## Minimal Unavoidable Set with 49 permutations

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

### Re: 21

Hi, dobrichev!
dobrichev wrote:21 permutations.
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`........1.....2.3...435.6........5...6..7..2.8..1....9.....3.4..5....7..4.19....8 #puzzle with 21 solutions93684725.58761.9.421...9.87743298.161.95.48.3.25.3647.69278.1.53.8461.92.7..2536. #weakly minimal UA set`

I confirm that this UA set is 21-valent weakly minimal UA59 set. Impressive work! I've checked your last posted 15-valent and 19-valent UA sets too. They are weakly minimal 15-valent and 19-valent UA59 sets.

I think your outstanding puzzle (not valid, having 21 solutions), producing 21-valent UA59 set, and UA59 set itself both are worth publishing. Here they are.
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`+-----+-----+-----+     Puzzle producing 21-valent UA59 set|. . .|. . .|. . 1||. . .|. . 2|. 3 .||. . 4|3 5 .|6 . .|+-----+-----+-----+|. . .|. . .|5 . .||. 6 .|. 7 .|. 2 .||8 . .|1 . .|. . 9|+-----+-----+-----+|. . .|. . 3|. 4 .||. 5 .|. . .|7 . .||4 . 1|9 . .|. . 8|+-----+-----+-----++-----+-----+-----+     21-valent UA59 set (minimal permutation)|9 3 6|8 4 7|2 5 .||5 8 7|6 1 .|9 . 4||2 1 .|. . 9|. 8 7|+-----+-----+-----+|7 4 3|2 9 8|. 1 6||1 . 9|5 . 4|8 . 3||. 2 5|. 3 6|4 7 .|+-----+-----+-----+|6 9 2|7 8 .|1 . 5||3 . 8|4 6 1|. 9 2||. 7 .|. 2 5|3 6 .|+-----+-----+-----+`

New maximal valency record for minimal UA sets is 21. Great!
Serg
Serg
2018 Supporter

Posts: 690
Joined: 01 June 2010
Location: Russia

### Re: puzzle difficulty & largest UA hit by a single clue

dobrichev wrote:
RW wrote:When you present the UA details, would it also be possible to give the amount of unsolved cells that are not included in any unhit UA's? In other words, the amount of cells that can be solved even with the clue removed. I think the sum of these numbers should have some correlation to the difficulty of the puzzle. If the number is large, then you can solve a lot by focusing on only part of the puzzle. If it is very small, then you have to gather information from all over the puzzle to solve any cell.

Will do it.

Done. See this post.
dobrichev
2016 Supporter

Posts: 1777
Joined: 24 May 2010

### 33

Here are 3 monsters with 33 permutations.
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`.......12.....34.5.45.6......1.7..2.4....23..52.8....6..4....7..38..4..96..1..... #puzzle with 33 solutions8935476..71628..9.2..9.178336.4.59.8.8961..57..7.3914.95.3268.11..75.26..72.98534 #weakly minimal UA set........1.....2.3...4..56........4...42.3..7..8.1....9..3.7..2..6...15..4..9....8 #puzzle with 33 solutions62849375.59176.8.473.81..92176259.839..6.81.53.5.4726.81.5.49.62.938..47.57.2631. #weakly minimal UA set........1.....2.3...4..56........4...42.3..7..8.1....9.3..7..2..6...15..4..9....8 #puzzle with 33 solutions62849375.19576.8.437.81..92716259.839..6.81.55.3.4726.8.15.49.62.938..47.57.2631. #weakly minimal UA set`

Second and third puzzles are similar. They are decomposed to quite symmetrical family of UA, all of size 6.
The first puzzle is decomposed to a family of UA4 and UA6.

