The New Sudoku Players' Forum

[ronk]

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.....8.6.19.....7...5.2.4..93.....4....2.7....6.....89..3.7.8...5.....26.1.3.....

23j4   2l47a 2Ê4ê     |1457A9b 1345  8      |1r259B 6     1235   
1      9     2T4Y6m8c |456     3456  3456   |2Ë5ë   7     23X58C
3J6M8  7A8a  5        |167a9   2     1369é  |4      13E9b 138c   
----------------------------------------------------------------
9      3     12u7Å    |1568d   1568D 156    |1257   4     12v57i
45k8   4a8A  1K48     |2       9     7      |6      13e5Â 13E5   
2w5K7H 6     127      |145     13f45 13F45  |12u57Å 8     9     
----------------------------------------------------------------
246Ä   2L4l  3        |4569    7     2o4569 |8      1g59è 1G4p5 
47h8   5     47H89n   |148D9   148   149    |3      2     6     
2468   1     246M89N  |3       4568Æ 2O4569 |57i9b  5Ì9ì  4P57I 

[] 9r9c3.N - 9r9c78 = AC: r9c13569(9r9c36 - 7r9c9) = 7r4c9 - 2r4c9 = 2r46c7
- 2r2c7 = AC: r2c39(2r2c39 - 6r2c3) = 6r9c3 - 9r9c3.N

[] AC: r7c4689(2r7c6.o - 6r7c46) = 6r7c1 - 6r9c3 = AC: r2c39(6r2c3 - 2r2c39) = 2r2c7 - 2r46c7 = AC: r49c9(2r4c9 - 4r9c9) = 4r7c9 - 4r7c2 = 2r7c2 - 2r7c6.o

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[] AHS:(9r9c3.N - 2468r9c3 = 2468r9c1569 - 7r9c9) = (7-2)r4c9 = 2r46c7
   - AHS:(2r2c7 = 28r2c39 - 6r2c3) = 6r9c3 - 9r9c3.N

[] AHS:(2r7c6.o - 459r7c6 = 459r7c489 - 6r7c46) = 6r7c1 - 6r9c3 = AHS:(6r2c3 - 8r2c3 = 8r2c9 - 2r2c39)
   = 2r2c7 - 2r46c7 = AHS:(2r4c9 - 7r4c9 - r9c9 - 4r9c9) = 4r7c9 - (4=2)r7c2 - 2r7c6.o

My translations to NL notation (using ALSs instead of AHSs):

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r9c3 -9- (ALS:r9c789 =9|4= r9c9) =7= r4c9 =2= r46c7 -2- (ALS:r2c7 =2|6= r2c4567) -6- r2c3 =6= r9c3, implies r9c3<>9

r7c6 -2- (ALS:r7c12 =2|6= r7c1) -6- r9c3 =6= r2c3 -6- (ALS:r2c4567 =6|2=  r2c7) -2- r46c7 =2= r4c9 =7= r9c9 =4= r7c9 -4- r7c2 -2- r7c6, implies r7c6<>2

[Mike Barker]

Ron, I know for every ALS there is a corresponding AHS and vice-versa, but is it true that for every AHS chain there is a corresponding ALS chain?

Champagne, AHS are relatively unexplored which is why I'm so interested in your solutions. The best link I know of is Myth's original one: http://forum.enjoysudoku.com/viewtopic.php?p=34826 . Preliminary thoughts were that either AHS or ALS could be used to solve a puzzle and there are many examples of that, maybe with one simpler than the other. With some eliminations Steve and you have used I'm not so certain.

[ronk]

I know for every ALS there is a corresponding AHS and vice-versa, but is it true that for every AHS chain there is a corresponding ALS chain?

I'd be quite surprised if it wasn't true. I've converted quite a few of AHS chains to ALS chains, and don't recall a failed attempt to do so.

[champagne]

Mike Barker wrote:
I know for every ALS there is a corresponding AHS and vice-versa, but is it true that for every AHS chain there is a corresponding ALS chain?

Ronk wrote
I'd be quite surprised if it wasn't true. I've converted quite a few of AHS chains to ALS chains, and don't recall a failed attempt to do so.

I have a slightly different opinion.

First of all, I checked the link given by Mike, AC and AHS is strictly the same object.

To come to the point, We have here examples of AICs crossing an AC/AHS but not crossing directly an ALS (entry in one ALS, exit in another one). It would be somehow artificial to search for a translation in a chain crossing ALS.

[ronk]

We have here examples of AICs crossing an AC/AHS but not crossing directly an ALS (entry in one ALS, exit in another one). It would be somehow artificial to search for a translation in a chain crossing ALS.

"Artificial?" I have no idea what you mean by that. Take the second of your three most recent examples ...

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...8...7.3...1.8..1.5..96......74...2.6...29......83....4..1.1...9.4...6...3..

1      23x4À6l 5      |4m6Ê79a 24679A   27     |8     236    23r67e
4D9d   7       249D   |3       246È8c   2c8C   |4p5b6 1      25B6   
8      2346L   23Z4   |1       2467e    5      |4P67E 23x6   9     
-------------------------------------------------------------------
6      35Ã8    1s38   |589A    1589a    13n8   |2     7      4     
45Å7Í  3458    13478  |2       14á57ì8Î 6      |1o59f 359F   3R5r   
2      9       134á7ì |4M57    1457     13N7   |1O56g 3y56G  8     
-------------------------------------------------------------------
3      258     2789d  |5678    2w56Ê78  4      |5679F 2568q9 1     
57     1       2h78i  |56é78I  3        9      |567   4      2H56ë7
4k579D 24K58   6      |578     1j2578   1J2u78 |3     258Q9d 257   

[]2r2c56 - 2r1c6 = 7r1c6 - 7r3c5 = 7r3c7 - 4r3c7 = AC: r2c79(4r2c7 - 6r2c79) = AC: r2c56(6r2c5 - 2r2c56)

The ALS equivalent in NL notation is ...

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r2c56 -2- r1c6 -7- r3c5 =7= r3c7 -7- ALS:[r1c9,r13c8] -6- ALS:r2c1379 -2- r2c56

... implying r2c56<>2. In what manner is this "artificial?"

[champagne]

Ronk wrote

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In what manner is this "artificial?"

Ok I studied your first "translation", I am convinced. As there is a little more than pure translation, I'll make a short comment.

