The New Sudoku Players' Forum

by **Serg** » Fri Sep 30, 2011 11:01 pm

Hi, all!
Now I am pushing step-by-step the proof of 16-clue valid puzzles impossibility. My work is going not so fast as I would like, but I got some intermediate results which can be interesting for this forum people. I investigated the map which boxes B1-B6 (2 upper bands) have exactly 1 clue each, but boxes B7-B9 (lowest band) have 9 clues each. Here is this map:

Code: Select all: 1 1 1 1 1 1 9 9 9

I've done exhaustive search (by specially written program) for all valid puzzles complying with this map and found that this map has no valid puzzles. Total CPU time - about 5 hours. I've found 22 non-isomorphic patterns having this map. Here they are.

Code: Select all: N1 N2 N3 N4 +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ |x . .|x . .|. . .| |x . .|x . .|. . .| |x . .|x . .|. . .| |x . .|x . .|. . .| |. . .|. . .|x . .| |. . .|. . .|x . .| |. . .|. . .|x . .| |. . .|. . .|x . .| |. . .|. . .|. . .| |. . .|. . .|. . .| |. . .|. . .|. . .| |. . .|. . .|. . .| +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ |x . .|x . .|. . .| |x . .|x . .|. . .| |x . .|. x .|. . .| |x . .|. x .|. . .| |. . .|. . .|x . .| |. . .|. . .|. x .| |. . .|. . .|x . .| |. . .|. . .|. x .| |. . .|. . .|. . .| |. . .|. . .|. . .| |. . .|. . .|. . .| |. . .|. . .|. . .| +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ N5 N6 N7 N8 +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ |x . .|x . .|. . .| |x . .|x . .|. . .| |x . .|x . .|. . .| |x . .|x . .|. . .| |. . .|. . .|x . .| |. . .|. . .|x . .| |. . .|. . .|x . .| |. . .|. . .|x . .| |. . .|. . .|. . .| |. . .|. . .|. . .| |. . .|. . .|. . .| |. . .|. . .|. . .| +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ |. x .|. x .|. . .| |. x .|. x .|. . .| |x . .|. . .|. . .| |x . .|. . .|. . .| |. . .|. . .|x . .| |. . .|. . .|. x .| |. . .|x . .|x . .| |. . .|x . .|. x .| |. . .|. . .|. . .| |. . .|. . .|. . .| |. . .|. . .|. . .| |. . .|. . .|. . .| +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ N9 N10 N11 N12 +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ |x . .|x . .|. . .| |x . .|x . .|. . .| |x . .|x . .|. . .| |x . .|x . .|. . .| |. . .|. . .|x . .| |. . .|. . .|x . .| |. . .|. . .|x . .| |. . .|. . .|x . .| |. . .|. . .|. . .| |. . .|. . .|. . .| |. . .|. . .|. . .| |. . .|. . .|. . .| +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ |x . .|. . .|. . .| |x . .|. . .|. . .| |. x .|. . .|. . .| |. x .|. . .|. . .| |. . .|. x .|x . .| |. . .|. x .|. x .| |. . .|x . .|. x .| |. . .|. x .|. x .| |. . .|. . .|. . .| |. . .|. . .|. . .| |. . .|. . .|. . .| |. . .|. . .|. . .| +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ N13 N14 N15 N16 +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ |x . .|x . .|. . .| |x . .|x . .|. . .| |x . .|x . .|. . .| |x . .|x . .|. . .| |. . .|. . .|x . .| |. . .|. . .|x . .| |. . .|. . .|x . .| |. . .|. . .|x . .| |. . .|. . .|. . .| |. . .|. . .|. . .| |. . .|. . .|. . .| |. . .|. . .|. . .| +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ |x . .|. . .|. . .| |x . .|. . .|. . .| |x . .|. . .|. . .| |x . .|. . .|. . .| |. . .|x . .|. . .| |. . .|x . .|. . .| |. . .|. x .|. . .| |. . .|. x .|. . .| |. . .|. . .|x . .| |. . .|. . .|. x .| |. . .|. . .|x . .| |. . .|. . .|. x .| +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ N17 N18 N19 N20 +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ |x . .|x . .|. . .| |x . .|x . .|. . .| |x . .|. . .|. . .| |x . .|. . .|. . .| |. . .|. . .|x . .| |. . .|. . .|x . .| |. . .|x . .|. . .| |. . .|x . .|. . .| |. . .|. . .|. . .| |. . .|. . .|. . .| |. . .|. . .|x . .| |. . .|. . .|x . .| +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ |. x .|. . .|. . .| |. x .|. . .|. . .| |x . .|. . .|. . .| |x . .|. . .|. . .| |. . .|. x .|. . .| |. . .|. x .|. . .| |. . .|x . .|. . .| |. . .|x . .|. . .| |. . .|. . .|x . .| |. . .|. . .|. x .| |. . .|. . .|x . .| |. . .|. . .|. x .| +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ +-----+-----+-----+ N21 N22 +-----+-----+-----+ +-----+-----+-----+ |x . .|. . .|. . .| |x . .|. . .|. . .| |. . .|x . .|. . .| |. . .|x . .|. . .| |. . .|. . .|x . .| |. . .|. . .|x . .| +-----+-----+-----+ +-----+-----+-----+ |x . .|. . .|. . .| |. x .|. . .|. . .| |. . .|. x .|. . .| |. . .|. x .|. . .| |. . .|. . .|. x .| |. . .|. . .|. x .| +-----+-----+-----+ +-----+-----+-----+ |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|x x x|x x x| |x x x|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 result can be formulated as follows. If any pattern contains two or more bands having boxes with 1 clue only or two or more stacks having boxes with 1 clue only, such pattern has no valid puzzles.

