The New Sudoku Players' Forum

[blue]

Hi Denis,

As for JExocet, I don't remember reading about this. I'll have a look.

(*: I also saw on the programmer's forum an alternative more formal definition proposed by Blue, but he received no answer).

I think the culmination of my ideas, was D.P. Bird's specification for the JExocet pattern. It extended them a little, to include some "easily recognized" cases that were not entirely describable in terms of "Finned Swordfish". I'll have to say (apologies to David), that like ronk, I was unsatisfied with David's specification, and would have preferred one that was more oriented towards sub-patterns that are strictly base/cover problems.

I would be curious if you can place the JExocet patterns in the "chains" or "subset" category.
My gut says no, but I'm open minded.

Like you (I'm sure), and David, I was not satisfied with champagne's definitions. It wasn't that there was a problem with the logic, but that, like with logel's definition, the valid instances of the "pattern", are not easily verified.
Note: I put quotes around 'pattern', to emphasize the fact that I wouldn't put it in the same class any of the "usual" patterns. I don't especially like calling it a pattern, but for lack of a better term, I'm using anyway (along with the quotation marks).

Best Regards,
Blue.

[champagne]

Like you (I'm sure), and David, I was not satisfied with champagne's definitions. It wasn't that there was a problem with the logic, but that, like with logel's definition, the valid instances of the "pattern", are not easily verified.

Hi blue,

May be an historical comment on that.

The Exocet general definition (and I am sure you are here thinking of a more restrictive one) is an attempt to define the wider specification to cover all patterns having the implicit logic shown by Allan BARKER in the puzzle "fata morgana" in a multi floor analysis.

At the start, I applied it to all AAHS, but,

1) I got a lot of redundancy,
2) It appeared that using only 2 cells belonging to a mini row, mini column the redundancy vanished and nearly nothing was lost
3) When, later, the twin Jexocet has been added, it appeared that nearly all EXOCET+ twin were located in a band (I have only one exception so far).

David, as manual player made a very good job focusing on the JEXOCET. It is by far the most common form of EXOCET when the base is a mini row/column and it is there in more than 50% of the potential hardest file.

The JEXOCET is "easy" to see for a manual player, it is also very fast to look directly for it with a computer (thousands times faster than using permutations). As in any pattern analysis, it is just requiring more code.

Last but not least, seen on the solving angle, but using the uniqueness property, the JEXOCET pattern finds its best efficiency when using the "abi loop" pattern.

It remains that the solving power of an EXOCET is highly variable form one puzzle to another one.

For sure, to-day one can think that the field has been deeply explored and the wider definition can be ignored.

Nevertheless, exploring all possibilities within a band has shown interesting situations and these are not JEXOCETS

regards

champagne

[denis_berthier]

I think the culmination of my ideas, was D.P. Bird's specification for the JExocet pattern.

I wasn't aware your ideas had been further developed by David on this forum.
In general, I can't read more than the first few posts of a thread before getting a headache, not being able to tell what is being spoken of; so I can't even imagine reading the 27 pages of the Exocet thread.

I was unsatisfied with David's specification, and would have preferred one that was more oriented towards sub-patterns that are strictly base/cover problems.
I would be curious if you can place the JExocet patterns in the "chains" or "subset" category.
My gut says no, but I'm open minded.

I'm not yet able to understand David's definition (due to a few ambiguities), let alone to answer these questions.
Considering only the pattern of cells, some kind of chain definition is not obvious. There might be a gSubset definition but I can't confirm this right now.

My rough classification of patterns into two broad categories may not be exhaustive. The fact is, all the known patterns fall into either of them (and sometimes in both, depending on one's POV) - except maybe a few exotic ones*. My main point in speaking of this rough classification was, and still is, the structures of these patterns carry the proofs of their eliminations. I'm completely open to any other pattern not falling in either of these two categories, provided that it also carries the proofs of its eliminations.
(* sk-loops can easily be considered as g-Subsets, although I don't think this is in the spirit of Steve's definition as a generalised chain pattern).

