The New Sudoku Players' Forum

by **JPF** » Fri Mar 10, 2017 10:18 pm

Hi Serg & coloin,

Serg wrote:It is not surprising that you didn't found subsets of Magic Pattern among F1-F27 patterns. Those subsets were filtered out (by "40 maximal patterns list") at earlier stages of my work. I started from 6016 ed fully symmetrical patterns and came to 16 "unknown" patterns list. The work is going to the end.
Serg

Thanks. I understand now how and why you came up with your 27F fully symmetric patterns.
When you are finished with your study, it would be nice to give, for each valid pattern, an example of a valid puzzle.

coloin wrote:Just to refer back to this thread Fully symmetrical puzzles
You may have seen this ...
it tried initially to do what you are doing , the numenclature is good i think
JPFpublished this
.....
from > 10 years ago !!!

Yes and we are all 10 years older

I am glad to know that now some of us are able to fully study a pattern in a couple of minutes/hours.

here, Serg wrote:Hi, JPF!
Your method is clear and straightforward, but I'd like to have mathematical calculated numbers for following reasons:
1. Neither program (tool) is guaranteed not to have bugs, so it is useful to have mathematically calculated results for proving program results.
2. ....

Well, it seems that the only technique curently used to check if a pattern is valid or not is to do it with computers and brute force.
but it's ok

JPF

by **Serg** » Sat Mar 11, 2017 9:48 am

Hi, JPF!

JPF wrote:When you are finished with your study, it would be nice to give, for each valid pattern, an example of a valid puzzle.

Yes, you are right - if someone says that a pattern has valid puzzles, he must demonstrate examples. But in this study a lot of patterns were determined as "valid", because there exist subset patterns with valid puzzle examples. So, I am planning to publish a list of such "irreducible" or "seed" valid patterns with examples. Other valid patterns will be supersets of these patterns. It would be more accurate to call these "irreducible" valid patterns as "symmetrically minimal", because they lose validity by removing any clue cells, provided that resulting pattern will remain fully symmetrical.

JPF wrote:
here, Serg wrote:Hi, JPF!
Your method is clear and straightforward, but I'd like to have mathematical calculated numbers for following reasons:
1. Neither program (tool) is guaranteed not to have bugs, so it is useful to have mathematically calculated results for proving program results.
2. ....
Well, it seems that the only technique curently used to check if a pattern is valid or not is to do it with computers and brute force.
but it's ok

Yes, we need mathematical foundations of such studies, but "brute force" search is unavoidable yet. It is interesting, that "classical" mathematics has very similar to sudoku area - Latin Squares. Mathematicians study them several hundred years, but know number of latin squares of given size up to 11 x 11 size only. They study also "critical sets" - sudoku valid puzzles analogue. For some kind of latin squares is known mathematically proved formula of "minimal critical sets cardinality". In sudoku world it would be "minimal number of clues in valid puzzle". For latin squares of odd size (for example, 9 x 9 latin squares) "minimal critical sets cardinality" (for "back circulant latin squares" only) is (n^2 - 1)/4. Back circulant latin square 9 x 9 has "minimal critical sets cardinality" 20. It is rather similar to sudoku "minimal number of clues in valid puzzle" - 17. Sudoku have more constraints than latin squares, so they need less clues to have unique solution.

Serg

[Edited. I corrected a typo in the formula.]

by **Afmob** » Sat Mar 11, 2017 4:19 pm

Here are all ED valid puzzles of F14:

Code: Select all: ...2.6.......1......7...8..6..1.2..9.2..3..1.1..4.5..7..4...2......8.......6.1... ...2.6.......1......7...8..6..1.2..9.2..3..1.1..4.5..7..5...2......8.......6.1...

And of F19 (isomorphic to coloin's puzzles):

Code: Select all: 12.3.4.566.......7.........3..1.5..4.........8..2.6..1.........7.......845.6.1.23 12.3.4.566.......7.........3..5.1..4.........8..6.2..1.........7.......845.1.6.23

by **coloin** » Sat Mar 11, 2017 10:26 pm

Code: Select all: +---+---+---+ |15.|3.4|.26| |9..|...|..7| |...|...|...| +---+---+---+ |3..|1.9|..5| |...|...|...| |6..|7.5|..1| +---+---+---+ |...|...|...| |7..|...|..9| |52.|6.1|.43| +---+---+---+

Code: Select all: 15.3.4.269.......7.........3..1.9..5.........6..7.5..1.........7.......952.6.1.43 15.3.4.269.......7.........6..1.9..5.........3..7.5..1.........7.......952.6.1.43

2 valid puzzles for F19
C

by **Serg** » Sat Mar 11, 2017 11:38 pm

Hi, Afmob!
Fantastic results in searching through F12-F27 patterns! Especially it's surprising existance of 2 (only!) valid puzzles for F14 pattern!

I hope you corrected my typos in F15 pattern description. Clues r5c1 and r5c9 were absent in my post with "16 unknown patterns" (I've already corrected this post.) Those typos were evident - they prevented F15 pattern from being fully symmetrical.

Serg

by **Serg** » Sun Mar 12, 2017 12:01 am

Hi, coloin!

coloin wrote:
Code: Select all
+---+---+---+ |15.|3.4|.26| |9..|...|..7| |...|...|...| +---+---+---+ |3..|1.9|..5| |...|...|...| |6..|7.5|..1| +---+---+---+ |...|...|...| |7..|...|..9| |52.|6.1|.43| +---+---+---+

Code: Select all
15.3.4.269.......7.........3..1.9..5.........6..7.5..1.........7.......952.6.1.43 15.3.4.269.......7.........6..1.9..5.........3..7.5..1.........7.......952.6.1.43

2 valid puzzles for F19

Wonderful! The last element of the puzzle went in place.

Of course, we must crosscheck Afmob's exhaustive search results for patterns having no valid puzzles. It's sufficient to check patterns: F11, F13, F17, F18, F20, F21, F22, F23, F26, F27 (10 patterns in total). I am planning to crosscheck these patterns, but it will take me 3-4 weeks to do it. So, I begin to process all collected data not waiting for invalidity crosscheck for these 10 patterns.

Serg

[Edited. I corrected typos.]

by **JPF** » Sun Mar 12, 2017 12:03 am

Here, 10 years ago, we had a fascinating debate with coloin about sex of angels and this puzzle:

Code: Select all: 1 . . | . . . | . . 2 . 3 . | 1 . 4 | . 5 . . . . | . . . | . . . -------+-------+------- . 5 . | 6 . 3 | . 4 . . . . | . . . | . . . . 6 . | 4 . 7 | . 3 . -------+-------+------- . . . | . . . | . . . . 4 . | 3 . 5 | . 6 . 8 . . | . . . | . . 9 3 solutions.

Adding an 8 in r1c2 makes the puzzle valid and therefore F19 also a valid pattern.
Edit: wrong statement ; the pattern with r1c2 is not a subset of F19
Congratulations to coloin for his quick findings!

JPF

by **Afmob** » Sun Mar 12, 2017 12:29 pm

Computation has finished. You can see the results in this table and also here. Of course, the results should be verified for the patterns having no valid puzzles as Serg said.

by **Serg** » Sun Mar 12, 2017 5:32 pm

Hi, all!
Afmob has done huge amount of work very, very fast, and coloin accelerated the work by his finding, so the first result of this project can be published. These results are preliminary, because not all Afmob's results are crosschecked yet.

It is known, that there are 6016 essentially different fully symmetrical patterns. It turns out, that 5145 of them have valid puzzles and 871 patterns have no valid puzzles. (85.5 % valid patterns.) I think, similar ratio should be for arbitrary (not fully symmetrical) patterns.

Collections of essentially different fully symmetrical valid and invalid patternsare attached to this post. All patterns are shown in "fully symmetrical essential form" or simply "essential form". Below is cells ordering schema (15 cells have hexadecimal sequence numbers: 1,2,...,9,A,B,C,D,E,F).

Code: Select all: A B C 1 2 . . . . . D E 3 4 . . . . . . F 5 6 . . . . . . . 7 8 . . . . . . . . 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Metric is 15-digit binary number (cell 1 is accounted as 2^14, cell F is accounted as 2^0 (1)). All isomorphic transformations preserving "full symmetry" are considered to maximize metric.

Here is distribution of valid patterns by number of clues.

