The New Sudoku Players' Forum

[mith]

(I'm running a check now for any other solution grids with nontrivial automorphisms; it's over halfway through the solution grids included in the 2022-11-06 update, and these are the only two found. So this may be a really small task for correcting the old list - literally just removing the puzzles Denis listed from the max-expand file. Given how infrequent this issue is, it should be quick to just add special handling when the automorphism count is >1, rather than changing the process for all trees.

[edit]Confirmed, these are the only two out of the 44251 solution grids in the 2022-11 update. I'll check the remainder of the database when I next run the update and determine the max-expands.[/edit])

[denis_berthier]

.
Hi mith
Thanks for your quick answer.
Fortunately, this will not change what I was doing with the max-expands (see the tridagon thread).
.

[mith]

Continued from hardest puzzle thread:

the 11.3 Denis posted earlier does *not* have a valid non-degenerate trivalue oddagon (at least by my definition); because r1c9 is limited to 123, the three marked cells in box 3 already cannot all contain 123, so the OR branching can be reduced to considering that r2c7 and r3c9 cannot both contain 123.

"My" 11.3 (indeed Paquita's) has a non-degenerate trivalue oddagon (it has indeed many different ones), unless one is willing to add to the definition conditions (such as those you're mentioning) that have never been defined and that are potentially in unlimited numbers. I think you're confusing the presence of the pattern (defined in the precise, simple way I gave in the tridagon thread) with its usefulness in specific circumstances.
In the present case, there are two independent circumstances that make it useless: the one you're mentioning and the very high number of guardians.
.

This condition was defined (insofar as it is part of the algorithm I use) in this thread: p328613 (fourth bullet point). This specific post does not exclude degenerate tridagons by your definition, though that is handled in a later algorithm (a trivalue oddagon can only contain all digits in all cells of the pattern if those digits are restricted as givens to a single box which is not aligned with the four boxes of the pattern; a given in any other box will necessarily remove that candidate from a cell of the pattern, since the pattern spans all rows and columns within each box it covers).

It is not merely a matter of usefulness; it is a matter of reduction to a smaller pattern. In this case, the set of cells r1c89+r2c7+r3c9 is a 4 cell pattern with a chromatic number of 4, and the guardians are a subset of the guardians in the trivalue oddagon. We can quickly identify such cases by noting that there is a cell in box 3 which is not part of the potential trivalue oddagon but which is limited to the three digits of the trivalue oddagon, and is quite limited in scope (I can think of one other similar limitation involving the rows/columns instead of the box, but it would be much more unlikely to occur; might add a check for that at some point anyway).

In fact, there is nothing precluding a case where the guardians of the smaller pattern are exactly the same as the guardians of the trivalue oddagon, and in such a case the trivalue oddagon would be exactly as useful as the smaller pattern. Even in cases where the guardians of the smaller pattern are a proper subset of the guardians of the trivalue oddagon, there is nothing precluding the smaller pattern from being useful.

You can choose to not define this case as degenerate, that's your prerogative. It's totally reasonable to draw the line at only considering cells and candidates which are part of the pattern. That said, one could perhaps make the argument that it is degenerate even by your definition in the sense that whatever (unknown) digit goes in r1c9 (which we know is from 123) is excluded from the other cells in the box (r1c8+r2c7+r3c9).

Anyway, I do consider this case degenerate, and specified in my post that my comment was according to my definition. And I have consistently defined it this way since starting to analyze the T&E(3) database for the pattern (it just happens to be the case that all puzzles in T&E(3) currently known satisfy this more restrictive condition as well).

[denis_berthier]

It is not merely a matter of usefulness; it is a matter of reduction to a smaller pattern. In this case, the set of cells r1c89+r2c7+r3c9 is a 4 cell pattern with a chromatic number of 4, and the guardians are a subset of the guardians in the trivalue oddagon. We can quickly identify such cases by noting that there is a cell in box 3 which is not part of the potential trivalue oddagon but which is limited to the three digits of the trivalue oddagon, and is quite limited in scope (I can think of one other similar limitation involving the rows/columns instead of the box, but it would be much more unlikely to occur; might add a check for that at some point anyway).
You can choose to not define such a case as degenerate, that's your prerogative. I do, and have done consistently since starting to analyze the T&E(3) database for the pattern (it just happens to be the case that all puzzles in T&E(3) currently known satisfy this more restrictive condition as well).

