The New Sudoku Players' Forum

[dobrichev]

Hi,
The pseudo-puzzle on the first row below has 13 solutions.
Its first solution contain a minimal unavoidable set of size 50.
In the rest 12 solutions, smaller minimal unavoidable sets are not covered by the givens.

Code: Select all

#The pseudo-puzzle
........1.6...2..3..4.5..6....21...6....7.15.18.9.57.4.2...36....5.8.249.7..2.31.

#solution 1, 1 non-hit minimal UA of size 50
897436521561792483234851967753214896649378152182965734928143675315687249476529318
89743652.5.179.48.23.8.19.7753..489.6493.8..2..2.6..3.9.814..7531.6.7...4.65.9..8       50

#solution 2, 4 non-hit minimal UA of size (4,4,4,6)
253867491968142573714359862547218936392476158186935724421593687635781249879624315
2.3.................................3.2..........................................       4
.........9.8............................................................8.9......       4
.......................................4.6.................................6.4...       4
.....7.9.9......7.7....9.........................................................       6

#solution 3, 3 non-hit minimal UA
253869471768142593914357862547218936392476158186935724421593687635781249879624315
2.3.................................3.2..........................................       4
.......................................4.6.................................6.4...       4
.....9.7.7......9.9....7.........................................................       6

#solution 4, 3 non-hit minimal UA
253867491869142573714359862547218936392476158186935724421593687635781249978624315
2.3.................................3.2..........................................       4
.........8.9............................................................9.8......       4
.......................................4.6.................................6.4...       4

#solution 5, 4 non-hit minimal UA
253867491968142573714359862547218936392674158186935724421593687635781249879426315
2.3.................................3.2..........................................       4
.........9.8............................................................8.9......       4
.......................................6.4.................................4.6...       4
.....7.9.9......7.7....9.........................................................       6

#solution 6, 3 non-hit minimal UA
253869471768142593914357862547218936392674158186935724421593687635781249879426315
2.3.................................3.2..........................................       4
.......................................6.4.................................4.6...       4
.....9.7.7......9.9....7.........................................................       6

#solution 7, 3 non-hit minimal UA
253867491869142573714359862547218936392674158186935724421593687635781249978426315
2.3.................................3.2..........................................       4
.........8.9............................................................9.8......       4
.......................................6.4.................................4.6...       4

#solution 8, 4 non-hit minimal UA
352867491968142573714359862547218936293476158186935724421593687635781249879624315
3.2.................................2.3..........................................       4
.........9.8............................................................8.9......       4
.......................................4.6.................................6.4...       4
.....7.9.9......7.7....9.........................................................       6

#solution 9, 3 non-hit minimal UA
352869471768142593914357862547218936293476158186935724421593687635781249879624315
3.2.................................2.3..........................................       4
.......................................4.6.................................6.4...       4
.....9.7.7......9.9....7.........................................................       6

#solution 10, 3 non-hit minimal UA
352867491869142573714359862547218936293476158186935724421593687635781249978624315
3.2.................................2.3..........................................       4
.........8.9............................................................9.8......       4
.......................................4.6.................................6.4...       4

#solution 11, 4 non-hit minimal UA
352867491968142573714359862547218936293674158186935724421593687635781249879426315
3.2.................................2.3..........................................       4
.........9.8............................................................8.9......       4
.......................................6.4.................................4.6...       4
.....7.9.9......7.7....9.........................................................       6

#solution 12, 3 non-hit minimal UA
352869471768142593914357862547218936293674158186935724421593687635781249879426315
3.2.................................2.3..........................................       4
.......................................6.4.................................4.6...       4
.....9.7.7......9.9....7.........................................................       6

#solution 13, 3 non-hit minimal UA
352867491869142573714359862547218936293674158186935724421593687635781249978426315
3.2.................................2.3..........................................       4
.........8.9............................................................9.8......       4
.......................................6.4.................................4.6...       4

Cheers,
MD

[daj95376]

A solution is also a UA ??? Hmmm!!!

[dobrichev]

A solution is also a UA ??? Hmmm!!!