A remark about the base/cover vocabulary. Even if "base" is convenient and I also use it to mean a set of CSP-variables in the general CSP context, i.e. a set of 2D-cells in the Sudoku context, I don't like "cover" because it supposes that one is dealing with a base-cover problem - which is not the case.
As you and everyone here know, it is not enough to cover the candidates in the base set - one must cover them with constraint sets (where a constraint set is a set of candidates with pairwise constraints between any two candidates in this set); due to additional conditions, I call them transversal sets or transversal constraint sets.
I think my general definition of a g-Subset - a straightforward extension of the most standard Subsets and of the Mutant Fish - covers (if I dare use this word in this context) all the clean part of Allan's definition. And it has the advantage of clearly showing that there is both a close relationship and an essential difference between Subset or g-Subset patterns and chain patterns (e.g. whips, braids, g-whips or g-braids) - see PBCS for details.

[denis_berthier]

Definition and discussion of the JExocet pattern has been moved to a new thread where it will fit better than here: http://forum.enjoysudoku.com/post226661.html#p226661
Please post any new comments on this topic there.

I'll continue to discuss here topics related to its classification.

[denis_berthier]

Exotic patterns

When I first introduced the notion of an exotic pattern (http://forum.enjoysudoku.com/pattern-based-classification-of-hard-puzzles-t30493-2.html), I din't say what I meant. The expression must have hit something, as it was almost immediately taken for the name of a new thread - but without further analysis of its meaning.

Using the sk-loop example, I can be more precise.

As I said before, there are two and as of now only two large families of rules:
- the chain-like ones: whips, braids, g-whips, g-braids, ...
- those based on Subsets or g-Subsets.
The patterns in each of these families depend on a single parameter (length or size) and this naturally leads to a rating of puzzles (according to the hardest step philosophy and the simplest first strategy).
Moreover, these families and ratings are not specific to Sudoku, they are meaningful for any finite CSP - in particular for many logic puzzles (see Kakuro examples here: http://forum.enjoysudoku.com/can-you-solve-this-without-trial-and-error-t30960.html).
When both types of rules are included in a resolution theory, general subsumption theorems (valid for any CSP) don't leave any choice on how to define a combined rating: length of chains and size of Subsets must be equated.


What I had in mind when I introduced the expression of an exotic pattern is a pattern (defined in a factual/descriptive way independent of any procedure) that can only have one (or two or a few) length(s)/size(s) and that can, in and of itself, lead to no classification system.
Given an exotic pattern, the best we can therefore hope is to see how adding it to a set of rules can modify the corresponding rating/classification system. For the sk-loop, I have done this in chapter 13 of PBCS.


Now, one could ask: as you have shown that sk-loops are gSubsets[16], they are no longer exotic.
But my answer is: yes, they remain exotic nevertheless. When we speak of sk-loops, we don't consider them as having to be looked for at the same time as all the other gSubsets or Subsets of same or smaller size. We somehow give them higher priority.
Said otherwise, we don't consider them mainly as gSubsets[16].
Actually, that's why I didn't write "they are Subsets/gSubsets" but "they can be considered as ...".

[David P Bird]

Denis, in addition to the 2 classes of elimination patterns that I've already mentioned (JExcocet, and Avoidable Patterns) there are two more that I would describe in everyday terms as exotic.

Reverse BUGs: which are again are based on the 'a given per UA" derived constraint. For manual solvers these are only practical for 2-digit UAs. If a digit placement would cause the UAs containing all the givens for a set of digits to be identifiable, that placement must be false if these UAs don't cover every house. (I'm sure you'll dismiss this again but I include it for completeness.)

Overlapping Oddagons: This is another one of Steve Kurzhals' amazing finds. An oddagon is the threat of a single digit conjugate chain with an odd number of nodes (eg a kite pattern). These must be disrupted by the digit being true in a cell that sees two of the nodes. The method locates the possible disrupting cells for sets of oddagons for different digits. When the total number of disrupting cells equals the number of digits in the set, any non-member digit in these cells can be eliminated. The different oddagons concerned needn't completely coincide but must certainly overlap.

Although locating one of these sets is hard work, I think it still qualifies as being a pattern based method. Steve's example was lost in the Eureka forum crash but possibly Don will have a copy.

[denis_berthier]

I'm sure you'll dismiss this again but I include it for completeness.

I dismiss nothing.
Let's say there are:
- regular patterns (those belonging to the large families) and independent of the assumption of uniqueness
- exotic patterns, as defined above and independent of the assumption of uniqueness
- U-patterns, depending on the assumption of uniqueness
- exotic U-patterns, as defined above but depending on the assumption of uniqueness

Probably all the U-patterns are exotic U-patterns, but I'm not very aware of all the U-patterns, so I leave this open.