Code: Select all: Clues Patterns 20 16 21 22 24 87 25 89 28 178 29 180 32 271 33 272 36 345 37 346 40 381 41 382 44 370 45 370 48 319 49 319 52 245 53 245 56 165 57 165 60 98 61 98 64 52 65 52 68 24 69 24 72 10 73 10 76 4 77 4 80 1 81 1

It's not surprise, that valid fully symmetrical puzzles must have not less than 20 clues. This study proves it.
There are 16 ed 20-clue valid fully symmetrical patterns. It looks like 2 or 3 new patterns are found (I'll analyze it in details in some time). Here are all 20-clue valid fully symmetrical patterns in "essential form".

Code: Select all: Patterns with 20 clues Pattern 1 . . . . x . . . . . x x . . . x x . . x . . . . . x . . . . x . x . . . x . . . . . . . x . . . x . x . . . . x . . . . . x . . x x . . . x x . . . . . x . . . . Pattern 2 . x . . x . . x . x . . . . . . . x . . x . . . x . . . . . x . x . . . x . . . . . . . x . . . x . x . . . . . x . . . x . . x . . . . . . . x . x . . x . . x . Pattern 3 x . . . x . . . x . x . . . . . x . . . x . . . x . . . . . x . x . . . x . . . . . . . x . . . x . x . . . . . x . . . x . . . x . . . . . x . x . . . x . . . x Pattern 4 . . . x . x . . . . x . . . . . x . . . x . . . x . . x . . . x . . . x . . . x . x . . . x . . . x . . . x . . x . . . x . . . x . . . . . x . . . . x . x . . . Pattern 5 . x . x . x . x . x . . . . . . . x . . . . . . . . . x . . . x . . . x . . . x . x . . . x . . . x . . . x . . . . . . . . . x . . . . . . . x . x . x . x . x . Pattern 6 x . . x . x . . x . x . . . . . x . . . . . . . . . . x . . . x . . . x . . . x . x . . . x . . . x . . . x . . . . . . . . . . x . . . . . x . x . . x . x . . x Pattern 7 . . . x . x . . . . x . . . . . x . . . x . . . x . . x . . x . x . . x . . . . . . . . . x . . x . x . . x . . x . . . x . . . x . . . . . x . . . . x . x . . . Pattern 8 . . . x . x . . . . . x . x . x . . . x . . . . . x . x . . . . . . . x . x . . . . . x . x . . . . . . . x . x . . . . . x . . . x . x . x . . . . . x . x . . . Pattern 9 . . . x . x . . . . x . . x . . x . . . x . . . x . . x . . . . . . . x . x . . . . . x . x . . . . . . . x . . x . . . x . . . x . . x . . x . . . . x . x . . . Pattern 10 . . x x . x x . . . . . . x . . . . x . . . . . . . x x . . . . . . . x . x . . . . . x . x . . . . . . . x x . . . . . . . x . . . . x . . . . . . x x . x x . . Pattern 11 . x . x . x . x . x . . . x . . . x . . . . . . . . . x . . . . . . . x . x . . . . . x . x . . . . . . . x . . . . . . . . . x . . . x . . . x . x . x . x . x . Pattern 12 x . . x . x . . x . . . . x . . . . . . x . . . x . . x . . . . . . . x . x . . . . . x . x . . . . . . . x . . x . . . x . . . . . . x . . . . x . . x . x . . x Pattern 13 x . . x . x . . x . x . . x . . x . . . . . . . . . . x . . . . . . . x . x . . . . . x . x . . . . . . . x . . . . . . . . . . x . . x . . x . x . . x . x . . x Pattern 14 . . . x . x . . . . . . . x . . . . . . x . . . x . . x . . . x . . . x . x . x . x . x . x . . . x . . . x . . x . . . x . . . . . . x . . . . . . . x . x . . . Pattern 15 . . . x . x . . . . x . . x . . x . . . . . . . . . . x . . . x . . . x . x . x . x . x . x . . . x . . . x . . . . . . . . . . x . . x . . x . . . . x . x . . . Pattern 16 . . . x x x . . . . . . . x . . . . . . x . . . x . . x . . . . . . . x x x . . . . . x x x . . . . . . . x . . x . . . x . . . . . . x . . . . . . . x x x . . .

Here is distribution of invalid patterns by number of clues.

Code: Select all: Clues Patterns 0 1 1 1 4 4 5 4 8 10 9 10 12 24 13 24 16 52 17 52 20 82 21 76 24 78 25 76 28 68 29 66 32 52 33 51 36 35 37 34 40 21 41 20 44 10 45 10 48 4 49 4 52 1 53 1

Here is invalid pattern with largest number of clues (53 clues):

Code: Select all: x x x . x . x x x x x x . x . x x x x x x . x . x x x . . . . x . . . . x x x x x x x x x . . . . x . . . . x x x . x . x x x x x x . x . x x x x x x . x . x x x

This is well known maximal pattern from "40 maximal patterns list".

I crosschecked numbers for valid and invalid patterns with JPF numbers, posted by coloin. Sums of valid and invalid patterns for given number of clues coincide with JPF numbers in all cases.

Continuation follows ...

Serg

by **coloin** » Sun Mar 12, 2017 6:11 pm

Great work !
For completeness, we could have the six 21-clue valid fully symmetrical patterns which don't have a 20 clue valid pattern by removing the central clue.
maybe Afmob can easily run through the 20- and the six 21-clue patterns with a puzzle count ...

F15plus1 = F13 - zero puzzles
F16plus1 = F12 - zero puzzles
F20plus1 = F14 - has 2 puzzles

The six are therefore - F17plus1,F18plus1,F20plus1,F21plus1,F26plus1,F27plus1 .... and they all have valid puzzles.

Code: Select all: F17plus1 F18plus1 F20plus1 F21plus1 F26plus1 F27plus1 +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ |.1.|3.6|.4.| |5..|3.1|..6| |...|2.6|...| |...|6.8|...| |...|8.6|...| |...|657|...| |2..|...|..8| |.8.|...|.4.| |...|.1.|...| |.3.|.1.|.7.| |.4.|.9.|.3.| |.4.|.2.|.3.| |...|...|...| |...|...|...| |..7|...|8..| |...|...|...| |...|.5.|...| |...|...|...| +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ |8..|1.2|..6| |6..|8.5|..2| |6..|1.2|..9| |4..|1.9|..8| |8..|...|..6| |5..|...|..6| |...|.3.|...| |...|.1.|...| |.2.|.3.|.1.| |.2.|.3.|.9.| |.12|.3.|45.| |12.|.3.|.45| |9..|4.5|..7| |3..|2.6|..1| |1..|4.5|..7| |3..|4.5|..6| |6..|...|..1| |7..|...|..1| +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ |...|...|...| |...|...|...| |..4|...|2..| |...|...|...| |...|.4.|...| |...|...|...| |7..|...|..2| |.7.|...|.9.| |...|.8.|...| |.9.|.7.|.2.| |.3.|.2.|.9.| |.3.|.8.|.9.| |.5.|2.1|.3.| |2..|6.3|..5| |...|6.1|...| |...|8.4|...| |...|5.7|...| |...|743|...| +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+ +---+---+---+

Code: Select all: F17plus1 has at least 57 puzzles F18plus1 has at least 7 puzzles F20plus1 has only 2 puzzles F21plus1 has at least 212 puzzles F26plus1 has at least 2222 puzzles F27plus1 has at least 1893 puzzles

updated

by **Serg** » Mon Mar 13, 2017 11:37 pm

Hi, people!
I should present a kind of proof that published above full lists of invalid and valid fully symmetrical patterns are correct.
The idea of proof:

1. I'll present "basic set" of invalid patterns (25 patterns in total), which are known to be invalid because an exhaustive search was done for it or it is a subset of another invalid pattern. Everyone will be able to check, that all fully symmetrical invalid patterns (871) are subsets of this "basic" invalid patterns.

2. I'll present "basic set" of valid patterns with examples of valid puzzles. Everyone will be able to check, that all fully symmetrical valid patterns (5145) are supersets of this "basic" valid patterns.

3. Then everyone will be able to check, that each essentially different fully symmetrical pattern (6016 patterns in total) is subset of an invalid pattern from "basic set" of invalid patterns (thus the pattern will be invalid) or is superset of a valid pattern from "basic set" of valid patterns (thus the pattern will be valid).

This is "basic set" of invalid patterns (strictly speaking they all are symmetrically maximal patterns).

