26)(18,27)(29,30)(32,33)(35,36)(38,39)(41,42)(44,45)(47,48)(49,52)(50,54)(51,53)
235,36,(5,6)(14,15)(23,24)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,45,38,33,44,39)(49,52)(50,53)(51,54)
236,72,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,24,14,6,23,15)(7,25,16)(8,26,17)(9,27,18)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,45,38,33,44,39)(49,52)(50,53)(51,54)
237,72,(1,3,2)(5,6)(10,12,11)(14,15)(19,21,20)(23,24)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,45,38,33,44,39)(46,48,47)(49,54,50,52,51,53)
238,144,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,24,14,6,23,15)(7,25,16)(8,26,17)(9,27,18)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,45,38,33,44,39)(46,48,47)(49,54,50,52,51,53)
239,108,(5,6)(10,19)(11,20)(12,21)(13,22)(14,24)(15,23)(16,25)(17,26)(18,27)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,45,38,33,44,39)(49,52)(50,53)(51,54)
240,216,(1,3,2)(5,6)(10,21,11,19,12,20)(13,22)(14,24)(15,23)(16,25)(17,26)(18,27)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,45,38,33,44,39)(46,48,47)(49,54,50,52,51,53)
241,108,(2,3)(5,6)(11,12)(14,15)(20,21)(23,24)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,45,38,33,44,39)(47,48)(49,52)(50,54)(51,53)
242,216,(1,19,10)(2,21,11,3,20,12)(4,22,13)(5,24,14,6,23,15)(7,25,16)(8,26,17)(9,27,18)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,45,38,33,44,39)(47,48)(49,52)(50,54)(51,53)
243,324,(2,3)(5,6)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,25)(17,26)(18,27)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,45,38,33,44,39)(47,48)(49,52)(50,54)(51,53)
244,54,(5,6)(14,15)(23,24)(32,33)(34,40)(35,41)(36,42)(37,43)(38,45)(39,44)(49,52)(50,53)(51,54)
245,108,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,24,14,6,23,15)(7,25,16)(8,26,17)(9,27,18)(32,33)(34,40)(35,41)(36,42)(37,43)(38,45)(39,44)(49,52)(50,53)(51,54)
246,108,(1,3,2)(5,6)(10,12,11)(14,15)(19,21,20)(23,24)(28,30,29)(32,33)(34,42,35,40,36,41)(37,43)(38,45)(39,44)(46,48,47)(49,54,50,52,51,53)
247,216,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,24,14,6,23,15)(7,25,16)(8,26,17)(9,27,18)(28,30,29)(32,33)(34,42,35,40,36,41)(37,43)(38,45)(39,44)(46,48,47)(49,54,50,52,51,53)
248,162,(5,6)(10,19)(11,20)(12,21)(13,22)(14,24)(15,23)(16,25)(17,26)(18,27)(32,33)(34,40)(35,41)(36,42)(37,43)(38,45)(39,44)(49,52)(50,53)(51,54)
249,324,(1,3,2)(5,6)(10,21,11,19,12,20)(13,22)(14,24)(15,23)(16,25)(17,26)(18,27)(28,30,29)(32,33)(34,42,35,40,36,41)(37,43)(38,45)(39,44)(46,48,47)(49,54,50,52,51,53)
250,162,(2,3)(5,6)(11,12)(14,15)(20,21)(23,24)(29,30)(32,33)(34,40)(35,42)(36,41)(37,43)(38,45)(39,44)(47,48)(49,52)(50,54)(51,53)
251,324,(1,19,10)(2,21,11,3,20,12)(4,22,13)(5,24,14,6,23,15)(7,25,16)(8,26,17)(9,27,18)(29,30)(32,33)(34,40)(35,42)(36,41)(37,43)(38,45)(39,44)(47,48)(49,52)(50,54)(51,53)
252,486,(2,3)(5,6)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,25)(17,26)(18,27)(29,30)(32,33)(34,40)(35,42)(36,41)(37,43)(38,45)(39,44)(47,48)(49,52)(50,54)(51,53)
253,12,(7,9,8)(16,18,17)(25,27,26)(49,52)(50,53)(51,54)
254,24,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,27,17)(8,25,18)(9,26,16)(49,52)(50,53)(51,54)
255,24,(1,3,2)(7,9,8)(10,12,11)(16,18,17)(19,21,20)(25,27,26)(28,30,29)(34,36,35)(40,42,41)(46,48,47)(49,54,50,52,51,53)
256,48,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,23,14)(6,24,15)(7,27,17)(8,25,18)(9,26,16)(28,30,29)(34,36,35)(40,42,41)(46,48,47)(49,54,50,52,51,53)
257,36,(7,9,8)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,27,17,25,18,26)(49,52)(50,53)(51,54)
258,72,(1,3,2)(7,9,8)(10,21,11,19,12,20)(13,22)(14,23)(15,24)(16,27,17,25,18,26)(28,30,29)(34,36,35)(40,42,41)(46,48,47)(49,54,50,52,51,53)
259,36,(2,3)(7,9,8)(11,12)(16,18,17)(20,21)(25,27,26)(29,30)(35,36)(41,42)(47,48)(49,52)(50,54)(51,53)
260,72,(1,19,10)(2,21,11,3,20,12)(4,22,13)(5,23,14)(6,24,15)(7,27,17)(8,25,18)(9,26,16)(29,30)(35,36)(41,42)(47,48)(49,52)(50,54)(51,53)