Adding conditions to my definition would:
1) weaken the result that the contradiction of the pattern (with no guardians) requires T&E(3)
2) lead to fewer puzzles having the pattern

What you're talking about here is the relation between two patterns, the tridagon and some (trivial) impossible pattern in the block b3.
As the second pattern is simpler than the first, it should have higher priority. Which in the present case would make the tridagon useless.
But I don't see this as a reason for restricting the definition of the tridagon itself.

Actually, when i try to identify puzzles with tridagons, I do something similar: I apply Triplets before. This eliminates a few trivial cases.
.

[mith]

To be clear, I'm not talking about adding to the definition of a tridagon/trivalue oddagon. The pattern is clearly there, regardless of whether there is a smaller pattern with higher priority that can be used instead. I am only discussing here the "degenerate" classification.

We are already restricting the use of the pattern a great deal. At the most basic level of being an impossible pattern, we can say:

For any graph of cells which is isomorphic to the 12-cell 4-chromatic graph (which can be defined independently in purely graph terms)
And for any three digits
1. We cannot place digits such that all 12 cells contain only those three digits (by the chromatic number of the graph)
2. Therefore, at least one "guardian" digit (any candidate in one of the 12 cells which is not from the three digits in question) is true.

This is the form in which the pattern was discovered: not as a rigorously defined resolution rule applicable to CSP as you later provided, but as a 4-chromatic pattern leading to the conclusion (2) of at least one true guardian (exactly comparable to how guardians are used in bivalue oddagons, which are 3-chromatic).
 
In specific puzzles it may in fact be trivially true without considering the pattern at all. The definition above places no restriction at all on our choice of digits (the requirement that all three digits be candidates of all 12 cells is a degeneracy requirement, not a pattern requirement). As an extreme example, all of the chosen 12 cells could be filled by givens which are all from the other six digits; the conclusion is nevertheless true, even if it is also trivially true that none of the 12 cells can be from the three digits. I don't think you would consider such a case to be a tridagon at all (correct me if that's wrong) but either way you certainly wouldn't consider it to be a non-degenerate tridagon - nevertheless the pattern exists and the conclusion (2) holds, even if it is completely pointless.

In no way does considering certain puzzles degenerate weaken the result regarding the contradiction; it is true that the pattern with no guardians is contradicted, and that it requires T&E(3) to contradict, regardless of any specific puzzle, choice of cells, choice of digits. The result holds regardless of whether a specific deduction in a puzzle can be achieved through some other (not T&E(3)) elimination.

I would argue it would also not lead to fewer puzzles having the pattern (by the above less restrictive definition every puzzle has many examples of the pattern, albeit in some wildly degenerate form), it only leads to fewer puzzles have non-degenerate forms of the pattern.

As you say, we are doing similar things here; I am just removing more cases (which I consider trivial, and which you may not) from the non-degenerate pile.

------

I edited the above post before you responded but apparently before you quoted it, so I'll reiterate: I think you could consider this type of puzzle degenerate even by your definition, on the basis that whatever digit is in r1c9 - which we know is from 123 - cannot be in the three pattern cells in box 3.

I'll also mention that the slightly broader "degenerate" check I alluded to in the previous post can be summarized as follows:

For each row or column intersecting the pattern cells, there must not be multiple cells in that row or column which:
1. Occupy the same box
2. Are restricted to the three digits of the pattern

(If the box is one of the four pattern boxes, then at least one of the cells must not be part of the pattern, since pattern cells in a box never share a row or column, and therefore we have the box degeneracy above. On the other hand, if the box is one of the other four boxes sharing a band or stack with the pattern boxes, then we have the similar row/column degeneracy: the two pattern cells in that row or column cannot both contain pattern digits, because two pattern digits appear outside the pattern in that row or column.)

This should be pretty simple to code so I may check if any of the ph2010 puzzles have the row/column degeneracy. I would assume it's much rarer since the requirement involves two cells outside the pattern rather than one, if there are any examples at all.

[denis_berthier]

We are already restricting the use of the pattern a great deal. At the most basic level of being an impossible pattern, we can say:
For any graph of cells which is isomorphic to the 12-cell 4-chromatic graph (which can be defined independently in purely graph terms)
And for any three digits
1. We cannot place digits such that all 12 cells contain only those three digits (by the chromatic number of the graph)
2. Therefore, at least one "guardian" digit (any candidate in one of the 12 cells which is not from the three digits in question) is true.

OK, but the question is, how can this "abstract" graph appear in Sudoku other than in the classical pattern of blocks, rows, columns? I don't think there's much restriction here.