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23j4   2l47a 2Ê4ê     |1457A9b 1345  8      |1r259B 6     1235   
1      9     2T4Y6m8c |456     3456  3456   |2Ë5ë   7     23X58C
3J6M8  7A8a  5        |167a9   2     1369é  |4      13E9b 138c   
----------------------------------------------------------------
9      3     12u7Å    |1568d   1568D 156    |1257   4     12v57i
45k8   4a8A  1K48     |2       9     7      |6      13e5Â 13E5   
2w5K7H 6     127      |145     13f45 13F45  |12u57Å 8     9     
----------------------------------------------------------------
246Ä   2L4l  3        |4569    7     2o4569 |8      1g59è 1G4p5 
47h8   5     47H89n   |148D9   148   149    |3      2     6     
2468   1     246M89N  |3       4568Æ 2O4569 |57i9b  5Ì9ì  4P57I

First AIC

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[] 9r9c3.N - 9r9c78 = AC: r9c13569(9r9c36 - 7r9c9) = 7r4c9 - 2r4c9 = 2r46c7 - 2r2c7 = AC: r2c39(2r2c39 - 6r2c3) = 6r9c3 - 9r9c3.N

First AC;

9r9c3 - 9r9c78 = AC: r9c13569(9r9c36 - 7r9c9) = 7r4c9

Your translation
r9c3 -9- (ALS:r9c789 =9|4= r9c9) =7= r4c9
(shown in the same form)
9r9c3 - ALS:r9c789(9r9c78|4r9c9) - 7r9c9 = 7r4c5

I would suggest, keeping the same couple ALS = AC

9r9c3 - ALS:r9c78(9r9c78|7r9c7) - 7r9c9 = 7r4c5


Second AC, source of my sceptisim;

- 2r2c7 = AC: r2c39(2r2c39 - 6r2c3) = 6r9c3

-2- (ALS:r2c7 =2|6= r2c4567) -6- r2c3 =6= r9c3

shown in the same form

- ALS:r2c4567(2r2c7 | 6r2c456) - 6r2c3 = 6r9c3

No reason to object on your statement.

[Mike Barker]

As I see it there are four types of AHS links. Let "H" be the set of cells containing the almost hidden set {ab...}, "W" be a cell in "H" which contains {a}, but not {b...}, "L" be the remaining cells making up the unit containing "H" ("L" is and ALS containing {1-9}-{ab...} = {pq...}), and "S" be a cell in "H" which contains "p". The we can define four types of AHS links as:

1) a - AHS:{aW = ab...(H-W) - q(H-W)} = q
2) p = AHS:{pS - ab...S = ab...(H-S) - q(H-S)} = q
3) a - AHS:{aW = ab...(H-W)} - b
4) p = AHS:{pS - ab...S = abc(H-S)} - b

As I see it the equivalent ALS links would look like:

1) a - ALS:{aW|pq...L = q(L+W)} - q(H-W) = q
2) p = pS - ALS:{pL = qL} - q(H-S) = q
3) a - ALS:{aW|pq...L = pq...(L+W)} - pq...(H-W) = ab...(H-W) - b
4) p = pS - ALS:{pL = q...L} - pq...(H-W) = ab...(H-S) - b

Maybe someone can come up with a better alternative for 3 and 4. As it stands I would argue that what I have is just a long way of writing the AHS and not truely and ALS alternative. Based on a quick survey it seems all of Champagne's AHS that I've looked at are of the first two types for which there are ALS alternatives. Its not obvious that types 3 and 4 shouldn't also exist in which case I would conclude that AHS and ALS are not equivalent.

[Edit: in fact I think types 3 and 4 reduce to a locked set or a grouped strong link which is why we don't have examples, thus it would appear that all non-degenerative AHS links have ALS alternatives.]

Turning the problem around there is really only one type of ALS link:
p - ALS:{pL = qL} - q

Its not really obvious how to always convert this into an AHS.

[champagne]

Hi Mike,

It seems that using Ronk translation we can answer your problem in an easier way, at least for my use of AC/AHS.

All ACs uses in AICs are of the form

- A = AC:xxx (a - b) = B - (tagging notation)


We have a binary split very simple

1) 'A' and 'B' belong to the same unit. Then they are in the ALS:yyy complementary to AC:xxx in that unit.
Translation - ALS:yyy(A|B) -

2) 'A' and 'B' are not in the same unit. This is the situation of the second example seen above.
Anyway, 'B' exists also in the same unit as 'A', so you'll have the translation :

- ALS:yyy(A|B) - b = B -

This is exactly what has done Ronk.

Code: Select all

23j4   2l47a 2Ê4ê     |1457A9b 1345  8      |1r259B 6     1235   
1      9     2T4Y6m8c |456     3456  3456   |2Ë5ë   7     23X58C
3J6M8  7A8a  5        |167a9   2     1369é  |4      13E9b 138c   
----------------------------------------------------------------
9      3     12u7Å    |1568d   1568D 156    |1257   4     12v57i
45k8   4a8A  1K48     |2       9     7      |6      13e5Â 13E5   
2w5K7H 6     127      |145     13f45 13F45  |12u57Å 8     9     
----------------------------------------------------------------
246Ä   2L4l  3        |4569    7     2o4569 |8      1g59è 1G4p5 
47h8   5     47H89n   |148D9   148   149    |3      2     6     
2468   1     246M89N  |3       4568Æ 2O4569 |57i9b  5Ì9ì  4P57I

- 2r2c7 = AC: r2c39(2r2c39 - 6r2c3) = 6r9c3 -


- ALS:r2c4567(2r2c7 | 6r2c456) - 6r2c3 = 6r9c3

ps: I have posted explainations on complementary tagging.

[StrmCkr]

unsolvable #33,

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  *-----------*
 |.1.|...|.5.|
 |7.8|...|...|
 |..2|6.8|3..|
 |---+---+---|
 |9..|3.5|..7|
 |...|...|...|
 |8..|4.2|..6|
 |---+---+---|
 |..9|7.1|2..|
 |..3|...|4..|
 |.8.|...|.9.|
 *-----------*

 
 *--------------------------------------------------------------------------------------*
 | 346      1        46       | 29       23479    3479     | 6789     5        2489     |
 | 7        34569    8        | 1259     123459   349      | 169      1246     1249     |
 | 45       459      2        | 6        14579    8        | 3        147      149      |
 |----------------------------+----------------------------+----------------------------|
 | 9        246      146      | 3        168      5        | 18       1248     7        |
 | 123456   234567   14567    | 189      16789    679      | 1589     12348    1234589  |
 | 8        357      157      | 4        179      2        | 159      13       6        |
 |----------------------------+----------------------------+----------------------------|
 | 456      456      9        | 7        34568    1        | 2        368      358      |
 | 1256     2567     3        | 2589     25689    69       | 4        1678     158      |
 | 12456    8        14567    | 25       23456    346      | 1567     9        135      |
 *--------------------------------------------------------------------------------------*


[denis_berthier]

I find a great way to compare different techniques is to be able to evaluate multiple solutions to the same puzzle. Here's an nrct-chain solution to U#26. It replaces the grouped and simple nice loops of my previous solution with nrct-chains. I've kept xy-chains and hxy-chains (x-cycles and advanced colouring) as these are part of nrct-chains. I hope others will post alternative solutions to the unsolvables, especially if they emphasize human solvability, some new technique, or some clever solution strategy.