Below are the decomposition details for the three puzzles.
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`.......12.....34.5.45.6......1.7..2.4....23..52.8....6..4....7..38..4..96..1.....   26   33   #pattern,nClues,nSolutions367485912892713465145269837981376524476952381523841796214598673738624159659137248   1   4   #solution,sNumber,nUnavoidables.67..................................76..........................................   4   n.........89................98....................................................   4   n.......................................95................59......................   4   n....8.9................98.................................98.....................   6   n367485912982713465145269837891376524476952381523841796214598673738624159659137248   2   4   #solution,sNumber,nUnavoidables.67..................................76..........................................   4   n.........98................89....................................................   4   n.......................................95................59......................   4   n....8.9................98.................................98.....................   6   n376485912892713465145269837981376524467952381523841796214598673738624159659137248   3   4   #solution,sNumber,nUnavoidables.76..................................67..........................................   4   n.........89................98....................................................   4   n.......................................95................59......................   4   n....8.9................98.................................98.....................   6   n376485912982713465145269837891376524467952381523841796214598673738624159659137248   4   4   #solution,sNumber,nUnavoidables.76..................................67..........................................   4   n.........98................89....................................................   4   n.......................................95................59......................   4   n....8.9................98.................................98.....................   6   n367495812892713465145268937981376524476952381523841796214589673738624159659137248   5   5   #solution,sNumber,nUnavoidables.67..................................76..........................................   4   n.........89................98....................................................   4   n..........................................................8...3.............3...8   4   n....95.................................95................5.9.....................   6   n....9.8................89.................................89.....................   6   n367495812982713465145268937891376524476952381523841796214589673738624159659137248   6   5   #solution,sNumber,nUnavoidables.67..................................76..........................................   4   n.........98................89....................................................   4   n..........................................................8...3.............3...8   4   n....95.................................95................5.9.....................   6   n....9.8................89.................................89.....................   6   n376495812892713465145268937981376524467952381523841796214589673738624159659137248   7   5   #solution,sNumber,nUnavoidables.76..................................67..........................................   4   n.........89................98....................................................   4   n..........................................................8...3.............3...8   4   n....95.................................95................5.9.....................   6   n....9.8................89.................................89.....................   6   n376495812982713465145268937891376524467952381523841796214589673738624159659137248   8   5   #solution,sNumber,nUnavoidables.76..................................67..........................................   4   n.........98................89....................................................   4   n..........................................................8...3.............3...8   4   n....95.................................95................5.9.....................   6   n....9.8................89.................................89.....................   6   n367495812892713465145268937981376524476952381523841796214539678738624159659187243   9   4   #solution,sNumber,nUnavoidables.67..................................76..........................................   4   n.........89................98....................................................   4   n..........................................................3...8.............8...3   4   n....95.................................95................5.9.....................   6   n367495812982713465145268937891376524476952381523841796214539678738624159659187243   10   4   #solution,sNumber,nUnavoidables.67..................................76..........................................   4   n.........98................89....................................................   4   n..........................................................3...8.............8...3   4   n....95.................................95................5.9.....................   6   n376495812892713465145268937981376524467952381523841796214539678738624159659187243   11   4   #solution,sNumber,nUnavoidables.76..................................67..........................................   4   n.........89................98....................................................   4   n..........................................................3...8.............8...3   4   n....95.................................95................5.9.....................   6   n376495812982713465145268937891376524467952381523841796214539678738624159659187243   12   4   #solution,sNumber,nUnavoidables.76..................................67..........................................   4   n.........98................89....................................................   4   n..........................................................3...8.............8...3   4   n....95.................................95................5.9.....................   6   n367458912892713465145269837981376524476592381523841796214985673738624159659137248   13   5   #solution,sNumber,nUnavoidables.67..................................76..........................................   4   n....58....................................................85.....................   4   n.....89................98........................................................   4   n.........89................98....................................................   4   n..........................................................8...3.............3...8   4   n367458912982713465145269837891376524476592381523841796214985673738624159659137248   14   5   #solution,sNumber,nUnavoidables.67..................................76..........................................   4   n....58....................................................85.....................   4   n.....89................98........................................................   4   n.........98................89....................................................   4   n..........................................................8...3.............3...8   4   n367459812892713465145268937981376524476592381523841796214985673738624159659137248   15   5   #solution,sNumber,nUnavoidables.67..................................76..........................................   4   n.....98................89........................................................   4   n.........89................98....................................................   4   n..........................................................8...3.............3...8   4   n....59.................................59................9.5.....................   6   n367459812982713465145268937891376524476592381523841796214985673738624159659137248   16   5   #solution,sNumber,nUnavoidables.67..................................76..........................................   4   n.....98................89........................................................   4   n.........98................89....................................................   4   n..........................................................8...3.............3...8   4   n....59.................................59................9.5.....................   6   n376458912892713465145269837981376524467592381523841796214985673738624159659137248   17   5   #solution,sNumber,nUnavoidables.76..................................67..........................................   4   n....58....................................................85.....................   4   n.....89................98........................................................   4   n.........89................98....................................................   4   n..........................................................8...3.............3...8   4   n376458912982713465145269837891376524467592381523841796214985673738624159659137248   18   5   #solution,sNumber,nUnavoidables.76..................................67..........................................   4   n....58....................................................85.....................   4   n.....89................98........................................................   4   n.........98................89....................................................   4   n..........................................................8...3.............3...8   4   n376459812892713465145268937981376524467592381523841796214985673738624159659137248   19   5   #solution,sNumber,nUnavoidables.76..................................67..........................................   4   n.....98................89........................................................   4   n.........89................98....................................................   4   n..........................................................8...3.............3...8   4   n....59.................................59................9.5.....................   6   n376459812982713465145268937891376524467592381523841796214985673738624159659137248   20   5   #solution,sNumber,nUnavoidables.76..................................67..........................................   4   n.....98................89........................................................   4   n.........98................89....................................................   4   n..........................................................8...3.............3...8   4   n....59.................................59................9.5.....................   6   n367458912892713465145269837981376524476592381523841796214935678738624159659187243   21   4   #solution,sNumber,nUnavoidables.67..................................76..........................................   4   n.....89................98........................................................   4   n.........89................98....................................................   4   n..........................................................3...8.............8...3   4   n367458912982713465145269837891376524476592381523841796214935678738624159659187243   22   4   #solution,sNumber,nUnavoidables.67..................................76..........................................   4   n.....89................98........................................................   4   n.........98................89....................................................   4   n..........................................................3...8.............8...3   4   n367459812892713465145268937981376524476592381523841796214935678738624159659187243   23   5   #solution,sNumber,nUnavoidables.67..................................76..........................................   4   n.....98................89........................................................   4   n.........89................98....................................................   4   n..........................................................3...8.............8...3   4   n....59.................................59................9.5.....................   6   n367459812982713465145268937891376524476592381523841796214935678738624159659187243   24   5   #solution,sNumber,nUnavoidables.67..................................76..........................................   4   n.....98................89........................................................   4   n.........98................89....................................................   4   n..........................................................3...8.............8...3   4   n....59.................................59................9.5.....................   6   n376458912892713465145269837981376524467592381523841796214935678738624159659187243   25   4   #solution,sNumber,nUnavoidables.76..................................67..........................................   4   n.....89................98........................................................   4   n.........89................98....................................................   4   n..........................................................3...8.............8...3   4   n376458912982713465145269837891376524467592381523841796214935678738624159659187243   26   4   #solution,sNumber,nUnavoidables.76..................................67..........................................   4   n.....89................98........................................................   4   n.........98................89....................................................   4   n..........................................................3...8.............8...3   4   n376459812892713465145268937981376524467592381523841796214935678738624159659187243   27   5   #solution,sNumber,nUnavoidables.76..................................67..........................................   4   n.....98................89........................................................   4   n.........89................98....................................................   4   n..........................................................3...8.............8...3   4   n....59.................................59................9.5.....................   6   n376459812982713465145268937891376524467592381523841796214935678738624159659187243   28   5   #solution,sNumber,nUnavoidables.76..................................67..........................................   4   n.....98................89........................................................   4   n.........98................89....................................................   4   n..........................................................3...8.............8...3   4   n....59.................................59................9.5.....................   6   n367485912892713465145269837981376524476592381523841796214958673738624159659137248   29   4   #solution,sNumber,nUnavoidables.67..................................76..........................................   4   n....85....................................................58.....................   4   n.........89................98....................................................   4   n.......................................59................95......................   4   n367485912982713465145269837891376524476592381523841796214958673738624159659137248   30   4   #solution,sNumber,nUnavoidables.67..................................76..........................................   4   n....85....................................................58.....................   4   n.........98................89....................................................   4   n.......................................59................95......................   4   n376485912892713465145269837981376524467592381523841796214958673738624159659137248   31   4   #solution,sNumber,nUnavoidables.76..................................67..........................................   4   n....85....................................................58.....................   4   n.........89................98....................................................   4   n.......................................59................95......................   4   n376485912982713465145269837891376524467592381523841796214958673738624159659137248   32   4   #solution,sNumber,nUnavoidables.76..................................67..........................................   4   n....85....................................................58.....................   4   n.........98................89....................................................   4   n.......................................59................95......................   4   n893547612716283495245961783361475928489612357527839146954326871138754269672198534   33   1   #solution,sNumber,nUnavoidables8935476..71628..9.2..9.178336.4.59.8.8961..57..7.3914.95.3268.11..75.26..72.98534   55   y`
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`........1.....2.3...4..56........4...42.3..7..8.1....9..3.7..2..6...15..4..9....8   23   33   #pattern,nClues,nSolutions359786241876412935214395687935867412142539876687124359593678124768241593421953768   1   7   #solution,sNumber,nUnavoidables359........................935...................................................   6   n35.........................93.........................59.........................   6   n.59........................9.5........................59.........................   6   n...786........................867................................................   6   n...7.6........................8.7........................6.8.....................   6   n....86........................86.........................6.8.....................   6   n.........8.6.................................6.7...............7.8...............   6   n539786241876412935214395687395867412142539876687124359953678124768241593421953768   2   7   #solution,sNumber,nUnavoidables539........................395...................................................   6   n53.........................39.........................95.........................   6   n5.9.........................95........................95.........................   6   n...786........................867................................................   6   n...7.6........................8.7........................6.8.....................   6   n....86........................86.........................6.8.....................   6   n.........8.6.................................6.7...............7.8...............   6   n359786241678412935214395687935867412142539876786124359593678124867241593421953768   3   7   #solution,sNumber,nUnavoidables359........................935...................................................   6   n35.........................93.........................59.........................   6   n.59........................9.5........................59.........................   6   n...786........................867................................................   6   n...7.6........................8.7........................6.8.....................   6   n....86........................86.........................6.8.....................   6   n.........6.8.................................7.6...............8.7...............   6   n539786241678412935214395687395867412142539876786124359953678124867241593421953768   4   7   #solution,sNumber,nUnavoidables539........................395...................................................   6   n53.........................39.........................95.........................   6   n5.9.........................95........................95.........................   6   n...786........................867................................................   6   n...7.6........................8.7........................6.8.....................   6   n....86........................86.........................6.8.....................   