I am only discussing here the "degenerate" classification.
[...]
(the requirement that all three digits be candidates of all 12 cells is a degeneracy requirement, not a pattern requirement).

This is our main disagreement, one of vocabulary.
For me, the main difference between the non-degenerate pattern and the degenerate ones is at which level of T&E the contradiction can be proven.
As I've proven in the tridagon thread, as soon as one of the three digits is missing in one of the twelve cells, the contradiction can be proven in T&E(2).
For me, (non-degenerate) tridagon and degenerate-cyclic tridagon are two different patterns (close to each other but different), based on the same pattern of cells but with different conditions on the 3 digits.
None of these two patterns involve any cell other than the 12 ones and, in particular, no condition about another cell in one of the 4 blocks having exactly the 3 digits can be part of their definition.

As an extreme example, all of the chosen 12 cells could be filled by givens which are all from the other six digits; the conclusion is nevertheless true, even if it is also trivially true that none of the 12 cells can be from the three digits. I don't think you would consider such a case to be a tridagon at all

First, don't confuse the abstract pattern (defined in terms of variables, not fixed rows, columns...) and its instantiations in a particular puzzle, in particular rows...
Your case is an instance of the non-degenerate tridagon pattern, with exactly 72 guardians, a totally useless instantiation and moreover an impossible situation.

In no way does considering certain puzzles degenerate

Puzzles can't be degenerate.

it is true that the pattern with no guardians is contradicted, and that it requires T&E(3) to contradict,

If one of the 3 digits is missing in one of the 12 cells, T&E(2) is enough to prove the contradiction of the abstract pattern, independently of any puzzle.

I'll also mention that the slightly broader "degenerate" check I alluded to in the previous post can be summarized as follows:

For each row or column intersecting the pattern cells, there must not be multiple cells in that row or column which:
1. Occupy the same box
2. Are restricted to the three digits of the pattern
(If the box is one of the four pattern boxes, then at least one of the cells must not be part of the pattern, since pattern cells in a box never share a row or column, and therefore we have the box degeneracy above. On the other hand, if the box is one of the other four boxes sharing a band or stack with the pattern boxes, then we have the similar row/column degeneracy: the two pattern cells in that row or column cannot both contain pattern digits, because two pattern digits appear outside the pattern in that row or column.)

But this would turn the 3-digit 12-cell pattern into something else, involving CSP-Variables that don't belong to the original pattern.This is a step I'm not ready to make.
There are smarter ways to keep my original formal definition (in the trdiagon thread) unchanged and to avoid the pattern to be used when other patterns make it useless. This is the general way resolution rules interact.
I've already given one example: apply Triplets before Tridagon rules. It doesn't take care of your cases, but similar rules can be written.

This should be pretty simple to code

No doubt, but see point above.
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[mith]

We are already restricting the use of the pattern a great deal. At the most basic level of being an impossible pattern, we can say:
For any graph of cells which is isomorphic to the 12-cell 4-chromatic graph (which can be defined independently in purely graph terms)
And for any three digits
1. We cannot place digits such that all 12 cells contain only those three digits (by the chromatic number of the graph)
2. Therefore, at least one "guardian" digit (any candidate in one of the 12 cells which is not from the three digits in question) is true.

OK, but the question is, how can this "abstract" graph appear in Sudoku other than in the classical pattern of blocks, rows, columns? I don't think there's much restriction here.

In classic sudoku, it can't. It can absolutely appear in different forms in variant sudoku.
But in classic sudoku, our nodes are cells, our edges are houses (blocks/boxes, rows, columns), so naturally any subgraph of the sudoku graph is going to relate to the abstract 4-chromatic graph via the houses.

And I think you misunderstood what I meant by "we are already restricting" here; I wasn't saying this limited definition of a trivalue oddagon was restrictive, I was saying we are restricting (adding restrictions) on top of that very much unrestricted abstract graph approach.

I am only discussing here the "degenerate" classification.
[...]
(the requirement that all three digits be candidates of all 12 cells is a degeneracy requirement, not a pattern requirement).

This is our main disagreement, one of vocabulary.

I don't disagree that our disagreement is one of vocabulary. To be clear, I am not trying to convince you to change your definition.

For me, the main difference between the non-degenerate pattern and the degenerate ones is at which level of T&E the contradiction can be proven.
As I've proven in the tridagon thread, as soon as one of the three digits is missing in one of the twelve cells, the contradiction can be proven in T&E(2).
For me, (non-degenerate) tridagon and degenerate-cyclic tridagon are two different patterns (close to each other but different), based on the same pattern of cells but with different conditions on the 3 digits.