"#solution,sNumber,nUnavoidables" is continuation of the previous row, correctly displayed on wide screen only. Will edit the table to avoid confusion.

[RW]

Nice one. This would be a valency 13 unavoidable set. Red Ed was exploring sets like this at some point, but I can't recall seeing any sets with valency >6 (posted by Red Ed here). But as that was back in 2006, perhaps someone has posted some higher valency sets somewhere else... Pat?

RW

[daj95376]

How I see your multi-solution puzzle.

Code: Select all

........1.6...2..3..4.5..6....21...6....7.15.18.9.57.4.2...36....5.8.249.7..2.31. # input
253867491869142573714359862547218936392476158186935724421593687635781249978624315
253867491869142573714359862547218936392674158186935724421593687635781249978426315
253867491968142573714359862547218936392476158186935724421593687635781249879624315
253867491968142573714359862547218936392674158186935724421593687635781249879426315
253869471768142593914357862547218936392476158186935724421593687635781249879624315
253869471768142593914357862547218936392674158186935724421593687635781249879426315
352867491869142573714359862547218936293476158186935724421593687635781249978624315
352867491869142573714359862547218936293674158186935724421593687635781249978426315
352867491968142573714359862547218936293476158186935724421593687635781249879624315
352867491968142573714359862547218936293674158186935724421593687635781249879426315
352869471768142593914357862547218936293476158186935724421593687635781249879624315
352869471768142593914357862547218936293674158186935724421593687635781249879426315
897436521561792483234851967753214896649378152182965734928143675315687249476529318
_________________________________________________________________________________________


Code: Select all

 First 12 solutions involve:   23 DP;   46 DP;   789 DP
 *-----------------------------------------------------*
 |  23   5    23   |  8    6    79   |  4    79   1    |
 |  789  6    89   |  1    4    2    |  5    79   3    |
 |  79   1    4    |  3    5    79   |  8    6    2    |
 |-----------------+-----------------+-----------------|
 |  5    4    7    |  2    1    8    |  9    3    6    |
 |  23   9    23   |  46   7    46   |  1    5    8    |
 |  1    8    6    |  9    3    5    |  7    2    4    |
 |-----------------+-----------------+-----------------|
 |  4    2    1    |  5    9    3    |  6    8    7    |
 |  6    3    5    |  7    8    1    |  2    4    9    |
 |  89   7    89   |  46   2    46   |  3    1    5    |
 * ----------------------------------------------------*


Code: Select all

 Last solution is just that!
 It contains the givens and "none" of the above DP values (in their cells)
 *-----------------------------------------------------*
 |  8    9    7    |  4    3    6    |  5    2    1    |
 |  5    6    1    |  7    9    2    |  4    8    3    |
 |  2    3    4    |  8    5    1    |  9    6    7    |
 |-----------------+-----------------+-----------------|
 |  7    5    3    |  2    1    4    |  8    9    6    |
 |  6    4    9    |  3    7    8    |  1    5    2    |
 |  1    8    2    |  9    6    5    |  7    3    4    |
 |-----------------+-----------------+-----------------|
 |  9    2    8    |  1    4    3    |  6    7    5    |
 |  3    1    5    |  6    8    7    |  2    4    9    |
 |  4    7    6    |  5    2    9    |  3    1    8    |
 *-----------------------------------------------------*


If anything, the minimal UA in the last solution is the 17 single values occupying the DP cells present in the first 12 solutions.

[RW]

If anything, the minimal UA in the last solution is the 17 single values occupying the DP cells present in the first 12 solutions.

How can that be a minimal UA if I can remove all those 17 clues and still have a unique solution?

RW

[dobrichev]

If anything, the minimal UA in the last solution is the 17 single values occupying the DP cells present in the first 12 solutions.

Code: Select all

........1.6...2..3..4.5..6....21...6....7.15.18.9.57.4.2...36....5.8.249.7..2.31. #   givens
89743652.5.179.48.23.8.19.7753..489.6493.8..2..2.6..3.9.814..7531.6.7...4.65.9..8 # + ANY of these 50 non-givens
897436521561792483234851967753214896649378152182965734928143675315687249476529318 # = this unique solution

This, along with "givens alone have multiple solutions" determines this minimal UA of size 50.
Adding any given from a smaller subset of these 50 non-gives is a weaker constraint and doesn't reduce the size of the UA.