6   n.........6.8.................................7.6...............8.7...............   6   n359687241876412935214395687935768412142539876687124359593876124768241593421953768   5   7   #solution,sNumber,nUnavoidables359........................935...................................................   6   n35.........................93.........................59.........................   6   n.59........................9.5........................59.........................   6   n...687........................768................................................   6   n...68..........................68........................8.6.....................   6   n...6.7........................7.8........................8.6.....................   6   n.........8.6.................................6.7...............7.8...............   6   n539687241876412935214395687395768412142539876687124359953876124768241593421953768   6   7   #solution,sNumber,nUnavoidables539........................395...................................................   6   n53.........................39.........................95.........................   6   n5.9.........................95........................95.........................   6   n...687........................768................................................   6   n...68..........................68........................8.6.....................   6   n...6.7........................7.8........................8.6.....................   6   n.........8.6.................................6.7...............7.8...............   6   n359687241678412935214395687935768412142539876786124359593876124867241593421953768   7   7   #solution,sNumber,nUnavoidables359........................935...................................................   6   n35.........................93.........................59.........................   6   n.59........................9.5........................59.........................   6   n...687........................768................................................   6   n...68..........................68........................8.6.....................   6   n...6.7........................7.8........................8.6.....................   6   n.........6.8.................................7.6...............8.7...............   6   n539687241678412935214395687395768412142539876786124359953876124867241593421953768   8   7   #solution,sNumber,nUnavoidables539........................395...................................................   6   n53.........................39.........................95.........................   6   n5.9.........................95........................95.........................   6   n...687........................768................................................   6   n...68..........................68........................8.6.....................   6   n...6.7........................7.8........................8.6.....................   6   n.........6.8.................................7.6...............8.7...............   6   n359867241876412935214395687935786412142539876687124359593678124768241593421953768   9   7   #solution,sNumber,nUnavoidables359........................935...................................................   6   n35.........................93.........................59.........................   6   n.59........................9.5........................59.........................   6   n...867........................786................................................   6   n...86..........................86........................6.8.....................   6   n...8.7........................7.6........................6.8.....................   6   n.........8.6.................................6.7...............7.8...............   6   n539867241876412935214395687395786412142539876687124359953678124768241593421953768   10   7   #solution,sNumber,nUnavoidables539........................395...................................................   6   n53.........................39.........................95.........................   6   n5.9.........................95........................95.........................   6   n...867........................786................................................   6   n...86..........................86........................6.8.....................   6   n...8.7........................7.6........................6.8.....................   6   n.........8.6.................................6.7...............7.8...............   6   n359867241678412935214395687935786412142539876786124359593678124867241593421953768   11   7   #solution,sNumber,nUnavoidables359........................935...................................................   6   n35.........................93.........................59.........................   6   n.59........................9.5........................59.........................   6   n...867........................786................................................   6   n...86..........................86........................6.8.....................   6   n...8.7........................7.6........................6.8.....................   6   n.........6.8.................................7.6...............8.7...............   6   n539867241678412935214395687395786412142539876786124359953678124867241593421953768   12   7  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 n5.3........................3.9........................9.5........................   6   n...867........................786................................................   6   n...86..........................86........................6.8.....................   6   n...8.7........................7.6........................6.8.....................   6   n.........8.6.................................6.7...............7.8...............   6   n593867241678412935214395687359786412142539876786124359935678124867241593421953768   14   7   #solution,sNumber,nUnavoidables593........................359...................................................   6   n59..........................59........................9.5........................   6   n5.3........................3.9........................9.5........................   6   n...867........................786................................................   6   n...86..........................86........................6.8.....................   6   n...8.7........................7.6........................6.8.....................   6   n.........6.8.................................7.6...............8.7...............   6   n593768241876412935214395687359687412142539876687124359935876124768241593421953768   15   7   #solution,sNumber,nUnavoidables593........................359...................................................   6   n59..........................59........................9.5........................   6   n5.3........................3.9........................9.5........................   6   n...768........................687................................................   6   n...7.8........................6.7........................8.6.....................   6   n....68........................68.........................8.6.....................   6   n.........8.6.................................6.7...............7.8...............   6   n593768241678412935214395687359687412142539876786124359935876124867241593421953768   16   7   #solution,sNumber,nUnavoidables593........................359...................................................   6   n59..........................59........................9.5........................   6   n5.3........................3.9........................9.5........................   6   n...768........................687................................................   6   n...7.8........................6.7........................8.6.....................   6   n....68........................68.........................8.6.....................   6   n.........6.8.................................7.6...............8.7...............   6   n359786241876412935214395687593867412142539876687124359935678124768241593421953768   17   7   #solution,sNumber,nUnavoidables359........................593...................................................   6   n3.9........................5.3........................9.5........................   6   n.59........................59.........................9.5........................   6   n...786........................867................................................   6   n...7.6........................8.7........................6.8.....................   6   n....86........................86.........................6.8.....................   6   n.........8.6.................................6.7...............7.8...............   6   n359786241678412935214395687593867412142539876786124359935678124867241593421953768   18   7   #solution,sNumber,nUnavoidables359........................593...................................................   6   n3.9........................5.3........................9.5........................   6   n.59........................59.........................9.5........................   6   n...786........................867................................................   6   n...7.6........................8.7........................6.8.....................   6   n....86........................86.........................6.8.....................   6   n.........6.8.................................7.6...............8.7...............   6   n359687241876412935214395687593768412142539876687124359935876124768241593421953768   19   7   #solution,sNumber,nUnavoidables359........................593...................................................   6   n3.9........................5.3........................9.5........................   6   n.59........................59.........................9.5........................   6   n...687........................768................................................   6   n...68..........................68........................8.6.....................   6   n...6.7........................7.8........................8.6.....................   6   n.........8.6.................................6.7...............7.8...............   6   n359687241678412935214395687593768412142539876786124359935876124867241593421953768   20   7   #solution,sNumber,nUnavoidables359........................593...................................................   6   n3.9........................5.3........................9.5........................   6   n.59........................59.........................9.5........................   6   n...687........................768................................................   6   n...68..........................68........................8.6.....................   6   n...6.7........................7.8........................8.6.....................   6   n.........6.8.................................7.6...............8.7...............   6   n359867241876412935214395687593786412142539876687124359935678124768241593421953768   21   7   #solution,sNumber,nUnavoidables359........................593...................................................   6   n3.9........................5.3........................9.5........................   6   n.59........................59.........................9.5........................   6   n...867........................786................................................   6   n...86..........................86........................6.8.....................   6   n...8.7........................7.6........................6.8.....................   6   n.........8.6.................................6.7...............7.8...............   6   n359867241678412935214395687593786412142539876786124359935678124867241593421953768   22   7   #solution,sNumber,nUnavoidables359........................593...................................................   6   n3.9........................5.3........................9.5........................   6   n.59........................59.........................9.5........................   6   n...867........................786................................................   6   n...86..........................86........................6.8.....................   6   n...8.7........................7.6........................6.8.....................   6   n.........6.8.................................7.6...............8.7...............   6   n359768241876412935214395687593687412142539876687124359935876124768241593421953768   23   7   #solution,sNumber,nUnavoidables359........................593...................................................   6   n3.9........................5.3........................9.5........................   6   n.59........................59.........................9.5........................   6   n...768........................687................................................   6   n...7.8........................6.7........................8.6.....................   6   n....68........................68.........................8.6.....................   6   n.........8.6.................................6.7...............7.8...............   6   n359768241678412935214395687593687412142539876786124359935876124867241593421953768   24   7   #solution,sNumber,nUnavoidables359........................593...................................................   6   n3.9........................5.3........................9.5........................   6   n.59........................59.........................9.5........................   6   n...768........................687................................................   6   n...7.8........................6.7........................8.6.....................   6   n....68........................68.........................8.6.....................   6   n.........6.8.................................7.6...............8.7...............   6   n953786241876412935214395687395867412142539876687124359539678124768241593421953768   25   7   #solution,sNumber,nUnavoidables953........................395...................................................   6   n95..........................95........................5.9........................   6   n9.3........................3.5........................5.9........................   6   n...786........................867................................................   6   n...7.6........................8.7........................6.8.....................   6   n....86........................86.........................6.8.....................   6   n.........8.6.................................6.7...............7.8...............   6   n953786241678412935214395687395867412142539876786124359539678124867241593421953768   26   7   #solution,sNumber,nUnavoidables953........................395...................................................   6   n95..........................95........................5.9........................   6   n9.3........................3.5........................5.9........................   6   n...786........................867................................................   6   n...7.6........................8.7........................6.8.....................   6   n....86........................86.........................6.8.....................   6   n.........6.8.................................7.6...............8.7...............   6   n953687241876412935214395687395768412142539876687124359539876124768241593421953768   27   7   #solution,sNumber,nUnavoidables953........................395...................................................   6   n95..........................95........................5.9........................   6   n9.3........................3.5........................5.9........................   6   n...687........................768................................................   6   n...68..........................68........................8.6.....................   6   n...6.7........................7.8........................8.6.....................   6   n.........8.6.................................6.7...............7.8...............   6   n953687241678412935214395687395768412142539876786124359539876124867241593421953768   28   7   #solution,sNumber,nUnavoidables953........................395...................................................   6   n95..........................95........................5.9........................   6   n9.3........................3.5........................5.9........................   6   n...687........................768................................................   6   n...68..........................68........................8.6.....................   6   n...6.7........................7.8........................8.6.....................   6   n.........6.8.................................7.6...............8.7...............   6   n953867241876412935214395687395786412142539876687124359539678124768241593421953768   29   7   #solution,sNumber,nUnavoidables953........................395...................................................   6   n95..........................95........................5.9........................   6   n9.3........................3.5........................5.9........................   6   n...867........................786................................................   6   n...86..........................86........................6.8.....................   6   n...8.7........................7.6........................6.8.....................   6   n.........8.6.................................6.7...............7.8...............   6   n953867241678412935214395687395786412142539876786124359539678124867241593421953768   30   7   #solution,sNumber,nUnavoidables953........................395...................................................   6   n95..........................95........................5.9........................   6   n9.3........................3.5........................5.9........................   6   n...867........................786................................................   6   n...86..........................86........................6.8.....................   6   n...8.7........................7.6........................6.8.....................   6   n.........6.8.................................7.6...............8.7...............   6   n953768241876412935214395687395687412142539876687124359539876124768241593421953768   31   7   #solution,sNumber,nUnavoidables953........................395...................................................   6   n95..........................95........................5.9........................   6   n9.3........................3.5........................5.9........................   6   n...768........................687................................................   6   n...7.8........................6.7........................8.6.....................   6   n....68........................68.........................8.6.....................   6   n.........8.6.................................6.7...............7.8...............   6   n953768241678412935214395687395687412142539876786124359539876124867241593421953768   32   7   #solution,sNumber,nUnavoidables953........................395...................................................   6   n95..........................95........................5.9........................   6   n9.3........................3.5........................5.9........................   6   n...768........................687................................................   6   n...7.8........................6.7........................8.6.....................   6   n....68........................68.........................8.6.....................   6   n.........6.8.................................7.6...............8.7...............   6   n628493751195762834374815692716259483942638175583147269831574926269381547457926318   33   1   #solution,sNumber,nUnavoidables62849375.19576.8.437.81..92716259.839..6.81.55.3.4726.8.15.49.62.938..47.57.2631.   58   y`
dobrichev
2016 Supporter