That's fair. I do not consider them to be different patterns, myself. For me, they are the same pattern, but examples of that pattern may or may not be degenerate in some way. (Calling them the same pattern or not isn't really a relevant distinction for me, though I appreciate that it may be for you and how you define pattern.)

None of these two patterns involve any cell other than the 12 ones and, in particular, no condition about another cell in one of the 4 blocks having exactly the 3 digits can be part of their definition.

I would argue that your degenerate tridagon pattern (under your definition), or at least its instantiation, does involve other cells; how else are you getting the restriction on a pattern cell that one of the digits is excluded as a candidate? We can certainly talk about the pattern in the abstract, and show that if we are missing a candidate the contradiction is in T&E(2), but such a pattern can't exist in the (classic) sudoku grid unless some given digit in another cell is imposing it.

As an extreme example, all of the chosen 12 cells could be filled by givens which are all from the other six digits; the conclusion is nevertheless true, even if it is also trivially true that none of the 12 cells can be from the three digits. I don't think you would consider such a case to be a tridagon at all

First, don't confuse the abstract pattern (defined in terms of variables, not fixed rows, columns...) and its instantiations in a particular puzzle, in particular rows...
Your case is an instance of the non-degenerate tridagon pattern, with exactly 72 guardians, a totally useless instantiation and moreover an impossible situation.

My example is an instance of a maximally degenerate (in that all 12 cells have none of the three pattern digits) trivalue oddagon with exactly 12 guardians (precisely the given digits in the 12 cells). It is (obviously) completely useless. The conclusion that at least one guardian is true is immediately proving by looking at the smaller pattern of any single cell. It is nevertheless a trivalue oddagon under my definition.

The non-degenerate tridagon with 72 guardians would appear in an empty sudoku grid (or empty in all boxes except for one). It's impossible in classic sudoku insofar as you can't have a unique solution with so much of the grid empty. However, it's also not an impossible situation in variant sudoku (I doubt it would be difficult to construct a sukaku with an example of this, say). And again, totally useless, sure. Maximally useless. But nevertheless a trivalue oddagon under my definition.

In no way does considering certain puzzles degenerate

Puzzles can't be degenerate.

Can we not snipe at little word choice things like this? I have no doubt you understood that my meaning was "considering certain puzzles to have degenerate trivalue oddagons"; I was responding to "lead to fewer puzzles having the pattern" and it's clear from the context of the rest of the sentence what I meant. Just as I understood you in the other thread to be saying "42,097 more tridagons in puzzles between 100,001 and 200,000, all [puzzles] from Paquita".

If I were publishing a paper and had sent it to you as an editor, this would have been a helpful and wanted comment. We're posting on a web forum. Shorthand and outright omissions are to be expected.

But this would turn the 3-digit 12-cell pattern into something else, involving CSP-Variables that don't belong to the original pattern.This is a step I'm not ready to make.

Wasn't asking you to. Keep in mind, I am responding to this:

unless one is willing to add to the definition conditions (such as those you're mentioning) that have never been defined and that are potentially in unlimited numbers

by pointing out that these conditions have been defined and have been part of my definition for degeneracy all along. Doesn't bother me if you don't want to include that in your definition of degeneracy, but at the same time your definition is not constraining how I choose to engage with this.

[denis_berthier]

We are already restricting the use of the pattern a great deal. At the most basic level of being an impossible pattern, we can say:
For any graph of cells which is isomorphic to the 12-cell 4-chromatic graph (which can be defined independently in purely graph terms)
And for any three digits
1. We cannot place digits such that all 12 cells contain only those three digits (by the chromatic number of the graph)
2. Therefore, at least one "guardian" digit (any candidate in one of the 12 cells which is not from the three digits in question) is true.

OK, but the question is, how can this "abstract" graph appear in Sudoku other than in the classical pattern of blocks, rows, columns? I don't think there's much restriction here.

In classic sudoku, it can't. It can absolutely appear in different forms in variant sudoku.
But in classic sudoku, our nodes are cells, our edges are houses (blocks/boxes, rows, columns), so naturally any subgraph of the sudoku graph is going to relate to the abstract 4-chromatic graph via the houses.

And I think you misunderstood what I meant by "we are already restricting" here; I wasn't saying this limited definition of a trivalue oddagon was restrictive, I was saying we are restricting (adding restrictions) on top of that very much unrestricted abstract graph approach.

I wouldn't call this a restriction. It's just an injection of the abstract graph into the Sudoku graph. It can easily be checked that it defines a bijection of the abstract graph to the sudoku subgraph defined by the 12 cells.
It doesn't add more conditions to the pattern.