Code: Select all: F1 F2 F3 F4 F5 x . . . . . . . x x x x . . . x x x . x x . . . x x . x x . . . . . x x . . . x x x . . . . x . . . . . x . x . . . . . . . x x . . . . . . . x x x . . . . . x x . x . . . . . x . . . x . . . x . . x . . . . . . . x x . . . . . . . x . . . . . . . . . . . . . . . . . . . . . x x x . . . . . . x . x . . . . . . x x x . . . . . . x x x . . . x . . x x x . . x . . . x x x . . . . . . . x . . . . . . . x x x . . . . . . x x x . . . x . . x x x . . x . . . x x x . . . . . . x . x . . . . . . x x x . . . . . . x x x . . . x . . x x x . . x . . x . . . x . . x . . . . . . . x x . . . . . . . x . . . . . . . . . . . . . . . . . . . x . . . . . x . x . . . . . . . x x . . . . . . . x x x . . . . . x x . x . . . . . x . x . . . . . . . x x x x . . . x x x . x x . . . x x . x x . . . . . x x . . . x x x . . . F6 F7 F8 F11 F12 x x . . . . . x x x . . . x . . . x . . . . x . . . . x . . . x . . . x x x . . x . . x x x . . . . . . . x . x . . x . . x . . x . . . . . x . . x . . x . . x . x . . . . . . . x . . x . . . x . . . . . . . . . . . . . x . . . x . . . . . . x . . . . . . . . . . . . . . . . x . x . . . . . . x x x . . . . . . x x x . . . . . . x . x . . . . . . x . x . . . . . . . x . . . . x x . x x x . x x x . . x x x . . x x x x . x . x x x x . . . x . . . x . . . x . x . . . . . . x x x . . . . . . x x x . . . . . . x . x . . . . . . x . x . . . . . x . . . x . . . . . . . . . . . . . x . . . x . . . . . . x . . . . . . . . . . . . . x . . . . . . . x . x . . x . . x . . x . . . . . x . . x . . x . . x . x . . . . . . . x x x . . . . . x x x . . . x . . . x . . . . x . . . . x . . . x . . . x x x . . x . . x x F13 F17 F18 F20 F21 . x . . x . . x . . x . x . x . x . x . . x . x . . x . . . x . x . . . . . . x . x . . . x x . . . . . x x x . . . . . . . x . x . . . . . x . . . . . x . . . . . x . . x . . x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x . . . x . . . . . . . . . . . . . . x . x . . . x . . x . x . . x x . . x . x . . x x . . x . x . . x x . . x . x . . x x . . . x . . . x . . . . . . . . . . . . . . . . . . . x . . . . . x . . x . . . . . x . . . . x . x . . . x . . x . x . . x x . . x . x . . x x . . x . x . . x x . . x . x . . x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x . . . x . . . . . . . . . . . x x . . . . . x x x . . . . . . . x . x . . . . . x . . . . . x . . . . . x . . x . . x . . x . . x . . x . . x . x . x . x . x . . x . x . . x . . . x . x . . . . . . x . x . . . F22 F23 F26 F27 F28 . . . x . x . . . . . . x . x . . . . . . x . x . . . . . . x x x . . . x . . x . x . . x . . . . x . . . . . x . . x . . x . . x . . x . . x . . x . . x . . x . . . . x . x . . . . . x . . . x . . . . . . . . . . . . . . . x . . . . . . . . . . . . . . . . x . x . . . x . . . . . . . x x . . . . . . . x x . . . . . . . x x . . . . . . . x x x x x x x x x x . x . . x . . x . . x . . x . . x . . x x . . . x x . x x . . . . . x x . . . x x x . . . x . . . . . . . x x . . . . . . . x x . . . . . . . x x . . . . . . . x x x x x x x x x x . . x . . . x . . . . . . . . . . . . . . . x . . . . . . . . . . . . . . . . x . x . . . . . . . x . . . . . x . . x . . x . . x . . x . . x . . x . . x . . x . . . . x . x . . . . . . x . x . . . . . . x . x . . . . . . x . x . . . . . . x x x . . . x . . x . x . . x F29 F30 F31 F32 F33 x . . x x x . . x x x x . . . x x x x x x x x x x x x x x x . x . x x x . . . x x x . . . . . . . x . . . . x x x . . . x x x x x x . . . x x x x x x . x . x x x . . . x x x . . . . . . . x . . . . x x x . . . x x x x x x . . . x x x x x x . x . x x x . . . x x x . . . x . . x x x . . x . . . x . x . . . x . . . . . . . x . . . . x . . . . x x x x x x x x x x x x x x x x x x . . . . . . . . . x . . . x . . . x x x x x x x x x x x x x x x x x x x x . . x x x . . x . . . x . x . . . x . . . . . . . x . . . . x . . . . x x x x x x x x x . . . . x . . . . x x x . . . x x x x x x . . . x x x x x x . x . x x x . . . x x x . . . . . . . x . . . . x x x . . . x x x x x x . . . x x x x x x . x . x x x . . . x x x . . . x . . x x x . . x x x x . . . x x x x x x x x x x x x x x x . x . x x x . . . x x x . . .

All patterns have independent confirmation of their invalidity by two (or more) persons who have done exhaustive search for these patterns.
One can make sure that all 871 invalid fully symmetrical patterns reported before are subsets of some patterns from this list.

First 20 patterns from this "basic set" of invalid patterns can be used for filtering out invalid patterns in addition to "40 patterns list" (we can say now about '60 patterns list").

Serg

[Edited. I stated all patterns have independent confirmation of their invalidity after finishing my exhaustive search.]

by **Serg** » Tue Mar 14, 2017 11:19 pm

Hi, all!
"Basic set" for fully symmetrical valid patterns consists of 49 patterns. All 5145 fully symmetrical valid patterns published above are supersets of some patterns from the list posted below. (Patterns from this list are symmetrically minimal, removal of several clues makes such patterns invalid, provided that resulting pattern remains fully symmetrical.)

Code: Select all: Distribution of patterns by number of clues Clues Patterns 20 16 21 6 24 23 25 2 28 2

Here are all patterns from this list with examples of valid puzzles.

