The New Sudoku Players' Forum

[ronk]

What players want is a set of independent rules that can be applied individually.

Did you conduct a survey to arrive at this conclusion?

Did Denis solicit feedback from (on-line communities of) Sudoku players and programmers before publishing his book?

[denis_berthier]

Has anybody anything interesting to say about my results or should we always speak only of the secondary matters?

Champagne
For the rules, I meant do you have any NEW rule.

For the lengths, if you knew they don't correspond to mine, why didn't you state it in your post??? On the contrary, by making such a comparison, you suggested they were the same, even writing that "the length can be significantly reduced".
As I understand it now, the length can be reduced only by changing their definition.

Can you try your algorithm on my Sudogen0 collection (see my web pages) and state your results - with the definition you use for the lenghts of your chains?


Ruud
Has anybody conducted any survey of the players?
In the context of my post, it should have been clear that I was just considering the topics being discussed in the forums in the domain of solving techniques - not in general.


Ronk
Should a book be submitted to the Inquisition before being published?

My book and the rules defined in it have been the topic of so much discussion in the forums that I can't understand you even thought of asking this.

[Ruud]

Has anybody conducted any survey of the players?

Strange question. I noticed your remark about "what players want" and I was really interested in the source of that information. Your return question indicates there is no such source. A simple "No" would have sufficed.

Since you use this claim to support your framework of rules in favor of "only algorithms", I can hardly call it a secundary issue. When you post your ideas here and on other forums, you should be careful what you write. Everything will be scrutinized. By evading my (and other people's) questions, you are undermining your own credibility. Your excellent work in researching and classifying chains, using RN and CN space and the logic foundation for solving techniques has received little criticism, but you need to clarify what you consider "secondary matters" to enable others to give it a place in their universe.

[edit]
Ronk's question was aimed at me rather than you. He kindly brought it to my attention that there may be other sources to obtain this information. Your response to him was inappropriate.

Ruud

[champagne]

For the rules, I meant do you have any NEW rule.

I have been clear. All rules I am using are described in my summary (if you include my last post /edit in the “full tagging thread”).

After one of your questions made before, I intended to end my post by a kind of “summary of rules applied”. I gave up when I saw that it was in fact a rewriting of the text.

For the lengths, if you knew they don't correspond to mine, why didn't you state it in your post??? On the contrary, by making such a comparison, you suggested they were the same, even writing that "the length can be significantly reduced".
As I understand it now, the length can be reduced only by changing their definition.

Denis, did you read my post? Here is the place where comparison is made in detail

Reduced to tags, an alternate chain should be something as :

A/a_A/B_b/B_b/C or A_a_A/B_b_B_b/C

here
- two "weak links" have been identified : A/B and b/C
- two "strong links" have been use in place of two "weak links".

Our "canonized form" for such a chain will be : [] AB bC
In such a chain, both end are defining a 'distant weak link'. [AC] is a weak link.

One remark to answer a question raised by Ronk.
. in Canonized form, weak links and strong links are fully balanced.
. the list of weak link is necessary and sufficient to work out possible chains.
this is done sorting weak links in the appropriate way.(a kind of graph oriented structure)


We could also have the "canonic form" [] a _ AB bC _ c
In fact, we have full equivalence for action with the following logic rule :
=>If we have a "weak link" [AB] then we have a logic OR {ab}. (a, b or both)
I am using the combination
. []AB bC and
. [AC] implies {ac}.

I don’t know how you count the length for the chain A_a_A/B_b_B_b/C.

What is for sure, handling the canonical form in tagging, it is a two terms chain. AB bC

As I told in my post, finding chains in full tagging is reduced to classifying some tens of relevant weak links. That’s why it is so easy to achieve it.

Can you try your algorithm on my Sudogen0 collection (see my web pages) and state your results - with the definition you use for the lenghts of your chains?

To save time, can you give me the link to that list.