This statement can be formulated in more general way. Let's define sparse box for sudoku puzzle (pattern). A box is sparse if it contains not more than 1 clue (0 or 1 clues). A band (stack) is sparse if it contains sparse boxes only. If any puzzle (pattern) contains two or more sparse bands (two or more sparse stacks), such puzzle (pattern) has no valid puzzles.

Please, pay attention - this result is universal and applicable for arbitrary sudoku puzzles (patterns), not only for 16-clue puzzles.

Serg

[Comment: this thread name should be read as "One-clue-boxes patterns" instead of "One-box-clue patterns". My mistake.]Fixed, JasonLion
[Edited: I corrected a typo in pattern N11. (Thanks to ronk, who found this mistake.)]

by **ronk** » Sat Oct 01, 2011 12:56 am

Serg wrote:I've found 22 non-isomorphic patterns having this map. Here they are.

You posted 21 non-isomorphic patterns (N4 and N11 are isomorphic).

by **Serg** » Sat Oct 01, 2011 7:00 am

Hi, ronk!

ronk wrote:
Serg wrote:I've found 22 non-isomorphic patterns having this map. Here they are.

You posted 21 non-isomorphic patterns (N4 and N11 are isomorphic).

Thank you for your cross-checking of my result. You are right, patterns N4 and N11 are isomorphic, but because of my typo. (I am sorry.) This is the right form of pattern N11:

Code: Select all: N11 +-----+-----+-----+ |x . .|x . .|. . .| |. . .|. . .|x . .| |. . .|. . .|. . .| +-----+-----+-----+ |. x .|. . .|. . .| |. . .|x . .|. x .| |. . .|. . .|. . .| +-----+-----+-----+ |x x x|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'll correct this typo in my previous post.

Thanks, Serg

by **ronk** » Mon Oct 31, 2011 3:12 pm

Serg, I can confirm your 22 non-isomorphic patterns. Made the run on Oct 2nd, but forgot to post. Have you made further progress?

by **Serg** » Tue Nov 01, 2011 7:26 am

Hi, ronk!

ronk wrote:Serg, I can confirm your 22 non-isomorphic patterns. Made the run on Oct 2nd, but forgot to post. Have you made further progress?

