The New Sudoku Players' Forum

[JPF]

To avoid any ambiguity, here are some definitions:

• Invalid pattern: A pattern that does not contain a valid puzzle.
• Maximal invalid pattern: An invalid pattern P such that for every cell c not belonging to P, the pattern P+c is valid.
  This concept was introduced by Serg and blue.

Now the question is: what is the smallest number of clues in a maximal invalid pattern?

JPF

[Serg]

Hi, JPF!

Now the question is: what is the smallest number of clues in a maximal invalid pattern?

I don't know exact answer to your question. My candidate - "Crab pattern":

Code: Select all

Crab pattern (23 clues)

x . . . . . . . x
. x . . . . . x .
. . x . x . x . .
. . . x x x . . .
. . . x x x . . .
. . . x x x . . .
. . x . x . x . .
. x . . . . . x .
x . . . . . . . x

Serg

[JPF]

That's a good start, but I think we can do better. In any case, I have something better in my bag

JPF

[m_b_metcalf]

Invalid pattern: A pattern that does not contain a valid puzzle.

Here is an invalid pattern, with 16 clues, P:

Code: Select all

 . . . . . . . . .
 4 . . . . . . . .
 . 2 . . . . . . .
 . . . . 5 . 4 . 7
 . . 8 . . . 3 . .
 . . 1 . 9 . . . .
 3 . . 4 . . 2 . .
 . 5 . 1 . . . . .
 . . . 8 . 6 . . .

Maximal invalid pattern: An invalid pattern P such that for any cell c not belonging to P, the pattern P+c is valid.

Here is the pattern P+c, forming a valid pattern:

Code: Select all

 . . . . . . . 1 .
 4 . . . . . . . .
 . 2 . . . . . . .
 . . . . 5 . 4 . 7
 . . 8 . . . 3 . .
 . . 1 . 9 . . . .
 3 . . 4 . . 2 . .
 . 5 . 1 . . . . .
 . . . 8 . 6 . . .

Now the question is: what is the smallest number of clues in a maximal invalid pattern?

Well, 16. But I've clearly misunderstood!

Mike

[JPF]

Well, your 16-clues pattern P is invalid:

Code: Select all

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


It's true.

However, to prove that it is a maximal invalid pattern, you need to show that all 65 (= 81-16) patterns created by P + one clue are valid.

JPF

[m_b_metcalf]

However, to prove that it is a maximal invalid pattern, you need to show that all 65 (= 81-16) patterns created by P + one clue are valid.

In other words, rather than:

Maximal invalid pattern: An invalid pattern P such that for any cell c not belonging to P, the pattern P+c is valid.

Maximal invalid pattern: An invalid pattern P such that for every cell c not belonging to P, the pattern P+c is valid.

[JPF]

Thank you.
Definition edited.

JPF

[Serg]

Hi, Mike!

However, to prove that it is a maximal invalid pattern, you need to show that all 65 (= 81-16) patterns created by P + one clue are valid.

In other words, rather than:

Maximal invalid pattern: An invalid pattern P such that for any cell c not belonging to P, the pattern P+c is valid.

Maximal invalid pattern: An invalid pattern P such that for every cell c not belonging to P, the pattern P+c is valid.

I think, "for any" and "for every" are synonyms in mathematics. (See, for example, an article Universal quantification.)

Serg

[JPF]

I should have written:

P is maximal invalid if ∀c ∉P ∶ P+c is valid

but not everyone is familiar with these elementary mathematical notations.
Using LaTeX would be very useful in certain cases.

JPF

[blue]

Here are two size 18 cases.
Each one has 12 automorphisms and 8 "+1" clue types:

Code: Select all

. 1 . . . . 1 . .    a . b c d b . e f
1 . . . . 1 . . .    . g d d g . e h e
. . . . 1 . . . 1    b d c b . a f e .
. . . 1 . . . 1 .    c d b . e f a . b
. . 1 . . . 1 . .    d g . e h e . g d
. 1 . . . 1 . . .    b . a f e . b d c
1 . . . 1 . . . .    . e f a . b c d b
. . . 1 . . . . 1    e h e . g d d g .
. . 1 . . . . 1 .    f e . b d c b . a

puzzles:
62....1..9....8.......4...6...7...1...4...5...6...3...1...6.......5....3..3....8. (a)
.82...9..1....3.......7...5...4...7...6...2...9...5...7...2.......8....9..5....3. (b)
.2.1..6..5....7.......9...4...6...5...4...2...9...3...6...4.......2....9..7....3. (c)
.9..4.8..3....6.......1...2...7...3...4...6...7...2...6...8.......9....7..2....5. (d)
.4....72.3....4.......8...9...2...4...9...5...1...6...4...5.......7....2..8....3. (e)
.3....5.87....9.......1...2...5...4...8...3...6...4...1...3.......8....6..2....9. (f)
.4....1..76...9.......3...5...7...4...1...3...7...4...3...8.......1....7..9....6. (g)
.6....1..7....8.2.....4...9...7...1...3...6...4...5...1...3.......6....4..8....5. (h)