[denis_berthier]

Hinged nrczt-chains
As I already mentioned, finding interesting extensions of nrczt chains is difficult. I've recently tried of few ones, but as I'm currently lacking time for Sudoku, I couldn't conduct the systematic experiments I did up to now.
Nevertheless, it may be interesting to notice the following.
I tried to replace all the links that were defined as mere nrc-links (both even links in the chains AND links between additional candidates and their justifying candidate AND links with the target) with hinges (direct nrc-links being still allowed, of course), thus getting hinged nrczt-chains.
Result: I could find no new puzzle that this allowed to solve.
I have no explanation. Although it is obvious that some types of hinges can merely be fused into the nrczt-chain (making it a longer ordinary one), this is not obvious at all in the general case. I don't think this could be proven in full generality.
Any counter example would therefore be greatly appreciated.

[denis_berthier]

nrct- AND nrczt- LASSOS

Once we have built a partial nrct- or nrczt- chain, it is normally ended on the right when its last right-linking candidate can be nrc-linked to a target. But there appears to be two other ways of getting a contradiction on the target. Notice that the following remarks are not useful for nrc- or nrcz- chains, due to the no-loop theorems that can be proven for them (as for xy-chains).

The first case is when there is already somewhere in the partial chain a left-linking candidate C that might be taken as a right-linking candidate of a later part of the chain if we had not excluded loops. In this case, the target of the partial chain can be eliminated (for the same reason as usual: this situation leads to a contradiction). Notice that, when this happens, the target can be eliminated but nothing can be said directly about C; this is because the part of the chain before C cannot be excised, due to the t-candidates it may be used to justify in further cells. We call this case an rl-lasso ("rl" because a right-linking candidate is equal to a previous left-linking candidate). (Notice that there is no full chain in this case and that a target of an rl-lasso does not have to be linked to the last candidate.)

The second case is when there is already somewhere in the chain a right-linking candidate C that might be taken as a left-linking candidate of a later part of the chain if we had not excluded loops. As in the previous case, the target can be eliminated (for the same reason and with the same other remarks applying). We call this case an lr-lasso ("lr" because a left-linking candidate is equal to a previous right-linking candidate). (Again, there is no full chain in this case and that a target of an lr-lasso does not have to be linked to the last candidate.)

Generally, these lassos lead to slightly shorter partial nrc(z)t-chains and faster solutions, and they are interesting for this reason, but they do not lead to eliminations that could not have been obtained without them, as shown by the following classification results.
More precisely, when applicable, they generally shorten the chains by only 1 cell and very rarely by 2 cells. (This has been tested on the full Sudogen0 collection of 10,000 puzzles).
We can therefore consider using them as a minor improvement over nrc(z)(t) chains alone.

Updated classification results
The following table gives the number of puzzles solved at each level, in the L, M and N classifications (elaborating on the results already given in a previous post):
Level n in the N classification allows lassos of length n.

level.......4..........5............6...........7
L.........8271.....8959......9326.......9472
M........9638.....9893......9964.......9984
N........9658.....9913......9975.......9991

(Notice that the 9 remaining puzzles are also solved by SudoRules, but with longer chains).

[denis_berthier]

The (currently) longest nrczt-chain
The classification results in my previous post show that long chains are very rarely useful.
But I've also been looking for cases for which they would be.
Until now, the longest nrczt-chain I had found in the Sudogen0 collection had length 13 (in puzzle Sudogen0-707, a very hard one). In Ruud's top 10000, I had found one of length 17. As SudoRules had chains programmed for lengths <= 20 (and, on any puzzle, it can check that longer chains would be useless), I didn't care for increasing this maximal length.

Nevertheless (in the little time I currently have for Sudoku), I still tried to find longer nrczt-chains (or lassos).
I was naturally led to try the hardest puzzles and I came upon the following monster, which rang the bell in SudoRules indicating that longer chains might be useful for it. I therefore extended the maximal length to 30.