None of these two patterns involve any cell other than the 12 ones and, in particular, no condition about another cell in one of the 4 blocks having exactly the 3 digits can be part of their definition.

I would argue that your degenerate tridagon pattern (under your definition), or at least its instantiation, does involve other cells; how else are you getting the restriction on a pattern cell that one of the digits is excluded as a candidate? We can certainly talk about the pattern in the abstract, and show that if we are missing a candidate the contradiction is in T&E(2), but such a pattern can't exist in the (classic) sudoku grid unless some given digit in another cell is imposing it.

The 'why" of candidates being present or absent in a grid has nothing to do with the presence or not of a pattern.
Do you ask why in a Triplets there are only 3 candidates?

As an extreme example, all of the chosen 12 cells could be filled by givens which are all from the other six digits; the conclusion is nevertheless true, even if it is also trivially true that none of the 12 cells can be from the three digits. I don't think you would consider such a case to be a tridagon at all

First, don't confuse the abstract pattern (defined in terms of variables, not fixed rows, columns...) and its instantiations in a particular puzzle, in particular rows...
Your case is an instance of the non-degenerate tridagon pattern, with exactly 72 guardians, a totally useless instantiation and moreover an impossible situation.

My example is an instance of a maximally degenerate (in that all 12 cells have none of the three pattern digits) trivalue oddagon with exactly 12 guardians (precisely the given digits in the 12 cells). It is (obviously) completely useless. The conclusion that at least one guardian is true is immediately proving by looking at the smaller pattern of any single cell. It is nevertheless a trivalue oddagon under my definition.

I had misunderstood your example. If all the 3x12 tridagon candidates are absent, for me it's not a tridagon at all... for the same reason that two Pairs in the same row are not a Quad.
This extreme case is a perfect example of why I've defined the degenerate cyclic tridagon pattern: in order to put some limit on allowed degeneracy, before we fall into absurd situations.
I'm surprised you add conditions involving other cells (which I would indeed call "restrictions"), but you don't add obvious ones that involve only the 12 cells and that can be defined at the level of the abstract graph.

unless one is willing to add to the definition conditions (such as those you're mentioning) that have never been defined and that are potentially in unlimited numbers

by pointing out that these conditions have been defined and have been part of my definition for degeneracy all along. Doesn't bother me if you don't want to include that in your definition of degeneracy, but at the same time your definition is not constraining how I choose to engage with this.

My point was not about your specific conditions, but about the potentially unlimited number of such conditions (plus the already mentioned fact that they involve other cells).
.

[mith]

I'm surprised you add conditions involving other cells (which I would indeed call "restrictions"), but you don't add obvious ones that involve only the 12 cells and that can be defined at the level of the abstract graph.

I do add the obvious ones. The "conditions involving other cells" are not conditions on the pattern being a trivalue oddagon, they are conditions on degeneracy of the pattern. My focus has been largely on non-degenerate trivalue oddagons, and the first pass filter I mentioned in the other thread is already a sufficient condition to guarantee that a trivalue oddagon found in (2) below must be contain all three candidates in all twelve cells.

Non-degenerate trivalue oddagon algorithm:

1. A trivalue oddagon can only be non-degenerate in a classic sudoku puzzle if none of the three candidates are present in any of the eight boxes seen by the cell pattern; therefore, if a puzzle has a non-degenerate trivalue oddagon, there must be three digits which are limited as givens to one box.
a. Check the given count; if there are not at least three digits appearing at most once in the puzzle, then the puzzle cannot have a non-degenerate trivalue oddagon.
b. Check the box where the limited digits appear. If there are not three digits appearing at most once in the puzzle such that they appear in the same box or not at all, then the puzzle cannot have a non-degenerate trivalue oddagon.
[Note that these first pass checks don't actually check for trivalue oddagons at all. They check for conditions required for the appearance of a non-degenerate trivalue oddagon.]

2. After applying some set of resolution rules (either singles or singles+subsets), find 12 cells which contain the digits under consideration (in the case of non-degenerate trivalue oddagons, any cell containing one of the digits as a candidate must contain all of the digits as candidates, due to the first pass filter).
3. Check for other degeneracy conditions (currently the one implemented is the one under discussion: there must not be a cell in one of the four boxes which is not part of the pattern but which only contains the three digits as candidates).