I don't know if the following meets any of these criteria but here is a solution using only nrczt-chains and their specialisations (in addition to basic rules). The maximal length of all the chains involved is 6 (remember that length is the number of cells, i.e. half the number of linking candidates, and that the target is not included in the chain).
This may be considered an illustration of the fact that the z-extension allows shorter chains (it also allows the solution of puzzles that would not be solvable without it).
For chains, it is also an illustration of the general trade between the complexity of their types and their lengths.


The puzzle is UN26

***** SudoRules version 13 *****
026000900
005060400
000907000
900100005
040000280
500003004
000602000
009050700
001000850
hidden-singles ==> r2c2 = 9, r3c7 = 5, r7c2 = 5
column c7 interaction-with-block b6 ==> r6c8 <> 6, r5c9 <> 6, r4c8 <> 6
block b1 interaction-with-column c1 ==> r9c1 <> 7, r7c1 <> 7, r5c1 <> 7
nrc2-chain n1{r3c2 r6c2} - n1{r5c1 r5c9} ==> r3c9 <> 1
nrct4-chain {n8 n1}r2c6 - n1{r1c5 r7c5} - n7{r7c5 r7c3} - n8{r7c3 r7c1} ==> r2c1 <> 8
nrczt4-chain {n1 n8}r2c6 - {n8 n4}r8c6 - n4{r8c4 r1c4} - n5{r1c4 r1c6} ==> r1c6 <> 1
nrczt4-chain {n8 n1}r2c6 - {n1 n4}r8c6 - n4{r8c4 r1c4} - n5{r1c4 r1c6} ==> r1c6 <> 8
nrct5-chain n4{r3c3 r7c3} - n4{r7c8 r8c8} - n6{r8c8 r3c8} - n2{r3c8 r2c8} - n2{r2c4 r3c5} ==> r3c5 <> 4
row r3 interaction-with-block b1 ==> r1c1 <> 4
nrczt5-chain n4{r9c6 r9c1} - n2{r9c1 r8c1} - n6{r8c1 r5c1} - n6{r5c6 r4c6} - n4{r4c6 r4c5} ==> r7c5 <> 4
nrczt5-chain n7{r9c5 r9c4} - n7{r9c2 r6c2} - n6{r6c2 r6c7} - {n6 n3}r4c7 - {n3 n7}r4c8 ==> r4c5 <> 7
nrczt5-chain {n8 n1}r2c6 - n1{r1c5 r7c5} - n7{r7c5 r7c3} - n8{r7c3 r7c1} - n8{r3c1 r3c3} ==> r3c5 <> 8
nrczt6-chain {n3 n6}r4c7 - n6{r6c7 r6c2} - {n6 n8}r8c2 - n8{r8c6 r7c5} - n7{r7c5 r7c3} - {n7 n3}r9c2 ==> r4c2 <> 3
nrczt5-chain n7{r7c3 r7c5} - n8{r7c5 r7c1} - {n8 n6}r8c2 - n6{r9c1 r5c1} - n3{r5c1 r4c3} ==> r7c3 <> 3
nrczt6-chain {n8 n1}r2c6 - {n1 n4}r8c6 - n4{r9c4 r9c1} - n2{r9c1 r8c1} - n6{r8c1 r5c1} - n6{r5c6 r4c6} ==> r4c6 <> 8
hidden-pairs-in-a-column {n1 n8}{r2 r8}c6 ==> r8c6 <> 4
nrct6-chain {n9 n4}r9c6 - n4{r4c6 r4c5} - n2{r4c5 r4c3} - n8{r4c3 r4c2} - {n8 n7}r6c3 - n7{r7c3 r7c5} ==> r7c5 <> 9
row r7 interaction-with-block b9 ==> r9c9 <> 9
nrczt6-chain n6{r6c2 r6c7} - n1{r6c7 r6c8} - n9{r6c8 r5c9} - {n9 n7}r5c5 - n7{r7c5 r7c3} - n7{r6c3 r6c2} ==> r6c2 <> 8
nrczt6-chain {n3 n7}r4c8 - {n7 n1}r1c8 - {n1 n2}r2c8 - n2{r3c9 r3c5} - n1{r3c5 r7c5} - {n1 n3}r7c7 ==> r7c8 <> 3, r8c8 <> 3
nrczt6-chain n6{r4c2 r5c1} - n1{r5c1 r6c2} - n6{r6c2 r6c7} - {n6 n3}r4c7 - {n3 n7}r4c8 - n7{r4c2 r9c2} ==> r9c2 <> 6
hidden-pairs-in-a-row {n2 n6}r9{c1 c9} ==> r9c9 <> 3, r9c1 <> 4
row r9 interaction-with-block b8 ==> r8c4 <> 4
hidden-pairs-in-a-row {n2 n6}r9{c1 c9} ==> r9c1 <> 3
nrc3-chain n4{r8c8 r8c1} - n2{r8c1 r9c1} - {n2 n6}r9c9 ==> r8c8 <> 6
hidden-single-in-a-column ==> r3c8 = 6
nrct5-chain n4{r8c8 r8c1} - n2{r8c1 r9c1} - n6{r9c1 r5c1} - n1{r5c1 r5c9} - n1{r6c7 r7c7} ==> r8c8 <> 1
nrczt3-chain n1{r8c9 r8c6} - n1{r2c6 r2c1} - n1{r5c1 r5c9} ==> r1c9 <> 1
nrct5-chain {n1 n8}r8c6 - {n8 n3}r8c4 - {n3 n6}r8c2 - n6{r4c2 r5c1} - n1{r5c1 r5c9} ==> r8c9 <> 1
naked and hidden singles ==> r8c6 = 1, r2c6 = 8
nrc4-chain n4{r8c1 r8c8} - n2{r8c8 r2c8} - {n2 n3}r2c4 - {n3 n8}r8c4 ==> r8c1 <> 8
nrct4-chain {n3 n2}r2c4 - n2{r3c5 r3c9} - {n2 n6}r9c9 - {n6 n3}r8c9 ==> r8c4 <> 3
naked-single ==> r8c4 = 8
nrc3-chain {n3 n7}r7c5 - {n7 n9}r5c5 - n9{r5c9 r7c9} ==> r7c9 <> 3
nrc3-chain {n6 n3}r8c2 - n3{r8c9 r7c7} - {n3 n6}r4c7 ==> r4c2 <> 6
hidden-pairs-in-a-block {n1 n6}{r5c1 r6c2} ==> r6c2 <> 7
naked-pairs-in-a-row {n1 n6}r6{c2 c7} ==> r6c8 <> 1
hidden-pairs-in-a-block {n1 n6}{r5c1 r6c2} ==> r5c1 <> 3
block b4 interaction-with-column c3 ==> r3c3 <> 3
xy4-chain {n9 n1}r7c9 - {n1 n3}r7c7 - {n3 n7}r7c5 - {n7 n9}r5c5 ==> r5c9 <> 9
hidden-single-in-a-block ==> r6c8 = 9, r7c9 = 9
nrczt2-chain n7{r6c4 r6c3} - n7{r7c3 r7c5} ==> r5c5 <> 7
naked and hidden singles ==> r5c5 = 9, r9c6 = 9
nrc3-chain n1{r2c9 r5c9} - n1{r6c7 r7c7} - n3{r7c7 r8c9} ==> r2c9 <> 3
nrc4-chain n1{r3c2 r6c2} - n1{r6c7 r5c9} - n7{r5c9 r4c8} - {n7 n8}r4c2 ==> r3c2 <> 8
hidden singles ==> r4c2 = 8, r6c5 = 8, r9c2 = 7, r7c5 = 7
nrc4-chain n3{r8c2 r3c2} - n1{r3c2 r6c2} - n1{r6c7 r7c7} - n3{r7c7 r7c1} ==> r8c1 <> 3
nrct4-chain n2{r3c5 r3c9} - {n2 n6}r9c9 - {n6 n3}r8c9 - n3{r8c2 r3c2} ==> r3c5 <> 3
xy3-chain {n3 n2}r2c4 - {n2 n1}r3c5 - {n1 n3}r3c2 ==> r2c1 <> 3
nrc3-chain {n7 n1}r2c1 - n1{r2c9 r5c9} - n7{r5c9 r4c8} ==> r2c8 <> 7
nrc4-chain n7{r5c4 r6c4} - n2{r6c4 r2c4} - n3{r2c4 r2c8} - {n3 n7}r4c8 ==> r5c9 <> 7
hidden-single-in-a-block ==> r4c8 = 7
column c8 interaction-with-block b3 ==> r3c9 <> 3
row r3 interaction-with-block b1 ==> r1c1 <> 3
column c8 interaction-with-block b3 ==> r1c9 <> 3
hidden-pairs-in-a-row {n7 n8}r1{c1 c9} ==> r1c1 <> 1
nrc3-chain n4{r3c3 r3c1} - n3{r3c1 r7c1} - n8{r7c1 r7c3} ==> r7c3 <> 4
naked-singles ==> r7c3 = 8, r3c3 = 4
nrc4-chain n2{r3c9 r3c5} - n2{r4c5 r4c3} - n3{r4c3 r4c7} - n3{r7c7 r8c9} ==> r8c9 <> 2
naked-pairs-in-a-row {n3 n6}r8{c2 c9} ==> r8c1 <> 6
hxy-rn5-chain {c1 c9}r3n8 - {c9 c5}r3n2 - {c5 c3}r4n2 - {c3 c7}r4n3 - {c7 c1}r7n3 ==> r3c1 <> 3
hidden-single-in-a-block ==> r3c2 = 3
...(naked singles)...