Code: Select all

. . . 1 . . 1 . .    a b b . c c . d d
. . 1 . . 1 . . .    b e . c f . d g h
. 1 . . 1 . . . .    b . e c . f d h g
1 . . 1 . . . . .    . c c . d d a b b
. . 1 . . . . . 1    c f . d g h b e .
. 1 . . . . . 1 .    c . f d h g b . e
1 . . . . . 1 . .    . d d a b b . c c
. . . . . 1 . . 1    d g h b e . c f .
. . . . 1 . . 1 .    d h g b . e c . f

puzzles:
4..1..2....3..5....9..6....2..4.......5.....9.7.....6.1.....4.......7..6....9..3. (a)
.3.9..7....8..4....9..5....7..4.......2.....4.6.....8.6.....9.......8..6....2..4. (b)
...49.2....7..3....1..2....2..9.......8.....1.5.....7.4.....9.......7..3....8..1. (c)
...8..69...1..9....3..7....6..2.......2.....1.4.....3.8.....2.......3..4....4..7. (d)
...2..5...79..1....1..4....3..5.......1.....9.6.....7.2.....8.......7..1....9..6. (e)
...9..3....8.17....4..2....3..5.......9.....1.2.....7.9.....5.......4..2....7..1. (f)
...2..8....7..5.1..3..9....8..3.......5.....2.6.....7.2.....3.......7..8....1..6. (g)
...7..8....2..1..9.6..4....8..3.......4.....6.7.....4.7.....5.......9..1....6..2. (h)

There are no size 16 cases, since we know all of the valid 17-clue patterns and the "-1" patterns that can be produced from them, and none of those satisfies the property that each of its "+1" extensions, is a valid 17-clue pattern.

Size 17 seems possible.

[JPF]

Here is my candidate (18 clues):

Code: Select all

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

...11......1..1....1....1..1......1.1.......1.1......1..1....1....1..1......11...


valid puzzles:

Code: Select all

1..23......4..5....6....7..2......3.3.......4.1......2..6....2....8..5......14...
.1.23......4..5....6....7..8......2.6.......5.7......3..5....3....1..4......76...
..123......4..5....6....4..7......5.3.......2.8......3..5....9....8..6......72...
...123.....4..5....6....7..8......5.7.......3.2......6..3....2....7..8......98...
...12.3....4..5....3....6..7......5.8.......9.6......4..5....1....6..2......98...
...12..3...4..5....6....7..8......2.6.......1.3......8..5....1....3..4......96...
...12...3..4..5....6....7..8......2.3.......1.1......5..6....4....7..9......83...
...12.....34..5....6....4..7......2.8.......5.9......8..6....4....9..6......81...
...12......34.5....5....6..7......4.6.......3.8......2..2....1....7..9......68...
...12......1.34....5....6..7......4.8.......1.3......2..4....5....6..2......78...
...12......3..45...6....7..7......2.8.......4.5......1..2....6....7..9......45...
...12......3..4.5..6....7..2......1.8.......4.9......3..5....4....7..9......96...
...12......3..4....56...7..1......6.7.......2.4......8..5....9....7..3......81...
...12......3..4....5.6..7..5......4.6.......8.7......3..8....6....5..2......37...
...12......3..4....5..3.6..7......4.4.......6.2......8..4....1....9..7......76...
...12......3..4....5...36..1......7.8.......9.7......1..5....3....7..5......19...
...12......3..4....5....6..5..7...1.2.......4.8......5..4....9....6..8......32...
...12......3..4....5....6..7...3..2.1.......4.6......1..2....3....6..8......57...
...12......3..4....5....6..7....1.4.6.......2.8......3..4....1....9..8......65...
...12......3..4....5....6..7......1.6...3...4.8......2..4....6....7..3......53...


JPF

[Serg]

Hi, Blue and JPF!
Impressive work! You discovered new family of "linear" maximal invalid patterns. Congratulations!
But I don't understand at the monent - can I verify your discoveries or not.

Serg

[JPF]

Size 17 seems possible.