The puzzle is the following - our friend EasterMonster, with the additional value r4c8=7 (randomly chosen among the non nrczt backdoors):
1.......2
.9.4...5.
..6...7..
.5.9.3.7.
....7....
...85..4.
7.....6..
.3...9.8.
..2.....1

The partial resolution path is:
hidden-single-in-a-column ==> r8c9 = 7
nrczt4-chain n2{r7c8 r8c7} - n2{r4c7 r4c1} - n2{r6c2 r6c6} - n2{r2c6 r2c5} ==> r7c5 <> 2
nrczt25-chain {n8 n6}r4c9 - n6{r5c8 r1c8} - {n6 n3}r2c9 - {n3 n9}r6c9 - {n9 n5}r5c9 - {n5 n4}r7c9 - n4{r3c9 r1c7} - n9{r1c7 r1c5} - n9{r3c5 r3c8} - n9{r7c8 r7c3} - n9{r9c1 r9c7} - n5{r9c7 r8c7} - n5{r8c3 r1c3} - n3{r1c3 r1c4} - n3{r3c5 r3c1} - n4{r3c1 r3c2} - n2{r3c2 r2c1} - {n2 n6}r6c1 - {n6 n4}r8c1 - {n4 n1}r8c3 - {n1 n8}r7c2 - n8{r1c2 r2c3} - {n8 n4}r4c3 - n4{r4c5 r9c5} - n8{r9c5 r3c5} ==> r3c9 <> 8

Although the puzzle is not solved, this example is interesting for several reasons:
- an nrczt25-chain appears immediately after an nrczt4, with nothing of moderate complexity in between;
- after this, nothing else applies - and there's no possibility for a longer nrczt-chain to appear;
- the computation time needed for finding this monster chain is much shorter than that for some much shorter chains in other examples. I think the reason is there are not many possibilities for building each step in the chain, because when the elimination allowed by this chain is done, the longest remaining partial chain has length 9. This indicates how hard it may be to define a complexity measure for chains.

Red Ed if you are still interested in such complexity matters, this example may be useful for you.

[Red Ed]

Thanks, Denis. That chain is a monster, isn't it?

I've stopped looking into complexity now; there just didn't seem to be enough interest. But thanks for thinking of me.

[denis_berthier]

Thanks, Denis. That chain is a monster, isn't it?

I haven't been able to find another one so ugly.
Steve's chain (or whatever it is) for EM is also very long, but it has some very nice symmetry properties. That one is long and ugly - and it doesn't lead to any interesting eliminations.

I've stopped looking into complexity now; there just didn't seem to be enough interest. But thanks for thinking of me.

As for me, I'm still interested (abstractly). But I can't find any effective entry point into the problem. And this chain, relatively easy to find, tends to deter me from any further effort.

[denis_berthier]

In the first post of this thread, I mentioned that SudoRules (version 13.0, with 3D chains) could solve 24 out of the first 30 puzzles in Ruud's top 10,000 list. (Remember that this list is ordered with the harderst puzzles first).

Since then, I hadn't tried again. But it can now solve all of them. How can it be so? There are two reasons:
1) As I reported in a previous post, I have introduced lassos, allowing some partial chains to lead to an elimination; as a result, some chains in the resolution path are shortened by one or two cells.
Let me take this opportunity to add a comment about lassos. I think they are a good illustration of my general principle that the target doesn't belong to the chain; in this case, the chain closes on itself for a contradiction, but the target is not the place where this closing happens.
2) There has been a new (beta) release of the inference engine (CLIPS) I use to run my rules (remember that SudoRules is not a program in the usual sense, but just a set of logical rules (more specifically, resolution rules) and that it needs an inference engine to run). This new release is much more efficient in terms both of computation time and memory management.
This is enough to unblock the situation (CLIPS memory overflows or my patience overflow) for the 6 remaining puzzles.

Notice that the longest chain used in Ruud's list is 15 - very long for a human solver, but still short compared to the 25 I found for EasterMonster.
Notice also that the length of the longest chain used to solve a puzzle is only slightly correlated with its place in Ruud's classification.

These new results should allow a better evaluation of the power of nrczt chains for puzzles in the upper range of complexity (at least from a theoretical POV - because, from a practical one, very long chains, be they nrczt or AIC, may be too hard to find for a human solver).
As the EasterMonster example in the "resolution rule" thread shows, there remains a very small % (< 1/10,000) of minimal puzzles that cannot be solved with basic rules plus nrczt chains alone - unless one uses some form of focused T&E as I did for EasterMonster or uncontrolled propagation of constraints through very complex nets (either explicit or implicit in some generalised tagging scheme).