Hidden Text: Show

Code: Select all: Patterns with 20 clues (16 patterns) Pattern F34 Old example (ab) Original shape . . . . x . . . . . . . . 1 . . . . 7 9 . . . . . 5 4 . x x . . . x x . . 7 9 . . . 5 4 . 5 . . . . . . . 7 . x . . . . . x . . 5 . . . . . 7 . . . . . 1 . . . . . . . x . x . . . . . . 1 . 2 . . . . . . 1 . 2 . . . x . . . . . . . x 8 . . . . . . . 1 . . 8 . . . 1 . . . . . x . x . . . . . . 5 . 7 . . . . . . 5 . 7 . . . . x . . . . . x . . 6 . . . . . 3 . . . . . 9 . . . . . x x . . . x x . . 2 4 . . . 7 9 . 6 . . . . . . . 3 . . . . x . . . . . . . . 9 . . . . 2 4 . . . . . 7 9 Pattern F35 Old example (ab) Original shape . x . . x . . x . . 7 . . 5 . . 3 . . . 7 . 5 . 3 . . x . . . . . . . x 8 . . . . . . . 9 . 6 . . . . . 5 . . . x . . . x . . . . 6 . . . 5 . . 8 . . . . . . . 9 . . . x . x . . . . . . 8 . 5 . . . . . . 8 . 5 . . . x . . . . . . . x 9 . . . . . . . 2 9 . . . . . . . 2 . . . x . x . . . . . . 4 . 3 . . . . . . 4 . 3 . . . . . x . . . x . . . . 2 . . . 4 . . 4 . . . . . . . 8 x . . . . . . . x 4 . . . . . . . 8 . 2 . . . . . 4 . . x . . x . . x . . 5 . . 3 . . 7 . . . 5 . 3 . 7 . . Pattern F36 Old example (ab) Original shape x . . . x . . . x 7 . . . 2 . . . 9 . 3 . . . . . 4 . . x . . . . . x . . 6 . . . . . 3 . 6 . . . . . . . 3 . . x . . . x . . . . 3 . . . 4 . . . . 7 . 2 . 9 . . . . . x . x . . . . . . 3 . 8 . . . . . . 3 . 8 . . . x . . . . . . . x 9 . . . . . . . 7 . . 9 . . . 7 . . . . . x . x . . . . . . 5 . 6 . . . . . . 5 . 6 . . . . . x . . . x . . . . 9 . . . 7 . . . . 2 . 1 . 4 . . . x . . . . . x . . 5 . . . . . 6 . 5 . . . . . . . 6 x . . . x . . . x 2 . . . 1 . . . 4 . 9 . . . . . 7 . Pattern F37 Old example (ab) Original shape . . . x . x . . . . . . 2 . 5 . . . . . . 2 . 5 . . . . x . . . . . x . . 7 . . . . . 4 . . . 7 . . . 4 . . . . x . . . x . . . . 9 . . . 3 . . . 9 . . . . . 3 . x . . . x . . . x 2 . . . 7 . . . 1 2 . . . 7 . . . 1 . . . x . x . . . . . . 8 . 3 . . . . . . 8 . 3 . . . x . . . x . . . x 1 . . . 9 . . . 5 1 . . . 9 . . . 5 . . x . . . x . . . . 4 . . . 8 . . . 4 . . . . . 8 . . x . . . . . x . . 8 . . . . . 5 . . . 8 . . . 5 . . . . . x . x . . . . . . 7 . 1 . . . . . . 7 . 1 . . . Pattern F38 Old example (ab) Original shape . x . x . x . x . . 2 . 5 . 8 . 1 . . . . . . . . . . x . . . . . . . x 7 . . . . . . . 9 . . 2 5 . 8 1 . . . . . . . . . . . . . . . . . . . . . 7 . . . . . 9 . x . . . x . . . x 5 . . . 6 . . . 4 . 5 . . 6 . . 4 . . . . x . x . . . . . . 3 . 2 . . . . . . 3 . 2 . . . x . . . x . . . x 6 . . . 9 . . . 3 . 6 . . 9 . . 3 . . . . . . . . . . . . . . . . . . . . 9 . . . . . 7 . x . . . . . . . x 9 . . . . . . . 7 . . 8 2 . 1 3 . . . x . x . x . x . . 8 . 2 . 1 . 3 . . . . . . . . . . Pattern F39 Old example (tarek) Original shape x . . x . x . . x 3 . . 4 . 5 . . 7 3 . . 4 . 5 . . 7 . x . . . . . x . . 2 . . . . . 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 . . . 6 . . x . . . x . . . x 5 . . . 1 . . . 9 5 . . . 1 . . . 9 . . . x . x . . . . . . 6 . 8 . . . . . . 6 . 8 . . . x . . . x . . . x 9 . . . 7 . . . 4 9 . . . 7 . . . 4 . . . . . . . . . . . . . . . . . . . . 3 . . . 2 . . . x . . . . . x . . 3 . . . . . 2 . . . . . . . . . . x . . x . x . . x 4 . . 9 . 7 . . 5 4 . . 9 . 7 . . 5 Pattern F40 Old example (ab) Original shape . . . x . x . . . . . . 4 . 5 . . . . . . 4 . 5 . . . . x . . . . . x . . 9 . . . . . 8 . . . 9 . . . 8 . . . . x . . . x . . . . 7 . . . 4 . . . 7 . . . . . 4 . x . . x . x . . x 3 . . 9 . 8 . . 6 3 . . 9 . 8 . . 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . x . . x . x . . x 8 . . 3 . 2 . . 5 8 . . 3 . 2 . . 5 . . x . . . x . . . . 5 . . . 9 . . . 5 . . . . . 9 . . x . . . . . x . . 7 . . . . . 4 . . . 7 . . . 4 . . . . . x . x . . . . . . 5 . 1 . . . . . . 5 . 1 . . . Pattern F41 Old example (ab) Original shape . . . x . x . . . . . . 7 . 2 . . . . . . 7 . 2 . . . . . x . x . x . . . . 2 . 4 . 6 . . . . 2 . 4 . 6 . . . x . . . . . x . . 6 . . . . . 5 . . 6 . . . . . 5 . x . . . . . . . x 4 . . . . . . . 9 4 . . . . . . . 9 . x . . . . . x . . 1 . . . . . 3 . . 1 . . . . . 3 . x . . . . . . . x 3 . . . . . . . 7 3 . . . . . . . 7 . x . . . . . x . . 8 . . . . . 9 . . 8 . . . . . 9 . . . x . x . x . . . . 4 . 2 . 1 . . . . 4 . 2 . 1 . . . . . x . x . . . . . . 5 . 3 . . . . . . 5 . 3 . . . Pattern F42 Old example (ab) Original shape . . . x . x . . . . . . 4 . 3 . . . 6 . . . . . . . 5 . x . . x . . x . . 5 . . 7 . . 1 . . . . 4 . 3 . . . . . x . . . x . . . . 6 . . . 5 . . . . 5 . 7 . 1 . . x . . . . . . . x 2 . . . . . . . 7 . 2 . . . . . 7 . . x . . . . . x . . 7 . . . . . 2 . . . 7 . . . 2 . . x . . . . . . . x 8 . . . . . . . 3 . 8 . . . . . 3 . . . x . . . x . . . . 4 . . . 3 . . . . 6 . 2 . 5 . . . x . . x . . x . . 6 . . 2 . . 5 . . . . 7 . 8 . . . . . . x . x . . . . . . 7 . 8 . . . 4 . . . . . . . 3 Pattern F43 Old example (ab) Original shape . . x x . x x . . . . 6 1 . 3 5 . . . . . . 6 . . . . . . . . x . . . . . . . . 6 . . . . . . 6 1 . 3 5 . . x . . . . . . . x 4 . . . . . . . 2 . 4 . . . . . 2 . x . . . . . . . x 2 . . . . . . . 9 . 2 . . . . . 9 . . x . . . . . x . . 9 . . . . . 4 . 9 . . . . . . . 4 x . . . . . . . x 3 . . . . . . . 8 . 3 . . . . . 8 . x . . . . . . . x 9 . . . . . . . 1 . 9 . . . . . 1 . . . . . x . . . . . . . . 5 . . . . . . 5 6 . 7 3 . . . . x x . x x . . . . 5 6 . 7 3 . . . . . . 5 . . . . Pattern F44 Old example (Ocean) Original shape . x . x . x . x . . 4 . 1 . 5 . 6 . . . . . . . . . . x . . . x . . . x 1 . . . 2 . . . 3 . . 1 . 2 . 3 . . . . . . . . . . . . . . . . . . . . . 4 . 1 . 5 . 6 . x . . . . . . . x 3 . . . . . . . 7 . . 3 . . . 7 . . . x . . . . . x . . 8 . . . . . 4 . . 8 . . . . . 4 . x . . . . . . . x 9 . . . . . . . 2 . . 9 . . . 2 . . . . . . . . . . . . . . . . . . . . . 5 . 4 . 7 . 8 . x . . . x . . . x 2 . . . 1 . . . 9 . . 2 . 1 . 9 . . . x . x . x . x . . 5 . 4 . 7 . 8 . . . . . . . . . . Pattern F45 Old example (ab) Original shape x . . x . x . . x 1 . . 7 . 2 . . 3 . . . . 3 . . . . . . . . x . . . . . . . . 3 . . . . . 9 . . . . . 5 . . . x . . . x . . . . 9 . . . 5 . . . . 1 7 . 2 3 . . x . . . . . . . x 3 . . . . . . . 2 . . 3 . . . 2 . . . x . . . . . x . . 6 . . . . . 9 . 6 . . . . . . . 9 x . . . . . . . x 4 . . . . . . . 7 . . 4 . . . 7 . . . . x . . . x . . . . 2 . . . 3 . . . . 7 8 . 4 1 . . . . . . x . . . . . . . . 5 . . . . . 2 . . . . . 3 . x . . x . x . . x 7 . . 8 . 4 . . 1 . . . . 5 . . . . Pattern F46 Old example (ab) Original shape x . . x . x . . x 3 . . 9 . 8 . . 1 . . . . . . . . . . x . . x . . x . . 2 . . 7 . . 8 . . 3 . 9 . 8 . 1 . . . . . . . . . . . . . . . . . . . . . 2 . 7 . 8 . . x . . . . . . . x 1 . . . . . . . 3 . 1 . . . . . 3 . . x . . . . . x . . 5 . . . . . 7 . . . 5 . . . 7 . . x . . . . . . . x 9 . . . . . . . 6 . 9 . . . . . 6 . . . . . . . . . . . . . . . . . . . . . 7 . 5 . 2 . . . x . . x . . x . . 7 . . 5 . . 2 . . 6 . 1 . 7 . 9 . x . . x . x . . x 6 . . 1 . 7 . . 9 . . . . . . . . . Pattern F47 Old example (JPF) Original shape . . . x . x . . . . . . 6 . 7 . . . . . . . 1 . . . . . . . . x . . . . . . . . 1 . . . . . . . 6 . 7 . . . . . x . . . x . . . . 3 . . . 4 . . . . 3 . . . 4 . . x . . . x . . . x 6 . . . 9 . . . 8 . 6 . . 9 . . 8 . . x . x . x . x . . 7 . 8 . 2 . 1 . 7 . . 8 . 2 . . 1 x . . . x . . . x 1 . . . 5 . . . 3 . 1 . . 5 . . 3 . . . x . . . x . . . . 5 . . . 7 . . . . 5 . . . 7 . . . . . . x . . . . . . . . 2 . . . . . . . 1 . 6 . . . . . . x . x . . . . . . 1 . 6 . . . . . . . 2 . . . . Pattern F48 Old example (Ocean) Original shape . . . x . x . . . . . . 1 . 2 . . . . . . . . . . . . . x . . x . . x . . 3 . . 4 . . 5 . . . . 1 . 2 . . . . . . . . . . . . . . . . . . . . . . . 3 . 4 . 5 . . x . . . x . . . x 2 . . . 6 . . . 7 . 2 . . 6 . . 7 . . x . x . x . x . . 6 . 4 . 5 . 8 . . . 6 4 . 5 8 . . x . . . x . . . x 7 . . . 8 . . . 9 . 7 . . 8 . . 