261,108,(2,3)(7,9,8)(10,19)(11,21)(12,20)(13,22)(14,23)(15,24)(16,27,17,25,18,26)(29,30)(35,36)(41,42)(47,48)(49,52)(50,54)(51,53)
262,24,(7,9,8)(16,18,17)(25,27,26)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,44,38)(33,45,39)(49,52)(50,53)(51,54)
263,48,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,27,17)(8,25,18)(9,26,16)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,44,38)(33,45,39)(49,52)(50,53)(51,54)
264,48,(1,3,2)(7,9,8)(10,12,11)(16,18,17)(19,21,20)(25,27,26)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,44,38)(33,45,39)(46,48,47)(49,54,50,52,51,53)
265,96,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,23,14)(6,24,15)(7,27,17)(8,25,18)(9,26,16)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,44,38)(33,45,39)(46,48,47)(49,54,50,52,51,53)
266,72,(7,9,8)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,27,17,25,18,26)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,44,38)(33,45,39)(49,52)(50,53)(51,54)
267,144,(1,3,2)(7,9,8)(10,21,11,19,12,20)(13,22)(14,23)(15,24)(16,27,17,25,18,26)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,44,38)(33,45,39)(46,48,47)(49,54,50,52,51,53)
268,72,(2,3)(7,9,8)(11,12)(16,18,17)(20,21)(25,27,26)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,44,38)(33,45,39)(47,48)(49,52)(50,54)(51,53)
269,144,(1,19,10)(2,21,11,3,20,12)(4,22,13)(5,23,14)(6,24,15)(7,27,17)(8,25,18)(9,26,16)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,44,38)(33,45,39)(47,48)(49,52)(50,54)(51,53)
270,216,(2,3)(7,9,8)(10,19)(11,21)(12,20)(13,22)(14,23)(15,24)(16,27,17,25,18,26)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,44,38)(33,45,39)(47,48)(49,52)(50,54)(51,53)
271,24,(4,6,5)(7,9,8)(13,15,14)(16,18,17)(22,24,23)(25,27,26)(31,33,32)(37,39,38)(43,45,44)(49,52)(50,53)(51,54)
272,48,(1,19,10)(2,20,11)(3,21,12)(4,24,14)(5,22,15)(6,23,13)(7,27,17)(8,25,18)(9,26,16)(31,33,32)(37,39,38)(43,45,44)(49,52)(50,53)(51,54)
273,48,(1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,42,41)(43,45,44)(46,48,47)(49,54,50,52,51,53)
274,96,(1,21,11)(2,19,12)(3,20,10)(4,24,14)(5,22,15)(6,23,13)(7,27,17)(8,25,18)(9,26,16)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,42,41)(43,45,44)(46,48,47)(49,54,50,52,51,53)
275,72,(4,6,5)(7,9,8)(10,19)(11,20)(12,21)(13,24,14,22,15,23)(16,27,17,25,18,26)(31,33,32)(37,39,38)(43,45,44)(49,52)(50,53)(51,54)
276,144,(1,3,2)(4,6,5)(7,9,8)(10,21,11,19,12,20)(13,24,14,22,15,23)(16,27,17,25,18,26)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,42,41)(43,45,44)(46,48,47)(49,54,50,52,51,53)
277,72,(2,3)(4,6,5)(7,9,8)(11,12)(13,15,14)(16,18,17)(20,21)(22,24,23)(25,27,26)(29,30)(31,33,32)(35,36)(37,39,38)(41,42)(43,45,44)(47,48)(49,52)(50,54)(51,53)
278,144,(1,19,10)(2,21,11,3,20,12)(4,24,14)(5,22,15)(6,23,13)(7,27,17)(8,25,18)(9,26,16)(29,30)(31,33,32)(35,36)(37,39,38)(41,42)(43,45,44)(47,48)(49,52)(50,54)(51,53)
279,216,(2,3)(4,6,5)(7,9,8)(10,19)(11,21)(12,20)(13,24,14,22,15,23)(16,27,17,25,18,26)(29,30)(31,33,32)(35,36)(37,39,38)(41,42)(43,45,44)(47,48)(49,52)(50,54)(51,53)
280,48,(4,6,5)(7,9,8)(13,15,14)(16,18,17)(22,24,23)(25,27,26)(28,40,34)(29,41,35)(30,42,36)(31,45,38)(32,43,39)(33,44,37)(49,52)(50,53)(51,54)
281,96,(1,19,10)(2,20,11)(3,21,12)(4,24,14)(5,22,15)(6,23,13)(7,27,17)(8,25,18)(9,26,16)(28,40,34)(29,41,35)(30,42,36)(31,45,38)(32,43,39)(33,44,37)(49,52)(50,53)(51,54)
282,96,(1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,42,35)(29,40,36)(30,41,34)(31,45,38)(32,43,39)(33,44,37)(46,48,47)(49,54,50,52,51,53)
283,192,(1,21,11)(2,19,12)(3,20,10)(4,24,14)(5,22,15)(6,23,13)(7,27,17)(8,25,18)(9,26,16)(28,42,35)(29,40,36)(30,41,34)(31,45,38)(32,43,39)(33,44,37)(46,48,47)(49,54,50,52,51,53)
284,144,(4,6,5)(7,9,8)(10,19)(11,20)(12,21)(13,24,14,22,15,23)(16,27,17,25,18,26)(28,40,34)(29,41,35)(30,42,36)(31,45,38)(32,43,39)(33,44,37)(49,52)(50,53)(51,54)