Posts: 1777
Joined: 24 May 2010

### Re: 33

Hi, dobrichev!
dobrichev wrote:Here are 3 monsters with 33 permutations.

Glad to see your new impressive results. New minimal UA set valency record is set. Congratulations!
I need some time to cross-check your results.

Serg
Serg
2018 Supporter

Posts: 690
Joined: 01 June 2010
Location: Russia

### Re: Minimal Unavoidable Set with 13 permutations

I cross-checked your just published 33-valent UA sets (one UA55 and 2 UA58). All are weakly minimal UA sets. Well done!
It is interesting - can anyone set theoretically high limits to minimal UA set size and minimal UA set valency?

Serg
Serg
2018 Supporter

Posts: 690
Joined: 01 June 2010
Location: Russia

### UA limits

Hi Serg,
Thank you for the confirmation.
Serg wrote:It is interesting - can anyone set theoretically high limits to minimal UA set size and minimal UA set valency?

The known low-clue puzzles are a natural upper limit to the possible UA size.
If there is no valid 16 clue puzzle, then the absolute maximum is 81-(17-1)=65, i.e. a 16 clue multiple solution puzzle in which all 65 non-givens form a minimal UA set and respectively a family of 65 valid 17s with 16 common clues. There is no such animal too.
Once I checked the known 17s for extremely large UA and found nothing. Actually I compared 16s only to the respective known solution grid.