Here is one:

Code: Select all

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


This pattern is invalid, as it does not belong to the set of 33,884 ed-patterns out of the 49,158 puzzles with 17 clues.
Furthermore, for every cell c not belonging to this pattern P : P+c is valid.
Here are the 64 valid corresponding puzzles.

Code: Select all

1...............23..4.56........74...3.......82......7...5........28.5....9...7..
.1..............23..4.56........75...8.......93......8...2........39.1....7...8..
..1.............23..4.56........46...6.......72......8...3........24.7....5...1..
...1............23..4.56........25...7.......23......8...3........78.1....5...6..
....1...........23..1.45........64...3.......27......8...1........23.7....5...8..
.....1..........23..1.45........45...6.......23......6...6........72.8....5...9..
......1.........23..4.56........47...3.......82......9...9........32.5....1...4..
.......1........23..4.56........45...3.......17......8...8........31.7....6...2..
........1.......23..4.56........47...2.......83......9...1........38.9....5...6..
.........1......23..4.56........42...3.......78......1...1........37.5....6...9..
..........1.....23..4.56........47...6.......32......8...2........17.4....6...5..
...........1....23..4.56........15...2.......73......8...2........84.1....6...4..
............1...23..1.45........67...7.......23......5...3........47.8....9...6..
.............1..23..4.56........46...7.......23......8...3........76.9....8...4..
..............1.23..4.56........34...1.......32......5...2........57.8....8...9..
...............123..4.56........47...5.......32......8...3........87.4....5...6..
................123.4.56........43...7.......21......8...8........21.9....6...5..
................12.34.56........47...6.......12......8...2........98.3....5...4..
................12..3456........78...5.......42......6...6........21.5....7...9..
................12..3.456.......75...8.......21......7...4........18.2....5...3..
................12..3.45.6......75...8.......12......9...1........62.8....5...9..
................12..3.45..6.....74...5.......21......8...2........69.8....7...3..
................12..3.45...6....37...8.......12......5...2........85.4....9...3..
................12..3.45....5...64...1.......72......8...1........63.9....8...5..
................12..3.45.....4..67...7.......61......8...2........19.3....5...9..
................12..3.45......6.37...2.......81......9...9........28.4....4...3..
................12..3.45.......364...2.......71......6...4........71.3....5...8..
................12..3.45........463..7.......82......9...8........21.4....5...3..
................12..3.45........64.7.5.......81......9...2........19.7....6...3..
................12..3.45........64..78.......12......3...1........78.6....5...3..
................12..3.45........63...25......71......4...1........57.8....9...6..
................12..3.45........36...7.1.....18......4...2........68.7....8...3..
................12..3.14........56...2..6....71......8...8........12.9....4...3..
................12..3.45........36...7...4...18......9...1........86.4....7...3..
................12..3.45........34...6....5..71......8...2........71.8....5...9..
................12..3.45........36...7.....8.16......7...2........16.8....5...7..
................12..3.45........65...7......812......7...2........83.6....5...4..
................12..3.45........64...7.......218.....3...1........87.6....5...9..
................12..3.45........35...6.......17.2....8...1........75.6....8...3..
................12..3.45........67...8.......92..3...4...4........12.8....7...5..
................12..3.45........65...2.......47...8..9...9........12.7....5...4..
................12..3.45........67...5.......21....8.4...2........89.4....6...3..
................12..3.45........36...7.......12.....89...8........17.2....5...9..
................12..3.45........35...6.......14......77..2........15.6....8...3..
................12..3.45........35...6.......14......6.7.1........25.3....8...9..
................12..3.45........34...6.......12......7..82........56.9....7...3..
................12..3.45........46...7.......81......9...21.......79.4....6...3..
................12..3.45........35...2.......61......7...2.6......78.3....4...9..
................12..3.45........34...6.......72......6...1..6.....27.8....9...5..
................12..3.45........36...7.......21......8...8...7....21.9....5...4..
................12..3.45........36...7.......28......7...1....6...28.4....4...3..
................12..3.45........65...7.......21......8...8.....9..27.1....5...9..
................12..3.45........36...1.......27......8...1......6.72.4....4...9..
................12..3.45........36...7.......12......4...4.......826.5....9...3..
................12..3.45........36...2.......71......8...2........9684....4...5..
................12..3.45........36...5.......71......8...2........18.37...9...5..
................12..3.45........36...7.......82......4...1........28.4.9..9...5..
................12..3.45........35...6.......71......6...1........27.8..5.8...9..
................12..3.45........65...2.......71......4...2........48.3...36...9..
................12..3.45........36...5.......21......7...1........86.4....74..3..
................12..3.45........36...5.......12......7...2........71.3....8.9.4..
................12..3.45........36...2.......71......8...2........98.3....5..74..
................12..3.45........36...7.......18......7...2........19.7....5...84.
................12..3.45........36...7.......21......4...6........71.5....8...3.9


JPF

[Serg]

Hi, JPF!
Congratulations with your 17-clue maximal invalid pattern discovery!
I've done some checks and can confirm, that

1. Your 17-clue pattern doesn't coincide with any isomorphs of 49158 17-clue known puzzles' patterns.
2. Your 64 1-clue extention patterns are all possible ED extentions of original 17-clue maximal invalid pattern.
3. All published 64 1-clue extention puzzles have unique solutions.