285,288,(1,3,2)(4,6,5)(7,9,8)(10,21,11,19,12,20)(13,24,14,22,15,23)(16,27,17,25,18,26)(28,42,35)(29,40,36)(30,41,34)(31,45,38)(32,43,39)(33,44,37)(46,48,47)(49,54,50,52,51,53)
286,144,(2,3)(4,6,5)(7,9,8)(11,12)(13,15,14)(16,18,17)(20,21)(22,24,23)(25,27,26)(28,40,34)(29,42,35,30,41,36)(31,45,38)(32,43,39)(33,44,37)(47,48)(49,52)(50,54)(51,53)
287,288,(1,19,10)(2,21,11,3,20,12)(4,24,14)(5,22,15)(6,23,13)(7,27,17)(8,25,18)(9,26,16)(28,40,34)(29,42,35,30,41,36)(31,45,38)(32,43,39)(33,44,37)(47,48)(49,52)(50,54)(51,53)
288,432,(2,3)(4,6,5)(7,9,8)(10,19)(11,21)(12,20)(13,24,14,22,15,23)(16,27,17,25,18,26)(28,40,34)(29,42,35,30,41,36)(31,45,38)(32,43,39)(33,44,37)(47,48)(49,52)(50,54)(51,53)
289,36,(7,9,8)(16,18,17)(25,27,26)(34,40)(35,41)(36,42)(37,43)(38,44)(39,45)(49,52)(50,53)(51,54)
290,72,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,27,17)(8,25,18)(9,26,16)(34,40)(35,41)(36,42)(37,43)(38,44)(39,45)(49,52)(50,53)(51,54)
291,72,(1,3,2)(7,9,8)(10,12,11)(16,18,17)(19,21,20)(25,27,26)(28,30,29)(34,42,35,40,36,41)(37,43)(38,44)(39,45)(46,48,47)(49,54,50,52,51,53)
292,144,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,23,14)(6,24,15)(7,27,17)(8,25,18)(9,26,16)(28,30,29)(34,42,35,40,36,41)(37,43)(38,44)(39,45)(46,48,47)(49,54,50,52,51,53)
293,108,(7,9,8)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,27,17,25,18,26)(34,40)(35,41)(36,42)(37,43)(38,44)(39,45)(49,52)(50,53)(51,54)
294,216,(1,3,2)(7,9,8)(10,21,11,19,12,20)(13,22)(14,23)(15,24)(16,27,17,25,18,26)(28,30,29)(34,42,35,40,36,41)(37,43)(38,44)(39,45)(46,48,47)(49,54,50,52,51,53)
295,108,(2,3)(7,9,8)(11,12)(16,18,17)(20,21)(25,27,26)(29,30)(34,40)(35,42)(36,41)(37,43)(38,44)(39,45)(47,48)(49,52)(50,54)(51,53)
296,216,(1,19,10)(2,21,11,3,20,12)(4,22,13)(5,23,14)(6,24,15)(7,27,17)(8,25,18)(9,26,16)(29,30)(34,40)(35,42)(36,41)(37,43)(38,44)(39,45)(47,48)(49,52)(50,54)(51,53)
297,324,(2,3)(7,9,8)(10,19)(11,21)(12,20)(13,22)(14,23)(15,24)(16,27,17,25,18,26)(29,30)(34,40)(35,42)(36,41)(37,43)(38,44)(39,45)(47,48)(49,52)(50,54)(51,53)
298,72,(4,6,5)(7,9,8)(13,15,14)(16,18,17)(22,24,23)(25,27,26)(31,33,32)(34,40)(35,41)(36,42)(37,45,38,43,39,44)(49,52)(50,53)(51,54)
299,144,(1,19,10)(2,20,11)(3,21,12)(4,24,14)(5,22,15)(6,23,13)(7,27,17)(8,25,18)(9,26,16)(31,33,32)(34,40)(35,41)(36,42)(37,45,38,43,39,44)(49,52)(50,53)(51,54)
300,144,(1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,30,29)(31,33,32)(34,42,35,40,36,41)(37,45,38,43,39,44)(46,48,47)(49,54,50,52,51,53)
301,288,(1,21,11)(2,19,12)(3,20,10)(4,24,14)(5,22,15)(6,23,13)(7,27,17)(8,25,18)(9,26,16)(28,30,29)(31,33,32)(34,42,35,40,36,41)(37,45,38,43,39,44)(46,48,47)(49,54,50,52,51,53)
302,216,(4,6,5)(7,9,8)(10,19)(11,20)(12,21)(13,24,14,22,15,23)(16,27,17,25,18,26)(31,33,32)(34,40)(35,41)(36,42)(37,45,38,43,39,44)(49,52)(50,53)(51,54)
303,432,(1,3,2)(4,6,5)(7,9,8)(10,21,11,19,12,20)(13,24,14,22,15,23)(16,27,17,25,18,26)(28,30,29)(31,33,32)(34,42,35,40,36,41)(37,45,38,43,39,44)(46,48,47)(49,54,50,52,51,53)
304,216,(2,3)(4,6,5)(7,9,8)(11,12)(13,15,14)(16,18,17)(20,21)(22,24,23)(25,27,26)(29,30)(31,33,32)(34,40)(35,42)(36,41)(37,45,38,43,39,44)(47,48)(49,52)(50,54)(51,53)
305,432,(1,19,10)(2,21,11,3,20,12)(4,24,14)(5,22,15)(6,23,13)(7,27,17)(8,25,18)(9,26,16)(29,30)(31,33,32)(34,40)(35,42)(36,41)(37,45,38,43,39,44)(47,48)(49,52)(50,54)(51,53)
306,648,(2,3)(4,6,5)(7,9,8)(10,19)(11,21)(12,20)(13,24,14,22,15,23)(16,27,17,25,18,26)(29,30)(31,33,32)(34,40)(35,42)(36,41)(37,45,38,43,39,44)(47,48)(49,52)(50,54)(51,53)
307,36,(5,6)(7,9,8)(14,15)(16,18,17)(23,24)(25,27,26)(32,33)(38,39)(44,45)(49,52)(50,53)(51,54)
308,72,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,24,14,6,23,15)(7,27,17)(8,25,18)(9,26,16)(32,33)(38,39)(44,45)(49,52)(50,53)(51,54)
309,72,(1,3,2)(5,6)(7,9,8)(10,12,11)(14,15)(16,18,17)(19,21,20)(23,24)(25,27,26)(28,30,29)(32,33)(34,36,35)(38,39)(40,42,41)(44,45)(46,48,47)(49,54,50,52,51,53)