[David P Bird]

I'm sure you'll dismiss this again but I include it for completeness.

I dismiss nothing.
Let's say there are:
- regular patterns (those belonging to the large families) and independent of the assumption of uniqueness
- exotic patterns, as defined above and independent of the assumption of uniqueness
- U-patterns, depending on the assumption of uniqueness
- exotic U-patterns, as defined above but depending on the assumption of uniqueness

Probably all the U-patterns are exotic U-patterns, but I'm not very aware of all the U-patterns, so I leave this open.

Denis, sorry 'dismiss' was too strong a word.

However your categories don't make it clear which one includes Avoidable Patterns.

As you're aware I think intuitively rather than mathematically or in terms of formal logic al theory, but childlike questions can sometimes be great eye openers regarding our un-stated conditions and assumptions.

'Daddy, are you saying that everything's made up out of atoms?'
'Yes sweetie'
'What about shadows?'

Your justification for refusing to assume uniqueness, (because publishers never claim this), don't carry over to proven unique sub-puzzles. Your reasons there just appear to be that they're inconvenient to you. To my simple mind it appears that your constantly combing inferred constraints from mixed chain and subset classes of inference patterns to reach conclusions, and I still fail to see a solid reason for excluding 'proven uniqueness' as a further class.

Your arguments imply that you'd be prepared to use uniqueness if a trustworthy publisher claimed it, and therefore I just feel that the discussion your leading here is rather incomplete.

I should quickly add that I'm only considering Avoidable Patterns here, not Reverse BUGs because they do assume uniqueness.

[denis_berthier]

David, it's just that I don't claim dealing with everything in the world.

[denis_berthier]

Can a JExocet be seen as a Subset or a gSubset? - classification of JExocet

I would be curious if you can place the JExocet patterns in the "chains" or "subset" category.

Now that the JExocet definition is clear, I can try to tackle this question.

I'll use the definition and notations here http://forum.enjoysudoku.com/jexocet-pattern-defintion-t31133-12.html. There may appear extensions of this core pattern, but that shouldn't change the following remarks.

Let p (= 3 or 4) be the number of base digits.


1) Can a JExocet be considered as a Subset (or a gSubset)?

Suppose we try to see the JExocet as a Subset or as a gSubset (according to my general definitions in "Pattern-Based Constraint Satisfaction - PBCS").

Then it is somehow obvious that the following 2+3p [= 11 or 14] CSP-variables/2D-cells must be part of the pattern:
- the two B rc-cells
- the 3p cn-cells: for each of the 3 S columns ci and for each of the p base digits nj, the cn-cell cinj
At this point, the situation is rather simple, as these 2+3p CSP-variables are pairwise disjoint.

It is also obvious that the following must be part of the transversal sets:
- for each of the base digits nj, the rn constraint r1nj (r1 is the row holding the B cells)
- for each of the base digits ni and for each of the two rows ri and r'i in which it can be present in the S cells, the rn constraint rini / r'ini.
This makes up 3p transversal sets.

We also have to deal with the Q and R pairs. As long as we are concerned only with base digits, we can discard the companion cells. There remains only one possibility, i.e. taking two more transversal sets: the two rc constraints defined by the two target cells. As the complement of the intersections of these constraints with the base CSP-variables is exactly the set of non-base digits in the target cells, it would be OK for the targets of a Subset or gSubset.
We now have 2+3p transversal sets and they are pairwise disjoint.

Is therefore everything OK for a Subset[2+3p]?

Nope. The base candidates that, by the general JExocet definition, are allowed in the intersection of block b1 and column c3 are not yet covered by these transversal sets.
Note: in a first version of this post, I had forgotten that there may be base digits in the intersection of block b1 and column 3. But Blue (many thanks to him) pointed it out in a PM.

If this was the only problem, we could consider a special case and try to conclude that
- any JExocet with 3 or 4 base digits and no base digit in the intersection of the base block and the S column intersecting it can be considered as a Subset[11 or 14].
By replacing the p rn-constraints r1nj by p bn-constraints b1nj, we would get similarly:
- any JExocet with 3 or 4 base digits and no base digit in the intersection of the base row and the two S columns not intersecting the base block can be considered as a Subset[11 or 14].
(I don't know if such special cases have already been considered.)