826514937
795368412
134927568
982146375
643795281
517283694
358672149
469851723
271439856

[champagne]

up to now I did not do more that the puzzle discussed with Mike in that file of 11 unsolved.
I checked this morning the entire file. Here below a quick ranking.

#23 1s750 level 2
#24 0s937 level 2
#25 0s531 level 1+
#26 0s485 level 1+
#27 0s437 level 1+
#28 0s563 level 1+
#29 0s469 level 1+
#30 0s625 level 1+
#31 0s437 level 1
#32 0s641 level 1+
#33 1s656 level 2

One puzzle #31 can be solved with bi values includng groups.
Seven are requesting a little more, but are still in the area of hand solvers.
Three are requesting use of ALS ACs and the difficulty exceeds likely human capacity.

If I am write, #23, the hardest of that file has not yet been published.
I will show how the solver cracked it

[Steve K]

Here is a link to my attempt at un23. I may update with the path at this location later.
http://sudoku.com.au/Unsolvable23Page1.aspx

[champagne]

Hi steve,

I had a look on your solution.
Showing solutions thru drawings is very nice, but it is a huge task.

I compared up to the point where we follow different paths.

It seems to me that my solver comes to a quicker fix.
The sequence is the following (next post will give details )

#7r9c5 #8r9c1 #5r8c9 #9r5c1 #4r8c7 #6r8c7
#6r7c8 #6r9c5 #5r5c7 #9r2c1
then a UR pattern clearing 9r3c7
then cell r9c5 Hh cleaned
#3r4c1 #7r8c6 #9r3c2 #6r6c9 #3r5c9
#6r7c7 #9r1c9 #8r4c2 #6r5c9

7 r2c7=9 8 r4c1=8

You reached more or less the same point, but you could not conclude, may be due to the UR clearing I did not see in your solution.

[champagne]

The solver proposal for number 23

...8.21...6..4..5.1......72..59.67......8......13.48..25......1.1..3..9...34.1... 23

Code: Select all

5      3m479  479      |8     6x7a9    2      |1         346    3p469ç
3o7a9Î 6      2        |1     4        3h7A9Í |3È9è      5      8     
1      348b9  48B9     |5à6À  569      3H59   |34q6x9    7      2     
----------------------------------------------------------------------
348d   2c348D 5        |9     1        6      |7         2C34   3É4é   
34679  23479  46Z79    |2G57Ã 8        5Ê7ê   |2l345w69ç 1      34569 
6Y79   279    1        |3     2g5e7Á   4      |8         2Ë6ë   5E69Ä 
----------------------------------------------------------------------
2      5      4s67Â89Å |6Ì7ì  679      78i9Æ  |3f46      3F468j 1     
4T678  1      4678     |2g567 3        578I   |2G456     9      4567k 
6789   789    3        |4     2G5V679Å 1      |256       2l68J  567K   

Code: Select all

[]7r9c5 - 7r7c456 = AC:r7c378(7r7c3 - 8r7c38) = 8r7ci - 9r7c6 = AC:r23c6(9r23c6 - 7r2c6) = 7r1c5 - 7r9c5
[]8r9c1 - 8r9c8 = AC:r147c8(8r7c8 - 2r4c8) = 2r4c2 - 8r4c2 = 8r4c1 - 8r9c1
[]5r8c9 - 5r8c46 = 5r9c5 - 5r6c5 = 5r6c9 - 5r8c9