310,144,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,24,14,6,23,15)(7,27,17)(8,25,18)(9,26,16)(28,30,29)(32,33)(34,36,35)(38,39)(40,42,41)(44,45)(46,48,47)(49,54,50,52,51,53)
311,108,(5,6)(7,9,8)(10,19)(11,20)(12,21)(13,22)(14,24)(15,23)(16,27,17,25,18,26)(32,33)(38,39)(44,45)(49,52)(50,53)(51,54)
312,216,(1,3,2)(5,6)(7,9,8)(10,21,11,19,12,20)(13,22)(14,24)(15,23)(16,27,17,25,18,26)(28,30,29)(32,33)(34,36,35)(38,39)(40,42,41)(44,45)(46,48,47)(49,54,50,52,51,53)
313,108,(2,3)(5,6)(7,9,8)(11,12)(14,15)(16,18,17)(20,21)(23,24)(25,27,26)(29,30)(32,33)(35,36)(38,39)(41,42)(44,45)(47,48)(49,52)(50,54)(51,53)
314,216,(1,19,10)(2,21,11,3,20,12)(4,22,13)(5,24,14,6,23,15)(7,27,17)(8,25,18)(9,26,16)(29,30)(32,33)(35,36)(38,39)(41,42)(44,45)(47,48)(49,52)(50,54)(51,53)
315,324,(2,3)(5,6)(7,9,8)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,27,17,25,18,26)(29,30)(32,33)(35,36)(38,39)(41,42)(44,45)(47,48)(49,52)(50,54)(51,53)
316,72,(5,6)(7,9,8)(14,15)(16,18,17)(23,24)(25,27,26)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,45,38,33,44,39)(49,52)(50,53)(51,54)
317,144,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,24,14,6,23,15)(7,27,17)(8,25,18)(9,26,16)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,45,38,33,44,39)(49,52)(50,53)(51,54)
318,144,(1,3,2)(5,6)(7,9,8)(10,12,11)(14,15)(16,18,17)(19,21,20)(23,24)(25,27,26)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,45,38,33,44,39)(46,48,47)(49,54,50,52,51,53)
319,288,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,24,14,6,23,15)(7,27,17)(8,25,18)(9,26,16)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,45,38,33,44,39)(46,48,47)(49,54,50,52,51,53)
320,216,(5,6)(7,9,8)(10,19)(11,20)(12,21)(13,22)(14,24)(15,23)(16,27,17,25,18,26)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,45,38,33,44,39)(49,52)(50,53)(51,54)
321,432,(1,3,2)(5,6)(7,9,8)(10,21,11,19,12,20)(13,22)(14,24)(15,23)(16,27,17,25,18,26)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,45,38,33,44,39)(46,48,47)(49,54,50,52,51,53)
322,216,(2,3)(5,6)(7,9,8)(11,12)(14,15)(16,18,17)(20,21)(23,24)(25,27,26)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,45,38,33,44,39)(47,48)(49,52)(50,54)(51,53)
323,432,(1,19,10)(2,21,11,3,20,12)(4,22,13)(5,24,14,6,23,15)(7,27,17)(8,25,18)(9,26,16)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,45,38,33,44,39)(47,48)(49,52)(50,54)(51,53)
324,648,(2,3)(5,6)(7,9,8)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,27,17,25,18,26)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,45,38,33,44,39)(47,48)(49,52)(50,54)(51,53)
325,108,(5,6)(7,9,8)(14,15)(16,18,17)(23,24)(25,27,26)(32,33)(34,40)(35,41)(36,42)(37,43)(38,45)(39,44)(49,52)(50,53)(51,54)
326,216,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,24,14,6,23,15)(7,27,17)(8,25,18)(9,26,16)(32,33)(34,40)(35,41)(36,42)(37,43)(38,45)(39,44)(49,52)(50,53)(51,54)
327,216,(1,3,2)(5,6)(7,9,8)(10,12,11)(14,15)(16,18,17)(19,21,20)(23,24)(25,27,26)(28,30,29)(32,33)(34,42,35,40,36,41)(37,43)(38,45)(39,44)(46,48,47)(49,54,50,52,51,53)
328,432,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,24,14,6,23,15)(7,27,17)(8,25,18)(9,26,16)(28,30,29)(32,33)(34,42,35,40,36,41)(37,43)(38,45)(39,44)(46,48,47)(49,54,50,52,51,53)
329,324,(5,6)(7,9,8)(10,19)(11,20)(12,21)(13,22)(14,24)(15,23)(16,27,17,25,18,26)(32,33)(34,40)(35,41)(36,42)(37,43)(38,45)(39,44)(49,52)(50,53)(51,54)
330,648,(1,3,2)(5,6)(7,9,8)(10,21,11,19,12,20)(13,22)(14,24)(15,23)(16,27,17,25,18,26)(28,30,29)(32,33)(34,42,35,40,36,41)(37,43)(38,45)(39,44)(46,48,47)(49,54,50,52,51,53)
331,324,(2,3)(5,6)(7,9,8)(11,12)(14,15)(16,18,17)(20,21)(23,24)(25,27,26)(29,30)(32,33)(34,40)(35,42)(36,41)(37,43)(38,45)(39,44)(47,48)(49,52)(50,54)(51,53)
332,648,(1,19,10)(2,21,11,3,20,12)(4,22,13)(5,24,14,6,23,15)(7,27,17)(8,25,18)(9,26,16)(29,30)(32,33)(34,40)(35,42)(36,41)(37,43)(38,45)(39,44)(47,48)(49,52)(50,54)(51,53)