Thank you for your confirmation. I am investigating all possible maps having one clue or 9 clues in each box. If I'll perform this job, number of possible distributions for 16-clue valid puzzles of type 1 (the most probable type of 16-clue valid puzzles) will be decreased from 10 to 6.

It turns out, that map

Code: Select all: 1 1 1 1 1 9 9 9 9

has valid puzzles.
Now I am investigating map

Code: Select all: 1 1 1 1 1 9 9 9 1

It is very difficult to do exhaustive search for this map, because it has not so many symmetries. According to my calculations, this map has 40 non-isomorphic patterns.

The task is difficult, but interesting.

Serg

by **Smythe Dakota** » Wed Nov 02, 2011 1:43 am

I think I heard, some years ago, that the 4-color map conjecture was finally proved, but in a way so inelegant that it displeased mathematicians. Apparently they first proved that, if there was a map that required more than 4 colors, then there was one with fewer than 563 countries (or some such absurd number). This reduced it to a finite problem, so then they just ran it through a computer.

Somehow, the methods described here remind me of the above story.

Bill Smythe

by **Serg** » Wed Nov 02, 2011 7:51 am

Hi, Bill!

Smythe Dakota wrote:I think I heard, some years ago, that the 4-color map conjecture was finally proved, but in a way so inelegant that it displeased mathematicians. Apparently they first proved that, if there was a map that required more than 4 colors, then there was one with fewer than 563 countries (or some such absurd number). This reduced it to a finite problem, so then they just ran it through a computer.

Somehow, the methods described here remind me of the above story.

Bill Smythe

You are right - I'd like to see elegant mathematical proof of 16-clue valid puzzles non-existance too, but I think it would be important to check by "brute force" method - do 16-clue puzzle really exist? If they will not be found by computer search, mathematicians can be sure 16-clue valid puzzles don't exist - this fact can assist them to search mathematical proof. The next reason for me to do computer search for 16-clue valid puzzles is possibility of finding universal "composition rules", that can be used elsewhere - for finding new 17-clue valid puzzles, etc. There are experimental and theoretical physics. I think computer search for 16-clue valid puzzles is somewhat like experimental physics.

Serg

by **Serg** » Tue Nov 15, 2011 9:35 pm

Hi, all!
Map

Code: Select all: 1 1 9 1 9 1 9 1 1

has valid puzzles. Here is an example.

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

JPF published in 2006 such "slash pattern" puzzle (see the thread Sudokus with an original rare shape). This pattern is subset of the map considered. Hence the map has valid puzzles.

So, I continue investigation of one-clue-boxes patterns.

Serg
[Edited: I added link to the first publication of 3-diagonal puzzle.]

by **Serg** » Sun Jan 08, 2012 10:47 pm

Hi, all!
Map

Code: Select all: 1 1 1 1 1 9 9 9 1

has valid puzzles. Here is an example.

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

The same in linear form:

Code: Select all: 1..5...........3...........2..8..651......284......9379312547..842673...567189...

CPU consumption time - 4 hours.

So, I finished investigation of one-clue-boxes patterns. I studied all possible (26) one-clue-boxes maps and shall post my results in some time.

Serg

by **JPF** » Mon Jan 09, 2012 1:18 pm

Serg wrote: It is curious, but I found this puzzle occasionally in my own (unpublished) book "Unusual Sudoku", which was written 2 years ago. I am not sure I was the first who invented such tridiagonal puzzle design, but I invented it by my own mind.

See here

JPF

by **Serg** » Mon Jan 09, 2012 1:36 pm

Hi, people!
Here are my results of one-clue-boxes patterns investigation.
I consider patterns, having 1 or 9 clues in each box. There are 26 non-isomorphic maps only for such patterns. Here they are.