Code: Select all

5      3m479  479      |8     6x7a9   2      |1         346    3p469ç
3o7a9Î 6      2        |1     4       3i7A9Í |3È9è      5      8     
1      348b9  48B9     |5à6À  569     3I59   |34q6x9    7      2     
---------------------------------------------------------------------
348d   2c348D 5        |9     1       6      |7         2C34   3É4é   
34679  23479  46Z79    |2G57Ã 8       5Ê7ê   |2l345w69ç 1      34569 
6Y79   279    1        |3     2g5e7Á  4      |8         2Ë6ë   5E69Ä 
---------------------------------------------------------------------
2      5      4s67Â89Å |6Ì7ì  679     78j9Æ  |3f46      3F468h 1     
4T678D 1      4678     |2g567 3       578J   |2G45V6    9      467k   
679    78h9   3        |4     2G5V69Å 1      |256       2l68H  5w67K 

Code: Select all

[]9r5c1 - 9r5c79 = 9r6c9 - 5r6c9 = 5r6c5 - 5r5c6 = 7r5c6 - 7r2c6 = 7r2c1 - 3r2c1 = AC:r458c1(3r45c1 - 9r5c1)

then
with two intermediate chains

Code: Select all

#[]5r5c7.w  - 46r8c7    []5r5c7.w - 5r6c9 = 5r6c5 - 2r6c5 = 2r8c7 - 46r8c7
#[]2r5c7 - 46r8c7   []2r5c7==2r9c8.l - 8r9c8 = 8r9c2 - 8r8c13 = AC:r8c467(8r8c6 - 46r8c47)

Code: Select all

#4r8c7
   []2r5c7 - 4r8c7|#
   []5r5c7.w -  4r8c7[#
   []AC:r589c7(9r5c7.ç - 4r58c7) = 4r37c7 - 4r8c7
   []AC:r589c7(3r5c7 - 4r58c7) = 4r37c7 - 4r8c7
   []4r5c7 - 4r8c7
   []6r5c7 - 6r6c8 = 2r6c8 - 2r6c5.g = 2r8c7.G - 4r8c7

Code: Select all

#6r8c7   
   []2r5c7 - 6r8c47   6r8c7
   []5r5c7.w - 6r8c7|#
   []AC:r589c7(9r5c7.ç - 6r589c7) = 6r37c7 - 6r8c7
   []AC:r589c7(3r5c7 - 6r589c7) = 6r37c7 - 6r8c7
   []4r5c7 - 4r37c7 = AC:r589c7(4r58c7 - 6r589c7) = 6r37c7 - 6r8c7
   []6r5c7 - 6r8c7

#[]7r2c6 - 6r7c378 []AC:r23c6(7r2c6 - 9r23c6) = 9r7c6 - 8r7c6 = AC:r7c378(8r7c38 - 6r7c378)
#[]3r2c1 - 6r7c378 []3r2c1 - 7r2c1 = 7r2c6 - 6r7c378|#

Code: Select all

#6r7c8
   []8r8c1==8r4c2.D - 2r4c2 = 2r4c8 - 2r6c8 = 6r6c8 - 6r7c8
   []4r8c1 - 4r7c3.s = AC:r7c78(4r7c78 - 6r7c78) = 6r7c345 - 6r7c8
   []6r8c1 - 6r69c1 = AC:r458c1(6r58c1 - 3r45c1) = 3r2c1 -  6r7c378|# = 6r7c45 - 6r7c8
   []7r8c1 - 7r2c1 = 7r2c6 - 6r7c378|# = 6r7c45 - 6r7c8

Code: Select all

#6r9c5
   []6r6c1==6r5c79.Y - 6r6c8 = 2r6c8 - 2r6c5 = 2r9c5 - 6r9c5
   []AC:r458c1(6r58c1 - 3r45c1) = 3r2c1 - 6r7c378|# = 6r7c45 - 6r9c5
   []6r9c1 - 6r9c5

Code: Select all

#w   
   []7r7c3==7r8c46. - 7r8c9 = 7r9c9 - 5r9c9.w
   []7r1c3 - 7r2c1 = 7r2c6 - 7r5c6 = 5r5c6 - 5r5c7.w
   []7r8c3 - 7r8c9 = 7r9c9 - 5r9c9.w
   []7r5c3 - 7r5c6 = 5r5c6 - 5r5c7.w

Code: Select all

#9r2c1   
   []9r7c3 - 9r7c6 = AC:r23c6(9r23c6 - 7r2c6) = 7r2c1 - 9r2c1
   []9r1c3 - 9r2c1
   []9r5c3 - 9r5c79 = 9r6c9 - 5r6c9 = 5r6c5 - 5r5c6 = 7r5c6 - 7r2c6 = 7r2c1 - 9r2c1
   []9r3c3 - 9r2c1

here a UR pattern clearing 9r3c7 P1=r2c7 39 P2=r2c6 379 P3=r3c7 346 P4=r3c6 359

Code: Select all

5      3o479  479      |8      6w7a9  2      |1       346   3q469b
3A7a   6      2        |1      4      3k7A9b |3b9B    5     8     
1      348c9  48C9     |5à6À   569    3K59   |34r6w   7     2     
------------------------------------------------------------------
348e   2d348E 5        |9      1      6      |7       2D34  3Ê4ê   
3467   23479  46Z79    |2H57Ä  8      5Ë7ë   |2n3469b 1     345f69
6X79j  279    1        |3      2h5f7Â 4      |8       2Ì6ì  5F69Æ 
------------------------------------------------------------------
2      5      4t67Ã89Ç |6Í7í   6À79   78l9È  |3g46    3G48i 1     
4U678E 1      4678     |2h56á7 3      578L   |2H5h    9     4t67m 
6y79J  78i9   3        |4      2H5h9Ç 1      |25H6    2n68I 6m7M 

cell r9c5 Hh cleaned

Code: Select all

5      3p479  479    |8      6x7a9  2      |1       346   3r469b
3A7a   6      2      |1      4      3l7A9b |3b9B    5     8     
1      348c9  48C9   |5á6Á   5Ã69Π 3L5Í9  |34s6x   7     2     
----------------------------------------------------------------
348e   2d348E 5      |9      1      6      |7       2D34  3É4é   
3467   23479  46à79è |2I57Å  8      5Ê7ê   |2o3469b 1     345f69
6Y79K  279    1      |3      2i5f7Ã 4      |8       2Ë6ë  5F69Ç 
----------------------------------------------------------------
2      5      4u67Ä8 |6Ì7ì   6Á79h  78m9H  |3g46    3G48j 1     
4V678E 1      4678   |2i56â7 3      578M   |2I5i    9     4u67n 
6z79k  78j9K  3      |4      2I5i   1      |25I6o   2o68J 6n7N   

Code: Select all

#3r4c1   
   []6r1c5 - 7r1c5 = 3r2c1 - 3r4c1
   []6r1c8 - 6r6c8 = 2r6c8 - 2r4c8 = 2r4c2 - 8r4c2 = 8r4c1 - 3r4c1
   []6r1c9 - 6r89c9 = 4r8c9 - 4r4c9 = 3r4c9 - 3r4c1