For high valency limit I have no idea at all. Once I read a discussion where someone said that the existence of 17 clue puzzles is in fact a consequence of the huge amount of the possible combinations and not so much a result of a systematical clue placements. The practice suggests that this is very likely to be true, and also applicable to many other exotic phenomenons like weakly minimal UA sets.
I can't see a reason for non-existence of some large UA which is decomposed to several UA having high valency itself, this reaching some large number. There are some limitations coming from overall size, but the absence of requirement other UA to be disjoint complicates the calculations.
As expected, statistically the valency trends to be 1 + any multiple of 2 and 3. Each disjoint "regular" UA adds 2 choices, and two joint UA add 3 choices. Below is the valency distribution at some stage of large UA generation process.
Hidden Text: Show
Code: Select all
`  Count Exemplar                                                                            Size Valency     42 ................12..3..4........53...1..6....2..7.......5..34...7.2...8.6...1...9   19   2      2 ........1.......2...3..4..5.....34.6.7..1....8..2.......6...3.9.2.8.....1...7....   19   3   7240 .................1..2..3.45.....256..7..1.8..9..8.......5..63...1..9....8..7.....   20   2    496 .................1..2..3.4......2.56.1..7...89..8..7....5..6.2..8.1.....7...9....   20   3     66 ................12..3..45.......3.56.1..7...82..8.......6..5.3..8.2.....7...1....   20   4     64 ................12..3..4........56....4.7...3.8.1...5...6..3..4.1..5..8.2.....7..   20   5      6 ..............1..2..3....4...2.4.....5...6..17..3....8..4.8..3..6...5...1..7...9.   20   6      2 ..............1.23..4..256...1....4..7..3....8..5.......6..4.1..3......85.....7..   20   7      5 ................12..3..45......1...6...2...7..8...54....1.6.....2.7.....4.5..83..   20   9      1 ........1.....2.3...4..56......1...4..2.7.5...8...3.7...94..2...3.....8.5........   20   13 179576 .................1...234.5...1..6.....27....8.5..9..3...61..2...3..5....9....8.4.   21   2  27323 .................1...234........5.2...1.6...7.8.9..4....3.8...62..1...9.4....3.5.   21   3   3881 .................1..1..2345.....1.53.4..6..7.8..7.......5..3..2.6.8.....7...4....   21   4   1713 .................1....23.4....5....6..1..78...4..9..3...61....5.2..4..9.3....87..   21   5    240 .................1..1..2345.....1.56.4..7..8.9..8.......5..6..2.7.9.....8...4....   21   6    258 .................1..2..3.45.....24.6.1..7..8.9..8....7..3..62...8..1....7..9.....   21   7     22 ..............1..2.23....4...1.5.62..7..4..3.6....8.....53......4..7....8....9..1   21   8     41 ................12..3..45......1...6..5..73...7.2...8..1..8.....68..54..2..6.....   21   9      9 ..............1.23..4.5.6.......2..7..8.6.5...1.3...9...7.4.8...6...7.1.3........   21   10     10 ...............123..1..45.6..4..7..1.8..2....9..3.......7..6.1..3..8....2..9.....   21   11      2 ........1.......2...3.14.........5...1..36..42..7...1.....2...3..4..8..6.8.9...7.   21   12      7 ..............1.23..4.56.........7...1...2.6.5.3.....4..7.4...6.8.....1.2...9...5   21   13      3 ..............1..2.34....5.....3.....652...4.7....4..1..15....4.8.6...3.9....7...   21   15 635661 .................1.....234...1.5...2.3.6...7.7....89....23..8....41...6.9...7...5   22   2 207134 .................1.....234...1.5...6.2.3..7..8....4.9...4..9.2..1.7..8...3..6...5   22   3  27223 .................1....2345...14..2....6..7..3.5.8...9..9...5.7.3.......84...1.6..   22   4  12144 .................1.....234...1.5...6.2...7.8.3..4..9....8.6....4..1....59.6..32..   22   5   1439 .................1..1..2345.....1.56.4..7..8.9...8......5.36..2.7.9.....8...4....   22   6   1772 .................1.....234...1.3...5.4.6...2.7....89....91...3..2...78..5...4...6   22   7    139 .................1..2..345.........2..3..46....617..8..1..2...74....69..5..8...3.   22   8    288 .................1..2..345........6..1..7...88..9.23.....1...9...36..2..5...89..4   22   9     76 .................1.....234...1.5...6.3...78..4..9...2...81...5..6..4...92....37..   22   10     43 .................1..2..345.....1......3..6.2..7.4.5..8.1.8....64...7.3..5....2.9.   22   11      2 ..............1..2..3.4.56......2.....4.5..3..1.7..6....5.3..4...78..1..2....69..   22   12     48 ................12..1.345....6..17...2..5..4.8.......9..79..3...4.6......5..2...8   22   13      3 ..............1..2..3.4..5...16...3..4..7....7....2..8..6......2....9.7181..3..4.   22   14      4 ..............1.23..4.5.6....7..6..4.1....8..5....2.9...9.4...7.2.9..1..8......5.   22   15      3 ..............1..2.34....5...1..6..46...5.7..8..9...3...6.74..1.5.3.....4.....2..   22   16      5 ..............1..2..3....4...4.3..2...56..1...7...8..5..9.4..3..1...9..85..7..6..   22   17      1 ........1.....2.....3.4.56...17....8.5...92..6...3...4.4...5....9.....3.2...6.4..   22   2510704662 .................1.....1234..1..4..5.6..2....7..3......3..7.6..2..6...7.674..5..8   23   22840882 .................1.....234...1....5..2...36..7.648...9..3..67...1.5....24...9..8.   23   3 347243 .................1.....234...1.5...2.3...6.7.8..9..6....6.1.....24..38.5.7...4.9.   23   4 162365 .................1.....234...1.2...5.6...7.2.23.8..9....2.1...4.8..5..6.9....37..   23   5  16963 .................1....23.45..16..5...2..7...84....9.3...51...6..7..9...23....4.5.   23   6  20973 .................1.....234...1.....525...36..67..4..8...58....9.4..1..7.3..6..2..   23   7   1636 .................1....2345...1..6..3.4..7.2..5..8...9...3..5..6.2..4.8..9..1...7.   23   8   4070 .................1.....234...1.5..6..6...78.273.6.......6.1...5.8.3..9..2....4.7.   23   9    703 .................1...234.5...1..6....7..8..9.58.4..2....4.1...6.9...7.8.2..5..3..   23   10    598 .................1....2345...1..6..2..4.7.3...5.8...1...94...8..2..5.7..3....9..6   23   11     73 ................12...134.....25....3.3...16...7..8..4...3.2..8..4...67..7..9....5   23   12    359 .................1...12345...16....7.4....2..8....9.3...57....62...4.8..3....1.9.   23   13     19 ..............1..2....3.45...1.4.6...3...2..78..5..9...9...7..33.8.9..1.5.....8..   23   14     38 .................1...23456...1..5..4.6..7....8..3..2....4..1..3.8..6..7.2..8..9..   23   15     13 ................12.....34.5..1..43...2..6..7.8..9...5...37....1.6..2...94....85..   23   16     40 .................1..1.2345....4..6....2.5..7..3...8..9..5.1..2..7...9..34..6..8..   23   17      3 ..............1..2..3...45...1..2.6...7.4.8...5.9....3..2..6.1..3.5....94...8.7..   23   18     12 ................12.....13....2..4..5..6.3.1...7.1...8...34.26...9.8...4.4....5..7   23   19      1 ........1....23.....45..67...1..8.3..6..4.2..7..9....5..3..7.9.4..1.....9...6....   23   20      4 ................12..3..45......6...7.2.1...6.68...74....6.5.9...7.2....84....1.3.   23   21      2 ..............1..2....2.34...35.....14.....6.27....8.9..9.3...5.8.2..7..6....4.1.   23   23      2 ..............1.23245..617...3..8....6..7...21......4...8..3..1.7..6....9..4.....   23   25      2 ..............1..2....3.45...1.6..3..5......78.7..49....3..9.1..2..7...64..5..8..   23   27      1 ..............1..2....3.45...3.2..6..7...41..8..5....9.14...7...26....385...9....   23   31      2 ........1.....2.3...4..56........4...42.3..7..8.1....9..3.7..2..6...15..4..9....8   23   33`

For UA58 distribution becomes linear up to valency of 23 only after log(log(num_ua)) transformation is applied, further it asymptotically makes zig-zag between 0..few..0..few for even and odd valency respectively.
There could exist some odd/even affinity in large UA sizes too.

Currently my collection consists of 69 UA62, 14972 UA61, and 737701 UA60 generating pseudo-puzzles.
The actual count of the UA sets is less than two times the number of the generation puzzles due to the morphs and weakly minimals.