But even this doesn't work. Indeed, if it worked, the Subset rule would also eliminate (in the first case) all the base digits in the first row non-S columns outside the B cells, which is obviously incorrect.


At this point, we must also notice that the condition of having no base digit in the companion cells has not yet been taken into account. We should therefore introduce two more CSP-variables corresponding to these cells. But then, we would also have to introduce (9-p) or 2(9-p) rn transversal sets [depending on whether they are in the same row or nor] to cover them. Finally, we would get many more transversal sets than CSP-variables.



I'm fully aware that this is not a full proof that a JExocet cannot be seen as a Subset or as a gSubset, but if there is some smarter way of proving it, it still eludes me. As I'm a little lazy about exotic patterns on Sundays, it's enough to deter me from looking longer for a proof that a JExocet can be considered as a Subset or as a gSubset.

To answer the second part of Blue's question, trying to see a JExocet as some kind of chain doesn't seem much more promising.



2) Classification of the JExocet

The above analysis isn't completely negative and that's why I posted it. Even though I couldn't prove that a JExocet is a Subset or a gSubset, the analysis shows that this pattern involves (at least) 13 or 16 CSP-variables.

As a result, in a systematic search for patterns of increasing size, JExocets would appear very late.
As the complexity of a systematic search for Subsets (and a fortiori for general patterns) increases very fast with their size (much faster than for whips or braids), it is probably out of reach of ordinary computers and JExocets can only be found if patterns with special properties (instead of all patterns) are looked for.


Finally, I wonder:
1) what's the easiest known puzzle (say, with smallest SER) having a JExocet? Did anyone already investigate this?
2) has anyone tried to express JExocet in Allan's formalism? What's the base set? Can any fixed "rank" be assigned to it?
3) how would it appear in logel's approach?

[champagne]

Can a JExocet be seen as a Subset or a gSubset? - classification of JExocet...


2) Classification of the JExocet

The above analysis isn't completely negative and that's why I posted it. Even though I couldn't prove that a JExocet is a Subset or a gSubset, the analysis shows that this pattern involves (at least) 13 or 16 CSP-variables.

As a result, in a systematic search for patterns of increasing size, JExocets would appear very late.
As the complexity of a systematic search for Subsets (and a fortiori for general patterns) increases very fast with their size (much faster than for whips or braids), it is probably out of reach of ordinary computers and JExocets can only be found if patterns with special properties (instead of all patterns) are looked for.


Finally, I wonder:
1) what's the easiest known puzzle (say, with smallest SER) having a JExocet? Did anyone already investigate this?
2) has anyone tried to express JExocet in Allan's formalism? What's the base set? Can any fixed "rank" be assigned to it?
3) how would it appear in logel's approach?

I understand here that in the CSP view the JEXOCET is nearly impossible to produce.

However, on a practical view,
a) it is a very common pattern in what appear as the "potential hardest" puzzles using a chain oriented set of rules
b) it is easy to detect for a manual player.
c) Many of these puzzles collapses when that pattern is applied (with the associated "abi loop" pattern)

May-be this proves that the CSP analysis has some limits.

answering your point 1)
I did not explore low ratings, but some examples have been shown with rating around SER 10.0
I'll try to find a pattern game data base stored (out of the recent games) giving EXOCETS to try to cover lower ratings.

answering your point 2)
The first EXOCET is directly coming form Allan Barker analysis of the puzzle "fata morgana".here and here
In that Truths/Links analysis, we find 6 triple points (link mode) attached to 2 sets.
The rank 0 area is not so easy to describe.

[denis_berthier]

I understand here that in the CSP view the JEXOCET is nearly impossible to produce.
[...]May-be this proves that the CSP analysis has some limits.

Totally wrong interpretation of my last post:
- writing JExocet as a resolution rule is very easy (now that there is a clear definition); what I say is, it is (apparently) not feasible to write it as a special kind of resolution rule: chain or Subset; but there can be many other types of resolution rules;
- my analysis of complexity in the above post has nothing to do with a CSP view or any other specific view; it has to do with systematic search along patterns of increasing logical complexity; it would apply to Allan's or logel's approaches as well, if they ever cared about considering ALL the patterns they define;
- once written, the JExocet is easy to find provided that you are not looking for all the logically simpler patterns before it.