333,972,(2,3)(5,6)(7,9,8)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,27,17,25,18,26)(29,30)(32,33)(34,40)(35,42)(36,41)(37,43)(38,45)(39,44)(47,48)(49,52)(50,54)(51,53)
334,9,(8,9)(17,18)(26,27)(49,52)(50,53)(51,54)
335,36,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,27,17,9,26,18)(49,52)(50,53)(51,54)
336,36,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,27,17,9,26,18)(28,30,29)(34,36,35)(40,42,41)(46,48,47)(49,54,50,52,51,53)
337,54,(8,9)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,27)(18,26)(49,52)(50,53)(51,54)
338,108,(1,3,2)(8,9)(10,21,11,19,12,20)(13,22)(14,23)(15,24)(16,25)(17,27)(18,26)(28,30,29)(34,36,35)(40,42,41)(46,48,47)(49,54,50,52,51,53)
339,81,(2,3)(8,9)(10,19)(11,21)(12,20)(13,22)(14,23)(15,24)(16,25)(17,27)(18,26)(29,30)(35,36)(41,42)(47,48)(49,52)(50,54)(51,53)
340,36,(8,9)(17,18)(26,27)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,44,38)(33,45,39)(49,52)(50,53)(51,54)
341,72,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,27,17,9,26,18)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,44,38)(33,45,39)(49,52)(50,53)(51,54)
342,72,(1,3,2)(8,9)(10,12,11)(17,18)(19,21,20)(26,27)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,44,38)(33,45,39)(46,48,47)(49,54,50,52,51,53)
343,144,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,27,17,9,26,18)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,44,38)(33,45,39)(46,48,47)(49,54,50,52,51,53)
344,108,(8,9)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,27)(18,26)(28,40,34)(29,41,35)(30,42,36)(31,43,37)(32,44,38)(33,45,39)(49,52)(50,53)(51,54)
345,216,(1,3,2)(8,9)(10,21,11,19,12,20)(13,22)(14,23)(15,24)(16,25)(17,27)(18,26)(28,42,35)(29,40,36)(30,41,34)(31,43,37)(32,44,38)(33,45,39)(46,48,47)(49,54,50,52,51,53)
346,108,(2,3)(8,9)(11,12)(17,18)(20,21)(26,27)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,44,38)(33,45,39)(47,48)(49,52)(50,54)(51,53)
347,216,(1,19,10)(2,21,11,3,20,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,27,17,9,26,18)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,44,38)(33,45,39)(47,48)(49,52)(50,54)(51,53)
348,324,(2,3)(8,9)(10,19)(11,21)(12,20)(13,22)(14,23)(15,24)(16,25)(17,27)(18,26)(28,40,34)(29,42,35,30,41,36)(31,43,37)(32,44,38)(33,45,39)(47,48)(49,52)(50,54)(51,53)
349,36,(4,6,5)(8,9)(13,15,14)(17,18)(22,24,23)(26,27)(28,40,34)(29,41,35)(30,42,36)(31,45,38)(32,43,39)(33,44,37)(49,52)(50,53)(51,54)
350,144,(1,19,10)(2,20,11)(3,21,12)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,27,17,9,26,18)(28,40,34)(29,41,35)(30,42,36)(31,45,38)(32,43,39)(33,44,37)(49,52)(50,53)(51,54)
351,144,(1,21,11)(2,19,12)(3,20,10)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,27,17,9,26,18)(28,42,35)(29,40,36)(30,41,34)(31,45,38)(32,43,39)(33,44,37)(46,48,47)(49,54,50,52,51,53)
352,216,(4,6,5)(8,9)(10,19)(11,20)(12,21)(13,24,14,22,15,23)(16,25)(17,27)(18,26)(28,40,34)(29,41,35)(30,42,36)(31,45,38)(32,43,39)(33,44,37)(49,52)(50,53)(51,54)
353,432,(1,3,2)(4,6,5)(8,9)(10,21,11,19,12,20)(13,24,14,22,15,23)(16,25)(17,27)(18,26)(28,42,35)(29,40,36)(30,41,34)(31,45,38)(32,43,39)(33,44,37)(46,48,47)(49,54,50,52,51,53)
354,324,(2,3)(4,6,5)(8,9)(10,19)(11,21)(12,20)(13,24,14,22,15,23)(16,25)(17,27)(18,26)(28,40,34)(29,42,35,30,41,36)(31,45,38)(32,43,39)(33,44,37)(47,48)(49,52)(50,54)(51,53)
355,54,(8,9)(17,18)(26,27)(34,40)(35,41)(36,42)(37,43)(38,44)(39,45)(49,52)(50,53)(51,54)
356,108,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,27,17,9,26,18)(34,40)(35,41)(36,42)(37,43)(38,44)(39,45)(49,52)(50,53)(51,54)
357,108,(1,3,2)(8,9)(10,12,11)(17,18)(19,21,20)(26,27)(28,30,29)(34,42,35,40,36,41)(37,43)(38,44)(39,45)(46,48,47)(49,54,50,52,51,53)
358,216,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,27,17,9,26,18)(28,30,29)(34,42,35,40,36,41)(37,43)(38,44)(39,45)(46,48,47)(49,54,50,52,51,53)