It would be nice to check your 17-clue maximal invalid pattern - is it really invalid, but it's rather difficult task...

Serg

[coloin]

Well done for finding those examples

It would be nice to check your 17-clue maximal invalid pattern - is it really invalid, but it's rather difficult task...
Serg

Indeed knowing that a pattern is invalid is a tricky one...but not in JPF's case as all the valid 17C patterns are known
Assuming the pattern is invalid, the more clues it has makes it more likely that all +1 patterns will have a puzzle.

I started with a 20C pattern from which I found 6714 puzzles - and only this number seem to exist.
This is unusually low for a 20C valid pattern

Code: Select all

1..234....23....5...6......4...1......5.....6....47......5.........6..1...73....8-20C example
x..xxx....xx....x...x......x...x......x.....x....xx......x.........x..x...xx....x-20C

There were 190 19C puzzles with 12 valid patterns with puzzles and 8 invalid patterns with no puzzles

Hidden Text: Show

Code: Select all

...234....56....1...7......2...1......3.....4....58......9.........2..5...45....8
...234....56....2...7......3...5......8.....9....23......7.........6..5...18....4
...234....56....3...4......2...7......8.....1....93......5.........6..5...24....8
...234....56....3...7......2...8......1.....6....49......3.........2..8...97....5
...234....56....3...7......4...5......8.....1....41......7.........6..5...18....2
...234....56....3...7......4...8......9.....1....13......6.........5..4...19....8
...234....56....4...7......2...4......1.....8....56......6.........2..3...81....7
...234....56....4...7......2...4......1.....8....78......1.........2..3...95....1
...234....56....4...7......2...4......8.....1....51......9.........2..3...18....7
...234....56....4...7......2...4......8.....1....61......6.........2..3...18....7
...234....56....4...7......2...6......8.....1....41......7.........2..6...18....5
...234....56....4...7......2...7......8.....1....45......5.........6..5...18....2
...234....56....4...7......3...4......1.....8....56......6.........2..3...81....7
...234....56....4...7......3...4......1.....8....78......1.........2..3...95....1
...234....56....4...7......3...4......1.....8....78......1.........2..3...95....2
...234....56....4...7......3...4......1.....8....78......6.........5..3...21....9
...234....56....4...7......3...4......1.....8....98......6.........5..3...91....7
...234....56....4...7......3...4......8.....1....25......8.........6..2...15....9
...234....56....4...7......3...4......8.....1....25......8.........9..2...15....6
...234....56....4...7......3...4......8.....1....51......9.........2..3...18....6
...234....56....4...7......3...4......8.....1....61......6.........2..3...18....7
...234....56....4...7......3...4......8.....1....61......7.........5..3...19....2
...234....56....4...7......3...4......8.....1....91......6.........5..3...18....5
...234....56....4...7......3...4......8.....7....51......9.........2..3...18....6
...234....56....4...7......3...4......8.....9....25......9.........6..2...17....8
...234....56....4...7......3...6......8.....1....41......8.........5..3...17....2
...234....56....4...7......3...6......8.....1....47......8.........5..3...17....2
...234....56....4...7......3...6......8.....9....41......8.........5..3...17....2
...234....56....4...7......3...7......1.....2....48......6.........5..3...29....1
...234....56....4...7......3...7......8.....1....49......6.........5..3...18....5
...234....56....4...7......3...7......8.....1....49......8.........5..3...16....2
...234....56....4...7......3...7......8.....2....48......6.........5..3...21....9
...234....56....4...7......3...7......8.....9....48......6.........5..3...19....2
...234....56....4...7......3...8......9.....1....41......7.........5..3...19....8
...234....56....4...7......3...8......9.....1....46......7.........5..3...19....6
...234....56....4...7......4...6......8.....1....15......8.........4..6...19....2
...234....56....4...7......4...6......8.....1....15......8.........4..6...19....7
...234....56....4...7......4...6......8.....1....51......8.........4..6...19....7