The degenerate trivalue oddagon algorithm works as described in p328613. The point is not to find every degenerate trivalue oddagon; under my definition, that would include very possible cell pattern (5832) and choice of digits (84). I filter highly degenerate puzzles first (bullet point three): the three digits under consideration must be naively placeable in each box of the pattern.

After this, I have also chosen to filter other types of degeneracy to make the results more manageable; this includes the "other cell" restriction (bullet point four) as well as expanding on "naively placeable" in consider rows/columns instead of just boxes (bullet point five). Bullet point five can only filter puzzles which are already degenerate (you can't have "fixed digits" in this sense if all the cells in the pattern contain all three digits). Bullet point four is where our definitions differ; you wouldn't consider these patterns degenerate at all, I filtered them out entirely because I wasn't interested in including them in already large lists of degenerate patterns.

[mith]

Continuing from the other thread:

These are the first T&E(3) puzzles found in my old "hardest" databases, by clue count:

Code: Select all

24c
(still running)
25c
........1.....2.34....4.56.....2478...41......7.8.9.....9..1....81.....724...7..9  ED=10.4/1.2/1.2 (DCFC+FC)
26c
(still running)
27c
.............12.34..135.6.2....34.65...2....1.5.16..2...5.4....6.7.....38.9....4.  ED=10.5/1.2/1.2 (DCFC+FC)
28c
.......12.....34......5..36....15....137.2.8.7.583..2..78......1.2..87..35...1...  ED=10.6/1.2/1.2 (DCFC+FC)
29c
...........1..2.34..2.1356....5..34..16.37....4..89.....5...62.12......56.4....13  ED=10.5/1.2/1.2 (DCFC+FC)
30c
........1.....2.34....356.7..28......31..9...98..23..5.19..8.5.2.3.51...85..9..1.  ED=10.3/1.2/1.2 (DRFC+FC)

The 27c is the earliest of these, found sometime in August/September 2021 (this is based puzzles at this clue count with high SER posted in the "hardest" thread; the T&E(3) puzzle is between, in the database, puzzles posted on August 13 and on September 29).

The 30c is not currently in the T&E(3) database. When I started the T&E(3) database, I only checked for depth 3 puzzles at SER >= 11.3 to use as a starting point, so these were not among the original seeds; however, the others were eventually found again (not terribly surprising, since the seeds in the T&E(3) database come from neighborhood searches on the ph databases, and the T&E(3) database is grown by neighborhood searches).

The next step (once the 24c and 26c scripts finish) is to find all depth 3 puzzles in the old databases. This should be pretty quick, since I only have to consider puzzles after the above puzzles in their respective databases (and apart from 27c these are already quite late in the database). Once that's done, I'll check against the T&E(3) database.

I'm overdue for an update anyway, but once these are found I'll add them to the database and run the expander/transformer/minimizer scripts on these new puzzles. I also still need to check the new puzzles posted by hendrik and Paquita since my last update. So it'll be a bit before the update is ready, but at least I have momentum on it again.

[denis_berthier]

I'm surprised you add conditions involving other cells (which I would indeed call "restrictions"), but you don't add obvious ones that involve only the 12 cells and that can be defined at the level of the abstract graph.

I do add the obvious ones. The "conditions involving other cells" are not conditions on the pattern being a trivalue oddagon, they are conditions on degeneracy of the pattern.

That's our main difference. For me, both the (non-degenerate) tridagon and the degenerate-cyclic-tridagon are defined in pure logic terms that refer only to the 12 CSP-Variables involved (they are patterns in the precise sense defined and used in all my publications).

As for your extra condition on the block opposite the pattern, they are consequences of the definition in the non-degenerate case. Adding them to the algorithm is a matter of programming, not of defining the pattern. In SudoRules, in which "the program is the rules", I tried when you first mentioned these conditions but it doesn't make tridagons faster.

Of course, the degenerate-cyclic case (precisely defined here: http://forum.enjoysudoku.com/the-tridagon-rule-t39859-118.html) doesn't satisfy these extra conditions and that's why I consider it an interesting pattern in and of itself - as a potential precursor of the non-degenerate one. In particular, it is present in about 0.27% of the controlled-bias puzzles (while the non-degenerate case is absent of all of them): http://forum.enjoysudoku.com/the-tridagon-rule-t39859-119.html.
.

[mith]

My scripts finished checking for the first depth 3 puzzle at each clue count, with puzzles found at 25c-30c. As mentioned above, the first of these was the 27c; however, after looking at it again this puzzle was never used as a seed, so Loki (for example) cannot trace back directly to this puzzle. I'll have to take some care in my backwards neighborhood search to ensure only puzzles which were used as a seed for the neighboorhood under consideration are kept as potential first seeds.