9 . . . . . . . . . . . . . . . . . . . . . 5 . 3 . 6 . . . x . . x . . x . . 5 . . 3 . . 6 . . . . 9 . 7 . . . . . . x . x . . . . . . 9 . 7 . . . . . . . . . . . . Pattern F49 Old example (Ocean) Original shape . . . x x x . . . . . . 2 3 4 . . . . . . . 1 . . . . . . . . x . . . . . . . . 1 . . . . . . . 2 3 4 . . . . . x . . . x . . . . 1 . . . 5 . . . . 1 . . . 5 . . x . . . . . . . x 2 . . . . . . . 6 . 2 . . . . . 6 . x x . . . . . x x 3 6 . . . . . 4 2 6 3 . . . . . 2 4 x . . . . . . . x 4 . . . . . . . 7 . 4 . . . . . 7 . . . x . . . x . . . . 5 . . . 8 . . . . 5 . . . 8 . . . . . . x . . . . . . . . 4 . . . . . . . 9 6 7 . . . . . . x x x . . . . . . 9 6 7 . . . . . . . 4 . . . . Patterns with 21 clues (6 patterns) Pattern F50 Old example (JPF) Original shape . x . x . x . x . . 8 . 6 . 3 . 9 . . . 1 . . . 7 . . x . . . . . . . x 1 . . . . . . . 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 . . 6 . 3 . . 9 x . . x . x . . x 6 . . 7 . 9 . . 4 . . 6 7 . 9 4 . . . . . . x . . . . . . . . 3 . . . . . . . . 3 . . . . x . . x . x . . x 2 . . 5 . 8 . . 1 . . 2 5 . 8 1 . . . . . . . . . . . . . . . . . . . . 3 . . 9 . 5 . . 7 x . . . . . . . x 4 . . . . . . . 2 . . . . . . . . . . x . x . x . x . . 3 . 9 . 5 . 7 . . . 4 . . . 2 . . Pattern F51 Old example (JPF) Original shape x . . x . x . . x 1 . . 5 . 9 . . 7 3 . . . . . . . 4 . x . . . . . x . . 3 . . . . . 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 5 . 9 7 . . x . . x . x . . x 5 . . 3 . 1 . . 9 . . 5 3 . 1 9 . . . . . . x . . . . . . . . 6 . . . . . . . . 6 . . . . x . . x . x . . x 7 . . 9 . 2 . . 1 . . 7 9 . 2 1 . . . . . . . . . . . . . . . . . . . . . . 9 1 . 5 8 . . . x . . . . . x . . 8 . . . . . 3 . . . . . . . . . . x . . x . x . . x 9 . . 1 . 5 . . 8 8 . . . . . . . 3 Pattern F14 New example (Afmob) . . . x . x . . . . . . 2 . 6 . . . . . . . x . . . . . . . . 1 . . . . . . x . . . x . . . . 7 . . . 8 . . x . . x . x . . x 6 . . 1 . 2 . . 9 . x . . x . . x . . 2 . . 3 . . 1 . x . . x . x . . x 1 . . 4 . 5 . . 7 . . x . . . x . . . . 4 . . . 2 . . . . . . x . . . . . . . . 8 . . . . . . . x . x . . . . . . 6 . 1 . . . Pattern F52 Old example (ab) Original shape . . . x . x . . . . . . 5 . 1 . . . . . . . . . . . . . x . . x . . x . . 4 . . 6 . . 9 . . 4 . . 6 . . 9 . . . . . . . . . . . . . . . . . . . . . . 5 . 1 . . . x . . x . x . . x 5 . . 9 . 3 . . 8 . . 5 9 . 3 8 . . . x . . x . . x . . 9 . . 4 . . 2 . . 9 . . 4 . . 2 . x . . x . x . . x 1 . . 8 . 7 . . 5 . . 1 8 . 7 5 . . . . . . . . . . . . . . . . . . . . . . . 7 . 8 . . . . x . . x . . x . . 6 . . 9 . . 3 . . 6 . . 9 . . 3 . . . . x . x . . . . . . 7 . 8 . . . . . . . . . . . . Pattern F53 Old example (JPF) Original shape . . . x . x . . . . . . 4 . 7 . . . . . . . 1 . . . . . x . . x . . x . . 9 . . 5 . . 3 . . 9 . . 5 . . 3 . . . . . x . . . . . . . . 1 . . . . . . . 4 . 7 . . . x . . . . . . . x 8 . . . . . . . 4 . . 8 . . . 4 . . . x x . x . x x . . 6 5 . 4 . 9 1 . 5 6 . . 4 . . 1 9 x . . . . . . . x 7 . . . . . . . 8 . . 7 . . . 8 . . . . . . x . . . . . . . . 7 . . . . . . . 2 . 8 . . . . x . . x . . x . . 1 . . 9 . . 6 . . 1 . . 9 . . 6 . . . . x . x . . . . . . 2 . 8 . . . . . . . 7 . . . . Pattern F54 Old example (ab) Original shape . . . x x x . . . . . . 6 2 3 . . . . . . . . . . . . . x . . x . . x . . 5 . . 4 . . 8 . . . . 6 2 3 . . . . . . . . . . . . . . . . . . . . . . . 5 . 4 . 8 . . x . . . . . . . x 7 . . . . . . . 3 . 7 . . . . . 3 . x x . . x . . x x 8 9 . . 5 . . 7 4 . 8 9 . 5 . 7 4 . x . . . . . . . x 3 . . . . . . . 2 . 3 . . . . . 2 . . . . . . . . . . . . . . . . . . . . . 6 . 8 . 9 . . . x . . x . . x . . 6 . . 8 . . 9 . . . . 4 7 2 . . . . . . x x x . . . . . . 4 7 2 . . . . . . . . . . . . Patterns with 24 clues (23 patterns) Pattern F55 New example (Serg) x x . . . . . x x 1 7 . . . . . 5 3 x . . . . . . . x 5 . . . . . . . 6 . . x . . . x . . . . 2 . . . 8 . . . . . x x x . . . . . . 1 2 3 . . . . . . x . x . . . . . . 4 . 6 . . . . . . x x x . . . . . . 7 8 9 . . . . . x . . . x . . . . 8 . . . 9 . . x . . . . . . . x 6 . . . . . . . 5 x x . . . . . x x 3 1 . . . . . 6 4 Pattern F56 Old example (gsf) Original shape . x . . x . . x . . 3 . . 6 . . 7 . . . 8 . . . 4 . . x . x . . . x . x 4 . 2 . . . 3 . 1 . . 3 . 6 . 7 . . . x . . . . . x . . 8 . . . . . 4 . 2 4 . . . . . 1 3 . . . x . x . . . . . . 6 . 2 . . . . . . 6 . 2 . . . x . . . . . . . x 8 . . . . . . . 9 . 8 . . . . . 9 . . . . x . x . . . . . . 7 . 4 . . . . . . 7 . 4 . . . . x . . . . . x . . 4 . . . . . 3 . 7 9 . . . . . 5 6 x . x . . . x . x 9 . 7 . . . 6 . 5 . . 1 . 2 . 9 . . . x . . x . . x . . 1 . . 2 . . 9 . . . 4 . . . 3 . . Pattern F57 New example (Serg) . x x . x . x x . . 2 9 . 5 . 4 1 . x . . . . . . . x 4 . . . . . . . 5 x . . . . . . . x 6 . . . . . . . 9 . . . x . x . . . . . . 1 . 2 . . . x . . . . . . . x 9 . . . . . . . 4 . . . x . x . . . . . . 3 . 4 . . . x . . . . . . . x 3 . . . . . . . 6 x . . . . . . . x 2 . . . . . . . 7 . x x . x . x x . . 8 7 . 6 . 2 9 . Pattern F58 Old example (Ocean) Original shape x x . . x . . x x 5 6 . . 2 . . 4 1 . . . . . . . . . x x . . . . . x x 4 2 . . . . . 9 5 . 2 4 . . . 5 9 . . . . . . . . . . . . . . . . . . . . 6 5 . 2 . 1 4 . . . . x . x . . . . . . 7 . 3 . . . . . . 7 . 3 . . . x . . . . . . . x 1 . . . . . . . 3 . . 1 . . . 3 . . . . . x . x . . . . . . 1 . 8 . . . . . . 1 . 8 . . . . . . . . . . . . . . . . . . . . . . 8 2 . 5 . 9 1 . x x . . . . . x x 9 7 . . . . . 6 2 . 7 9 . . . 2 6 . x x . . x . . x x 2 8 . . 5 . . 1 9 . . . . . . . . . Pattern F59 New example (Serg) . x . . x . . x . . 2 . . 1 . . 5 . x x . . . . . x x 7 1 . . . . . 4 6 . . . . . . . . . . . . . . . . . . . . . x x x . . . . . . 1 2 3 . . . x . . x . x . . x 9 . . 4 . 6 . . 7 . . . x x x . . . . . . 7 8 9 . . . . . . . . . . . . . . . . . . . . . x x . . . . . x x 6 4 . . . . . 1 8 . x . . x . . x . . 3 . . 9 . . 2 . Pattern F60 New example (Serg) x x . . x . . x x 1 7 . . 3 . . 2 8 x . . . . . . . x 6 . . . . . . . 4 . . . . . . . . . . . . . . . . . . . . . x x x . . . . . . 1 2 3 . . . x . . x . x . . x 3 . . 4 . 6 . . 9 . . . x x x . . . . . . 7 8 9 . . . . . . . . . . . . . . . . . . . . . x . . . . . . . x 9 . . . . . . . 6 x x . . x . . x x 7 2 . . 1 . . 8 5 Pattern F61 New example (Serg) x x . . x . . x x 3 7 . . 1 . . 8 9 x . . . x . . . x 8 . . . 6 . . . 7 . . . . . . . . . . . . . . . . . . . . . x . x . . . . . . 1 . 2 . . . x x . . . . . x x 5 9 . . . . . 3 2 . . . x . x . . . . . . 3 . 4 . . . . . . . . . . . . . . . . . . . . . x . . . x . . . x 7 . . . 2 . . . 8 x x . . x . . x x 6 1 . . 7 . . 9 3 Pattern F62 New example (Serg) . . x . x . x . . . . 2 . 4 . 8 . . . . . . x . . . . . . . . 1 . . . . x . . . . . . . x 6 . . . . . . . 4 . . . x x x . . . . . . 1 2 3 . . . x x . x . x . x x 7 9 . 4 . 6 . 2 1 . . . x x x . . . . . . 7 8 9 . . . x . . . . . . . x 4 . . . . . . . 7 . . . . x . . . . . . . . 6 . . . . . . x . x . x . . . . 5 . 3 . 2 . . Pattern F63 New example (Serg) x . . . x . . . x 6 . . . 1 . . . 3 . . . . x . . . . . . . . 9 . . . . . . x . . . x . . . . 8 . . . 7 . . . . . x x x . . . . . . 1 2 3 . . . x x . x . x . x x 1 3 . 4 . 6 . 7 8 . . . x x x . . . . . . 7 8 9 . . . . . x . . . x . . . . 9 . . . 2 . . . . . . x . . . . . . . . 6 . . . . x . . . x . . . x 4 . . . 3 . . . 1 Pattern F64 Old example (Ocean) Original shape . x . x . x . x . . 1 . 2 . 3 . 4 . . . 1 2 . 3 4 . . x . . . . . . . x 7 . . . . . . . 8 . 5 . . . . . 6 . . . x . . . x . . . . 5 . . . 6 . . 7 . . . . . . . 8 x . . x . x . . x 1 . . 5 . 8 . . 6 1 . . 5 . 8 . . 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . x . . x . x . . x 4 . . 3 . 2 . . 9 4 . . 3 . 2 . . 9 . . x . . . x . . . . 8 . . . 4 . . 2 . . . . . . . 7 x . . . . . . . x 2 . . . . . . . 7 . 8 . . . . . 4 . . x . x . x . x . . 3 . 1 . 6 . 9 . . . 3 1 . 