Code: Select all: 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 9 1 1 1 1 1 1 2 1 1 1 1 1 9 1 9 9 1 9 1 1 1 1 1 1 1 1 1 1 1 1 9 3 1 1 1 1 1 9 1 1 9 1 9 1 9 9 9 1 9 9 9 9 1 9 1 1 1 1 1 1 1 1 1 1 1 1 1 9 1 1 9 4 1 1 9 1 9 9 1 9 9 1 9 1 1 1 9 9 9 9 1 9 9 9 1 9 9 1 9 9 9 1 1 1 1 1 1 9 1 1 9 1 1 9 9 1 1 5 1 9 9 1 1 9 1 9 1 1 9 9 1 9 9 9 9 9 9 9 9 9 9 9 9 9 1 1 9 9 1 1 1 1 1 9 1 1 9 1 9 9 6 9 9 9 1 9 9 9 9 1 9 1 9 9 9 9 9 9 9 9 9 9 9 9 1 1 1 9 1 9 9 7 9 9 9 9 1 9 9 9 9 9 9 9 1 9 9 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

I'll use this diagram in reduced form, where each map will be replaced by "?", "+" or "-". Symbol "?" means uncertainty - it is unknown, has that map any valid puzzles? Symbol "+" means the map has valid puzzles, symbol "-" means the map has no valid puzzles. We started from full uncertainty:

Code: Select all: 0 ? 1 ? 2 ? ? 3 ? ? ? ? 4 ? ? ? ? ? 5 ? ? ? ? ? 6 ? ? ? ? 7 ? ? 8 ? 9 ?

Map

Code: Select all: 1 1 1 1 9 9 1 9 9

has no valid puzzles (see proof in the thread How to prove non-existance of 8-clue valid puzzles?). So, we can mark by "-" sign this map in the diagram (and 5 other maps being subsets of this map) as maps not having valid puzzles.

Code: Select all: 0 - 1 - 2 - - 3 ? - ? ? 4 ? - ? ? ? 5 ? ? ? ? ? 6 ? ? ? ? 7 ? ? 8 ? 9 ?

It turns out, that map

Code: Select all: 1 1 1 1 1 9 9 9 9

has valid puzzles. Here is an example.

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

So, we can mark by "+" sign ("map has valid puzzle") this map and all other maps (10 maps) containing this map as subset.

Code: Select all: 0 - 1 - 2 - - 3 ? - ? ? 4 + - ? ? ? 5 + + + ? ? 6 + + + ? 7 + + 8 + 9 +

Map

Code: Select all: 1 1 1 1 1 1 9 9 9

has no valid puzzles (see the beginning of this thread). Let's mark this map by "-" sign.

Code: Select all: 0 - 1 - 2 - - 3 - - ? ? 4 + - ? ? ? 5 + + + ? ? 6 + + + ? 7 + + 8 + 9 +

Map

Code: Select all: 1 1 1 1 1 9 9 9 1

has valid puzzles (see my previous post). So, we can mark by "+" sign ("map has valid puzzle") this map and all other maps (5 maps) containing this map as subset.

Code: Select all: 0 - 1 - 2 - - 3 - - + ? 4 + - + ? + 5 + + + + + 6 + + + + 7 + + 8 + 9 +

Now let's consider the map

Code: Select all: 1 1 9 1 9 1 9 1 1

Mike Metcalf published in 2006 "band plus dot" puzzle (see the thread Sudokus with an original rare shape), which is subset of this map. Hence this map has valid puzzles. So, we can mark remaining 2 "uncertain" maps by "+" sign ("map has valid puzzle").

Code: Select all: 0 - 1 - 2 - - 3 - - + + 4 + - + + + 5 + + + + + 6 + + + + 7 + + 8 + 9 +

It is detailed maps diagram, where asterisk line separates maps not having valid puzzles (they are above the line) and maps having valid puzzles (they are below the line).