I'm now running a full check on each clue count starting at the first found and going to the end of the database. At 25c for example since it just finished running:

Code: Select all

;........1.....2.34....4.56.....2478...41......7.8.9.....9..1....81.....724...7..9;0.0;0.0;0.0;;mith;;;25;C;940;2003;10.2;1.2;1.2;;;0;0;0;0;4010553;3
;........1.....2.34....4.56.....63.....14.5..2.461......7.2......893.....6.4....25;0.0;0.0;0.0;;mith;;;25;C;916;2072;10.2;1.2;1.2;;;0;0;0;0;4010669;3
;...........1..2..3.2..41.56....65..2...12.54....3....1..72.......84.3...51....6..;11.7;1.2;1.2;DRFC+DFC;mith;;;25;C;30018;59729;11.5;1.2;1.2;;;1;1;1;1;4012210;3
;...........1..2..3.2..41.56....2.54....3....11...65..2..72.......84.3...51....6..;11.7;1.2;1.2;DRFC+DFC;mith;;;25;C;30018;59729;11.6;1.2;1.2;;;1;1;1;1;4012211;3
;........1.....2.3....456.7.....258.....6......5681.2...1..4.....48.....65.2..8..4;11.7;1.2;1.2;DRFC+DFC;mith;;;25;C;1500;880;11.6;1.2;1.2;;;1;1;1;1;4017334;3
;........1.....2.3....456.7.....258.....6......5618.2...1..4.....48.....65.2..8..4;11.7;1.2;1.2;DCFC+DFC;mith;;;25;C;1479;1016;11.6;1.2;1.2;;;1;1;1;1;4017413;3
;........1....1234...3.5.6.2....78.35...3..4.....62.....16...5..2.5......43.....1.;0.0;0.0;0.0;;mith;;;25;C;669;844;10.9;1.2;1.2;;;0;0;0;0;4024523;3
;................12....34..5..2..6....51.7....86.....7..7.6...5112.8..6..5..7..28.;11.7;1.2;1.2;DRFC+DFC;mith;;;25;C;1535;1065;11.7;1.2;1.2;;;0;0;1;1;4063568;3
;........1.....2.34....56.....1.257...2.68.....567..8...87.1....5.2....1.6.......7;11.7;1.2;1.2;DRFC+DFC;mith;;;25;C;1109;1906;11.7;1.2;1.2;;;0;0;1;1;4063577;3
;........1.....2.......3..45..1.236...276..8...3.7.8....86..1...2.3..7.1.7.......6;11.7;1.2;1.2;DRFC+DFC;mith;;;25;C;1318;560;11.7;1.2;1.2;;;0;0;1;1;4063588;3

The last columns are most relevant; in reverse order: depth (3), rowid in the database (not the same as the position in the database, there are gaps for various reasons), and then the four neighborhood scripts. We can see here that the third puzzle in the 25c database was the first depth 3 puzzle to be used as a seed (by all scripts). All of the 11.7 puzzles were used as a seed (though note the last three puzzles were not used by all neighborhood scripts, because I stopped running these scripts when I switched from using SER to T&E depth).

Any puzzle 11.3+ should already be present in the T&E(3) database. Some of the others will have been found again later in the depth search.

[mith]

This is the start of the 27c list:

Code: Select all

;.............12.34..135.6.2....34.65...2....1.5.16..2...5.4....6.7.....38.9....4.;0.0;0.0;0.0;;mith;;;27;C;59892;95108;10.3;1.2;1.2;;;0;0;0;0;2288138;3
;...........1.23.45.2.41.36......4.1....63.52.2...514......6.....57....3..89...6..;0.0;0.0;0.0;;mith;;;27;C;57561;92592;10.3;1.2;1.2;;;0;0;0;0;4487632;3
;........1.....2..3....3.45...3...26..728.....61..23.8...6..1....81.6...732...7..6;11.7;1.2;1.2;DRFC+DFC;mith;;;27;C;2370;1466;11.5;1.2;1.2;;;1;1;1;1;4631832;3

So we're looking at a huge gap between when the first T&E(3) puzzle was found in the database, and when the first T&E(3) puzzle was actually used as a seed (which is now looking much closer to when Loki was found and Denis identified it as T&E(3)).

Anyway, the 27c run should be done soon, and that's the last of them. I'll compile the results and check dates again later today.