As the full 30 resolution paths would take too much space in this thread, I've put them on my web pages, in the "supplements" section
(direct link: http://www.carva.org/denis.berthier/HLS/Ruud-top30.txt).

[denis_berthier]

An nrczt-lasso of length 28

In a previous post, I described an nrczt-chain of length 25. Here is now an nrczt-lasso of length 28.
It is found in puzzle #3 in the magictour/top1465 collection.
I think this example is interesting for the same reasons as the previous one:
- an nrczt28-lasso appears immediately after an nrczt11-chain and an nrczt6-lasso, with nothing of moderate length in between;
- after this, nothing else (in my set of rules) applies - and there's no possibility for a longer nrczt-chain to appear;
- the computation time needed for finding this monster chain is much shorter than that for some much shorter chains (of length only 6 or 7) in other examples.

In particular, this confirms what was already hinted at in the previous example: complexities of chains are not related to their lengths in a systematic way but they largely depend on the particularities of the puzzles. This makes somewhat hopeless any effort to define an priori complexity measure for chains based on the length.

The partial resolution path is:
(solve-nth-grid-from-text-file (str-cat ?*GridsDir* "Magictour/magictour-top1465.txt") 3)
***** SudoRules version 13 *****
708000300000601000500000000040000026300080000000100090090200004000070500000000000
hidden-single-in-a-block ==> r6c9 = 3
row r2 interaction-with-block b3 ==> r3c9 <> 8, r3c8 <> 8, r3c7 <> 8, r3c9 <> 7, r3c8 <> 7, r3c7 <> 7
block b6 interaction-with-column c7 ==> r9c7 <> 8, r7c7 <> 8, r2c7 <> 8
block b6 interaction-with-row r5 ==> r5c6 <> 5, r5c4 <> 5, r5c3 <> 5, r5c2 <> 5
hidden-pairs-in-a-row {n7 n8}r3{c4 c6} ==> r3c6 <> 9, r3c6 <> 4, r3c6 <> 3, r3c6 <> 2, r3c4 <> 9, r3c4 <> 4, r3c4 <> 3
block b2 interaction-with-column c5 ==> r9c5 <> 3, r7c5 <> 3, r4c5 <> 3
nrczt4-chain n9{r2c1 r4c1} - {n9 n5}r4c5 - n5{r2c5 r2c8} - n8{r2c8 r2c9} ==> r2c9 <> 9
nrczt4-chain n6{r1c2 r3c3} - n6{r5c3 r5c6} - n6{r6c5 r7c5} - n6{r7c7 r9c7} ==> r9c2 <> 6
nrczt6-lr-lasso n7{r9c3 r9c2} - n7{r9c9 r2c9} - n8{r2c9 r2c8} - n5{r2c8 r2c5} - {n5 n9}r4c5 - n9{r4c1 r4c3} ==> r5c3 <> 7
nrczt11-chain n6{r9c1 r6c1} - n6{r6c5 r9c5} - n1{r9c5 r7c5} - {n1 n8}r7c1 - n8{r4c1 r6c2} - n5{r6c2 r9c2} - n7{r9c2 r5c2} - n7{r9c2 r9c3} - n7{r9c9 r2c9} - n7{r2c8 r7c8} - {n7 n6}r7c7 ==> r7c3 <> 6
nrczt28-lr-lasso n5{r6c3 r4c3} - n5{r7c3 r7c5} - n1{r7c5 r9c5} - n6{r9c5 r6c5} - n2{r6c5 r5c6} - n4{r5c6 r5c4} - n4{r6c6 r6c7} - n8{r6c7 r4c7} - n1{r4c7 r4c1} - n9{r4c1 r2c1} - n9{r3c3 r5c3} - n6{r5c3 r5c2} - n6{r1c2 r1c8} - n6{r3c7 r3c3} - n4{r3c3 r2c3} - n4{r2c8 r3c8} - n4{r3c5 r1c5} - {n4 n9}r1c6 - {n9 n5}r1c4n5{r9c4 r9c2} - n7{r9c2 r6c2} - n8{r6c2 r8c2} - {n8 n6}r7c1 - n6{r7c7 r9c7} - n9{r9c7 r3c7} - n2{r3c7 r2c7} - {n2 n1}r3c9 - {n1 n2}r1c9 ==> r6c6 <> 5
GRID 3 NOT SOLVED. 62 VALUES MISSING.