6 9 . . Pattern F65 Old example (Ocean) Original shape . x . x . x . x . . 9 . 5 . 1 . 3 . . . . . . . . . . x x . . . . . x x 2 6 . . . . . 8 1 . 6 2 . . . 1 8 . . . . . . . . . . . . . . . . . . . . 9 . 5 . 1 . 3 . x . . x . x . . x 7 . . 3 . 8 . . 2 . . 7 3 . 8 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x . . x . x . . x 6 . . 4 . 7 . . 5 . . 6 4 . 7 5 . . . . . . . . . . . . . . . . . . . . . 5 . 7 . 4 . 9 . x x . . . . . x x 3 2 . . . . . 6 8 . 2 3 . . . 8 6 . . x . x . x . x . . 5 . 7 . 4 . 9 . . . . . . . . . . Pattern F19 New example (coloin) x x . x . x . x x 1 5 . 3 . 4 . 2 6 x . . . . . . . x 9 . . . . . . . 7 . . . . . . . . . . . . . . . . . . x . . x . x . . x 3 . . 1 . 9 . . 5 . . . . . . . . . . . . . . . . . . x . . x . x . . x 6 . . 7 . 5 . . 1 . . . . . . . . . . . . . . . . . . x . . . . . . . x 7 . . . . . . . 9 x x . x . x . x x 5 2 . 6 . 1 . 4 3 Pattern F66 Old example (Afmob) Original shape . . . x . x . . . . . . 2 . 3 . . . . . . . 1 . . . . . x . . x . . x . . 1 . . 4 . . 2 . . . . 2 . 3 . . . . . . . x . . . . . . . . 1 . . . . . . 1 . 4 . 2 . . x . . x . x . . x 1 . . 3 . 2 . . 4 . 1 . 3 . 2 . 4 . . x x . . . x x . . 5 2 . . . 7 6 . 2 . 5 . . . 6 . 7 x . . x . x . . x 7 . . 8 . 5 . . 3 . 7 . 8 . 5 . 3 . . . . . x . . . . . . . . 2 . . . . . . 4 . 5 . 9 . . . x . . x . . x . . 4 . . 5 . . 9 . . . . 6 . 7 . . . . . . x . x . . . . . . 6 . 7 . . . . . . . 2 . . . . Pattern F67 Old example (JPF) Original shape . . x x . x x . . . . 3 2 . 7 4 . . . . . 9 . 3 . . . . . . x . x . . . . . . 9 . 3 . . . . . 3 2 . 7 4 . . x . . . . . . . x 7 . . . . . . . 8 . 7 . . . . . 8 . x x . . . . . x x 8 7 . . . . . 4 3 7 8 . . . . . 3 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . x x . . . . . x x 4 6 . . . . . 1 5 6 4 . . . . . 5 1 x . . . . . . . x 1 . . . . . . . 4 . 1 . . . . . 4 . . . . x . x . . . . . . 8 . 9 . . . . . 7 3 . 6 9 . . . . x x . x x . . . . 7 3 . 6 9 . . . . . 8 . 9 . . . Pattern F68 Old example (gsf) Original shape x . . x . x . . x 4 . . 3 . 1 . . 5 2 . . . . . . . 9 . . . x . x . . . . . . 4 . 2 . . . . . . 4 . 2 . . . . . x . . . x . . . . 2 . . . 9 . . . . 4 3 . 1 5 . . x x . . . . . x x 9 3 . . . . . 5 6 . 3 9 . . . 6 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . x x . . . . . x x 6 7 . . . . . 2 3 . 7 6 . . . 3 2 . . . x . . . x . . . . 1 . . . 8 . . . . 7 9 . 6 1 . . . . . x . x . . . . . . 5 . 3 . . . . . . 5 . 3 . . . x . . x . x . . x 7 . . 9 . 6 . . 1 1 . . . . . . . 8 Pattern F69 Old example (ab) Original shape x . . x . x . . x 5 . . 9 . 6 . . 7 . . . . . . . . . . x . x . x . x . . 9 . 8 . 5 . 6 . . . 9 8 . 5 6 . . . . . . . . . . . . . . . . . . . . . 5 . 9 . 6 . 7 . x x . . . . . x x 8 3 . . . . . 9 1 . 8 3 . . . 9 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . x x . . . . . x x 6 1 . . . . . 8 4 . 6 1 . . . 8 4 . . . . . . . . . . . . . . . . . . . . 7 . 1 . 4 . 6 . . x . x . x . x . . 8 . 6 . 9 . 5 . . . 8 6 . 9 5 . . x . . x . x . . x 7 . . 1 . 4 . . 6 . . . . . . . . . Pattern F70 New example (Serg) . . . x . x . . . . . . 1 . 2 . . . . . . x . x . . . . . . 3 . 4 . . . . . x . x . x . . . . 1 . 5 . 6 . . x x . . . . . x x 6 4 . . . . . 3 8 . . x . . . x . . . . 8 . . . 9 . . x x . . . . . x x 7 3 . . . . . 5 4 . . x . x . x . . . . 2 . 9 . 8 . . . . . x . x . . . . . . 4 . 7 . . . . . . x . x . . . . . . 8 . 1 . . . Pattern F71 Old example (ab) Original shape x . . x . x . . x 1 . . 5 . 9 . . 7 . . . . 6 . . . . . . . x . x . . . . . . 3 . 8 . . . . . . 3 . 8 . . . . . . . x . . . . . . . . 6 . . . . . . 1 5 . 9 7 . . x x . . . . . x x 5 6 . . . . . 4 1 . 6 5 . . . 1 4 . . . x . . . x . . . . 9 . . . 2 . . 9 . . . . . . . 2 x x . . . . . x x 2 7 . . . . . 6 3 . 7 2 . . . 3 6 . . . . . x . . . . . . . . 2 . . . . . . 4 7 . 5 9 . . . . . x . x . . . . . . 8 . 1 . . . . . . 8 . 1 . . . x . . x . x . . x 4 . . 7 . 5 . . 9 . . . . 2 . . . . Pattern F72 Old example (Ocean) Original shape . x . x x x . x . . 3 . 1 8 9 . 7 . . . . . . . . . . x . . . . . . . x 5 . . . . . . . 6 . . 3 1 8 9 7 . . . . . . . . . . . . . . . . . . . . . 5 . . . . . 6 . x . . x . x . . x 7 . . 4 . 1 . . 5 . 7 . 4 . 1 . 5 . x . . . . . . . x 2 . . . . . . . 4 . 2 . . . . . 4 . x . . x . x . . x 6 . . 9 . 7 . . 3 . 6 . 9 . 7 . 3 . . . . . . . . . . . . . . . . . . . . 3 . . . . . 1 . x . . . . . . . x 3 . . . . . . . 1 . . 9 6 5 2 8 . . . x . x x x . x . . 9 . 6 5 2 . 8 . . . . . . . . . . Pattern F73 Old example (JPF) Original shape x . . x x x . . x 3 . . 4 5 6 . . 1 1 . . . . . . . 2 . x . . . . . x . . 1 . . . . . 2 . . 3 . 4 5 6 . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . x . . x . x . . x 1 . . 7 . 2 . . 6 . 1 . 7 . 2 . 6 . x . . . . . . . x 2 . . . . . . . 4 . 2 . . . . . 4 . x . . x . x . . x 5 . . 6 . 8 . . 9 . 5 . 6 . 8 . 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x . . . . . x . . 6 . . . . . 1 . . 4 . 3 2 7 . 5 . x . . x x x . . x 4 . . 3 2 7 . . 5 6 . . . . . . . 1 Pattern F74 New example (Serg) . . . x x x . . . . . . 1 8 7 . . . . x . . x . . x . . 2 . . 9 . . 7 . . . . . . . . . . . . . . . . . . . x . . x . x . . x 6 . . 4 . 8 . . 1 x x . . . . . x x 4 7 . . . . . 3 5 x . . x . x . . x 8 . . 3 . 1 . . 9 . . . . . . . . . . . . . . . . . . . x . . x . . x . . 9 . . 3 . . 4 . . . . x x x . . . . . . 8 6 2 . . . Pattern F75 Old example (Ocean) Original shape . . . x x x . . . . . . 4 5 7 . . . . . . . . . . . . . x . x . x . x . . 2 . 9 . 6 . 4 . . . . 4 5 7 . . . . . . . . . . . . . . . . . . . . . . . 2 9 . 6 4 . . x x . . . . . x x 9 6 . . . . . 1 2 . 9 6 . . . 1 2 . x . . . . . . . x 3 . . . . . . . 8 . 3 . . . . . 8 . x x . . . . . x x 5 4 . . . . . 7 9 . 5 4 . . . 7 9 . . . . . . . . . . . . . . . . . . . . . 7 2 . 1 9 . . . x . x . x . x . . 7 . 2 . 1 . 9 . . . . 6 3 5 . . . . . . x x x . . . . . . 6 3 5 . . . . . . . . . . . . Pattern F76 New example (Serg) x . . x x x . . x 4 . . 1 2 3 . . 5 . . . x . x . . . . . . 4 . 5 . . . . . . . . . . . . . . . . . . . . . x x . . . . . x x 5 6 . . . . . 2 9 x . . . . . . . x 7 . . . . . . . 3 x x . . . . . x x 2 8 . . . . . 4 1 . . . . . . . . . . . . . . . . . . . . . x . x . . . . . . 8 . 7 . . . x . . x x x . . x 3 . . 2 1 9 . . 4 Patterns with 25 clues (2 patterns) Pattern F77 New example (Serg) x x . . . . . x x 6 3 . . . . . 1 9 x . x . . . x . x 7 . 9 . . . 3 . 2 . x . . . . . x . . 8 . . . . . 6 . . . . x . x . . . . . . 9 . 7 . . . . . . . x . . . . . . . . 8 . . . . . . . x . x . . . . . . 6 . 5 . . . . x . . . . . x . . 2 . . . . . 8 . x . x . . . x . x 4 . 3 . . . 7 . 6 x x . . . . . x x 1 7 . . . . . 2 5 Pattern F78 Old example (blue) Original shape x x . . . . . x x 1 5 . . . . . 6 3 1 2 . . . . . . . x x . . . . . x x 6 3 . . . . . 8 2 4 5 . . . . . . . . . x . . . x . . . . 7 . . . 5 . . . . 9 . . . . . . . . . x . x . . . . . . 1 . 2 . . . . . . . 6 3 . 5 1 . . . . x . . . . . . . . 9 . . . . . . . 5 . . 7 . . . . . x . x . . . . . . 4 . 5 . . . . . . . 8 2 . 3 6 . . x . . . x . . . . 9 . . . 3 . . . . . 3 . . 9 . . x x . . . . . x x 8 1 . . . . . 2 6 . . . . 2 6 . 1 8 x x . . . . . x x 7 4 . . . . . 9 1 . . . . 9 1 . 4 7 Patterns with 28 clues (2 patterns) Pattern F79 New example (Serg) x x x . . . x x x 5 8 7 . . . 9 6 4 x . . . . . . . x 9 . . . . . . . 8 x . . . . . . . x 6 . . . . . . . 1 . . . x x x . . . . . . 6 3 4 . . . . . . x . x . . . . . . 9 . 5 . . . . . . x x x . . . . . . 8 7 2 . . . x . . . . . . . x 8 . . . . . . . 6 x . . . . . . . x 1 . . . . . . . 2 x x x . . . x x x 7 2 9 . . . 1 4 3 Pattern F80 New example (Serg) . x x x . x x x . . 7 4 6 . 9 1 5 . x . . . . . . . x 1 . . . . . . . 6 x . . . . . . . x 5 . . . . . . . 8 x . . x . x . . x 6 . . 9 . 3 . . 1 . . . . . . . . . . . . . . . . . . x . . x . x . . x 4 . . 1 . 7 . . 3 x . . . . . . . x 3 . . . . . . . 7 x . . . . . . . x 9 . . . . . . . 4 . x x x . x x x . . 8 2 4 . 5 6 3 . Totally 49 patterns