However, on a practical view,
a) it is a very common pattern in what appear as the "potential hardest" puzzles using a chain oriented set of rules
b) it is easy to detect for a manual player.
c) Many of these puzzles collapses when that pattern is applied (with the associated "abi loop" pattern)

This is a very biased personal view:
- the "potential hardest" represent less than 1 in 30,000,000 puzzles (unbiased stats); they will never appear in any newspaper; calling this "very common" is at least unexpected; but you have so much submerged yourself in the hardest topic that you've forgotten the "normal" puzzles;
- you invoke a "practical view" but you don't know if there are JExocets in moderate puzzles (say with SER < 9.3), the only ones that any normal player will ever see;
- only a handful of people ever heard of Exocet or JExocet;
- the puzzles "collapse" to less hard ones that remain nevertheless beyond reach of most players (I've shown this for the sk-loop and I have no doubt that it is not different for the JExocet).

The first EXOCET is directly coming form Allan Barker analysis of the puzzle "fata morgana".here and here
In that Truths/Links analysis, we find 6 triple points (link mode) attached to 2 sets.
The rank 0 area is not so easy to describe.

OK, I understand that there's currently no description of JExocet in Allan's view.

[champagne]

The first EXOCET is directly coming form Allan Barker analysis of the puzzle "fata morgana".here and here
In that Truths/Links analysis, we find 6 triple points (link mode) attached to 2 sets.
The rank 0 area is not so easy to describe.

OK, I understand that there's currently no description of JExocet in Allan's view.

Once more the discussion is slipping in a dangerous slope, so I shall not comment your last post except on that point

If you read carefully the references, you will see that in "fata morgana" we have a JEXOCET. In that specific case, as in most of the JEXOCET with 3 digits, the puzzle really collapses applying the "abi loop" pattern;

For sure, Allan did not name that pattern, but found it in several puzzles including Golden Nugget.

[JasonLion-Admin]

A conversation about The Sudoku grey zone has been moved to a new topic.

[denis_berthier]

Classification and frequency of patterns/rules

As an illustration of the importance of the classification of patterns when one tries to estimate their frequency, I'll take the following example, the simplest in the short list of JExocets provided by Champagne here: http://forum.enjoysudoku.com/jexocet-pattern-defintion-t31133-37.html
Here is the puzzle:
5..........46..9...8..4..1..2.9.6.....6.7.3.....3...2..1..2..5...9..37..........8
It is mentioned as having a "classical JExocet form" with:
- base cells r4c5 r6c5
- target cells r2c6 r8c4
- and 3 base digits 158
This JExocet can therefore (potentially) eliminate candidates (n2r2c6 n2r8c4 n3r2c6 n3r8c4 n4r2c6 n4r8c4 n6r2c6 n6r8c4 n7r2c6 n7r8c4 n9r2c6 n9r8c4). In practice, most of them are eliminated during initialisation, as being in direct contradiction with a given. There remain the following 3: n2r2c6 n4r8c4 n7r2c6.

If we don't take the JExocet into account, we get the following resolution path (in W4):

