The New Sudoku Players' Forum

by **eleven** » Tue Mar 15, 2022 12:41 pm

denis_berthier wrote:Are you sure it satisfies the parity conditions??

There is a single rectangle 349 in r68c34.

by **eleven** » Tue Mar 15, 2022 1:39 pm

denis_berthier wrote:
Code: Select all
........1.....2.34....562....56..1...627.18..8...257...7.......52..6...86.1..8... ED=11.8/1.2/1.2 not in T&E(2) No Tridagon

Here the link between the 2 externals 1r8c4=7r9c5 can be used to reduce the ER to 10.1 (still a long puzzle).

Hidden Text: Show

Same with next:

Code: Select all: ........1.....2.34....562....567.1...628.17..7...258...87......52..6...76.1...... ED=11.8/1.2/1.2 not in T&E(2) No Tridagon

In the next, 5r7c5 implies 3r9c6, which makes it easy.

Code: Select all: ........1.....2.34235...6......78.6..8.23....7.35.6....26..78..3.8...7.657.......

Hidden Text: Show

[Edit:] So it seems to be worth to have the link between 2 externals in mind.

by **hendrik_monard** » Tue Mar 15, 2022 2:06 pm

Hi Hendrik,
did you check if some have the same minlex expanded version?

I have no script dedicated to such operation. However I tested manually numbers 3 and 4. As I still can't calculate minlex, I systematically convert all newcomers to maxlex:
nr 3: 9876.........958.......4...76.3..2..2.8.......392.678.3.6....97.92...6.8......32. ED=11.8/1.2/1.2
nr 4: 9876.........958.......4...76.3..2..2.8.......392.678.3......97.92...6.8....6.32. ED=11.8/1.2/1.2
Expansion leads to the same result for both:
exp. 9876.........958.......4...76.3..2..2.8.......392.678.3.6....97.92...6.8....6.32.

by **marek stefanik** » Tue Mar 15, 2022 4:13 pm

Impressive findings, mith!

Thanks Denis for identifying the puzzles without direct eliminations.
They really show that using the derived links from the TH, the puzzles can often be solved using very simple patterns.

denis_berthier wrote:........1.....2.34....562....56..1...627.18..8...257...7.......52..68..76.1..7... ED=11.8/1.2/1.2 not in T&E(2) No Tridagon

Rather than eliminating one of the internals as eleven did, let's consider the externals:

Code: Select all: .---------------------.------------------.----------------------. | 2 34589 346789 | 3489 3479 349 | 569 56789 1 | resolution state after basics | 19 589 6789 | 189 179 2 | 569 3 4 | # TH 349b4578, externals 349* | 1349 3489 34789 | 13489 5 6 | 2 789 89 | :---------------------+------------------+----------------------: | 7 #349 5 | 6 8 #349 | 1 249 239 | |#349 6 2 | 7 #349 1 | 8 459 359 | | 8 1 349 |#349 2 5 | 7 469 369 | :---------------------+------------------+----------------------: |#349 7 *3489 | 25 *1349 #349 | 56–349 12568–9 2568–9| | 5 2 #349 |#1349 6 8 | 349 19 7 | | 6 #3489 1 | 25 #349 7 | 3459 2589 2589 | '---------------------'------------------'----------------------'

Triple r7c16+* => -349r7c789, btte (S3 or lower)
Max. size overall: 12 EDIT: 14 (349b4578, 1b8, 8b7; without the last two you could fit another 349 in b78 not in r7 and thus have one in r7c789)

The rest of them can be solved with a short grouped AIC from the internals:

denis_berthier wrote:........1.....2.34....562....56..1...627.18..8...257...7.......52..6...86.1..8... ED=11.8/1.2/1.2 not in T&E(2) No Tridagon