In more details, for UA62 there are 69 generating pseudo-puzzles forming 101 UA62 sets in 97 solution grids. There are 4 grids with 2 UA62 each. UA62 occupy 41 different patterns with a maximum of 7 UA per pattern. Below are the patterns.
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`      7 ........1..1..1....1..1..1......11...1.......1...1...1....1..1...1...1...1.1....1      5 ..............1111..1..1..1.....1..1.1..1....1..1.........1..1....1..1....1.....1      4 ...............111..1..1..1.....1..1.1..1....1..1.........1..1....1..1....1..1..1      3 ........1..1..1....1..1..1......11...1.......1...1...1....1......1..11...1.1....1      3 ........1.......1...1.11........11...1......11......1...11.11...1..1....1...1....      3 ........1.......1...1..1.......1...1...1...1...1..11...1..1.....1.1.....11...11..      3 ...........1..1..1.1..1..1............1..1..1.1..1..1.......1....1..1..1.1..1..1.      3 ...............111..1..1.....1..1..1.1..1....1..1.......1..1..1.1.1.....1...1....      2 ........1.......1...1.11.....1..11...1......11......1...11..1...1..1....1...1....      2 ........1.......1...1..1.......1...1...1...1.11....1...1..1.....1.1......11..11..      2 ..............1111..1..1..1....1..1....1..1....1.....1..1.....1.1..1....1..1.....      2 ...............111..1..1.......1...1...1...1...1..11....1..11...1..1....1..1.....      2 ................11..1..1........11...1..1....1..1.......1..11...1.1....11...1..1.      1 .......11.....11..111...........11....1.1.....1.1.........1...1...1...1.1.....1..      1 ........1..1..1....1..1..1.........1..1..1...1..1...1.......1....1..1..1.1.1..1..      1 ........1.....1.1...1.1..........1...1..1..1.1..1....1...1....1..1....1..11..1...      1 ........1.....1.1...1..11.......111..1..1....1..1.......1...1...1..1....1..1.....      1 ........1.......11..1..1.....1..11...1..1....1..1.......1..11...1.1.....1...1....      1 ........1.......11..1..1.....1..1.1..1..1....1..1.......1..11...1.1.....1...1....      1 ........1.......1...1.111.......11...1......11......1...11..1...1..1....1...1....      1 ........1.......1...1.11.....11..1...1......11......1...11..1...1...1...1....1...      1 ........1.......1...1.11........11...1......11......1...111.1...1..1....1...1....      1 ........1.......1...1..11......1...1...1...1..1...11..1....11..1...1....1..1.....      1 ........1.......1...1..1..1.....11.1.1..1....1..1.......1...1.1.1.1.....1...1....      1 ........1.......1...1..1.......1...1...1...1..1...11..1...1....1..1.....1.1..11..      1 ........1........1..1..1..1..1..1.1..1..1....1..1.......1..11...1.1.....1...1....      1 ........1........1..1..1..1..1..1.1..1..1....1..1.......1..11...1..1....1..1.....      1 ........1........1..1..1..1.....111..1..1....1..1.......1..1.1..1.1.....1...1....      1 ........1........1..1..1..1.....111..1..1....1..1.......1..1.1..1..1....1..1.....      1 ........1........1..1..1........1.1..1..1..1.1..1...1...1..11...1..1....1..1.....      1 ...........1..1..1.1..1.11.....1..1....1.......1..1..1..1..1..1.1.....1.1........      1 ...........1..1..1.1..1..1.........1..11......1..1..1...11....1.1..1..1.1........      1 ...........1..1..1.1..1..1.........1..1..11...1..1.........11....11....1.1.....1.      1 ...........1..1..1.1..1..1............1..1..1.1..1.1.....1.......1..1..1.1..1..1.      1 ..............1111..1..1..1.....1..1.1.....1.1.....1....1.....1.1..1....1..1.....      1 ..............1.11..1.1.1.....1....1..1...1...1...1....1.....1.1.......11...1.1..      1 ..............1.11..1..11....1...1...1..1....1..1.......1..11...1..1....1..1.....      1 ..............1..1111.....1....1..1....1..1....1..1..1..1..1....1.....1.1.....1..      1 ..............1..1..1.1..1......1..1..11..1...1.........1.1.1...1...1..11..1.....      1 ...............111..1.11......1....1.1...1...1...1......11....1.1.....1.1.....1..      1 ...............111..1..1.....1..1..1.1..1....1..1.......1..1..1.1..1....1..1.....`

There is no much chance UA63 to appear, but who knows...

Cheers,
MD
dobrichev
2016 Supporter

Posts: 1777
Joined: 24 May 2010

### Re: UA limits

Hi, dobrichev!
dobrichev wrote:The known low-clue puzzles are a natural upper limit to the possible UA size.
If there is no valid 16 clue puzzle, then the absolute maximum is 81-(17-1)=65, i.e. a 16 clue multiple solution puzzle in which all 65 non-givens form a minimal UA set and respectively a family of 65 valid 17s with 16 common clues.

I think your consideration relates to strongly minimal UA sets, which can be destroyed by single hit. But weakly minimal UA sets can require several hits to be destroyed. So, weakly minimal UA set size can exceed 65. Maybe I am wrong.

Serg
Serg
2018 Supporter

Posts: 690
Joined: 01 June 2010
Location: Russia

### Re: UA limits

Serg wrote:... But weakly minimal UA sets can require several hits to be destroyed...

One of the permutations of the weakly minimal UA must still require one (and only one) hit to be destroyed.
dobrichev
2016 Supporter

Posts: 1777
Joined: 24 May 2010

### 2^5+1=33 permutations

This pseudo-puzzle has 33 solutions caused by a weakly-minimal UA55 which is decomposed to 5 disjoint UA4.
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`.......12..3..14...1.56.73...8.9.....4.2...5.7....3..1..9.8.3...2.4..1..6....7...`

Pencilmarks and unavoidable rectangles
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`548 879 76  | 397 43  984 | 65  1   225  697 3   | 879 72  1   | 4   986 568984 1   42  | 5   6   248 | 7   3   89------------+-------------+------------12  53  8   | 71  9   45  | 26  674 34739  4   61  | 2   17  86  | 98  5   737   96  25  | 68  54  3   | 829 492 1------------+-------------+------------41  75  9   | 16  8   52  | 3   247 67483  2   57  | 4   35  69  | 1   768 9866   38  14  | 93  21  7   | 592 829 45Five UA4 at:       r13c16, r12c24, r28c89, r47c89, r69c78Bi-value cells:    UA55 value, the single alternate valueThree-value cells: UA55 value, UA4 first permutation value, UA4 second permutation`