[mith]

1282 minimal puzzles in T&E(3) from local ph databases

Since all of the earliest puzzles used as seeds are 11.6+, they should have been posted on the forums at some point, and I may be able to identify the dates they were found to within a small window. They also would have been used as seeds very quickly after being found, since SER was the primary sorting parameter for selecting seeds. (At 29c, I may not be able to determine which was used as a seed first since they second in the database is very close to the first and has a higher SER.)

Earliest seeds by clue count and SER (if a higher rated puzzle is "close enough" in the database to a lower rated puzzle, it may have been used as a seed first):

Code: Select all

25c
(4012210-4012211)
...........1..2..3.2..41.56....65..2...12.54....3....1..72.......84.3...51....6..  ED=11.7/1.2/1.2
...........1..2..3.2..41.56....2.54....3....11...65..2..72.......84.3...51....6..  ED=11.7/1.2/1.2

26c
(4838755-4838762)
........1.....2.34....4.56.....2478...41......728.9....81......42...7..97.9..1...  ED=11.6/1.2/1.2
........1....23.45.25...36.....51......2.4.1.4..63..2..78...6..1...6....6.9....3.  ED=11.6/1.2/1.2
........1.....234.....51.62....2578...21......7589.....81......25..7...97.9......  ED=11.6/1.2/1.2
........1....23.45.25...36.....51......2.4...4..63..2..78...6..1...6....6.9...13.  ED=11.6/1.2/1.2
........1.....2.34....4.56.....2478...28.9.....417.....79..1...24...7..98.1......  ED=11.6/1.2/1.2
.......12.....134...4...5.6....15.6..2.7......3.89.....16.4..2524.1.....5.3......  ED=11.6/1.2/1.2
........1.....234.....51.62....2578...21.7.....589.....79......52..7...98.1......  ED=11.6/1.2/1.2
.......12......3.4..1..256......46...45.76....6.89.....5613....3.2.....541.......  ED=11.6/1.2/1.2
(4844194)
........1.....2..3....3.45...1.2367..267.8....3....2...17......3.2..6..868...1...  ED=11.7/1.2/1.2
Loki (4883671)
57....9..........8.1.........168..4......28.9..2.9416.....2.....6.9.82.4...41.6..  ED=11.9/1.2/1.2

27c
(4631832/4631853)
........1.....2..3....3.45...3...26..728.....61..23.8...6..1....81.6...732...7..6  ED=11.7/1.2/1.2
........1..2..3.4..4.56..2.....2..3....3.5..42...41.56..7...31...8...4..52..36...  ED=11.7/1.2/1.2
(4691076)
........1.....2.......3..45..1.23....267.81..73.61.8...17.6.....8.......2.3.87..6  ED=11.8/1.2/1.2

28c
(3473087-3473088)
........1.....2..3....3.45...3...26..728.....61..2378..81.6....32...7..67.6..1...  ED=11.7/1.2/1.2
........1..2..3.4..4.56..2.....36..4...25..3.2..4.1.65..7...31...8...45.62.3.....  ED=11.7/1.2/1.2
(3491728)
........1.....2.3.....4.56......7.....481.2..19..248...89..1..742..78..97.1.9....  ED=11.7/8.5/2.6
(3675804)
........1.....2.34....562....7.......61..8...25..67..8.7..25....856..1..6.28.17..  ED=11.8/1.2/1.2

29c
(3444467)
........1.....2.34..2.3156.....564...1.2.43...4.31..26..4....5..5..63...7.8..5...  ED=11.6/1.2/1.2
(3444855)
........1.....2.34....562....1.25....267.18..75.68.1...8.......5.2.68..761...7...  ED=11.8/1.2/1.2

30c
(2917477)
........1.....2.34....562....1.257...267.18...5.68.1...87......5.2.68..761...7...  ED=11.8/1.2/1.2

The first set of 26c puzzles only went through the -2+1 and singles expander scripts, the rest of these puzzles were seeds for all scripts.

[mith]

Compared the first 26c to Loki (using gsf's Hamming distance algorithm to morph Loki to the closest form):

Code: Select all

........1.....2.34....4.56.....2478...41......728.9....81......42...7..97.9..1...;1;1;1;1;1;n356;b4p357+b5p168+b7p168+b8p348;
........9.....2.35....4.....1..24...8.47..9...728.91....1......24..71..87.9..8...;1;1;1;1;1;n356;b4p357+b5p168+b7p168+b8p348;

Different guardians (8r9c5 vs. 9r8c4), but promising for there being a direct line between the two.

(The first 26c has the smallest distance, 18, to Loki.)