BTW, does anybody have a pure logic solution for this puzzle (I mean no T&E, no backtracking, no guessing, no Nishio, no tabling, no tagging, ... - or whatever names you call these)?

[champagne]

Hi Denis

Sorry, but none of the 1465 collection resists to my solver. A very small number only exceed level two of the process. None enter level four.

Side question, where do you see any hidden T&E in the tagging process.

Last but not least, any T chain is fully compatible with what is done in my tagging process (in fact with nets of AICs as noticed DXP in another forum).

The main difference between what you are doing and tagging process is not in the taging itself, but in the fact that I use groups, ALS AHS and AC2.

The tagging is just a performing way to extract AICs chains. Have a look on my examples and you'll see.

Excluding tagging of your competition is somehow an attempt to create a pool for poorly performing process

[champagne]

solved in 2.8 seconds in that way

1 r6c9=3
-> row 2 digit 8 in box 3
-> row 2 digit 7 in box 3
-> box 6 digit 5 in row 5
-> box 6 digit 8 in column 7
->LS2 digits 78 cells 46 row 3
-> box 2 digit 3 in column 5

#6r9c2 []2r6c3/2r5c23_2r5c6/6r5c6_6r5c23/6r6c1_6r789c1/6r9c2
[]5r6c3/5r6c2_5r9c2/6r9c2 []6r6c3/6r6c1... []7r6c3/7r56c2_7r9c2/6r9c2
#9r2c9 []5r1c89/5r1c456_5r2c5/5r4c5_9r4c5/9r4c1_9r2c1/9r2c9
[]5r2c9/9r2c9 []5r2c8/8r2c8_8r2c9/9r2c9

end of common path

[]9r2c7/5r2c89|*r2c789(AHS)_5r2c5/5r4c5_9r4c5/9r4c1_9r2c1/9r2c7
[]7r5c236/9r5c36|*r5c236_9r5c4/9r4c5_5r4c5/5r2c5_5r2c89/7r2c89|*r2c89_7r2c7/7r456c7_7r5c89/7r5c236

->LS2 digits 57 cells 69 column 2
2 r8c2=8 C
-> column 2 digit 3 in box 1
-> UniqueRectangle 3 P1=r2c2 23 P2=r2c5 23459 P3=r3c2 1236 P4=r3c5 349

#6r7c3 []6r7c6/6r7c3 []6r7c5/6r7c3 []6r89c6/6r5c6_6r5c23/6r6c1_6r789c1/6r7c3
[]6r9c5/1r9c5_1r7c5/1r7c1_6r7c1/6r7c3

#1r9c7 []6r7c7/6r7c1_1r7c1/1r7c5_1r9c5/1r9c7 []6r9c7/1r9c7
[]6r7c8/6r7c1_1r7c1... []6r89c8/6r79c7_6r3c7/9r3c7_9r9c7/1r9c7

and the last set of AICs

[]7r456c7/7r2c7_7r2c89/5r2c89|*r2c89_5r2c5/5r7c5_5r7c36/7r7c38|*r7c368_7r7c7/7r456c7
[]5r4c5/5r7c5_5r7c36/7r7c38|*r7c368_7r7c7/7r2c7_7r2c89/5r2c89|*r2c89_5r2c5/5r4c5
[]1r7c378/5r7c36|*r7c3678_5r7c5/5r2c5_5r2c89/7r2c89|*r2c89_7r2c7/2r2c7_2r39c7/6r39c7|*r39c7_6r7c7/6r7c1_1r7c1/1r7c378
[]6r7c68/7r7c38|*r7c368_7r7c7/7r2c7_7r2c89J/5r2c89|*r2c89_5r2c5/5r7c5_5r7c36/6r7c68|*r7c368
[]4r2c78/2r2c79|*r2c789_2r2c1235/2r2c7_2r39c7/6r39c7|*r39c7_6r7c7/6r7c15_6r7c678/5r7c36|*r7c3678_5r7c5/5r2c5_5r2c89/4r2c78|*r2c789
[]5r1c5/5r7c5_5r7c36/7r7c38|*r7c368_7r7c7/7r2c7_7r2c89/5r2c89|*r2c89_5r1c89/5r1c5
[]2r2c9/7r2c89|*r2c89_7r2c7/7r7c7_7r7c38/5r7c36|*r7c368_5r7c5/5r2c5_5r2c89/2r2c9|*r2c89
[]5r6c5/5r7c5_5r7c36/7r7c38|*r7c368_7r7c7/7r2c7_7r2c89J/5r2c89|*r2c89_5r2c5/5r6c5
[]7r9c7/7r2c7_7r2c89/5r2c89|*r2c89_5r2c5/5r7c5_5r7c36/7r7c38|*r7c368_7r7c7/7r9c7
[]5r9c5/5r7c5_5r7c36/7r7c38|*r7c368_7r7c7/7r2c7_7r2c89J/5r2c89|*r2c89_5r2c5/5r9c5