Here are examples of valid puzzles in the line form.

Code: Select all: ....1.....79...54..5.....7....1.2...8.......1...5.7....6.....3..24...79.....9.... #F34 ab .7..5..3.8.......9..6...5.....8.5...9.......2...4.3.....2...4..4.......8.5..3..7. #F35 ab 7...2...9.6.....3...3...4.....3.8...9.......7...5.6.....9...7...5.....6.2...1...4 #F36 ab ...2.5....7.....4...9...3..2...7...1...8.3...1...9...5..4...8...8.....5....7.1... #F37 ab .2.5.8.1.7.......9.........5...6...4...3.2...6...9...3.........9.......7.8.2.1.3. #F38 ab 3..4.5..7.2.....6..........5...1...9...6.8...9...7...4..........3.....2.4..9.7..5 #F39 tarek ...4.5....9.....8...7...4..3..9.8..6.........8..3.2..5..5...9...7.....4....5.1... #F40 ab ...7.2.....2.4.6...6.....5.4.......9.1.....3.3.......7.8.....9...4.2.1.....5.3... #F41 ab ...4.3....5..7..1...6...5..2.......7.7.....2.8.......3..4...3...6..2..5....7.8... #F42 ab ..61.35......6....4.......22.......9.9.....4.3.......89.......1....5......56.73.. #F43 ab .4.1.5.6.1...2...3.........3.......7.8.....4.9.......2.........2...1...9.5.4.7.8. #F44 Ocean 1..7.2..3....3......9...5..3.......2.6.....9.4.......7..2...3......5....7..8.4..1 #F45 ab 3..9.8..1.2..7..8..........1.......3.5.....7.9.......6..........7..5..2.6..1.7..9 #F46 ab ...6.7.......1......3...4..6...9...8.7.8.2.1.1...5...3..5...7......2.......1.6... #F47 JPF ...1.2....3..4..5..........2...6...7.6.4.5.8.7...8...9..........5..3..6....9.7... #F48 Ocean ...234.......1......1...5..2.......636.....424.......7..5...8......4.......967... #F49 Ocean .8.6.3.9.1.......7.........6..7.9..4....3....2..5.8..1.........4.......2.3.9.5.7. #F50 JPF 1..5.9..7.3.....4..........5..3.1..9....6....7..9.2..1..........8.....3.9..1.5..8 #F51 JPF ...2.6.......1......7...8..6..1.2..9.2..3..1.1..4.5..7..4...2......8.......6.1... #F14 Afmob ...5.1....4..6..9..........5..9.3..8.9..4..2.1..8.7..5..........6..9..3....7.8... #F52 ab ...4.7....9..5..3.....1....8.......4.65.4.91.7.......8....7.....1..9..6....2.8... #F53 JPF ...623....5..4..8..........7.......389..5..743.......2..........6..8..9....472... #F54 ab 17.....535.......6..2...8.....123......4.6......789.....8...9..6.......531.....64 #F55 Serg .3..6..7.4.2...3.1.8.....4....6.2...8.......9...7.4....4.....3.9.7...6.5.1..2..9. #F56 gsf .29.5.41.4.......56.......9...1.2...9.......4...3.4...3.......62.......7.87.6.29. #F57 Serg 56..2..4142.....95............7.3...1.......3...1.8............97.....6228..5..19 #F58 Ocean .2..1..5.71.....46............123...9..4.6..7...789............64.....18.3..9..2. #F59 Serg 17..3..286.......4............123...3..4.6..9...789............9.......672..1..85 #F60 Serg 37..1..898...6...7............1.2...59.....32...3.4............7...2...861..7..93 #F61 Serg ..2.4.8......1....6.......4...123...79.4.6.21...789...4.......7....6......5.3.2.. #F62 Serg 6...1...3....9......8...7.....123...13.4.6.78...789.....9...2......6....4...3...1 #F63 Serg .1.2.3.4.7.......8..5...6..1..5.8..6.........4..3.2..9..8...4..2.......7.3.1.6.9. #F64 Ocean .9.5.1.3.26.....81.........7..3.8..2.........6..4.7..5.........32.....68.5.7.4.9. #F65 Ocean 15.3.4.269.......7.........3..1.9..5.........6..7.5..1.........7.......952.6.1.43 #F19 coloin ...2.3....1..4..2.....1....1..3.2..4.52...76.7..8.5..3....2.....4..5..9....6.7... #F66 Afmob ..32.74.....9.3...7.......887.....43.........46.....151.......4...8.9.....73.69.. #F67 JPF 4..3.1..5...4.2.....2...9..93.....56.........67.....23..1...8.....5.3...7..9.6..1 #F68 gsf 5..9.6..7.9.8.5.6..........83.....91.........61.....84..........8.6.9.5.7..1.4..6 #F69 ab ...1.2......3.4.....1.5.6..64.....38..8...9..73.....54..2.9.8.....4.7......8.1... #F70 Serg 1..5.9..7...3.8.......6....56.....41..9...2..27.....63....2.......8.1...4..7.5..9 #F71 ab .3.189.7.5.......6.........7..4.1..52.......46..9.7..3.........3.......1.9.652.8. #F72 Ocean 3..456..1.1.....2..........1..7.2..62.......45..6.8..9..........6.....1.4..327..5 #F73 JPF ...187....2..9..7..........6..4.8..147.....358..3.1..9..........9..3..4....862... #F74 Serg ...457....2.9.6.4..........96.....123.......854.....79..........7.2.1.9....635... #F75 Ocean 4..123..5...4.5............56.....297.......328.....41............8.7...3..219..4 #F76 Serg 63.....197.9...3.2.8.....6....9.7.......8.......6.5....2.....8.4.3...7.617.....25 #F77 Serg 15.....6363.....82..7...5.....1.2.......9.......4.5.....9...3..81.....2674.....91 #F78 blue 587...9649.......86.......1...634......9.5......872...8.......61.......2729...143 #F79 Serg .746.915.1.......65.......86..9.3..1.........4..1.7..33.......79.......4.824.563. #F80 Serg