Code: Select all

#7r8c6   
   []6r1c5 - 7r1c5 = 7r2c6 - 7r8c6
   []6r1c8 - 6r6c8 = 2r6c8 - 2r6c5 = 2r5c4 - 7r5c4 = 7r78c4 - 7r8c6
   []6r1c9 - 6r89c9 = 4r7c3 - 7r7c3 = 7r7c456 - 7r8c6

Code: Select all

#9r3c2   
   []9r2c6==9r5c7.b - 9r5c3 = 9r13c3 - 9r3c2
   []9r1c23 - 9r3c2
   []9r1c5 - 7r1c5 = 7r2c6 - 7r5c6 = 5r5c6 - 5r6c5 = 5r6c9 - 9r6c9 = 9r5c79 - 9r5c3 = 9r13c3 - 9r3c2

Code: Select all

5      3p479è  479    |8       6x7a9  2      |1       346   3r469b
3A7a   6       2      |1       4      3l7A9b |3b9B    5     8     
1      348c    48C9Æ  |5á6Á    5Ã69Π 3L5Í9  |34s6x   7     2     
------------------------------------------------------------------
4E8e   2d3q48E 5      |9       1      6      |7       2D34  3É4é   
3a467  23479   46à79è |2I57Å   8      5Ê7ê   |2o3469b 1     345f69
6Y79K  279     1      |3       2i5f7Ã 4      |8       2Ë6ë  5F69Ç 
------------------------------------------------------------------
2      5       4u67Ä8 |6Ì7ì    6Á79h  78m9H  |3g46    3G48j 1     
4V678E 1       4678   |2i56â7Ä 3      5m8M   |2I5i    9     4u67n 
6z79k  78j9K   3      |4       2I5i   1      |25I6o   2o68J 6n7N   

Code: Select all

#6r6c9   
   []4r4c1==8r4c2.E - 2r4c2 = 2r4c8 - 2r6c8 = 6r6c8 - 6r6c9
   []4r8c1 - 4r7c3 = 6r89c9 - 6r6c9
   []4r5c1 - 3r5c1 = 7r2c6 - 7r5c6 = 5r5c6 - 5r6c5 = 5r6c9 - 6r6c9

Code: Select all

#3r5c9     
   []9r5c7==9r2c6.b - 7r2c6 = 3r5c1 - 3r5c9
   []9r5c23==9r6c9.Ç - 5r6c9 = 5r5c9 - 3r5c9
   []9r5c9 - 3r5c9

Code: Select all

5      3r479ç  479    |8        6y7a9  2      |1        346   3o469b
3A7a   6       2      |1        4      3m7A9b |3b9B     5     8     
1      348c    48C9Æ  |5á6Á     5Ã69Ë  3M5Ê9  |34t6y    7     2     
--------------------------------------------------------------------
4E8e   2d3s48E 5      |9        1      6      |7        2D34  3O4o   
3a467  23479   46à79ç |2J57Å    8      5È7è   |2q3s469b 1     45f69 
6g79L  279     1      |3        2j5f7Ã 4      |8        2g6G  5F9f   
--------------------------------------------------------------------
2      5       4v67Ä8 |6É7é     6Á79i  78n9I  |3h46     3H48k 1     
4W678E 1       4678   |2j5Î6â7Ä 3      5n8N   |2J5j     9     4v67p 
6z79l  78k9L   3      |4        2J5j   1      |25J6q    2q68K 6p7P   

Code: Select all

#6r7c7   
   []5r6c5==9r5c23.f - 9r6c1 = 9r9c1 - 6r9c1 = 6r9c789 - 6r7c7
   []2r6c5==2r8c4.j - 6r8c4 = 6r7c45 - 6r7c7
   []7r6c5==5r3c5.Ã - 5r3c4 = 6r7c5 - 6r7c7

Code: Select all

5      3r479æ  479     |8        6x7a9  2      |1        346   3o469b
3A7a   6       2       |1        4      3m7A9b |3b9B     5     8     
1      348c    48C9Å   |5á6Á     5Â69Ê  3M5É9  |34t6x    7     2     
---------------------------------------------------------------------
4E8e   2d3s48E 5       |9        1      6      |7        2D34  3O4o   
3a467  23479   46à79æ  |2J57Ä    8      5Ç7ç   |2q3s469b 1     45f69 
6g79L  279     1       |3        2j5f7Â 4      |8        2g6G  5F9f   
---------------------------------------------------------------------
2      5       4k6y7Ã8 |6È7è     6Á79i  78n9I  |3h4H     3H48k 1     
4V678E 1       4678    |2j5Í6y7Ã 3      5n8N   |2J5j     9     4k6z7p
6z79l  7Î8k9L  3       |4        2J5j   1      |25J6q    2q68K 6p7P

Code: Select all

#b   
   []4r1c3 - 4r1c89 = 4r3c7 - 4r7c7 = 3r7c7 - 3r2c7.b
   []7r1c3 - 7r2c1 = 7r2c6 - 9r2c6.b
   []9r1c3 - 9r1c9.b

Code: Select all

#E   
   []6r1c5.x - 7r1c5 = 7r2c6 - 7r5c6 = 5r5c6 - 5r8c6 = 8r8c6 - 8r8c1.E
   []6r1c8 - 6r6c8 = 2r6c8 - 2r4c8 = 2r4c2 - 8r4c2.E
   []6r1c9 - 6r89c9 = 8r9c2 - 8r4c2.E

Code: Select all

#6r5c9   
   []9r5c7==3r2c7.b - 3r7c7 = 3r7c8 - 8r7c8 = 6r89c9 - 6r5c9
   []9r5c23==5r5c9.f - 6r5c9
   []9r5c9 - 6r5c9

7 r2c7=9 B
8 r4c1=8 B

Code: Select all

5     3A479ã 479   |8    67A9 2    |1    346 346 
3a7A  6      2     |1    4    3A7a |9    5   8   
1     348    489   |56   569  3a59 |3A46 7   2   
-------------------------------------------------
8     234    5     |9    1    6    |7    234 34   
3A467 23479  4679ã |257  8    5Ä7ä |2346 1   5d9D
679   279    1     |3    25d7 4    |8    26  5D9d
-------------------------------------------------
2     5      4678  |67   679  789  |34   348 1   
467   1      4678  |2567 3    58   |25   9   467 
679   789    3     |4    25   1    |256  268 67   