...234....56....4...7......4...6......8.....9....25......7.........4..6...18....3
...234....56....4...7......4...6......8.....9....95......8.........4..6...27....8
...234....56....4...7......4...8......9.....1....57......9.........4..3...17....2
...234....56....4...7......5...4......1.....8....78......1.........5..3...96....2
...234....56....7...8......2...7......1.....4....86......9.........2..5...45....1
...234....56....7...8......3...2......1.....6....73......7.........9..2...45....8
...234....56....7...8......3...7......1.....6....23......7.........9..2...45....8
1...34....25....1...6......3...4......7.....8....21......8.........1..3...96....7
1...34....25....1...6......3...7......7.....6....48......6.........1..3...95....7
1...34....25....1...6......4...1......7.....8....23......8.........4..5...96....7
1...34....25....6...7......3...1......6.....5....83......6.........2..1...89....7
1...34....25....6...7......3...1......6.....7....83......5.........4..1...97....4
1...34....25....6...7......4...1......6.....8....49......8.........2..1...85....9
1...34....25....6...7......4...1......8.....5....93......8.........4..9...26....7
1...34....56....3...2......4...7......8.....5....13......6.........9..7...48....2
1...34....56....3...7......4...1......8.....5....23......6.........9..1...48....7
1...34....56....4...2......4...7......8.....5....19......9.........4..7...58....2
1...34....56....4...7......3...4......8.....2....52......6.........1..3...28....7
1...34....56....4...7......3...4......8.....2....52......7.........1..3...28....6
1..2.4....53....6...2......7...8......6.....3....97......3.........1..9...45....2
1..2.4....53....6...7......4...1......2.....8....48......3.........5..1...87....5
1..2.4....53....6...7......4...1......2.....8....48......6.........5..1...87....5
1..2.4....53....6...7......4...1......8.....2....48......6.........5..1...27....9
1..2.4....53....6...7......4...1......8.....5....49......6.........2..1...97....8
1..2.4....56....1...2......4...5......3.....7....81......5.........6..5...97....2
1..2.4....56....1...7......2...6......3.....8....59......1.........7..6...94....3
1..2.4....56....3...3......2...1......7.....6....48......3.........2..1...89....5
1..2.4....56....3...3......2...4......7.....6....18......3.........2..1...89....5
1..2.4....56....3...3......4...1......7.....5....48......3.........2..1...89....7
1..2.4....56....3...3......4...1......7.....6....48......3.........2..1...85....7
1..2.4....56....3...3......4...1......7.....6....48......3.........2..1...89....7
1..2.4....56....3...3......4...1......7.....6....48......9.........2..1...83....5
1..2.4....56....3...3......4...1......7.....8....48......7.........2..1...86....5
1..2.4....56....3...3......4...1......7.....8....48......7.........2..1...89....5
1..2.4....56....3...3......4...1......7.....8....49......7.........2..1...89....5
1..2.4....56....3...3......4...2......7.....6....48......9.........1..2...86....5
1..2.4....56....3...3......4...2......7.....8....49......7.........1..2...89....5
1..2.4....56....3...7......8...5......4.....1....93......8.........7..5...26....4
1..2.4....56....7...8......2...6......3.....9....51......1.........8..5...93....2
1..2.4....56....7...8......2...6......3.....9....51......3.........8..5...91....6
1..23.....45....1...6......3...5......7.....8....21......8.........1..3...96....7
1..23.....45....6...7......2...4......6.....7....75......6.........2..3...81....5
1..23.....54....1...6......3...1......7.....8....25......8.........4..2...96....7
1..23.....54....1...6......3...1......7.....8....52......8.........2..3...96....7
1..23.....54....1...6......3...1......7.....8....52......8.........2..5...96....7
1..23.....54....2...6......3...2......7.....8....15......4.........9..1...87....6
1..23.....54....2...6......3...2......7.....8....15......6.........9..1...87....4
1..23.....54....3...6......2...7......5.....6....83......1.........2..7...95....4
1..23.....56....1...4......2...5......7.....8....18......3.........7..5...84....6
1..23.....56....1...4......3...5......7.....8....21......4.........6..5...97....3
1..23.....56....2...4......3...2......7.....8....15......6.........4..1...87....9