Code: Select all: .--------------------.-------------------.--------------------. | 2 34589 34679 | 3489 3479 3479 | 569 56789 1 | resolution state after basics | 19 589 679 | 189 179 2 | 569 3 4 | # TH 349b4578, internals 1r8c4, 7r9c5 | 1349 3489 3479 | 13489 5 6 | 2 789 79 | :--------------------+-------------------+--------------------: | 7 #349 5 | 6 8 #349 | 1 249 239 | |#349 6 2 | 7 #349 1 | 8 459 359 | | 8 1 #349 |#349 2 5 | 7 469 369 | :--------------------+-------------------+--------------------: |#349 7 8 | 25 1349 #349 | 34569 12569 2569 | | 5 2 #349 |#349+1 6 3479 | 349 79–1 8 | | 6 #349 1 | 25 #349+7 8 | 3459 2579 2579 | '--------------------'-------------------'--------------------'

1r8c4 == 7r9c5 – 7r9c89 = 7r8c8 => –1r8c8, skfr 6.6 (BC4 or lower)
Max. size overall: 13 (TH cells + 7b9)

I believe that it's worth using the TH links, even if it may seem like they would lead to overcomplicated solutions, as that is rarely the case.

Marek

by **mith** » Tue Mar 15, 2022 4:50 pm

Sorry, I had written an explanation and somehow it got deleted. The most recent batch is all minimals during the period my internet was down. The 11.9s are the two minimals of the same 29c expanded form, as referenced here: the-hardest-sudokus-new-thread-t6539-1125.html#p318311

I’ll be updating the list of expanded forms sometime this week, I just haven’t fully automated that process yet.

by **eleven** » Tue Mar 15, 2022 5:44 pm

Ah, nice improvements of my quick tries, Marek !

by **mith** » Tue Mar 15, 2022 6:11 pm

denis_berthier wrote:Wow, 2 new 11.9s and a whole lot (38) of new 11.8s, many of which are not in T&E(2).

Are they all independent, i.e. did you stick to your decision to publish here only the minlex form of expanded puzzles?

Here are the expanded forms of these puzzles. I've also noted where smaller clue-count expanded forms are "sub-puzzles" of larger counts (e.g. the 11.9s expand to 29c, a couple of the 11.8s expand to 30c which is the 29c plus one more given):