The unavoidable rectangles occupy 5*4=20 cells and their 32 solution grids have the same values in the rest 61 cells - givens plus the second value of the bi-value cells.
Additionally, there exists one solution grid which has the same values in 26 of these 61 cells.
So, this 26-clue pseudo-puzzle has 33 solutions in total.
In order to define a grid uniquely (and this to become a valid puzzle), either one more clue must be added (the leftmost one from the pencilmarks above), or 5 more clues must be added (one per UA4, chosing between second and third value from the three-value cells).
Below is the list of all 33 solutions and respective UA.
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`.......12..3..14...1.56.73...8.9.....4.2...5.7....3..1..9.8.3...2.4..1..6....7...   26   33   #pattern,nClues,nSolutions587349612263871495914562738158794263346218957792653841479185326825436179631927584   1   1   #solution,sNumber,nUnavoidables5873496..26.87..959.4..2..815.7.42633.6.189.7.9265.84.47.1.5.268.5.36.79.3192.584   55   y476938512593721486812564739238195674941276853765843291159682347327459168684317925   2   5   #solution,sNumber,nUnavoidables4....8............8....4.........................................................   4   n.7.9......9.7....................................................................   4   n................86....................................................68.........   4   n..................................74.........................47..................   4   n...................................................29.........................92.   4   n496738512573921486812564739238195674941276853765843291159682347327459168684317925   3   5   #solution,sNumber,nUnavoidables4....8............8....4.........................................................   4   n.9.7......7.9....................................................................   4   n................86....................................................68.........   4   n..................................74.........................47..................   4   n...................................................29.........................92.   4   n876934512593721486412568739238195674941276853765843291159682347327459168684317925   4   5   #solution,sNumber,nUnavoidables8....4............4....8.........................................................   4   n.7.9......9.7....................................................................   4   n................86....................................................68.........   4   n..................................74.........................47..................   4   n...................................................29.........................92.   4   n896734512573921486412568739238195674941276853765843291159682347327459168684317925   5   5   #solution,sNumber,nUnavoidables8....4............4....8.........................................................   4   n.9.7......7.9....................................................................   4   n................86....................................................68.........   4   n..................................74.........................47..................   4   n...................................................29.........................92.   4   n476938512593721468812564739238195674941276853765843291159682347327459186684317925   6   5   #solution,sNumber,nUnavoidables4....8............8....4.........................................................   4   n.7.9......9.7....................................................................   4   n................68....................................................86.........   4   n..................................74.........................47..................   4   n...................................................29.........................92.   4   n496738512573921468812564739238195674941276853765843291159682347327459186684317925   7   5   #solution,sNumber,nUnavoidables4....8............8....4.........................................................   4   n.9.7......7.9....................................................................   4   n................68....................................................86.........   4   n..................................74.........................47..................   4   n...................................................29.........................92.   4   n876934512593721468412568739238195674941276853765843291159682347327459186684317925   8   5   #solution,sNumber,nUnavoidables8....4............4....8.........................................................   4   n.7.9......9.7....................................................................   4   n................68....................................................86.........   4   n..................................74.........................47..................   4   n...................................................29.........................92.   4   n896734512573921468412568739238195674941276853765843291159682347327459186684317925   9   5   #solution,sNumber,nUnavoidables8....4............4....8.........................................................   4   n.9.7......7.9....................................................................   4   n................68....................................................86.........   4   n..................................74.........................47..................   4   n...................................................29.........................92.   4   n476938512593721486812564739238195647941276853765843291159682374327459168684317925   10   5   #solution,sNumber,nUnavoidables4....8............8....4.........................................................   4   n.7.9......9.7....................................................................   4   n................86....................................................68.........   4   n..................................47.........................74..................   4   n...................................................29.........................92.   4   n496738512573921486812564739238195647941276853765843291159682374327459168684317925   11   5   #solution,sNumber,nUnavoidables4....8............8....4.........................................................   4   n.9.7......7.9....................................................................   4   n................86....................................................68.........   4   n..................................47.........................74..................   4   n...................................................29.........................92.   4   n876934512593721486412568739238195647941276853765843291159682374327459168684317925   12   5   #solution,sNumber,nUnavoidables8....4............4....8.........................................................   4   n.7.9......9.7....................................................................   4   n................86....................................................68.........   4   n..................................47.........................74..................   4   n...................................................29.........................92.   4   n896734512573921486412568739238195647941276853765843291159682374327459168684317925   13   5   #solution,sNumber,nUnavoidables8....4............4....8.........................................................   4   n.9.7......7.9....................................................................   4   n................86....................................................68.........   4   n..................................47.........................74..................   4   n...................................................29.........................92.   4   n476938512593721468812564739238195647941276853765843291159682374327459186684317925   14   5   #solution,sNumber,nUnavoidables4....8............8....4.........................................................   4   n.7.9......9.7....................................................................   4   n................68....................................................86.........   4   n..................................47.........................74..................   4   n...................................................29.........................92.   4   n496738512573921468812564739238195647941276853765843291159682374327459186684317925   15   5   #solution,sNumber,nUnavoidables4....8............8....4.........................................................   4   n.9.7......7.9....................................................................   4   n................68....................................................86.........   4   n..................................47.........................74..................   4   n...................................................29.........................92.   4   n876934512593721468412568739238195647941276853765843291159682374327459186684317925   16   5   #solution,sNumber,nUnavoidables8....4............4....8.........................................................   4   n.7.9......9.7....................................................................   4   n................68....................................................86.........   4   n..................................47.........................74..................   4   n...................................................29.........................92.   4   n896734512573921468412568739238195647941276853765843291159682374327459186684317925   17   5   #solution,sNumber,nUnavoidables8....4............4....8.........................................................   4   n.9.7......7.9....................................................................   4   n................68....................................................86.........   4   n..................................47.........................74..................   4   n...................................................29.........................92.   4   n476938512593721486812564739238195674941276853765843921159682347327459168684317295   18   5   #solution,sNumber,nUnavoidables4....8............8....4.........................................................   4   n.7.9......9.7....................................................................   4   n................86....................................................68.........   4   n..................................74.........................47..................   4   n...................................................92.........................29.   4   n496738512573921486812564739238195674941276853765843921159682347327459168684317295   19   5   #solution,sNumber,nUnavoidables4....8............8....4.........................................................   4   n.9.7......7.9....................................................................   4   n................86....................................................68.........   4   n..................................74.........................47..................   4   n...................................................92.........................29.   4   n876934512593721486412568739238195674941276853765843921159682347327459168684317295   20   5   #solution,sNumber,nUnavoidables8....4............4....8.........................................................   4   n.7.9......9.7....................................................................   4   n................86....................................................68.........   4   n..................................74.........................47..................   4   n...................................................92.........................29.   4   n896734512573921486412568739238195674941276853765843921159682347327459168684317295   21   5   #solution,sNumber,nUnavoidables8....4............4....8.........................................................   4   n.9.7......7.9....................................................................   4   n................86....................................................68.........   4   n..................................74.........................47..................   4   n...................................................92.........................29.   4   n476938512593721468812564739238195674941276853765843921159682347327459186684317295   22   5   #solution,sNumber,nUnavoidables4....8............8....4.........................................................   4   n.7.9......9.7....................................................................   4   n................68....................................................86.........   4   n..................................74.........................47..................   4   n...................................................92.........................29.   4   n496738512573921468812564739238195674941276853765843921159682347327459186684317295   23   5   #solution,sNumber,nUnavoidables4....8............8....4.........................................................   4   n.9.7......7.9....................................................................   4   n................68....................................................86.........   4   n..................................74.........................47..................   4   n...................................................92.........................29.   4   n876934512593721468412568739238195674941276853765843921159682347327459186684317295   24   5   #solution,sNumber,nUnavoidables8....4............4....8.........................................................   4   n.7.9......9.7....................................................................   4   n................68....................................................86.........   4   n..................................74.........................47..................   4   n...................................................92.........................29.   4   n896734512573921468412568739238195674941276853765843921159682347327459186684317295   25   5   #solution,sNumber,nUnavoidables8....4............4....8.........................................................   4   n.9.7......7.9....................................................................   4   n................68....................................................86.........   4   n..................................74.........................47..................   4   n...................................................92.........................29.   4   n476938512593721486812564739238195647941276853765843921159682374327459168684317295   26   5   #solution,sNumber,nUnavoidables4....8............8....4.........................................................   4   n.7.9......9.7....................................................................   4   n................86....................................................68.........   4   n..................................47.........................74..................   4   n...................................................92.........................29.   4   n496738512573921486812564739238195647941276853765843921159682374327459168684317295   27   5   #solution,sNumber,nUnavoidables4....8............8....4.........................................................   4   n.9.7......7.9....................................................................   4   n................86....................................................68.........   4   n..................................47.........................74..................   4   n...................................................92.........................29.   4   n876934512593721486412568739238195647941276853765843921159682374327459168684317295   28   5   #solution,sNumber,nUnavoidables8....4............4....8.........................................................   4   n.7.9......9.7....................................................................   4   n................86....................................................68.........   4   n..................................47.........................74..................   4   n...................................................92.........................29.   4   n896734512573921486412568739238195647941276853765843921159682374327459168684317295   29   5   #solution,sNumber,nUnavoidables8....4............4....8.........................................................   4   n.9.7......7.9....................................................................   4   n................86....................................................68.........   4   n..................................47.........................74..................   4   n...................................................92.........................29.   4   n476938512593721468812564739238195647941276853765843921159682374327459186684317295   30   5   #solution,sNumber,nUnavoidables4....8............8....4.........................................................   4   n.7.9......9.7....................................................................   4   n................68....................................................86.........   4   n..................................47.........................74..................   4   n...................................................92.........................29.   4   n496738512573921468812564739238195647941276853765843921159682374327459186684317295   31   5   #solution,sNumber,nUnavoidables4....8............8....4.........................................................   4   n.9.7......7.9....................................................................   4   n................68....................................................86.........   4   n..................................47.........................74..................   4   n...................................................92.........................29.   4   n876934512593721468412568739238195647941276853765843921159682374327459186684317295   32   5   #solution,sNumber,nUnavoidables8....4............4....8.........................................................   4   n.7.9......9.7....................................................................   4   n................68....................................................86.........   4   n..................................47.........................74..................   4   n...................................................92.........................29.   4   n896734512573921468412568739238195647941276853765843921159682374327459186684317295   33   5   #solution,sNumber,nUnavoidables8....4............4....8.........................................................   4   n.9.7......7.9....................................................................   4   n................68....................................................86.........   4   n..................................47.........................74..................   4   n...................................................92.........................29.   4   n`
dobrichev
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### Re: 2^5+1=33 permutations