enough to have immediatly 3 r2c1=9 4 r4c5=9 5 r5c3=9 6 r5c4=

game over.

[denis_berthier]

Hi Denis
Sorry, but none of the 1465 collection resists to my solver. A very small number only exceed level two of the process. None enter level four.

Hi Champagne,
No reason of being sorry for this. Congratulations to your computer.

Side question, where do you see any hidden T&E in the tagging process.

I didn't read the details of your tagging algorithm because I'm allergic to tagging (even though I published a tagging algorithm for nrc(z)(t) chains - for those who like it). I don't consider Sudoku playing as overloading the grid with thousands of tags. My interest in Sudoku is strictly focused on the pure logic approach and on clearly formulated resolution rules that can be used independently without tying the player to any specific algorithm. That's all. Nothing personal in this. If you like tagging, that's fine.
As for T&E, it is so easy to simulate T&E with tags that the burden of proving that your algorithm doesn't is on you. When a tagging algorithm is a clear implementation of a well defined resolution rule, things are clear, but your mixture is not.

Last but not least, any T chain is fully compatible with what is done in my tagging process (in fact with nets of AICs as noticed DXP in another forum).

That's where you're missing the point:
1) I'm not looking for the most general algorithm including nets, subsets and whatever one may imagine, as in your second post - resulting in something totally unmanageable by a human solver. I'm looking for the least general family of human oriented patterns that allow solving puzzles a human can solve. I don't count EasterMonster among these. I'm therefore not very impressed by your claims that your program can solve everything.
2) My chains correspond to a whole family of patterns of increasing complexities (xy, hxy, xyt, hxyt, xyz, hxyz, xyzt, hxyzt,nrc, nrct, nrcz, nrczt). In my solver, simple patterns in this family are searched for before more complex ones; this entails some redundancy and some inefficiency; but I prefer it that way; anyway, I'm not in this forum to speak of my solver but of my resolution rules.
3) My chains are not nets, they really are chains: when building them, one follows a single path; your computer may not see the difference but any human player will; if, as you suggest, your program doesn't make any difference between a chain and a net, that's another reason for me for not being interested. Again, nothing personal.

The main difference between what you are doing and tagging process is not in the taging itself,

It seems you don't understand the difference between a logic rule and an algorithm.

but in the fact that I use groups, ALS AHS and AC2.

You're forgetting nets. Fine for you if you like all this stuff. My results show that this is very rarely needed. Anyway, as my chains can be used independently of any algorithm, each player can freely use them when he wants and mix them with whatever he likes best.

The tagging is just a performing way to extract AICs chains.

Which brings us back to the question you never answered : have you devised any single NEW rule? If so, can you write it in a few lines? I'm not asking for being flooded with a listing of the previously known rules you have implemented, but for a single NEW rule.

Excluding tagging of your competition is somehow an attempt to create a pool for poorly performing process

Who spoke of a competition?
Who spoke of processes (or programs)?
Who spoke of computer efficiency?

I was just wondering if anybody has clearly defined resolution rules that can deal with the puzzle I mentioned. This question remains unanswered.
I recall the puzzle:
708000300
000601000
500000000
040000026
300080000
000100090
090200004
000070500
000000000

[Mike Barker]

Here's a solution using only grouped nice loops. Obviously these ALS are not something I would find by hand, of course, finding a 28 element nrczt chain might be challenging as well.