Code: Select all: ........1.....2.......3..45.16.23...27.81.6..3.87.61...32.67..81.7.8....68....... (sub-puzzle of e) ........1.....2.......3..45..1.23....2671.8..73.6.81...8.......2.3.86..761..7.... ED=11.9/1.2/1.2 not in T&E(2) ........1.....2.......3..45..6.......71.8....32..67..8.6..23....837..1..7.281.6.. ED=11.9/1.2/1.2 not in T&E(2) ........1.....2.34.35.......62......7.3.25..885...67...8753....3.62.8...52..67.8. (e) ........1.....2.34..2.3156.....564...1.2.43.6.4.31..2.....6......4....5.7.8..5... ED=11.8/1.2/1.2 not in T&E(2) ........1.....2.34..2.3156.....56.1..1.2.43.6.4.3...2.....6......4....5.7.8..5... ED=11.8/1.2/1.2 not in T&E(2) ........1.....2.34.23...5....6.78.5..5823.....7.6.5....67.....25.2..78..83..267.5 (d) ........1.....2.34.23...5....6.78....5823.....7.6.5....67.....25.2..78..83...67.5 ED=11.8/1.2/1.2 not in T&E(2) ........1.....2.34.23...5....6.78....5823.....7.6.5....67.....25....78..83..267.5 ED=11.8/1.2/1.2 not in T&E(2) ........1.....2.34.23...5....6.78.5...823.....7.6.5....67.....25....78..83..267.. ED=11.8/1.2/1.2 not in T&E(2) ........1.....2.34.23...5....6.78.5...823.....7.6.5....67.....25.2..78..83...67.. ED=11.8/1.2/1.2 not in T&E(2) ........1.....2.34.23...5...5623....2.7.86.5.38.7.5....78......5.2..86..63...78.5 (sub-puzzle of c) ........1.....2.34.23...5....623.....7.8.5...2.8.76.5..87......5.2..76..63...87.5 ED=11.8/1.2/1.2 not in T&E(2) ........1.....2.34.23...5....6.78.5...823....37.6.5....67......5.2..78..83...67.5 ED=11.8/1.2/1.2 not in T&E(2) ........1.....2.34....562....7.25....5867.1..62.8.17...16......2.5.68..787....... ED=11.8/1.2/1.2 not in T&E(2) ........1.....2.34....562....1.25....267.18..75.68.1...87......5.2.67..861....... ED=11.8/1.2/1.2 not in T&E(2) ........1.....2.34235...6...26..78..3.8...7.657........52.78.6.68.23....7.35.6... (sub-puzzle of b) ........1.....2.34....562....567.1...628.17..7...258...87......52..6...76.1...... ED=11.8/1.2/1.2 not in T&E(2) No Tridagon ........1.....2.34235...6....2.78.6..8.23....7..5.6....26..78..3.8...7.657....... ED=11.8/1.2/1.2 not in T&E(2) No Tridagon ........1.....2.34235...6......78.6..8.23....7.35.6....26..78..3.8...7.657....... ED=11.8/1.2/1.2 not in T&E(2) No Tridagon ........1.....2.34....562....1.257...267.18...5.68.1...87......5.2.6...861....... ED=11.8/1.2/1.2 not in T&E(2) No Tridagon ........1.....123...1.245.6..3...6...6..52...7.8..6....1..654.3.3.24.15..4.1.3.62 (sub-puzzle of a) ........1.....2.34....562....56..1...627.18..8...257...7.......52..68..76.1..7... ED=11.8/1.2/1.2 not in T&E(2) No Tridagon ........1.....2.34.23...5...5623....2.7.86.5.38.7.5....78......5.2..86..63..278.5 (c) ........1.....2.34....562....7.......52.67..816...8....8..25...2.67.18..57.6..1.. ED=11.8/1.2/1.2 not in T&E(2) ........1.....234...2.315.6....652...2.1.4..3.4.3..15...4...6...6..53...7.8..6... ED=11.8/1.2/1.2 not in T&E(2) ........1.....2.34.35...6...72......65...78..8.3.257.6.8653....3.72.6...52..78.6. (sub-puzzle of a) ........1.....2.34....562....1.25....267.18..75.68.1...8.......5.2.68..761...7... ED=11.8/1.2/1.2 not in T&E(2) ........1.....2.34....562....7.......61..8...25..67..8.7..25....856..1..6.28.17.. ED=11.8/1.2/1.2 not in T&E(2) ........1.....2.34235...6...26..78..3.8.2.7.657........52.78.6.68.23....7.35.6... (b) ........1.....2.34....562....56..1...627.18..8...257...7.......52..6...86.1..8... ED=11.8/1.2/1.2 not in T&E(2) No Tridagon ........1.....2.34235...6...72......65...78..8.3.257.6.8653....3.72.6...52..78.6. (a) ........1.....2.34....562....1.257...267.18...5.68.1...87......5.2.68..761...7... ED=11.8/1.2/1.2 not in T&E(2) ........1.....2.34235...6......78.6..8.53....3.72.6....72......65...78..8.3.257.6 ED=11.8/1.2/1.2 not in T&E(2)

Note that all but one of the puzzles you've marked as "no tridagon" (that is, ~~having more than one extra candidate as guardians for the pattern~~ having more than one guardian cell with extra candidate(s)) are related to each other. The outlier appears to be the first case of a puzzle with more than one guardian which can be expanded to a puzzle with only one guardian while remaining not in T&E(2).

I will work on automating this process so we can better see how these puzzles are related.

by **mith** » Tue Mar 15, 2022 8:20 pm

I hadn't checked this in a while, but there is a new lower bound for D*FC+DFC:

Code: Select all: ........1.....1....23.4..5...6..3.2..7.4...1.2...8.5....9..4.8..67....3.8...3...5 ED=11.2/1.2/1.2 DLFC+DFC

Huge amount of overlap now between +MFC (10.6-11.7) and +DFC (11.2-11.9).

by **mith** » Tue Mar 15, 2022 9:56 pm

I checked Loki and its 27c friend (rating 11.8 in gsf minlex form, but the same expanded form as Loki) with 1to9's fork of SE, to see what the node counts are (for all 11.8+ steps):

Loki:

Code: Select all: 57....9..........8.1.........168..4......28.9..2.9416.....2.....6.9.82.4...41.6.. 11.8, Contradiction Forcing Chain (w/303 nodes): R3C1.9 on ==> R4C1.7 both on & off: r3c1<>9 11.8, Contradiction Forcing Chain (w/320 nodes): R2C6.9 on ==> R6C4.3 both on & off: r2c6<>9 11.9, Contradiction Forcing Chain (w/400 nodes): R2C5.4 on ==> R4C6.3 both on & off: r2c5<>4 11.9, Contradiction Forcing Chain (w/423 nodes): R3C5.3 on ==> R8C5.7 both on & off: r3c5<>3 11.8, Region Forcing Chains (w/269 nodes): 5 in column ==> R3C3.6 off: r3c3<>6

27c 11.8:

Code: Select all: ........1.....2.......3..45..1.23....267.81..73.61.8...17.6.....8.......2.3.87..6 11.8, Contradiction Forcing Chain (w/303 nodes): R1C8.3 on ==> R5C8.5 both on & off: r1c8<>3 11.8, Contradiction Forcing Chain (w/320 nodes): R1C8.2 on ==> R5C5.5 both on & off: r1c8<>2 11.8, Contradiction Forcing Chain (w/351 nodes): R2C1.9 on ==> R9C4.4 both on & off: r2c1<>9

At some point I'd like to dig into the pruning mechanism and figure out why exactly that 11.8 is pruned in Loki but not in the 27c/other morphs.

And the newer 27c 11.9 (26c is the same):

Code: Select all: ........1.....2.......3..45..1.23....2671.8..73.6.81...8.......2.3.86..761..7.... 11.9, Contradiction Forcing Chain (w/410 nodes): R1C8.2 on ==> R2C2.5 both on & off: r1c8<>2 11.8, Contradiction Forcing Chain (w/283 nodes): R3C2.7 on ==> R8C4.9 both on & off: r3c2<>7

Website · by **denis_berthier** » Wed Mar 16, 2022 7:28 am

mith wrote:Note that all but one of the puzzles you've marked as "no tridagon" (that is, having more than one extra candidate as guardians for the pattern)

More precisely: having extra candidates in more than one cell (the target cell). The number of extra candidates in the target cell is irrelevant.

I'm aware (see also eleven's and Marek's posts) that there is a more general situation where no candidate can be eliminated but a "strong link" can instead be found. This is IMO a different rule. What I've studied is the "tridagon elimination rule", not such generalisations that have a very different kind of conclusion. That's what I meant by "no tridagon".

Website · by **denis_berthier** » Wed Mar 16, 2022 7:43 am

mith wrote:I checked Loki and its 27c friend (rating 11.8 in gsf minlex form, but the same expanded form as Loki) with 1to9's fork of SE, to see what the node counts are (for all 11.8+ steps):
Loki:
Code: Select all
57....9..........8.1.........168..4......28.9..2.9416.....2.....6.9.82.4...41.6.. 11.8, Contradiction Forcing Chain (w/303 nodes): R3C1.9 on ==> R4C1.7 both on & off: r3c1<>9 11.8, Contradiction Forcing Chain (w/320 nodes): R2C6.9 on ==> R6C4.3 both on & off: r2c6<>9 11.9, Contradiction Forcing Chain (w/400 nodes): R2C5.4 on ==> R4C6.3 both on & off: r2c5<>4 11.9, Contradiction Forcing Chain (w/423 nodes): R3C5.3 on ==> R8C5.7 both on & off: r3c5<>3 11.8, Region Forcing Chains (w/269 nodes): 5 in column ==> R3C3.6 off: r3c3<>6

27c 11.8:
Code: Select all
........1.....2.......3..45..1.23....267.81..73.61.8...17.6.....8.......2.3.87..6 11.8, Contradiction Forcing Chain (w/303 nodes): R1C8.3 on ==> R5C8.5 both on & off: r1c8<>3 11.8, Contradiction Forcing Chain (w/320 nodes): R1C8.2 on ==> R5C5.5 both on & off: r1c8<>2 11.8, Contradiction Forcing Chain (w/351 nodes): R2C1.9 on ==> R9C4.4 both on & off: r2c1<>9

At some point I'd like to dig into the pruning mechanism and figure out why exactly that 11.8 is pruned in Loki but not in the 27c/other morphs.