1..23.....56....2...4......3...2......7.....8....15......9.........4..1...87....6
1..23.....56....3...4......2...7......8.....9....53......1.........4..2...76....4
1..23.....56....4...2......3...1......4.....7....83......7.........2..8...96....1
1..23.....56....4...2......3...1......4.....7....83......7.........5..8...96....1
1..234.....5....4...2......6...4......7.....8....91......1.........6..1...87....5
1..234.....5....4...6......3...4......7.....8....98......5.........1..3...87....6
1..234....5.....6...7......8...6......9.....2....15......7.........8..5...24....9
1..234....5.....6...7......8...6......9.....2....51......7.........8..5...24....9
1..234....5.....6...7......8...7......4.....1....59......6.........9..5...28....4
1..234....56........3......4...1......7.....6....48......3.........2..1...89....5
1..234....56........8......2...4......5.....6....81......1.........2..1...76....5
1..234....56....1...4......3...1......7.....8.....5......9.........5..3...87....4
1..234....56....1...4......3...5......7.....8.....1......9.........1..3...87....4
1..234....56....1...4......3...5......7.....8....21......7.........1..3...8.....4
1..234....56....1...4......3...6......7.....8.....1......9.........5..3...87....4
1..234....56....1...4......6...1......7.....8.....5......6.........2..5...98....4
1..234....56....1...7......2...1......3.....8....25......7............2...98....4
1..234....56....1...7......2...1......8.....4....65......9.........2..5...4.....8
1..234....56....1...7......2...6......8.....4....15......8.........2..5...4.....9
1..234....56....1...7......2...6......8.....4....15......9.........2..5...4.....8
1..234....56....1...7......2...7......8.....4....1.......9.........2..5...45....8
1..234....56....1...7......3...4......2.....6.....5......7.........2..3...86....5
1..234....56....1...7......3...4......2.....6.....5......7.........2..8...96....5
1..234....56....1...7......3...4......2.....7....13......8.........7..4...8.....6
1..234....56....1...7......3...4......2.....7....13......8.........7..4...9.....6
1..234....56....1...7......3...4......2.....7....13......8.........9..4...8.....6
1..234....56....1...7......3...4......2.....7....13......8.........9..4...9.....6
1..234....56....1...7......3...4......8.....2....51......7.........6..4...2.....9
1..234....56....1...7......3...5......2.....8.....3......8.........1..5...47....9
1..234....56....1...7......3...5......8.....2....41................6..5...27....9
1..234....56....1...7......3...6......2.....8.....1......7.........5..3...48....9
1..234....56....1...7......3...6......8.....2.....1......7.........5..3...24....9
1..234....56....1...7......3...6......8.....2.....1......7.........5..3...28....9
1..234....56....1...7......3...6......8.....9.....1......7.........5..3...49....8
1..234....56....1...7......3...6......8.....9.....1......9.........5..3...47....8
1..234....56....1...7......4...5......2.....3.....1......7.........8..5...83....6
1..234....56....1...7......4...5......2.....8....71......3.........1..5...8.....2
1..234....56....1...7......4...6......8.....2....13................4..3...27....9
1..234....56....1...7......5...1......8.....9.....5......7.........5..3...49....2
1..234....56....1...7......6...7......8.....2....43................1..3...25....8
1..234....56....1...7......6...7......8.....2....43................1..4...25....8
1..234....56....1...7......7...6......8.....2....43................1..3...25....8
1..234....56....1...7......7...6......8.....2....43................1..4...25....8
1..234....56....1...7......8...6......9.....2....43................1..3...25....9
1..234....56....1...7......8...6......9.....2....43................1..4...25....9
1..234....56....1...7......8...7......9.....2....43................1..3...25....9
1..234....56....1...7......8...7......9.....2....43................1..4...25....9
1..234....56....2...7......3...2......1.....8....53......7............5...98....4
1..234....56....2...7......3...5......1.....8....23......7............5...91....4
1..234....56....3...7......2...4......3.....5.....7......3.........2..1...89....6