Code: Select all

Hidden Pair in a box: r3c46 => r3c4=78,r3c6=78,r3c7<>78,r3c8<>78,r3c9<>78
Locked Column Line/Box: r23c5 => r479c5<>3
Locked Column Line/Box: r46c7 => r279c7<>8
Locked Row Line/Box: r5c89 => r5c2346<>5
2-element Grouped Nice Loop: ALS:r4c456 -9- ALS:r5c4789 ~7~  => r5c6<>7
+--------------------------+-----------------------+----------------------+
|     7      126        8  |   459   2459    2459  |     3   1456   1259  |
|   249       23     2349  |     6  23459       1  |  2479   4578  25789  |
|     5     1236   123469  |    78   2349      78  | 12469    146    129  |
+--------------------------+-----------------------+----------------------+
|   189        4     1579  |  3579*    59*   3579* |   178      2      6  |
|     3     1267    12679  |  479*b     8  2469-7  |  147*b 1457*b  157*b |
|   268    25678     2567  |     1   2456   24567  |   478      9      3  |
+--------------------------+-----------------------+----------------------+
|   168        9    13567  |     2    156    3568  |   167  13678      4  |
| 12468    12368    12346  |  3489      7   34689  |     5   1368   1289  |
| 12468  1235678  1234567  | 34589  14569  345689  | 12679  13678  12789  |
+--------------------------+-----------------------+----------------------+
Overlap 3-element Grouped Nice Loop: ALS:r2c1237 -7- r456c7 =7= r5c89 -7- ALS:r1235c2 ~3~  => r3c3<>3
+--------------------------+-----------------------+----------------------+
|     7     126*d       8  |   459   2459    2459  |     3   1456   1259  |
|   249*     23*d    2349* |     6  23459       1  |  2479*  4578  25789  |
|     5    1236*d 12469-3  |    78   2349      78  | 12469    146    129  |
+--------------------------+-----------------------+----------------------+
|   189        4     1579  |  3579     59    3579  |  178*b     2      6  |
|     3    1267*d   12679  |   479      8    2469  |  147*b 1457*c  157*c |
|   268    25678     2567  |     1   2456   24567  |  478*b     9      3  |
+--------------------------+-----------------------+----------------------+
|   168        9    13567  |     2    156    3568  |   167  13678      4  |
| 12468    12368    12346  |  3489      7   34689  |     5   1368   1289  |
| 12468  1235678  1234567  | 34589  14569  345689  | 12679  13678  12789  |
+--------------------------+-----------------------+----------------------+
UR+3C/2SL (r23c25=23)  => r3c5<>2
4-element Grouped Nice Loop: r5c89 =7= r456c7 -7- ALS:r2c12357 -5- r4c5 -9- r4c13 =9= r5c3 ~7~ r5c89 => r5c3<>7
+--------------------------+------------------------+----------------------+
|     7      126        8  |   459    2459    2459  |     3   1456   1259  |
|  249*c     23*c   2349*c |     6  23459*c      1  | 2479*c  4578  25789  |
|     5     1236    12469  |    78     349      78  | 12469    146    129  |
+--------------------------+------------------------+----------------------+
|  189*d       4    1579*d |  3579      59*   3579  |  178*b     2      6  |
|     3     1267   1269-7* |   479       8    2469  |  147*b  1457*   157* |
|   268    25678     2567  |     1    2456   24567  |  478*b     9      3  |
+--------------------------+------------------------+----------------------+
|   168        9    13567  |     2     156    3568  |   167  13678      4  |
| 12468    12368    12346  |  3489       7   34689  |     5   1368   1289  |
| 12468  1235678  1234567  | 34589   14569  345689  | 12679  13678  12789  |
+--------------------------+------------------------+----------------------+
Overlap 4-element Grouped Nice Loop: ALS:r2c1235 -5- r4c5 -9- r4c13 =9= r5c3 -9- r3c3 =9= r2c13 ~9~  => r2c79<>9
+--------------------------+-----------------------+-----------------------+
|     7      126        8  |   459   2459    2459  |     3   1456    1259  |