The Deadly Pattern formed by this UA has 17 cells with more than two choices, that is correct.
The DP Identity is the set of the 50 non-givens from the UA.
The rest 12 permutations are non-minimal UA sets, and could be decomposed to several combinations of smaller minimal UA sets.

[daj95376]

I was wrong to limit myself to the 17 DP cells. I agree that all 50 non-givens are significant in my solution #13.

I think of UAs in terms of DPs. (This is probably a mistake.) Specifically:

If I add givens r1c1=2, r2c1=7, and r5c4=4 to the original puzzle ... then the DPs are killed and the resulting puzzle has a unique solution. All 12 permutations of givens for these three cells result in my first 12 solutions to the original puzzle. As far as I'm concerned, the three DPs represent three separate UAs.

In my solution #13, I don't see a DP. What I see is 50 individual values that force the 49 other values. Specifically:

If I add given r1c1=8 to the original puzzle (from solution #13) ... then the resulting puzzle has a unique solution. The same holds true for the 49 other non-givens in my solution #13. The fact that they all result in solution #13 is inconsequential. Listing the 50 non-givens as a set for solution #13 made sense to me, but I'm only now beginning to accept that the set, instead of the individual cells, can be considered a UA.

[dobrichev]

Here is another one

Code: Select all

....85..1.5.9.2.....3.47.594...76....35...7.6876.5.9...2...83..34..6....5..7..... #pseudo-puzzle with 13 solutions
9643..27.7.8.1.46321.6..8...928..1351..429.8....1.3.246.159..47..72.1598.89.34612 #+ any of these non-givens
964385271758912463213647859492876135135429786876153924621598347347261598589734612 #= this unique solution

Both UA have similar topology.
The rest of solutions are caused by 3 uncovered UA of size 4, plus one UA of size 6. The U6 and one of the U4 have one cell in common.
One of the differences is the shape of the U6.

[Serg]

Congratulations, dobrichev!

Hi,
The pseudo-puzzle on the first row below has 13 solutions.
Its first solution contain a minimal unavoidable set of size 50.
In the rest 12 solutions, smaller minimal unavoidable sets are not covered by the givens.

Code: Select all

#The pseudo-puzzle
........1.6...2..3..4.5..6....21...6....7.15.18.9.57.4.2...36....5.8.249.7..2.31.

#solution 1, 1 non-hit minimal UA of size 50
897436521561792483234851967753214896649378152182965734928143675315687249476529318
89743652.5.179.48.23.8.19.7753..489.6493.8..2..2.6..3.9.814..7531.6.7...4.65.9..8       50
...
#solution 13, 3 non-hit minimal UA
352867491869142573714359862547218936293674158186935724421593687635781249978426315

You set new record by finding UA set(s) with highest valency ever observed (at least - ever published)!
I've cross-checked your first UA set. It looks like OK. It is weakly minimal unavoidable set according to Red Ed's terminology (old thread How close together are the isomorphs of two random grids ?). Let me cite Red Ed's post in that thread:

Imagine listing all the permutations (that have the same footprint) of a given unavoidable as I did in the previous post. An unavoidable in that list is minimal if & only if for each of its digits d@(r,c) it's the only unavoidable in the list that has d@(r,c).

If all the permuted unavoidables are minimal, I call them strongly minimal. Otherwise, the minimal ones (if any) are only weakly minimal. IIRC, most unavoidables are strongly minimal; only a few of the bigger (18+ cells) unavoidables are weakly minimal.

Serg

[Serg]

Hello, dobrichev!