359,162,(8,9)(10,19)(11,20)(12,21)(13,22)(14,23)(15,24)(16,25)(17,27)(18,26)(34,40)(35,41)(36,42)(37,43)(38,44)(39,45)(49,52)(50,53)(51,54)
360,324,(1,3,2)(8,9)(10,21,11,19,12,20)(13,22)(14,23)(15,24)(16,25)(17,27)(18,26)(28,30,29)(34,42,35,40,36,41)(37,43)(38,44)(39,45)(46,48,47)(49,54,50,52,51,53)
361,162,(2,3)(8,9)(11,12)(17,18)(20,21)(26,27)(29,30)(34,40)(35,42)(36,41)(37,43)(38,44)(39,45)(47,48)(49,52)(50,54)(51,53)
362,324,(1,19,10)(2,21,11,3,20,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,27,17,9,26,18)(29,30)(34,40)(35,42)(36,41)(37,43)(38,44)(39,45)(47,48)(49,52)(50,54)(51,53)
363,486,(2,3)(8,9)(10,19)(11,21)(12,20)(13,22)(14,23)(15,24)(16,25)(17,27)(18,26)(29,30)(34,40)(35,42)(36,41)(37,43)(38,44)(39,45)(47,48)(49,52)(50,54)(51,53)
364,108,(4,6,5)(8,9)(13,15,14)(17,18)(22,24,23)(26,27)(31,33,32)(34,40)(35,41)(36,42)(37,45,38,43,39,44)(49,52)(50,53)(51,54)
365,216,(1,19,10)(2,20,11)(3,21,12)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,27,17,9,26,18)(31,33,32)(34,40)(35,41)(36,42)(37,45,38,43,39,44)(49,52)(50,53)(51,54)
366,216,(1,3,2)(4,6,5)(8,9)(10,12,11)(13,15,14)(17,18)(19,21,20)(22,24,23)(26,27)(28,30,29)(31,33,32)(34,42,35,40,36,41)(37,45,38,43,39,44)(46,48,47)(49,54,50,52,51,53)
367,432,(1,21,11)(2,19,12)(3,20,10)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,27,17,9,26,18)(28,30,29)(31,33,32)(34,42,35,40,36,41)(37,45,38,43,39,44)(46,48,47)(49,54,50,52,51,53)
368,324,(4,6,5)(8,9)(10,19)(11,20)(12,21)(13,24,14,22,15,23)(16,25)(17,27)(18,26)(31,33,32)(34,40)(35,41)(36,42)(37,45,38,43,39,44)(49,52)(50,53)(51,54)
369,648,(1,3,2)(4,6,5)(8,9)(10,21,11,19,12,20)(13,24,14,22,15,23)(16,25)(17,27)(18,26)(28,30,29)(31,33,32)(34,42,35,40,36,41)(37,45,38,43,39,44)(46,48,47)(49,54,50,52,51,53)
370,324,(2,3)(4,6,5)(8,9)(11,12)(13,15,14)(17,18)(20,21)(22,24,23)(26,27)(29,30)(31,33,32)(34,40)(35,42)(36,41)(37,45,38,43,39,44)(47,48)(49,52)(50,54)(51,53)
371,648,(1,19,10)(2,21,11,3,20,12)(4,24,14)(5,22,15)(6,23,13)(7,25,16)(8,27,17,9,26,18)(29,30)(31,33,32)(34,40)(35,42)(36,41)(37,45,38,43,39,44)(47,48)(49,52)(50,54)(51,53)
372,972,(2,3)(4,6,5)(8,9)(10,19)(11,21)(12,20)(13,24,14,22,15,23)(16,25)(17,27)(18,26)(29,30)(31,33,32)(34,40)(35,42)(36,41)(37,45,38,43,39,44)(47,48)(49,52)(50,54)(51,53)
373,81,(5,6)(8,9)(14,15)(17,18)(23,24)(26,27)(32,33)(34,40)(35,41)(36,42)(37,43)(38,45)(39,44)(49,52)(50,53)(51,54)
374,324,(1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,24,14,6,23,15)(7,25,16)(8,27,17,9,26,18)(32,33)(34,40)(35,41)(36,42)(37,43)(38,45)(39,44)(49,52)(50,53)(51,54)
375,324,(1,21,11)(2,19,12)(3,20,10)(4,22,13)(5,24,14,6,23,15)(7,25,16)(8,27,17,9,26,18)(28,30,29)(32,33)(34,42,35,40,36,41)(37,43)(38,45)(39,44)(46,48,47)(49,54,50,52,51,53)
376,486,(5,6)(8,9)(10,19)(11,20)(12,21)(13,22)(14,24)(15,23)(16,25)(17,27)(18,26)(32,33)(34,40)(35,41)(36,42)(37,43)(38,45)(39,44)(49,52)(50,53)(51,54)
377,972,(1,3,2)(5,6)(8,9)(10,21,11,19,12,20)(13,22)(14,24)(15,23)(16,25)(17,27)(18,26)(28,30,29)(32,33)(34,42,35,40,36,41)(37,43)(38,45)(39,44)(46,48,47)(49,54,50,52,51,53)
378,729,(2,3)(5,6)(8,9)(10,19)(11,21)(12,20)(13,22)(14,24)(15,23)(16,25)(17,27)(18,26)(29,30)(32,33)(34,40)(35,42)(36,41)(37,43)(38,45)(39,44)(47,48)(49,52)(50,54)(51,53)
379,216,(2,10)(3,19)(4,28)(5,34)(6,40)(7,46)(8,49)(9,52)(12,20)(13,29)(14,35)(15,41)(16,47)(17,50)(18,53)(22,30)(23,36)(24,42)(25,48)(26,51)(27,54)(32,37)(33,43)(39,44)
380,432,(1,19,21,12,11,2)(3,10,20)(4,28,22,30,13,29)(5,34,23,36,14,35)(6,40,24,42,15,41)(7,46,25,48,16,47)(8,49,26,51,17,50)(9,52,27,54,18,53)(32,37)(33,43)(39,44)
381,648,(2,19,3,10)(4,28)(5,34)(6,40)(7,46)(8,49)(9,52)(11,20,21,12)(13,29,22,30)(14,35,23,36)(15,41,24,42)(16,47,25,48)(17,50,26,51)(18,53,27,54)(32,37)(33,43)(39,44)
382,432,(2,10)(3,19)(4,40,6,34,5,28)(7,46)(8,49)(9,52)(12,20)(13,41,15,35,14,29)(16,47)(17,50)(18,53)(22,42,24,36,23,30)(25,48)(26,51)(27,54)(31,43,45,39,38,32)(33,37,44)
