## Minimal Unavoidable Set with 49 permutations

Everything about Sudoku that doesn't fit in one of the other sections

### Minimal Unavoidable Set with 49 permutations

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 5089743652156179248323485196775321489664937815218296573492814367531568724947652931889743652.5.179.48.23.8.19.7753..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)2538674919681425737143598625472189363924761581869357244215936876357812498796243152.3.................................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 UA2538694717681425939143578625472189363924761581869357244215936876357812498796243152.3.................................3.2..........................................       4.......................................4.6.................................6.4...       4.....9.7.7......9.9....7.........................................................       6#solution 4, 3 non-hit minimal UA2538674918691425737143598625472189363924761581869357244215936876357812499786243152.3.................................3.2..........................................       4.........8.9............................................................9.8......       4.......................................4.6.................................6.4...       4#solution 5, 4 non-hit minimal UA2538674919681425737143598625472189363926741581869357244215936876357812498794263152.3.................................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 UA2538694717681425939143578625472189363926741581869357244215936876357812498794263152.3.................................3.2..........................................       4.......................................6.4.................................4.6...       4.....9.7.7......9.9....7.........................................................       6#solution 7, 3 non-hit minimal UA2538674918691425737143598625472189363926741581869357244215936876357812499784263152.3.................................3.2..........................................       4.........8.9............................................................9.8......       4.......................................6.4.................................4.6...       4#solution 8, 4 non-hit minimal UA3528674919681425737143598625472189362934761581869357244215936876357812498796243153.2.................................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 UA3528694717681425939143578625472189362934761581869357244215936876357812498796243153.2.................................2.3..........................................       4.......................................4.6.................................6.4...       4.....9.7.7......9.9....7.........................................................       6#solution 10, 3 non-hit minimal UA3528674918691425737143598625472189362934761581869357244215936876357812499786243153.2.................................2.3..........................................       4.........8.9............................................................9.8......       4.......................................4.6.................................6.4...       4#solution 11, 4 non-hit minimal UA3528674919681425737143598625472189362936741581869357244215936876357812498794263153.2.................................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 UA3528694717681425939143578625472189362936741581869357244215936876357812498794263153.2.................................2.3..........................................       4.......................................6.4.................................4.6...       4.....9.7.7......9.9....7.........................................................       6#solution 13, 3 non-hit minimal UA3528674918691425737143598625472189362936741581869357244215936876357812499784263153.2.................................2.3..........................................       4.........8.9............................................................9.8......       4.......................................6.4.................................4.6...       4`

Cheers,
MD
Last edited by dobrichev on Fri Oct 19, 2012 8:14 pm, edited 2 times in total.
dobrichev
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### Re: Minimal Unavoidable Set with 13 permutations

A solution is also a UA ??? Hmmm!!!
daj95376
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### Re: Minimal Unavoidable Set with 13 permutations

daj95376 wrote: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.
dobrichev
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### Re: Minimal Unavoidable Set with 13 permutations

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
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### Re: Minimal Unavoidable Set with 13 permutations

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. # input253867491869142573714359862547218936392476158186935724421593687635781249978624315253867491869142573714359862547218936392674158186935724421593687635781249978426315253867491968142573714359862547218936392476158186935724421593687635781249879624315253867491968142573714359862547218936392674158186935724421593687635781249879426315253869471768142593914357862547218936392476158186935724421593687635781249879624315253869471768142593914357862547218936392674158186935724421593687635781249879426315352867491869142573714359862547218936293476158186935724421593687635781249978624315352867491869142573714359862547218936293674158186935724421593687635781249978426315352867491968142573714359862547218936293476158186935724421593687635781249879624315352867491968142573714359862547218936293674158186935724421593687635781249879426315352869471768142593914357862547218936293476158186935724421593687635781249879624315352869471768142593914357862547218936293674158186935724421593687635781249879426315897436521561792483234851967753214896649378152182965734928143675315687249476529318_________________________________________________________________________________________`

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.
daj95376
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### Re: Minimal Unavoidable Set with 13 permutations

daj95376 wrote: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
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### Re: Minimal Unavoidable Set with 13 permutations

daj95376 wrote: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. #   givens89743652.5.179.48.23.8.19.7753..489.6493.8..2..2.6..3.9.814..7531.6.7...4.65.9..8 # + ANY of these 50 non-givens897436521561792483234851967753214896649378152182965734928143675315687249476529318 # = 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.
dobrichev
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### Re: Minimal Unavoidable Set with 13 permutations

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.
daj95376
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### #2

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 solutions9643..27.7.8.1.46321.6..8...928..1351..429.8....1.3.246.159..47..72.1598.89.34612 #+ any of these non-givens964385271758912463213647859492876135135429786876153924621598347347261598589734612 #= 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.
dobrichev
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### Re: Minimal Unavoidable Set with 13 permutations

Congratulations, dobrichev!
dobrichev wrote: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 5089743652156179248323485196775321489664937815218296573492814367531568724947652931889743652.5.179.48.23.8.19.7753..489.6493.8..2..2.6..3.9.814..7531.6.7...4.65.9..8       50...#solution 13, 3 non-hit minimal UA352867491869142573714359862547218936293674158186935724421593687635781249978426315`

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:
Red Ed wrote: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
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### Re: #2

Hello, dobrichev!
dobrichev wrote: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 solutions9643..27.7.8.1.46321.6..8...928..1351..429.8....1.3.246.159..47..72.1598.89.34612 #+ any of these non-givens964385271758912463213647859492876135135429786876153924621598347347261598589734612 #= this unique solution`

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

Serg
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### list of 16 UA

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 1395386724.71694.5.828.31...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 1385396724.71684.5.929.31...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 1327583694.68.49.7.539.7..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 1327593684.69.48.7.538.7..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 1373582946.61.74.8958.9..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 133475968..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 133475968..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 134357689..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 138754963..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 138754963..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 138754963..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 138754963..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 133964587..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 133964587..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 135934678..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 135934678..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
dobrichev
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### Re: Minimal Unavoidable Set with 13 permutations

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
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### open issues

Hi Serg,

Thank you for the confirmation.

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
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### UA 58+

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.... 12A.......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 UA5836....................................................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=62.......1....23.4...56..2....7..8.3..4.1.....9...5..2...8..75...2.4....67...9....  1,A=2,identity3.......1....23.4...56..2....7..8.3..4.1.....9...5..2...8..75...2.4....67...9....  1,A=34.......1....23.4...56..2....7..8.3..4.1.....9...5..2...8..75...2.4....67...9....  46.......1....23.4...56..2....7..8.3..4.1.....9...5..2...8..75...2.4....67...9....  28.......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
dobrichev
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Posts: 1603
Joined: 24 May 2010

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