Here is another one

Code: Select all

....85..1.5.9.2.....3.47.594...76....35...7.6876.5.9...2...83..34..6....5..7..... #pseudo-puzzle with 13 solutions
9643..27.7.8.1.46321.6..8...928..1351..429.8....1.3.246.159..47..72.1598.89.34612 #+ any of these non-givens
964385271758912463213647859492876135135429786876153924621598347347261598589734612 #= this unique solution

I've verified your second UA50 is (weakly) minimal UA set having 1 minimal permutation (unavoidable). Nice job!

Serg

[dobrichev]

Here is the list of the 16 known UA with 13 permutations. This list include the previously published 2 UA.
The first line is the generating pseudo-puzzle with 13 solutions, in row minlex form.
The second line is the identity completion - any of these givens complete the pseudo-puzzle uniquely.

Code: Select all

#pattern                                                                      nClues nSolutions
#identity completion                                                          nClues unique
........1.....2.3...4..567.....2..17.2...83.65..6..92....48..6..4..7619.36..59.8. 31 13
95386724.71694.5.828.31...96395.48..4.179..5..78.31..4192..37.58.52....3..71..4.2 50 y

........1.....2.3...4..567.....7618....49..6.36..58.9..2...93.65..6..82.6...2..17 31 13
85396724.71684.5.929.31...89452....3182..37.5..71..4.24.178..5..79.31..4.385.49.. 50 y

........1..1..2.3...4.15....1267.5...3..21.6..68.49..2.461.....1.3..74868.9...... 32 13
27583694.68.49.7.539.7..6284....8.939.75..8.45..3..17.7...83259.2.95.....5.264317 49 y

........1..1..2.3...4.15....1267.5...3.521.6..6..48..2.461.....1.3..74969.8...... 32 13
27593684.69.48.7.538.7..6294....9.838.7...9.45.93..17.7...93258.2.85.....5.264317 49 y

........1..2..3....4.56...7.......58..3.5761..5.9....3...4951.6..4.123..19...8.42 31 13
73582946.61.74.8958.9..123.2671349..98.2....44.1.8672.328....7.57.6...89..637.5.. 50 y

.......12......345.152346......5.1...7.6..5..58....4..1.3..875.49..7..3.7583...6. 33 13
3475968..269817...8......796324.9.879.4.81.23..1723.96.2.96...4..61.52.8....429.1 48 y

.......12......345.152346......5.1...7.6..5..58....4..1.3..875.49..7.23.75.3...6. 33 13
3475968..269817...8......796324.9.879.4.81.23..1723.96.2.96...4..61.5..8..8.429.1 48 y

.......12.....13......45.6.....375....3156...57.8.4.36.9.4.26.33.8..94..64..7.... 32 13
4357689..78692..452193..7.88642...9192....874..1.9.2..1.7.8..5..5.61..27..25.3189 49 y

.......12.....3..4..4.15.....1...675.2..7..316..1..2.8.4....1...53.6.8.79..8...23 30 13
8754963..16278.95.39.2..78648.329...5.96.84...37.54.9.7.8532.692..9.1.4..16.475.. 51 n

.......12.....3..4..4.15.6...1...675.2..7..316..1..2...4....1.6.53.6.8.79..8...23 31 13
8754963..16278.95.39.2..7.848.329...5.96.84...37.54.897.8532.9.2..9.1.4..16.475.. 50 n

.......12.....3..4..4.25.....2...675.1..7..236..2..1.8.4....2...53.6.8.79..8...31 30 13
8754963..26178.95.39.1..78648.319...5.96.84...37.54.9.7.8531.691..9.2.4..26.475.. 51 n

.......12.....3..4..4.25.6...2...675.1..7..236..2..1...4....2.6.53.6.8.79..8...31 31 13
8754963..26178.95.39.1..7.848.319...5.96.84...37.54.897.8531.9.1..9.2.4..26.475.. 50 n

.......12.....34.5145.2.63....37.5...3....2..6...8.3....72.48.3.8.59...44..837..6 33 13
3964587..72816..9....7.9..8814..2.695.9641.87.729.5.4196..1..5.2.3..617..51...92. 48 y

.......12.....34.5145.2.63....37.5...3....2..6...8.3....72.48.3.8.59.1.44..83...6 33 13
3964587..72816..9....7.9..8814..2.695.9641.87.729.5.4196..1..5.2.3..6.7..51..792. 48 y