Code: Select all

#3r3c2   []3r3c2 - 8r3c2 = 4r7c3 - 4r8c1 = 4r5c1 - 3r5c1 = 3r2c1 - 3r3c2

Code: Select all

#ã     []9r5c3.ã - 9r5c9 = 5r6c5 - 5r5c6 = 7r5c6 - 7r2c6 = 3r1c2 - 9r1c2.ã

Code: Select all

5      3B47   479a    |8        6s7B9A 2      |1        346    3m46C 
3b7B   6      2       |1        4      3B7b   |9        5      8     
1      4C8c   48C9A   |5W6w     5x69à 3b5Â9h |3B4A6s   7      2     
---------------------------------------------------------------------
8      2d3p4q 5       |9        1      6      |7        2D3Ä4Å 3M4m   
3B4l67 23479e 46V7    |2I5Æ7z   8      5À7à   |2o3p4q6f 1      5e9E   
6f79K  279    1       |3        2i5e7x 4      |8        2f6F   5E9e   
---------------------------------------------------------------------
2      5      4C6t7y8 |6Á7á     6w79h  78j9H  |3g4G     3G48C  1     
4L67   1      4è678j  |2i5Ë6t7y 3      5j8J   |2I5i     9      4C6u7n
6u79k  7Ì8C9K 3       |4        2I5i   1      |25I6o    2o68c  6n7N   

Code: Select all

#C   
   []9r1c5==9r3c3.A - 8r3c3.C
   []7r1c5==3r3c7.B - 3r7c7 = 3r7c8 - 8r7c8.C
   []6r1c5.s - 6r1c9.C

Code: Select all

#5r5c4   
   []6r5c7==2r6c8.f - 2r6c5 = 2r5c4 - 5r5c4
   []6r9c7==2r9c8.o - 8r9c8 = 8r7c8 - 3r7c8 = 3r7c7 - 3r3c7 = 7r2c6 - 7r5c6 = 5r5c6 - 5r5c4
   []6r3c7==6r1c5.s - 6r3c4 = 5r3c4 - 5r5c4

Code: Select all

#7r7c6   
   []6r5c7==2r6c8.f - 2r6c5 = 2r5c4 - 7r5c4 = 7r78c4 - 7r7c6
   []6r9c7==2r9c8.o - 8r9c8 = 8r7c8 - 3r7c8 = 3r7c7 - 3r3c7 = 7r2c6 - 7r7c6
   []6r3c7.s - 3r3c7 = 7r2c6 - 7r7c6

9 r3c2=8 B
10 r9c8=8 B

->UR P1=r9c9 67 P2=r9c1 679 P3=r8c9 67 P4=r8c1 467

Code: Select all

5      3C47   479b   |8     6a7C9B 2     |1      346A 34   
3c7C   6      2      |1     4      3C7c  |9      5    8   
1      8      4b9B   |56    569    3c59F |3C4B6a 7    2   
----------------------------------------------------------
8      2A34   5      |9     1      6     |7      2a34 34   
3C4G67 23479C 467    |27    8      5c7C  |346A   1    5C9c
6A79I  279    1      |3     25C7   4     |8      2A6a 5c9C
----------------------------------------------------------
2      5      67r8f  |67    679F   8F9f  |34     34   1   
4g6G   1      4G678F |2567r 3      5F8f  |25     9    67   
679i   7i9I   3      |4     25     1     |25     8    67 

Code: Select all

[]4r8c3.G - 4r3c3 = 9r1c5 - 6r1c5 = 6r6c1 - 6r8c1.G
[]7r7c3.r - 7r9c2 = 9r9c2 - 9r5c2 = 3r3c6 - 9r3c6 = 8r7c3 - 7r7c3.r

11 r8c1=4 B

Code: Select all

5      3C4m7q    4à7m9b |8      6a7C9B 2      |1      3o4z6A 3j4J
3c7C   6         2      |1      4      3C7c   |9      5      8   
1      8         4b9B   |5i6I   5n6Y9s 3c5r9F |3C4B6a 7      2   
----------------------------------------------------------------
8      2A3k4l    5      |9      1      6      |7      2a3t4u 3J4j
3C6w7å 2d3á4â79C 4m6g7È |2D7d   8      5c7C   |3k4l6A 1      5C9c
6A7ã9H 2x7Æ9ç    1      |3      2d5C7n 4      |8      2A6a   5c9C
----------------------------------------------------------------
2      5         6F8f   |6d7D   6I7d9F 8F9f   |3e4E   3E4e   1   
4      1         6Ä7g8F |2d5I6F 3      5F8f   |2D5d   9      6g7G
6g7v9h 7h9H      3      |4      2D5d   1      |2d5D   8      6G7g

note in r13c3 7r1c3 = 4r13c3 = 4r1c5 = 7r58c5

Code: Select all

[]7r5c2 - 7r5c46 = 5r3c5 - 5r3c4 = 6r3c4 - 6r1c5 = 6r6c1 - 9r6c1 = 7r9c2 - 7r5c2

Code: Select all

#g   
 []3r5c1==7r1c5.C - 6r1c5 = 6r6c1 - 6r9c1.g
 []6r5c1.w - 6r9c1.g
 []7r5c1 - 7r5c46 = 5r3c5 - 5r3c4 = 6r3c4 - 6r1c5 = 6r6c1 - 6r9c1.g

then

Code: Select all

[]7r5c6.C - 7r5c3 =  7r1c3 - 7r2c1.C    'C' is dead

then

Code: Select all

[]6r1c5.a - 9r1c5 = 9r1c3 - 7r1c3 = 7r5c3 - 7r5c46 = 5r3c5 - 5r3c4 = 6r3c4 - 6r1c5.a
'a' is dead

12 r1c2=4
the game is over

[denis_berthier]

SudoRules solution path for UN23
The longest chain or lasso has length 8.