Hidden Text: Show

Code: Select all

*****  SudoRules 16.2 based on CSP-Rules 1.2, config: W-SFin   *****
naked-pairs-in-a-block: b9{r7c7 r8c8}{n4 n6} ==> r9c8 <> 6, r9c8 <> 4, r9c7 <> 6, r9c7 <> 4, r8c9 <> 6, r8c9 <> 4, r7c9 <> 6, r7c9 <> 4
hidden-pairs-in-a-block: b1{r1c2 r3c1}{n6 n9} ==> r3c1 <> 7, r3c1 <> 3, r3c1 <> 2, r1c2 <> 7, r1c2 <> 3
finned-x-wing-in-columns: n3{c2 c8}{r9 r2} ==> r2c9 <> 3, r8c1 <> 6
biv-chain[2]: b1n6{r1c2 r3c1} - r7n6{c1 c7} ==> r1c7 <> 6
biv-chain[2]: r3n3{c3 c9} - b9n3{r7c9 r9c8} ==> r9c3 <> 3
swordfish-in-columns: n3{c2 c5 c8}{r9 r2 r1} ==> r9c1 <> 3, r2c1 <> 3, r1c9 <> 3, r1c3 <> 3
swordfish-in-rows: n6{r3 r6 r7}{c1 c9 c7} ==> r9c1 <> 6, r1c9 <> 6
biv-chain[3]: c8n9{r5 r9} - r7n9{c9 c6} - r3n9{c6 c1} ==> r5c1 <> 9
biv-chain[3]: c5n9{r1 r9} - r9c8{n9 n3} - r1n3{c8 c5} ==> r1c5 <> 8, r1c5 <> 1
biv-chain[3]: b8n6{r8c5 r9c5} - c5n9{r9 r1} - r1c2{n9 n6} ==> r8c2 <> 6
biv-chain[3]: r9c8{n3 n9} - c5n9{r9 r1} - r1n3{c5 c8} ==> r2c8 <> 3
biv-chain[3]: c5n9{r1 r9} - r9n6{c5 c2} - r1c2{n6 n9} ==> r1c6 <> 9
hidden-triplets-in-a-row: r1{n3 n6 n9}{c5 c8 c2} ==> r1c8 <> 8, r1c8 <> 7
whip[2]: c8n7{r4 r2} - b1n7{r2c1 .} ==> r4c3 <> 7
hidden-triplets-in-a-row: r1{n3 n6 n9}{c5 c8 c2} ==> r1c8 <> 4
biv-chain[3]: r9n3{c2 c8} - r1c8{n3 n6} - c2n6{r1 r9} ==> r9c2 <> 7
finned-x-wing-in-columns: n7{c2 c8}{r2 r6} ==> r6c9 <> 7
whip[1]: b6n7{r4c8 .} ==> r4c1 <> 7
biv-chain[3]: r9n3{c2 c8} - r1c8{n3 n6} - c2n6{r1 r9} ==> r9c2 <> 5, r9c2 <> 4
whip[3]: b5n4{r5c6 r6c6} - c2n4{r6 r8} - c8n4{r8 .} ==> r5c9 <> 4
whip[3]: r4n4{c9 c1} - c8n4{r4 r8} - b7n4{r8c1 .} ==> r6c7 <> 4, r6c9 <> 4
whip[3]: c5n9{r9 r1} - r1c2{n9 n6} - r9n6{c2 .} ==> r9c5 <> 5, r9c5 <> 1, r9c6 <> 9
whip[4]: r4n4{c9 c1} - c1n3{r4 r7} - r7n6{c1 c7} - r8c8{n6 .} ==> r5c8 <> 4
whip[1]: b6n4{r4c7 .} ==> r4c1 <> 4
whip[4]: c8n7{r2 r4} - c8n4{r4 r8} - c8n6{r8 r1} - b3n3{r1c8 .} ==> r3c9 <> 7
whip[4]: r2c8{n7 n8} - r5c8{n8 n9} - r9c8{n9 n3} - c2n3{r9 .} ==> r2c2 <> 7
... easy end

Note that the finned-x-wing, swordfish, biv-chains, ... are special cases of whips[<= 3]

Considering the recent post in which I showed that a JExocet involves at least 13 or 16 CSP-variables (depending on whether it has 3 or 4 base digits), we could already conclude that it would never be found in this puzzle if, in the rules hierarchy, JExocet was classified at a place consistent with this high count.


However, one can still want to see what happens if we use it nevertheless. So, let's give it the highest priority and apply it right at the start. Do we get a simpler solution? NO. Here is the new resolution path.
As JExocet is currently not programmed in SudoRules, I use a special rule that allows me to make "simulated eliminations" of any list of candidates. Only the effective ones are displayed (here only 3 in the list of 12 potential ones).