.......12.....3456146.257.....58.6...5....2..7...3.5....82.43.5.3.6....44..358..7 33 13
5934678..82719.......8...93314..2.796.9741.38.829.6.4197..1..6.2.5.7918..61...92. 48 y

.......12.....3456146.257.....58.6...5....2..7...3.5....82.43.5.3.6..1.44..35...7 33 13
5934678..82719.......8...93314..2.796.9741.38.829.6.4197..1..6.2.5.79.8..61..892. 48 y


The 4 completions which don't uniquely identify the solution (w/o the pseudo-puzzle) contain uncovered U6. This generates a related UA set. See the pairs for U50 and U51 with "n" in the last column.

7 from the UA have been found by scanning 17-clue puzzles by removing one clue.
2 of these 7 are found in one 17-clue puzzle, with the same clue removed.
8-th UA is related to one of the first 7.
Other 8 are found by applying {-2} to the first 8 pseudo-puzzles on several passes.

The complete results of scanning the 17-clue puzzles are in file ua17m1.zip.

Below is the number of 17-clue puzzles and maximal valency UA covered by a single clue.

Code: Select all

      5 2
  11851 3
  15802 4
  19122 5
    865 6
   1278 7
     20 8
    187 9
     11 10
      4 11
      6 13

Note the rarity of UA of valency 6, 8 and 11.

Here are the 5 puzzles that have only bivalue UA covered by a single clue.

Code: Select all

................12..3..4.........5......364..17..........12..7...8......6.4...3..
................12..3.45.........4.6.7.......81.2........1........83.....46...5..
..............1..2.34....5.......63.1........7....8.......5...7.6.43....2.......1
........1.......23..4..5........26...7.......13..........38...7..5...4....61.....
........1.......23..4.56......7.3.....8.......56...4.....86....1........3......7.


The average number of UA hit by a single clue in 17-clue puzzles is 2000.
The minimum UA hit separately by all 17 givens in a puzzle is 2541, the maximum is 301833.
The minimum of the UA with valency > 2 hit by a single clue, summarized to whole puzzle, is 0. The maximum is 6685, the average is 302.
Both maximums for any UA and UA with valency > 2 are in this puzzle, discovered by Kohei Noshita in January 2008.

Code: Select all

........1.....2.....3.4..5......61...3....7.68...5.........83...4.......5..7.....

The same puzzle has UA with valency of 13, it is the #2 published previously in this topic.

Cheers,
MD

[Serg]

Hi, dobrichev!
I've just verified all your 16 13-valent UA sets. They all are weakly minimal UA sets having only one minimal unavoidable (permutation) each. These data can be very useful for UA set properties investigations. Good job!

I have several question concerning UA sets. Can you answer some of them?

1. Do there exist strongly minimal multivalent UA sets (valency > 2)?
2. Do there exist weakly minimal multivalent UA sets having more then 1 minimal unavoidables (permutation)?
3. Do there exist minimal UA sets of size greater than 60?

Serg

[dobrichev]

Hi Serg,

Thank you for the confirmation.

Unfortunately, AFAIK your questions have no answer yet.

I would add fourth question.

4. Do there exist multivalent UA set with a permutation decomposed to a smaller multivalent UA set?

You motivated me to take a look at the largest UA of size 60. Details in the next post.

MD

[dobrichev]

In 2006 Ocean found this UA of size 60.

I assembled a small gotchi to check what happens around it.
First, applied {-2} to the pseudo-puzzle, and checked all puzzles for UA of size 58 or more.
Next iterations were {-1} to the puzzles from next generations, expanded to maximal size - all cell values except these from the UA are given.

Surprisingly, after few generations several new UA of size 60 appeared.
After several hours of processing and about 10 iterations, the list of UA60s growth to 68.
The whole cache consists of 298 689 known UA of size 58, 59 and 60, and 3 394 678 processed pseudo-puzzles.
The number of UA of size 59 is 1559.
In the last 2 generations no new 60s, and only few 59s appeared. 58s are still almost doubled at each iteration.