383,864,(1,19,21,12,11,2)(3,10,20)(4,40,24,36,14,29)(5,28,22,42,15,35)(6,34,23,30,13,41)(7,46,25,48,16,47)(8,49,26,51,17,50)(9,52,27,54,18,53)(31,43,45,39,38,32)(33,37,44)
384,1296,(2,19,3,10)(4,40,6,34,5,28)(7,46)(8,49)(9,52)(11,20,21,12)(13,41,24,36,14,29,22,42,15,35,23,30)(16,47,25,48)(17,50,26,51)(18,53,27,54)(31,43,45,39,38,32)(33,37,44)
385,648,(2,10)(3,19)(4,28)(5,40,6,34)(7,46)(8,49)(9,52)(12,20)(13,29)(14,41,15,35)(16,47)(17,50)(18,53)(22,30)(23,42,24,36)(25,48)(26,51)(27,54)(32,43,33,37)(38,44,45,39)
386,1296,(1,19,21,12,11,2)(3,10,20)(4,28,22,30,13,29)(5,40,24,36,14,41,6,34,23,42,15,35)(7,46,25,48,16,47)(8,49,26,51,17,50)(9,52,27,54,18,53)(32,43,33,37)(38,44,45,39)
387,1944,(2,19,3,10)(4,28)(5,40,6,34)(7,46)(8,49)(9,52)(11,20,21,12)(13,29,22,30)(14,41,24,36)(15,35,23,42)(16,47,25,48)(17,50,26,51)(18,53,27,54)(32,43,33,37)(38,44,45,39)
388,432,(2,10)(3,19)(4,28)(5,34)(6,40)(7,52,9,49,8,46)(12,20)(13,29)(14,35)(15,41)(16,53,18,50,17,47)(22,30)(23,36)(24,42)(25,54,27,51,26,48)(32,37)(33,43)(39,44)
389,864,(1,19,21,12,11,2)(3,10,20)(4,28,22,30,13,29)(5,34,23,36,14,35)(6,40,24,42,15,41)(7,52,27,51,17,47)(8,46,25,54,18,50)(9,49,26,48,16,53)(32,37)(33,43)(39,44)
390,1296,(2,19,3,10)(4,28)(5,34)(6,40)(7,52,9,49,8,46)(11,20,21,12)(13,29,22,30)(14,35,23,36)(15,41,24,42)(16,53,27,51,17,47,25,54,18,50,26,48)(32,37)(33,43)(39,44)
391,864,(2,10)(3,19)(4,40,6,34,5,28)(7,52,9,49,8,46)(12,20)(13,41,15,35,14,29)(16,53,18,50,17,47)(22,42,24,36,23,30)(25,54,27,51,26,48)(31,43,45,39,38,32)(33,37,44)
392,1728,(1,19,21,12,11,2)(3,10,20)(4,40,24,36,14,29)(5,28,22,42,15,35)(6,34,23,30,13,41)(7,52,27,51,17,47)(8,46,25,54,18,50)(9,49,26,48,16,53)(31,43,45,39,38,32)(33,37,44)
393,2592,(2,19,3,10)(4,40,6,34,5,28)(7,52,9,49,8,46)(11,20,21,12)(13,41,24,36,14,29,22,42,15,35,23,30)(16,53,27,51,17,47,25,54,18,50,26,48)(31,43,45,39,38,32)(33,37,44)
394,1296,(2,10)(3,19)(4,28)(5,40,6,34)(7,52,9,49,8,46)(12,20)(13,29)(14,41,15,35)(16,53,18,50,17,47)(22,30)(23,42,24,36)(25,54,27,51,26,48)(32,43,33,37)(38,44,45,39)
395,2592,(1,19,21,12,11,2)(3,10,20)(4,28,22,30,13,29)(5,40,24,36,14,41,6,34,23,42,15,35)(7,52,27,51,17,47)(8,46,25,54,18,50)(9,49,26,48,16,53)(32,43,33,37)(38,44,45,39)
396,3888,(2,19,3,10)(4,28)(5,40,6,34)(7,52,9,49,8,46)(11,20,21,12)(13,29,22,30)(14,41,24,36)(15,35,23,42)(16,53,27,51,17,47,25,54,18,50,26,48)(32,43,33,37)(38,44,45,39)
397,648,(2,10)(3,19)(4,28)(5,34)(6,40)(7,46)(8,52,9,49)(12,20)(13,29)(14,35)(15,41)(16,47)(17,53,18,50)(22,30)(23,36)(24,42)(25,48)(26,54,27,51)(32,37)(33,43)(39,44)
398,1296,(1,19,21,12,11,2)(3,10,20)(4,28,22,30,13,29)(5,34,23,36,14,35)(6,40,24,42,15,41)(7,46,25,48,16,47)(8,52,27,51,17,53,9,49,26,54,18,50)(32,37)(33,43)(39,44)
399,1944,(2,19,3,10)(4,28)(5,34)(6,40)(7,46)(8,52,9,49)(11,20,21,12)(13,29,22,30)(14,35,23,36)(15,41,24,42)(16,47,25,48)(17,53,27,51)(18,50,26,54)(32,37)(33,43)(39,44)
400,1296,(2,10)(3,19)(4,40,6,34,5,28)(7,46)(8,52,9,49)(12,20)(13,41,15,35,14,29)(16,47)(17,53,18,50)(22,42,24,36,23,30)(25,48)(26,54,27,51)(31,43,45,39,38,32)(33,37,44)
401,2592,(1,19,21,12,11,2)(3,10,20)(4,40,24,36,14,29)(5,28,22,42,15,35)(6,34,23,30,13,41)(7,46,25,48,16,47)(8,52,27,51,17,53,9,49,26,54,18,50)(31,43,45,39,38,32)(33,37,44)
402,3888,(2,19,3,10)(4,40,6,34,5,28)(7,46)(8,52,9,49)(11,20,21,12)(13,41,24,36,14,29,22,42,15,35,23,30)(16,47,25,48)(17,53,27,51)(18,50,26,54)(31,43,45,39,38,32)(33,37,44)
403,1944,(2,10)(3,19)(4,28)(5,40,6,34)(7,46)(8,52,9,49)(12,20)(13,29)(14,41,15,35)(16,47)(17,53,18,50)(22,30)(23,42,24,36)(25,48)(26,54,27,51)(32,43,33,37)(38,44,45,39)
404,3888,(1,19,21,12,11,2)(3,10,20)(4,28,22,30,13,29)(5,40,24,36,14,41,6,34,23,42,15,35)(7,46,25,48,16,47)(8,52,27,51,17,53,9,49,26,54,18,50)(32,43,33,37)(38,44,45,39)
405,5832,(2,19,3,10)(4,28)(5,40,6,34)(7,46)(8,52,9,49)(11,20,21,12)(13,29,22,30)(14,41,24,36)(15,35,23,42)(16,47,25,48)(17,53,27,51)(18,50,26,54)(32,43,33,37)(38,44,45,39)