Code: Select all: 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 9 1 1 1 1 1 1 2 1 1 1 1 1 9 1 9 9 1 9 1 *********************** 1 1 1 1 1 1 * 1 1 1 1 1 9 3 1 1 1 1 1 9 * 1 1 9 1 9 1 9 9 9 1 9 9 * 9 9 1 9 1 1 ********** * 1 1 1 * 1 1 1 * 1 1 1 1 1 9 1 1 9 4 1 1 9 * 1 9 9 * 1 9 9 1 9 1 1 1 9 9 9 9 * 1 9 9 * 9 1 9 9 1 9 9 9 1 ********* 1 1 1 1 1 9 1 1 9 1 1 9 9 1 1 5 1 9 9 1 1 9 1 9 1 1 9 9 1 9 9 9 9 9 9 9 9 9 9 9 9 9 1 1 9 9 1 1 1 1 1 9 1 1 9 1 9 9 6 9 9 9 1 9 9 9 9 1 9 1 9 9 9 9 9 9 9 9 9 9 9 9 1 1 1 9 1 9 9 7 9 9 9 9 1 9 9 9 9 9 9 9 1 9 9 8 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9

Serg

by **Serg** » Mon Jan 09, 2012 2:03 pm

Hi, JPF!

JPF wrote:
Serg wrote: It is curious, but I found this puzzle occasionally in my own (unpublished) book "Unusual Sudoku", which was written 2 years ago. I am not sure I was the first who invented such tridiagonal puzzle design, but I invented it by my own mind.

See here

Thank you for your link. I haven't seen your post during publishing by previuos post.
I found myself the post you cited in a short time after my publication of 3-diagonal shape. I was planning to write about it in my resulting (previous) post. But I noted at the moment of resulting post publication, that "band plus dot" pattern designed by Mike Metcalf (3-diagonal pattern plus 1 cell) is enough to prove that "diagonal" map has valid puzzles. Because Mike published his post earlier, I decided to use his pattern in my proof.

But I agree, you are 3-diagonal shape inventor. What a nice shape! It is hard to believe that this shape can have valid puzzles, but it really has them. I called it "Milky Way".

Serg

by **Serg** » Mon Jan 09, 2012 7:07 pm

Hi, people!
I'd like to post some comments - how results of this thread can be used, and why did I investigate one-clue-boxes pattern.

I think the main result is that any map (pattern, puzzle) containing 7 or more sparse boxes (boxes containing 0 or 1 clue) definitly has no valid puzzles. If any map (pattern, puzzle) contains 6 sparse boxes, it can have valid puzzles when all 3 boxes, containning more than 1 clue, occupies 3 different bands (stacks) AND 2 different stacks (bands).

The first statement could be derived from the knowlege that "Magic map"

Code: Select all: 1 1 1 1 9 9 1 9 9

has no valid puzzles.

I am planning to use these results for exhaustive search of 16-clue or 17-clue puzzles. For this purpose only new map, not having valid puzzles,

Code: Select all: 1 1 1 1 1 1 9 9 9

is of some interest. For my method of exhaustive search ("Composition rules" method) maps having no valid puzzles are much more valueable, because if any map has no valid puzzles, we have exact knowlege that any pattern belonging to such map has no valid puzzles. But if a map has valid puzzles, is not known whether given pattern has valid puzzles.

At the moment of the thread beginning I hoped to find that all one-clue-boxes maps having 6 one-clue boxes have no valid puzzles. If it would be so, number of possible maps of 16-clue and 17-clue valid puzzles would be decreased substantially. But it turns out that 2 such maps only have no valid puzzles, but remaining 2 maps do have valid puzzles.

The goal of this thread is achieved, so I am closing this thread.

Thanks, Serg

[Edited: I corrected a typo.]

by **Afmob** » Thu Apr 26, 2012 6:38 am

As I try to confirm you results, I would like to know the details of your exhaustive search you did for each pattern. Did you go through all essentially different puzzles?

by **Serg** » Thu Apr 26, 2012 10:07 am

Hi, Afmob!

Afmob wrote:As I try to confirm you results, I would like to know the details of your exhaustive search you did for each pattern. Did you go through all essentially different puzzles?

Yes, I went through all essentially different puzzles. All 22 e-d patterns for the map

Code: Select all: 1 1 1 1 1 1 9 9 9

I used are posted in the beginning of this thread. Do you want to see my algorithm description?

Serg

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