Here is the distribution by valency.

Code: Select all

    #UA Valency
 247451  2
  43163  3
   5126  4
   2372  5
    256  6
    241  7
     22  8
     39  9
      3 10
     10 11
      1 12
      5 13
=======
 298689

This UA with valency of 12 fills the gap in the observed valency values.

Code: Select all

#pseudo puzzle                                                                    nSolutions
........1....23.4...56..2....7..8.3..4.1.....9...5..2...8..75...2.4....67...9.... 12
A.......1....23.4...56..2....7..8.3..4.1.....9...5..2.B.8..75...2.4....67...9....

28497536.1698..7.537..14.9865.24.1.98.2.39657.317.68.449.36..125.3.8197..165.2483  identity UA58
36....................................................63.........................  alternative UA4

........1....23.4...56..2....7..8.3..4.1.....9...5..2.1.8..75...2.4....67...9....  5
........1....23.4...56..2....7..8.3..4.1.....9...5..2.3.8..75...2.4....67...9....  5
........1....23.4...56..2....7..8.3..4.1.....9...5..2.4.8..75...2.4....67...9....  1,B=4,identity
........1....23.4...56..2....7..8.3..4.1.....9...5..2.6.8..75...2.4....67...9....  1,B=6

2.......1....23.4...56..2....7..8.3..4.1.....9...5..2...8..75...2.4....67...9....  1,A=2,identity
3.......1....23.4...56..2....7..8.3..4.1.....9...5..2...8..75...2.4....67...9....  1,A=3
4.......1....23.4...56..2....7..8.3..4.1.....9...5..2...8..75...2.4....67...9....  4
6.......1....23.4...56..2....7..8.3..4.1.....9...5..2...8..75...2.4....67...9....  2
8.......1....23.4...56..2....7..8.3..4.1.....9...5..2...8..75...2.4....67...9....  4


It has interesting properties.
The large UA is decomposed to several U4, but one of them solves the grid uniquely.
Additionally, the leftmost cell (labelled A) allows 5 possible values, which is uncommon.

Below is the list of the UA with valency of 10+.
Pseudo-puzzles are given, one of solutions is the identity UA.
Column 2 is the number of givens = 81 - UA size. Column 3 is the valency.

Code: Select all

................12..3.145....4..2....6..7.3...7.8...9...5.3...4.2...6..71..9...8.   23   10
........1.....2.3....34.5....2..3.6..4...17..5...8...9..6...4....86.....91...7.2.   23   10
........1.....2.3...4.5.6....1..74...2...3...8..9....6.3...6.2.1...7.5..9..4....8   23   10

........1.....2.3...4.5.6.....7..4.3.6...1.8.8...9.1....73..8...1...6.2.9...4....   23   11
........1.....2.3...456........3......1..7.4..3.1..8.9..3..4.2.1..8..5..6...9...7   23   11
........1..2..3.....4.5.........6.72...4..3...1..8...9.7.8..5..3....5.6.9.5.1...8   23   11
........1..2..3.4..4..5.6....3..7....8..6.4..7..1.......7..5.3..6..2.5..1..4....9   23   11
........1..2..3.4..5..6.7.......5.3.....8.6..1..3....9..38...2..9...75..6..1....7   23   11
........1..2..3.4..5.16.7........6...3..81.9.4..67.1....1....2..6....5..7..4.....   23   11
........1..2..3.4..5.16.7........6...8..91.3.4..67.1....1....2..6....5..7..4.....   23   11
.....1..2....3..4...5...1....65..7...2...8..19...6..3...76..5...9.....1.3...9..7.   23   11
.....1..2....3..4...5...6....2.4.5.7.3.2...8.4...9......7..5..1.2.6..8..9......3.   23   11
.....1..2....3..4..56...1....4...6...7.8..9..8...2......9..6..1.8.7....52...8..3.   23   11

........1....23.4...56..2....7..8.3..4.1.....9...5..2...8..75...2.4....67...9....   23   12

........1.....2....34...56.....7...8..2..1.3..6.5..9...5.9..2..1...8...76....3.4.   23   13
........1.....2.3..45.6..2...2..35...7.8.....6...1......9..5..6.8.7...9.1...4...2   23   13
........1.....234...5.6..7...3.5.....1...42..6..8....7..6.3..5..8...17..2..9.....   23   13
........1..2..3....1..4.56......1..7..3....8..9..582....8..2.3..6....9..5..6....4   23   13
.....1..2..2....3..4..5......34...6..7..2.5..8....9..1..67...2..3..9.4..1....8...   23   13


Below are the UA of size 60, including the one from Ocean.