Hi, dobrichev!
dobrichev wrote:This pseudo-puzzle has 33 solutions caused by a weakly-minimal UA55 which is decomposed to 5 disjoint UA4.
Code: Select all
`.......12..3..14...1.56.73...8.9.....4.2...5.7....3..1..9.8.3...2.4..1..6....7...`

I confirm you found weakly-minimal UA55. Very interesting! It is possible to say, that this UA has 2 basic permutations - the first is minimal, the second contains 5 disjoint UAs. We have rare possibility to compute manually number of UA set permutations.

Serg
Serg
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### 49

Code: Select all
`........1.....1.2..13..45....1..3..2.6..4..1.5..17.8....2..7..3.8...9.6.1..5...4.`

The above has 49 permutations.

It is time to rename the thread, "13 permutations" now sounds naive
dobrichev
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### 2^6+1=65

Code: Select all
`........1....12....134..56...4.5.....2.1.64..7..8.4.....57...3..6...12..8...4..9.`

disjoint 5*UA4 + 1*UA6
dobrichev
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### Minimal UA sets with 49 and 65 permutations

Hi, dobrichev!
Congratulations! You set new record for minimal UA set valency! Now 65 is maximal valency of (weakly) minimal UA set.
I confirm that your 2 last published UA55 are weakly minimal UA sets. They have 49 and 65 permutations.

I have some considerations about maximal valency of strongly minimal UA sets. So far bi-valent strongly minimal UA sets were observed only. Do multi-valent strongly minimal UA sets exist? This question remains open.

Let's consider a cell participating strongly minimal UA set. This cell must get new unique digit while passing through all UA set permutations. For each given permutation, the same digits must be observed in the cells participating this UA set, located in the same row (as starting cell), in the same column and located in the same box. It follows, strongly minimal UA set with maximal possible valency should be 9-valent UA81, containing all grid cells. But each strongly minimal UA permutation should be broken by single clue. So, if 9-valent strongly minimal UA81 would exist, should exist valid sudoku puzzle having 1 clue only. It is impossible, hence 9-valent strongly minimal UA sets do not exist.

But maybe 8-valent strongly minimal UA set exist? If it would be so, 8-valent strongly minimal UA72 should exist, having 8 cells in each row/column/box. If am not mistaken, there is only grid cells subset (up to isomorphisms), having 8 cells in each row/column/box:
Code: Select all
`+-----+-----+-----+     Hypothetical 8-valent strongly minimal UA72 (crosses denote UA set's cells)|x x x|x x x|x x .||x x x|x x .|x x x||x x .|x x x|x x x|+-----+-----+-----+|x x x|x x x|x . x||x x x|x . x|x x x||x . x|x x x|x x x|+-----+-----+-----+|x x x|x x x|. x x||x x x|. x x|x x x||. x x|x x x|x x x|+-----+-----+-----+`

Let's assume 8-valent strongly minimal UA72 does exist. Then 9-clue pseudo-puzzle must exist, producing it:
Code: Select all
`+-----+-----+-----+     Hypothetical pattern, producing 8-valent strongly minimal UA72|. . .|. . .|. . x||. . .|. . x|. . .||. . x|. . .|. . .|+-----+-----+-----+|. . .|. . .|. x .||. . .|. x .|. . .||. x .|. . .|. . .|+-----+-----+-----+|. . .|. . .|x . .||. . .|x . .|. . .||x . .|. . .|. . .|+-----+-----+-----+`

Pseudo-puzzle having this pattern must have 8 solutions exactly and must be turned into valid puzzle by adding one clue (this additional clue hit every strongly minimal UA set's permutation). It follows, 10-clue valid sudoku puzzles must exist. But such puzzles don't exist, so 8-valent strongly minimal UA sets don't exist too.

7-valent strongly minimal UA set claims that 18-clue pseudo-puzzle, producing this UA set, must exist. This psedo-puzzle must have 7 solutions exactly, and must contain 2 cells in each row/column/box and must be turned into valid puzzle by adding one clue. (19-clue valid sudoku puzzles do exist.) Here is example of possible pattern for this pseudo-puzzle.
Code: Select all
`+-----+-----+-----+     Hypothetical pattern, which could produce 7-valent strongly minimal UA63|x . x|. . .|. . .||. . .|x . .|x . .||. . .|. . x|. x .|+-----+-----+-----+|. x .|. . x|. . .||. . .|x . .|. . x||. . x|. . .|. . x|+-----+-----+-----+|. . .|. x .|. x .||x x .|. . .|. . .||. . .|. x .|x . .|+-----+-----+-----+`

So, we should study patterns, having 2 clues in each row/column/box to clarify 7-valent strongly minimal UA63 existence. (It looks like 2-rookery.)

Serg
Serg
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### Re: Minimal Unavoidable Set with 49 permutations

If a UA has 2 digits involved, then it occupies either the entire 2-rookery or a subset a the 2-rookery.
In both cases the rookery is isolated from the rest of the grid, as always rookeries are.
If the 2-rookery is splittable to smaller minimal UA, then they are isolated form each other too, occupying 2 cells in a row/column/box.

If a UA has 3 digits, then it can't occupy entire 3-rookery. If so, it wouldn't be minimal since it is a superset of the shorter UA coming from the 2-rookeries.
Is it possible a 3-rookery to be split into 2 or more parts so that
- each part occupies 3 cells in a row/column/box and is isolated from the other parts, and
- some part can't be split into smaller 2-digit valid UA?
dobrichev
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### Re: Minimal Unavoidable Set with 49 permutations

Hi, dobrichev!
dobrichev wrote:If a UA has 2 digits involved, then it occupies either the entire 2-rookery or a subset a the 2-rookery.

Old discussion - are relabellings or VPTs kinds of UA sets? I think they are kinds of UA sets, but trivial, not interesting. (2-rookery presents 2 digits relabelling scheme and must be broken by clue if we want to have valid puzzle.)
dobrichev wrote:If a UA has 3 digits, then it can't occupy entire 3-rookery. If so, it wouldn't be minimal since it is a superset of the shorter UA coming from the 2-rookeries.
Is it possible a 3-rookery to be split into 2 or more parts so that
- each part occupies 3 cells in a row/column/box and is isolated from the other parts, and
- some part can't be split into smaller 2-digit valid UA?

Difficult questions. I can state only that any cell of 3-valent strongly minimal UA sets must have not less than 2 neighbours in its row/column/box.

Serg

[Edited. I corrected mistake - "3 neighbours in its row/column/box".]
Serg
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