|  249*c      23*   2349*c |     6  23459*      1  | 247-9   4578  2578-9  |
|     5     1236    12469* |    78    349      78  | 12469    146     129  |
+--------------------------+-----------------------+-----------------------+
|  189*b       4    1579*b |  3579     59*   3579  |   178      2       6  |
|     3     1267     1269* |   479      8    2469  |   147   1457     157  |
|   268    25678     2567  |     1   2456   24567  |   478      9       3  |
+--------------------------+-----------------------+-----------------------+
|   168        9    13567  |     2    156    3568  |   167  13678       4  |
| 12468    12368    12346  |  3489      7   34689  |     5   1368    1289  |
| 12468  1235678  1234567  | 34589  14569  345689  | 12679  13678   12789  |
+--------------------------+-----------------------+-----------------------+
4-element Grouped Nice Loop ( -7- r5c45789 -5- r2c5 -2349- r2c1237 -7- r456c7 =7= r5c89 -7-) => r5c2<>7
+--------------------------+-----------------------+--------------------------+
|      7     126        8  |   459   2459    2459  |      3     1456    1259  |
|    249b     23b    2349b |     6  23459*      1  |    247b    4578    2578  |
|      5    1236    12469  |    78    349      78  |  12469      146     129  |
+--------------------------+-----------------------+--------------------------+
|    189       4     1579  |  3579     59e   3579  |    178b       2       6  |
|      3   126-7     1269  |   479e     8    2469  |    147be   1457be   157be|
|    268   25678     2567  |     1   2456   24567  |    478b       9       3  |
+--------------------------+-----------------------+--------------------------+
|    168       9    13567  |     2    156    3568  |    167    13678       4  |
|  12468   12368    12346  |  3489      7   34689  |      5     1368    1289  |
|  12468  123578  1234567  | 34589  14569  345689  |  12679    13678   12789  |
+--------------------------+-----------------------+--------------------------+
Hidden Column Pair: r69c2 => r6c2=57,r9c2=57
Locked Column Line/Box: r789c3 => r2c3<>3
B=2 cell ALS xy-rule: r2c1235 -5- r7c15 -6- r4567c7 => r2c7<>4
B=1 cell ALS xy-mer: r7c157 -7- r2c7 -2- r2c1235 -5- r7c157|r9c5 => r7c38<>1,r7c368<>6,r1469c5<>5,r2c8<>4
+----------------------+------------------------+-----------------------+
|    7   126        8  |   459   249-5    2459  |     3    1456   1259  |
|  249c   23c     249c |     6   23459c      1  |    27*  578-4   2578  |
|    5  1236    12469  |    78     349      78  | 12469     146    129  |
+----------------------+------------------------+-----------------------+
|  189     4     1579  |  3579     9-5    3579  |   178       2      6  |
|    3   126     1269  |   479       8    2469  |   147    1457    157  |
|  268    57     2567  |     1   246-5   24567  |   478       9      3  |
+----------------------+------------------------+-----------------------+
|   16b    9   357-16  |     2     156b  358-6  |   167b 378-16      4  |
| 1246     8    12346  |   349       7    3469  |     5     136    129  |
| 1246    57  1234567  | 34589  1469-5  345689  | 12679   13678  12789  |
+----------------------+------------------------+-----------------------+
Locked Column Line/Box: r23c3 => r89c3<>4
Locked Row Line/Box: r1c46 => r1c9<>9
Hidden Row Pair: r5c26 => r5c2=26,r5c6=26
Locked Row Line/Box: r4c13 => r4c7<>1
Locked Column Box/Box: r4789c1|r489c3 => r3c3<>1
4-element Grouped Advanced Colouring: r3c5 =3= r3c2 =1= r1c2 =6= r1c8 =4= r1c456 ~4~ r3c5 => r3c5<>4
B=2 cell ALS xy-mer: r2c7 -2- r2c35 -5- r7c157 -7- r2c7|r89c1 => r4569c7<>7
Locked Row Line/Box Pair: r1c56 => r1c2<>2
Naked Block Pair: r89c1 => r89c3<>2
Row Finned X-Wing: r6c26|r7c36 => r9c2<>5