***** SudoRules version 13 *****
000802100
060040050
100000072
005906700
000080000
001304800
250000001
010030090
003401000
hidden-singles ==> r2c3 = 2, r2c4 = 1, r4c5 = 1, r5c8 = 1, r1c1 = 5, r2c9 = 8
nrczt2-chain n5{r6c9 r6c5} - n5{r9c5 r9c9} ==> r8c9 <> 5
nrct4-chain n8{r4c1 r4c2} - n2{r4c2 r4c8} - {n2 n6}r6c8 - {n6 n8}r9c8 ==> r9c1 <> 8
nrczt4-chain n7{r1c5 r2c6} - {n7 n5}r5c6 - n5{r8c6 r8c4} - n2{r8c4 r9c5} ==> r9c5 <> 7
nrct5-chain n5{r6c9 r6c5} - {n5 n7}r5c6 - n7{r2c6 r2c1} - n7{r5c1 r6c2} - n7{r9c2 r9c9} ==> r9c9 <> 5
column c9 interaction-with-block b6 ==> r5c7 <> 5
nrczt6-chain n9{r5c7 r6c9} - n5{r6c9 r6c5} - {n5 n7}r5c6 - n7{r2c6 r2c1} - {n7 n6}r6c1 - {n6 n9}r9c1 ==> r5c1 <> 9
nrczt6-chain n2{r8c7 r8c4} - n5{r8c4 r8c6} - n8{r8c6 r7c6} - n8{r7c8 r9c8} - n2{r9c8 r9c7} - n5{r9c7 r8c7} ==> r8c7 <> 6
nrczt5-chain n9{r9c2 r9c5} - n5{r9c5 r9c7} - n2{r9c7 r9c8} - {n2 n4}r8c7 - n4{r8c3 r7c3} ==> r7c3 <> 9
row r7 interaction-with-block b8 ==> r9c5 <> 9
nrczt5-chain n9{r9c1 r9c2} - n9{r6c2 r6c9} - n5{r6c9 r6c5} - {n5 n7}r5c6 - n7{r2c6 r2c1} ==> r2c1 <> 9
nrczt6-chain n9{r3c3 r5c3} - n9{r6c2 r6c9} - n5{r6c9 r5c9} - {n5 n7}r5c6 - n7{r2c6 r1c5} - n9{r1c5 r1c3} ==> r3c2 <> 9
nrczt6-chain n5{r5c9 r6c9} - n9{r6c9 r5c7} - n9{r2c7 r2c6} - n7{r2c6 r1c5} - {n7 n2}r6c5 - {n2 n6}r6c8 ==> r5c9 <> 6
nrczt6-chain n2{r8c7 r8c4} - n5{r8c4 r8c6} - n8{r8c6 r7c6} - n8{r7c8 r9c8} - n2{r9c8 r9c7} - n5{r9c7 r8c7} ==> r8c7 <> 4
nrc3-chain n2{r9c5 r8c4} - {n2 n5}r8c7 - n5{r9c7 r9c5} ==> r9c5 <> 6
nrczt6-chain n6{r9c9 r9c1} - n6{r6c1 r6c9} - n5{r6c9 r5c9} - n9{r5c9 r1c9} - {n9 n3}r2c7 - n3{r7c7 r7c8} ==> r7c8 <> 6
nrczt6-lr-lasso {n9 n3}r2c7 - {n3 n7}r2c1 - {n7 n4}r1c3 - n4{r1c8 r3c7} - n4{r7c7 r7c8} - n3{r7c8 r7c7} ==> r1c9 <> 9
column c9 interaction-with-block b6 ==> r5c7 <> 9
hidden-pairs-in-a-block {n5 n9}{r5c9 r6c9} ==> r6c9 <> 6, r5c9 <> 4, r5c9 <> 3
nrczt6-chain n3{r4c9 r1c9} - n3{r2c7 r2c6} - n7{r2c6 r1c5} - n6{r1c5 r1c8} - n6{r6c8 r5c7} - n3{r5c7 r5c2} ==> r4c1 <> 3
nrczt8-chain n7{r2c6 r1c5} - n7{r6c5 r5c4} - n2{r5c4 r6c5} - {n2 n6}r6c8 - n6{r1c8 r1c9} - {n6 n4}r8c9 - n4{r7c7 r7c3} - n7{r7c3 r7c6} ==> r8c6 <> 7
nrczt8-chain n6{r7c5 r8c4} - n2{r8c4 r9c5} - n5{r9c5 r8c6} - {n5 n7}r5c6 - n7{r2c6 r2c1} - n7{r6c1 r6c2} - n2{r6c2 r6c8} - n6{r6c8 r5c7} ==> r7c7 <> 6
nrczt8-chain n6{r7c5 r8c4} - n2{r8c4 r9c5} - n5{r9c5 r8c6} - {n5 n7}r5c6 - n7{r2c6 r1c5} - n7{r1c3 r8c3} - n8{r8c3 r8c1} - n4{r8c1 r7c3} ==> r7c3 <> 6
row r7 interaction-with-block b8 ==> r8c4 <> 6
nrczt8-lr-lasso {n4 n3}r7c7 - {n3 n8}r7c8 - n8{r9c8 r9c2} - n8{r4c2 r4c1} - n4{r4c1 r5c1} - n3{r5c1 r5c2} - {n3 n4}r3c2 - n4{r3c7 r5c7} ==> r7c3 <> 4
row r7 interaction-with-block b9 ==> r8c9 <> 4
naked-pairs-in-a-block {n6 n7}{r8c9 r9c9} ==> r9c8 <> 6, r9c7 <> 6
block b9 interaction-with-column c9 ==> r1c9 <> 6
naked-pairs-in-a-block {n2 n5}{r8c7 r9c7} ==> r9c8 <> 2
naked-single ==> r9c8 = 8
column c8 interaction-with-block b6 ==> r5c7 <> 2
nrc4-chain n2{r5c4 r6c5} - {n2 n6}r6c8 - n6{r1c8 r1c5} - {n6 n5}r3c4 ==> r5c4 <> 5
nrct4-chain n8{r3c2 r4c2} - {n8 n4}r4c1 - n4{r4c9 r1c9} - n4{r1c2 r3c2} ==> r3c2 <> 3
nrct5-chain n7{r2c6 r1c5} - n6{r1c5 r1c8} - {n6 n2}r6c8 - {n2 n5}r6c5 - {n5 n7}r5c6 ==> r7c6 <> 7
nrct5-chain n6{r1c5 r1c8} - n6{r6c8 r6c1} - n9{r6c1 r9c1} - {n9 n7}r9c2 - n7{r6c2 r6c5} ==> r1c5 <> 7
naked and hidden singles ==> r2c6 = 7, r5c6 = 5, r8c6 = 8, r7c6 = 9, r3c6 = 3, r5c9 = 9, r6c9 = 5, r2c1 = 3, r2c7 = 9, r7c3 = 8, r3c2 = 8, r4c1 = 8
row r7 interaction-with-block b8 ==> r8c4 <> 7
column c3 interaction-with-block b1 ==> r1c2 <> 9
nrc3-chain n4{r3c3 r3c7} - n6{r3c7 r5c7} - n6{r5c3 r8c3} ==> r8c3 <> 4
hidden-single-in-a-block ==> r8c1 = 4
nrc3-chain {n4 n3}r7c8 - n3{r1c8 r1c9} - {n3 n4}r4c9 ==> r4c8 <> 4
nrc4-chain n6{r5c7 r3c7} - n6{r3c4 r7c4} - n7{r7c4 r5c4} - {n7 n6}r5c1 ==> r5c3 <> 6
naked and hidden singles ==> r8c3 = 6, r8c9 = 7, r9c9 = 6
nrc4-chain n6{r7c5 r7c4} - n7{r7c4 r5c4} - n7{r5c3 r1c3} - n9{r1c3 r1c5} ==> r1c5 <> 6
...(naked and hidden singles)...
547892163
362147958
189653472
825916734
734285619
691374825
258769341
416538297
973421586