1..234....56....3...7......2...4......8.....6.....9......3.........2..1...56....7
1..234....56....3...7......2...4......8.....7.....5......3.........2..1...57....6
1..234....56....3...7......2...8......9.....5....43................1..4...56....7
1..234....56....4...2......3...4......7.....8....5.......8.........1..3...97....6
1..234....56....4...2......3...5......7.....6....4.......6.........1..3...87....9
1..234....56....4...7......2...6......1.....7....53......7.........2..3...8.....1
1..234....56....4...7......2...8......9.....5....43................1..3...56....7
1..234....56....4...7......3..........2.....7....48......7.........1..3...89....5
1..234....56....4...7......3..........2.....7....48......7.........1..3...89....6
1..234....56....4...7......3..........2.....7....48......9.........1..3...87....5
1..234....56....4...7......3..........2.....8....49......6.........1..3...98....2
1..234....56....4...7......3...4......2.....6.....7......5.........1..3...86....7
1..234....56....4...7......3...4......2.....6.....7......5.........1..3...86....9
1..234....56....4...7......3...6......8.....7....4.......7.........1..3...58....6
1..234....56....7...2......3...1......7.....2....48................8..1...35....6
1..234....56....7...2......3...1......7.....2....48................8..1...35....7
1..234....56....7...2......3...1......7.....6....48................8..1...36....2
1..234....56....7...2......3...4......7.....2....18................8..4...35....6
1..234....56....7...2......3...4......7.....2....18................8..4...35....7
1..234....56....7...2......3...4......7.....2....81................1..4...35....6
1..234....56....7...2......3...4......7.....2....81................1..4...35....7
1..234....56....7...2......3...4......7.....5....18................6..4...95....6
1..234....56....7...2......3...4......7.....5....18................7..4...95....6
1..234....56....7...2......3...4......7.....6....18................8..4...36....2
1..234....56....7...2......3...4......7.....6....81................1..4...36....2
1..234....56....7...2......4..........7.....6....18......6.........4..1...95....2
1..234....56....7...2......4...8......8.....2....53................1..3...74....6
1..234....56....7...3......2..........8.....5....46......1.........2..1...73....6
1..234....56....7...4......2..........8.....5....63......1.........2..1...74....6
1..234....56....7...4......2...4......8.....4....53................1..3...76....8
1..234....56....7...4......2...8......9.....5....13......6.........2..3...5.....4
1..234....56....7...4......3...8......7.....6.....1......3.........2..1...96....5
1..234....56....7...4......7...5......1.....4.....7......4.........8..5...91....6
1..234....56....7...8......2...4......5.....6....8.......1.........2..1...76....5
1..234....56....7...8......2...6......1.....4....53......7.........2..3...9.....1
1..234....56....7...8......2...7......1.....4....53......6.........2..3...9.....1
1..234....56....7...8......2...7......1.....4....53......9.........2..3...4.....8
1..234....56....7...8......2...7......9.....4....53......9.........2..3...4.....8
1..234....56....7...8......3...4......2.....5....83................1..4...59....6
1..234....56....7...8......3...4......2.....5....83................1..4...97....8
1..234....56....7...8......3...4......2.....5....93................1..4...97....8
1..234....56....7...8......3...4......2.....6....73................1..4...79....8
1..234....56....7...8......3...4......2.....6....93................1..4...97....8
1..234....56....7...8......3...4......5.....6....83................1..4...76....5
1..234....56....7...8......3...8......2.....5....43................1..4...59....6
1..234....56....7...8......3...8......2.....5....43................1..4...59....8
1..234....56....7...8......4...1......2.....6.....9......5.........4..1...76....8
1..234....56....7...8......4...7......7.....2....53......9.........4..3...2.....1
1..234....56....7...8......4...7......9.....2....53......8.........4..3...2.....1
1..234....56....7...8......4...8......2.....5....43................1..3...59....6
1..234....56....7...8......4...8......2.....5....43................1..3...59....8