Hidden Text: Show

Code: Select all

#pseudo-puzzle in row minlex form                                                nSolutions
........1.....2.3...4.5......1.4.6...3.....5.7....8..2..6...4...5...3.8.2..7....6  3
........1.....2.3...4.5......6...4...2.7....85....3.9...7.4.6...1......23..8...5.  2
........1.....2.3...4.5......6...7...2.1....85....3.4...8.4.6...9......23..9...5.  4
........1.....2.3..45...6....1.5.....6...7.2.3..8.......4.1.5...2...64..7..3.....  2
........1.....2.3..45...6....1.5.....6...7.2.3..8....4..4.1.5...2...6...7..3.....  2
........1....23.4...56.......7...8...2...1..96...4..5...95..7...3......24......6.  4
.....1..2....2..3...4...5....35..4...1...6...7...8......54......6...3..18...7..2.  2
.....1..2....2..3...4...5....35..4...1...6...7...8......54......6...3..18...9..2.  2
.....1..2....2..3...4...5....54......6...3..17...8..2...95..4...1...6...8...7....  2
.....1..2....3..4...5...1....46......2..7..3.8....9..1..74..5...9..2....1....8...  2
.....1..2....3..4...5...6....12..5...3......67....4.....25......8..7..3.4....9..1  2
.....1..2....3..4...5...6....12..5...3..6....7....4.....25......6..7..3.4....8..1  2
.....1..2....3..4...5...6....12..5...3..7....6....4.....24......7..5..3.4....8..1  2
.....1..2....3..4...5...6....12..5...3..7....6....4.....24......7..6..3.4....8..1  2
.....1..2....3..4...5...6....12..5...3..7....6....4.....24......7..8..3.4....9..1  2
.....1..2....3..4...5...6....12..5...3..7....6....8.....28......7..5..3.8....9..1  2
.....1..2....3..4...5...6....12..5...3..7....8....4.....24......7..5..3.4....9..1  2
.....1..2....3..4...5...6....12..5...3..7....8....4.....24......7..6..3.4....9..1  2
.....1..2....3..4...5...6....12..5...3..7....8....4.....24......7..8..3.4....9..1  2
.....1..2....3..4...5...6....12..5...3..7....8....4.....25......7..8..3.4....9..1  2
.....1..2....3..4...5...6....16..5...3..7....8....4.....65......7..2..3.4....8..1  2
.....1..2....3..4...5...6....16..5...3..7....8....4.....65......7..8..3.4....9..1  2
.....1..2....3..4...5...6....16..5...3..7....8....4.....65......7..9..3.4....2..1  2
.....1..2....3..4...5...6....17..5...3..6....8....4.....75......6..2..3.4....7..8  2
.....1..2....3..4...5...6....17..5...3..6....8....4.....75......6..2..3.4....9..8  2
.....1..2....3..4...5...6....17..5...3..6....8....4.....75......6..8..3.4....2..1  2
.....1..2....3..4...5...6....17..5...3..6....8....4.....75......6..8..3.4....9..1  2
.....1..2....3..4...5...6....17..5...3..6....8....4.....75......6..9..3.4....2..1  2
.....1..2....3..4...5...6....17..5...3..6....8....4.....75......6..9..3.4....7..1  2
.....1..2....3..4...5...6....17..5...3..6....8....4.....75......6..9..3.4....7..8  4
.....1..2....3..4...5...6....17..5...3..8....9....4.....75......8..9..3.4....6..1  2
.....1..2....3..4...5...6....25......4...7..16...8..3...92..5...8...4...3...6....  2
.....1..2....3..4...5...6....26......4...7..18...5..3...72..5...9...4...3...8....  2
.....1..2....3..4...5...6....26......4...7..18...9..3...72..5...9...4...3...8....  2