Serg

[Edited. Examples (old and new) for all 49 valid patterns are published.]

by **Afmob** » Wed Mar 15, 2017 4:08 pm

I can confirm coloin's result regarding the six "plus1" patterns:

Code: Select all: F17plus1: 57 ED valid puzzles F18plus1: 7 ED valid puzzles F20plus1: 2 ED valid puzzles F21plus1: 212 ED valid puzzles F26plus1: 2222 ED valid puzzles F27plus1: 1893 ED valid puzzles

Serg, regarding the next steps, I would say it is more important to verify the invalid patterns. I think the task of finding at least one valid puzzle for the minimal 49 valid patterns can (easily?) be done by others and it's easy to verify once a valid puzzle has been found.

by **Serg** » Wed Mar 15, 2017 8:41 pm

Hi!

Serg wrote:This is the next candidate invalid pattern with 5 clues in the central box.
Code: Select all
F11 x . . . x . . . x . x . . x . . x . . . . . x . . . . . . . x . x . . . x x x . x . x x x . . . x . x . . . . . . . x . . . . . x . . x . . x . x . . . x . . . x

I've done ehaustive search for this pattern - no valid puzzles. The search took 68 hours (not all automorphisms were used).

Serg

by **Serg** » Wed Mar 15, 2017 9:03 pm

Hi, Afmob!

Afmob wrote:I can confirm coloin's result regarding the six "plus1" patterns:
Code: Select all
F17plus1: 57 ED valid puzzles F18plus1: 7 ED valid puzzles F20plus1: 2 ED valid puzzles F21plus1: 212 ED valid puzzles F26plus1: 2222 ED valid puzzles F27plus1: 1893 ED valid puzzles

It's amazing, but these 6 patterns are exactly in the 49 valid patterns list (there are no other 21-clue patterns in that list).
Afmob, did you check found puzzles for minimality? It would be nice to see all found puzzles lists, at least for the first 4 patterns.

Afmob wrote:Serg, regarding the next steps, I would say it is more important to verify the invalid patterns. I think the task of finding at least one valid puzzle for the minimal 49 valid patterns can (easily?) be done by others and it's easy to verify once a valid puzzle has been found.

I agree with you and I am searching through remaining invalid patterns. But this search runs in the background mode, so I have spare time to prepare examples.

Serg

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