These were all minimal - so there were no 18C puzzles
This means we have 190 invalid [with no puzzles] 18C patterns [20/2 x 19]

I took one of these [supposedly] invalid 18C patterns and by adding one clue found an example 19C puzzle in all but 2 of the 63 patterns with 19C.
I would guess that using one of the 19C invalid patterns you could easily find all 20C examples ...

Code: Select all

1..234....23....5...6......4...1......5.....6....47......5.........6..1...73....8-20C example
...xxx....xx....x...x......x...x......x.....x.....x......x.........x..x...xx....x-18

...134....56....7...4......7...5......2.....4.....7......2.........8..5...94...86
...134....56....7...4......7...5......2.....4.....7......2.........8..5...94..3.6
...234....15....7...9......2...4......8.....9.....1......9.........2..6...73.8..1
...234....56....4...7......2...4......8.....1.....1......6.........2..5...195...8
...234....56....7...9......3...4......2.....1.....6......1.........5..3..918....6
...832....56....7...4......8...5......1.....9.....7......9.........8..5.7.93....6
...231....56....4...9......2...4......3.....1.....5......9.........2..53..18....7
...134....62....5...7......3...6......1.....8.....2......7.........5.63...48....9
...254....36....4...7......2...4......8.....1.....1......6.........27.3...13....9
...234....19....3...4......2...5......1.....9.....8......9........32..5...86....1**** only one of these found
...138....29....4...6......3...4......7.....8.....9......2.......1.5..9...83....5
...234....56....4...7......3...4......9.....1.....1......9......7..5..3...18....2
...234....56....4...7......3...4......8.....9.....1......8.....5...7..3...19....8
...234....56....4...7......3...4......8.....9.....1......7....3....6..5...19....8
...134....57....2...9......3...4......8.....1.....7......9...4.....2..3...18....7
...134....57....2...9......3...4......8.....1.....7......9..5......2..3...18....7
...124....56....1...7......4...6......8.....3.....5......8.1.......4..6...37....2
...236....19....6...4......3...6......7.....5.....9......58........1..3...57....2
...134....56....2...7......3...5......9.....8.....2.....87.........9..5...48....9
...234....56....4...7......4...6......2.....1.....5....1.7.........4..6...91....2
...234....56....4...7......2...4......8.....1.....1...5..6.........2..5...19....8
...123....54....2...7......3...4......8.....1.....2..9...7.........5..3...18....5
...123....56....7...4......7...5......2.....4.....7.9....2.........8..5...94....6
...124....56....4...3......4...6......8.....1.....59.....9.........4..6...18....7
...124....56....4...7......4...6......8.....3....35......8.........4..6...39....7
...234....57....4...9......2...4......8.....1...6.7......9.........2..5...18....7
...138....56....4...9......3...4......8.....1..4..2......9.........5..3...18....2
...234....56....4...7......2...4......8.....1.7...1......9.........2..5...15....8
...134....56....2...7......2...4......1.....63....5......7.........1..3...96....5
...234....56....4...7......2...4......8....91.....1......6.........2..3...19....7
...124....56....1...7......1...6......8...9.3.....5......8.........1..6...39....2
...234....56....4...7......3...4......1..7..8.....1......6.........5..3...98....7
...234....56....4...7......2...4......8.9...1.....1......6.........2..5...19....8
...238....56....7...4......8...5......26....4.....7......4.........8..3...91....6
...234....56....4...7......4...6.....78.....9.....5......7.........4..6...18....3
...234....56....4...7......6...4....3.8.....1.....1......5.........2..3...18....9
...134....56....4...7......3...4...8..9.....1.....2......8.........5..3...89....2
...124....56....7...4......7...5..2...3.....4.....7......3.........8..5...94....6
...124....56....7...4......1...5.8....2.....4.....7......4.........8..5...93....6
...234....16....4...7......2...47.....8.....9.....5......6.........2..3...59....8
...231....87....5...9......2..45......8.....9.....7......9.........2..3...13....8

...xxx....xx....x...x......x.x.x......x.....x.....x......x.........x..x...xx....x  no puzzle found

...134....56....7...2......73..6......8.....1.....5......8.........4..6...19....2
...234....56....4...7.....12...4......8.....9.....1......9.........2..3...16....8
...234....56....4...7....8.3...4......8.....9.....1......8.........7..3...19....2
...234....56....4...7...2..3...4......8.....9.....1......9.........2..3...16....8
...234....56....4...7..5...2...4......8.....1.....1......6.........2..5...19....8
...234....56....4...7.8....3...4......8.....1.....1......6.........2..3...18....7
...234....56....4...78.....2...4......8.....9.....1......1.........2..3...19....8

...xxx....xx....x..xx......x...x......x.....x.....x......x.........x..x...xx....x  no puzzle found

...234....56....7.9.4......7...5......1.....4.....7......4.........8..5...91....6
...234....59....41..7......2...4......5.....9.....1......9.........2..3...16....5
...234....56...84...7......3...4......8.....1.....1......6.........2..3...18....7
...234....56..7.4...7......4...6......8.....1.....5......8.........4..6...19....7
...234....56.7..4...7......2...4......8.....1.....1......6.........2..5...19....8
...234....568...4...7......2...4......8.....7.....1......6.........2..3...97....1
...234...195....4...2......4...5......3.....7.....1......3.........4..5...87....6
...634..9.58....7...4......3...5......7.....4.....6......9.........8..5...21....6
...134.9..56....7...4......7...5......2.....4.....7......2.........8..5...94....6
...7342...56....4...7......3...4......8.....1.....1......8.........2..3...16....5
..9837....46....1...5......1...6......7.....8.....3......5.........1..4...87....2
.3.214....56....4...7......2...4......8.....7.....5......7.........2..1...58....6
2..134....56....2...7......3...4......1.....6.....5......7.........1..3...86....5
---------------------------------------------------------------------------------

Maybe someone can find a puzzle for those 2 pattern !! ?

The fact that there were no valid 18C patterns goes some way to explain why there are many more invalid 18C patterns than could be excluded with the magic 40 templates...