.....1..2....3..4...5...6....26..5...3..7....8....4.....65......7..2..3.4....8..1  2
.....1..2....3..4...5...6....26..5...3..7....8....4.....65......7..2..3.4....9..1  2
.....1..2....3..4...5...6....27......4...8..16...9..3...82..5...9...4...3...6....  2
.....1..2....3..4...5...6....27..5...1...4...3...6......75......4...8..16...2..3.  2
.....1..2....3..4...5...6....27..5...1...4...3...8......75......4...6..18...2..3.  2
.....1..2....3..4...5...6....27..5...1...6...8...9......74......6...5..19...2..3.  2
.....1..2....3..4...5...6....27..5...1...6...8...9......75......6...8..19...2..3.  2
.....1..2....3..4...5...6....27..5...3..6....7....4.....72......6..5..3.4....8..1  2
.....1..2....3..4...5...6....27..5...3..6....8....4.....75......6..8..3.4....7..1  2
.....1..2....3..4...5...6....27..5...3..6....8....4.....75......6..9..3.4....7..1  2
.....1..2....3..4...5...6....27..5...3..6....8....4.....78......6..2..3.4....7..1  2
.....1..2....3..4...5...6....27..5...3..6....8....4.....79......6..2..3.4....7..1  2
.....1..2....3..4...5...6....27..5...3..8....6....4.....75......8..9..3.4....7..1  2
.....1..2....3..4...5...6....62......4...7..18...5..3...76..5...6...4...3...8....  2
.....1..2....3..4...5...6....62......4...7..89...5..3...76..5...6...4...3...9....  2
.....1..2....3..4...5...6....62......4..5..3.7....8..9..86..5...3..4....6....7...  2
.....1..2....3..4...5...6....65......2..4..3.7....8..1..86..5...9..2....1....7...  2
.....1..2....3..4...5...6....65......2..7..3.8....4..1..96..5...7..2....1....8...  2
.....1..2....3..4...5...6....65......2..7..3.8....9..1..96..5...7..2....1....8...  2
.....1..2....3..4...5...6....67......2..8..3.9....4..1..86..5...6..2....1....9...  2
.....1..2....3..4...5...6....67......2..8..3.9....5..1..86..5...6..2....1....9...  2
.....1..2....3..4...5...6....67..5...1...8...4...2......79......8...5..12...7..3.  2
.....1..2....3..4...5...6....67..5...1...8...9...2......74......8...5..12...7..3.  2
.....1..2....3..4...5...6....72......4...7..16...8..3...87..5...9...4...3...6....  2
.....1..2....3..4...5...6....72......4...8..16...5..3...87..5...9...4...3...6....  2
.....1..2....3..4...5...6....75......2..7..3.6....4..1..87..5...3..2....1....6...  2
.....1..2....3..4...5...6....75......2..7..3.6....4..1..89..5...3..2....1....6...  2
.....1..2....3..4...5...6....75......2..7..3.6....8..1..97..5...3..2....1....6...  2
.....1..2....3..4...5...6....75......2..7..3.8....4..1..97..5...3..2....1....8...  2
.....1..2....3..4...5...6....75......2..7..3.8....6..1..97..5...3..2....1....8...  2
.....1..2....3..4...5...6....75......2..8..3.6....4..1..97..5...3..2....1....6...  2
.....1..2....3..4...5...6....75......2..8..3.6....4..1..97..5...8..2....1....6...  2
.....1..2....3..4...5...6....75......2..8..3.6....9..1..87..5...9..2....1....6...  2
.....1..2....3..4...5...6....75......2..8..3.9....6..1..87..5...6..2....1....9...  2

If somebody is interested I'll publish the whole list of large UA.

I can't predict the existence or non-existence of UA of size 61+ as well as the existence of UA of valency 15+.

MD