Hidden Text: Show

Code: Select all

*****  SudoRules 16.2 based on CSP-Rules 1.2, config: W-SFin   *****
Simulated elimination of n4r8c4, n7r2c6, n2r2c6
naked-pairs-in-a-block: b9{r7c7 r8c8}{n4 n6} ==> r9c8 <> 6, r9c8 <> 4, r9c7 <> 6, r9c7 <> 4, r8c9 <> 6, r8c9 <> 4, r7c9 <> 6, r7c9 <> 4
hidden-pairs-in-a-block: b1{r1c2 r3c1}{n6 n9} ==> r3c1 <> 7, r3c1 <> 3, r3c1 <> 2, r1c2 <> 7, r1c2 <> 3
x-wing-in-rows: n2{r2 r8}{c1 c9} ==> r9c1 <> 2, r3c9 <> 2, r1c9 <> 2 ;;; <<<<<<<<<<<<<<
finned-x-wing-in-columns: n3{c2 c8}{r9 r2} ==> r2c9 <> 3, r8c1 <> 6
biv-chain[2]: b1n6{r1c2 r3c1} - r7n6{c1 c7} ==> r1c7 <> 6
biv-chain[2]: r3n3{c3 c9} - b9n3{r7c9 r9c8} ==> r9c3 <> 3
swordfish-in-columns: n3{c2 c5 c8}{r9 r2 r1} ==> r9c1 <> 3, r2c1 <> 3, r1c9 <> 3, r1c3 <> 3
swordfish-in-rows: n6{r3 r6 r7}{c1 c9 c7} ==> r9c1 <> 6, r1c9 <> 6
biv-chain[3]: c8n9{r5 r9} - r7n9{c9 c6} - r3n9{c6 c1} ==> r5c1 <> 9
biv-chain[3]: c5n9{r1 r9} - r9c8{n9 n3} - r1n3{c8 c5} ==> r1c5 <> 8, r1c5 <> 1
biv-chain[3]: b8n6{r8c5 r9c5} - c5n9{r9 r1} - r1c2{n9 n6} ==> r8c2 <> 6
biv-chain[3]: r9c8{n3 n9} - c5n9{r9 r1} - r1n3{c5 c8} ==> r2c8 <> 3
biv-chain[3]: c5n9{r1 r9} - r9n6{c5 c2} - r1c2{n6 n9} ==> r1c6 <> 9
hidden-triplets-in-a-row: r1{n3 n6 n9}{c5 c8 c2} ==> r1c8 <> 8, r1c8 <> 7
whip[2]: c8n7{r4 r2} - b1n7{r2c1 .} ==> r4c3 <> 7
hidden-triplets-in-a-row: r1{n3 n6 n9}{c5 c8 c2} ==> r1c8 <> 4
biv-chain[3]: r9n3{c2 c8} - r1c8{n3 n6} - c2n6{r1 r9} ==> r9c2 <> 7
finned-x-wing-in-columns: n7{c2 c8}{r2 r6} ==> r6c9 <> 7
whip[1]: b6n7{r4c8 .} ==> r4c1 <> 7
biv-chain[3]: r9n3{c2 c8} - r1c8{n3 n6} - c2n6{r1 r9} ==> r9c2 <> 5, r9c2 <> 4
whip[3]: b5n4{r5c6 r6c6} - c2n4{r6 r8} - c8n4{r8 .} ==> r5c9 <> 4
whip[3]: r4n4{c9 c1} - c8n4{r4 r8} - b7n4{r8c1 .} ==> r6c7 <> 4, r6c9 <> 4
whip[3]: c5n9{r9 r1} - r1c2{n9 n6} - r9n6{c2 .} ==> r9c5 <> 5, r9c5 <> 1
naked-triplets-in-a-row: r9{c2 c5 c8}{n3 n6 n9} ==> r9c6 <> 9  ;;; <<<<<<<<<<<<<<
whip[4]: r4n4{c9 c1} - c1n3{r4 r7} - r7n6{c1 c7} - r8c8{n6 .} ==> r5c8 <> 4
whip[1]: b6n4{r4c7 .} ==> r4c1 <> 4
whip[4]: c8n7{r2 r4} - c8n4{r4 r8} - c8n6{r8 r1} - b3n3{r1c8 .} ==> r3c9 <> 7
whip[4]: r2c8{n7 n8} - r5c8{n8 n9} - r9c8{n9 n3} - c2n3{r9 .} ==> r2c2 <> 7
... easy end

As you can see, there's almost no difference. Not only is the W rating unchanged, but the resolution path itself is almost unchanged (differences are shown by the ";;; <<<<<<<<<<<<<<" sign).


What can we conclude? When one tries to estimate the frequency of a pattern or a resolution rule, at least three conditions of the estimate should be clearly stated:
- on which collection of puzzles it is based
- how the pattern/rule is classified in some hierarchy of patterns/rules (i.e. which rules are applied before it)
- whether what is being talked about is the pattern or the rule, i.e. whether the impact of the mere presence of the pattern on the rating of a puzzle is taken into account (e.g. is the puzzle still considered as "having the pattern" if applying the associated rule has no impact or no significant impact?)

You think all this is obvious? I fully agree. Why then is it (almost) never applied?