Possible cycle lengths = list of i such that m_i is non-zero: 1,2,3,4,6,12

To Blue: Which method did you use?

JPF

[blue]

I finally managed to find my old programs.
If the question is still relevant, I confirm the numbers provided by Blue.
Some elements for their calculation according to the method explained here:
Number of permutations: 2 x 6^6 = 93312
Number of conjugacy classes : 405
Class number, size, cycle decomposition:
(...)
Possible cycle lengths = list of i such that m_i is non-zero: 1,2,3,4,6,12

To Blue: Which method did you use?

Hi JPF,

Thank you very much, and special thanks for the link to the explanation.
The question was still relevant for me.
It's was work than I imagined, doing the conjugacy classes and all, so thanks again.

In principle, I had generated one shape from each equivalence class, using a canonical form that's equivalent to the "minlex" version of the shape, when the cell data is read out in the order shown here:

Code: Select all

+----------+----------+-------+
| 28 29 30 | 46 47 48 | 1 2 3 |
| 31 32 33 | 49 50 51 | 4 5 6 |
| 34 35 36 | 52 53 54 | 7 8 9 |
+----------+----------+-------+
| 37 38 39 | 19 20 21 |       |
| 40 41 42 | 22 23 24 |       |
| 43 44 45 | 25 26 27 |       |
+----------+----------+-------+
| 10 13 16 |          |       |
| 11 14 17 |          |       |
| 12 15 18 |          |       |
+----------+----------+-------+

It's highly "box oriented".
Taking every shortcut I could, I ended up generating ((36*35/2)*36 + 36*26)*512*512*512 = 3,169,685,864,448 test cases, and checking them for being canonical forms.
It was a ratio of ~7.33 test cases per actual canonical/minlex form.
[ 512 = # of box fills ]
[ 36 = # that are ED w/rsp to row & column perms ]
[ 26 = # that are ED w/rsp to diagonal reflection and row & column perms ]

The Wikipedia article on Burnside's Lemma, mentions using "enumeration/generation" with "minlex forms" as a way of verifying that "Burnside's lemma was correctly applied".
I was hoping for the opposite: to use a calculation from you or from Serg, as a check that I didn't take too many shortcuts, and (likely) didn't have bug(s) in by canonicalization code.

That and I thought it might be a fun exercise for you guys !

[blue]

Another "trivia/curiosity" table, with B123457 clue counts on the left:

Hidden Text: Show

Code: Select all

   |    ED shapes | with puzzles |  without
---+--------------+--------------+---------
 0 |           1  |           0  |        1
 1 |           4  |           0  |        4
 2 |          23  |           0  |       23
 3 |         125  |           0  |      125
 4 |         630  |           0  |      630
 5 |        2970  |           0  |     2970
 6 |       13089  |         322  |    12767
 7 |       53801  |        7931  |    45870
 8 |      206531  |       67945  |   138586
 9 |      739539  |      376905  |   362634  (crossover point)
10 |     2468234  |     1637582  |   830652
11 |     7674378  |     6000606  |  1673772
12 |    22221573  |    19242693  |  2978880
13 |    59917001  |    55207479  |  4709522
14 |   150468286  |   143809400  |  6658886
15 |   352048728  |   343570837  |  8477891
16 |   767736056  |   757955049  |  9781007
17 |  1561293914  |  1551012712  | 10281202
18 |  2962363840  |  2952476453  |  9887387
19 |  5246685692  |  5237962077  |  8723615
20 |  8678086628  |  8671014152  |  7072476
21 | 13410257969  | 13404986567  |  5271402
22 | 19368071661  | 19364460624  |  3611037
23 | 26152415035  | 26150143201  |  2271834
24 | 33024031396  | 33022720219  |  1311177
25 | 39006180915  | 39005487792  |   693123
26 | 43101117092  | 43100781986  |   335106
27 | 44559029938  | 44558881961  |   147977
28 | 43101117092  | 43101057515  |    59577
29 | 39006180915  | 39006159070  |    21845
30 | 33024031396  | 33024024108  |     7288
31 | 26152415035  | 26152412831  |     2204
32 | 19368071661  | 19368071057  |      604
33 | 13410257969  | 13410257820  |      149
34 |  8678086628  |  8678086596  |       32
35 |  5246685692  |  5246685686  |        6
36 |  2962363840  |  2962363839  |        1
37 |  1561293914  |  1561293914  |        0
38 |   767736056  |   767736056  |        0
39 |   352048728  |   352048728  |        0
40 |   150468286  |   150468286  |        0
41 |    59917001  |    59917001  |        0
42 |    22221573  |    22221573  |        0
43 |     7674378  |     7674378  |        0
44 |     2468234  |     2468234  |        0
45 |      739539  |      739539  |        0
46 |      206531  |      206531  |        0
47 |       53801  |       53801  |        0
48 |       13089  |       13089  |        0
49 |        2970  |        2970  |        0
50 |         630  |         630  |        0
51 |         125  |         125  |        0
52 |          23  |          23  |        0
53 |           4  |           4  |        0
54 |           1  |           1  |        0

[JPF]

Taking every shortcut I could, I ended up generating ((36*35/2)*36 + 36*26)*512*512*512 = 3,169,685,864,448 test cases, and checking them for being canonical forms.

just impressive

The Wikipedia article on Burnside's Lemma, mentions using "enumeration/generation" with "minlex forms" as a way of verifying that "Burnside's lemma was correctly applied".

Do you mean the lemma that is not Burnside's ?

JPF

[Serg]

Hi, JPF!

I finally managed to find my old programs.
If the question is still relevant, I confirm the numbers provided by Blue.
...
Number of permutations: 2 x 6^6 = 93312

Did you account for bands/stacks permutations? Some anticorner patterns having more than 3 filled boxes permit bands/stacks permutations. For example, these 2 anticorner patterns are equivalent because pattern B produced from pattern A by stacks B147/B258 swapping.

Code: Select all

          A                      B

+-----+-----+-----+     +-----+-----+-----+
|. . .|. . .|. . .|     |. . .|. . .|. . .|
|. . .|. . .|. . x|     |. . .|. . .|. . x|
|. . x|. . .|. . .|     |. . .|. . x|. . .|
+-----+-----+-----+     +-----+-----+-----+
|. . .|x x x|x x x|     |x x x|. . .|x x x|
|. . .|x x x|x x x|     |x x x|. . .|x x x|
|x x x|x x x|x x x|     |x x x|x x x|x x x|
+-----+-----+-----+     +-----+-----+-----+
|x x x|x x x|x x x|     |x x x|x x x|x x x|
|x x x|x x x|x x x|     |x x x|x x x|x x x|
|x x x|x x x|x x x|     |x x x|x x x|x x x|
+-----+-----+-----+     +-----+-----+-----+

Serg