I don't know what "pruning mechanisms" you're talking about and they may add there own parts to the problem.
But, generally speaking, depending on the order of eliminations, an elimination that required n inference steps (nodes) in one path may require fewer ones if one of the nodes has been deleted by previous steps.
That's the general problem of non-confluence of the underlying RSs for any definition of complexity based on the number of inferences (nodes).

In SE, if a puzzle is rated high, but the number of nodes in the hardest step is just above one of the arbitrary thresholds, it's more likely that a morph (which will use a different path) will be rated lower.

by **mith** » Wed Mar 16, 2022 3:23 pm

The pruning issue was discussed earlier in the thread as one of the reasons *why* SE doesn't have confluence. The search for chains isn't exhaustive, it is pruning the search tree at some (unknown to me) point.

Obviously the order of elimination matters in the complexity of subsequent steps, but this should only ever reduce the complexity (barring uniqueness considerations, which is a separate issue not relevant here). What appears to be happening here is that there is some difference in order of eliminations (occurring before the 11.8 steps even - I need to take a closer look to determine exactly where the differences are) which is resulting in choosing a different pair of 11.8 steps (despite the node counts being the same, we can see immediately that they are different - in Loki, same digit in two cells, giving an immediate single; in the 11.8 morph, two digits in same cell, resulting in a pointing step and an 11.1 chain before getting the single).

The point is that the two 11.8 steps in the latter case must still exist in Loki after the single - and in fact they do, but they are now rated 11.9. Even if there were some eliminations reducing the node count in the 11.8 puzzle that are missing from Loki (that doesn't appear to be happening from my analysis in the GUI - the 11.8 steps are available before the single) the steps leading to those eliminations should also be present in Loki, again at a lower complexity. Instead, what we are seeing is that something - presumably the pruning, but I don't know that for sure yet - is causing Loki to miss these lower complexity chains completely.

by **mith** » Wed Mar 16, 2022 3:41 pm

Only 11.6-11.7 for today's update (the last 11.6 is a 23c pearl though):

Hidden Text: Show

Code: Select all: ........1...234.....21.5.....6.7..28.87...91.9.......6.79...1.22.87...6961....87. ED=11.7/1.2/1.2 ........1...234.....21.5.....6.7..28.87...91.92......6..87...69.79...1.261....87. ED=11.7/1.2/1.2 ........1......234....25.......67.....6..8.79.2759..8..79.52...2.86.....65.87.9.. ED=11.7/1.2/1.2 ........1......234....25.......67.....6..8.79.2759..8..79.52...2.86.9...65.87.... ED=11.7/1.2/1.2 ........1..2..3....14.56..........74......8.9.48.7.12...17..9.8.79....4282.9..7.. ED=11.7/1.2/1.2 ..............1.23....45.67.26....8.83.9...7.9.78..3.2.69......38.......7.2.9..36 ED=11.7/1.2/1.2 ........1..2.34....56.17..........38......9.7.9..7341......87.48..34..9.9....1.83 ED=11.7/1.2/1.2 ..............1.23....45.67..6.......38....9.27..9..36.839...7.6.2....8.79.8..3.2 ED=11.7/1.2/1.2 ........1......234....25........6.....657..89..9..8.67.289.7...59.86....6.7.52... ED=11.7/1.2/1.2 ..............1.23....45..6..7.......38....9.26..9..37.839..76.69.8..3.27.2....8. ED=11.7/1.2/1.2 ........1......234....25........6.....657..89..9..8.67.289.....59.86.7..6.7.52... ED=11.7/1.2/1.2 ........1....23.....4..5.....6....47.78...91.9...7.6.8.89.....44.71....661.9..87. ED=11.7/1.2/1.2 ........1....23.....4..5.....6....47.78...91.9...7.6.8.8......44.71...9661.9..87. ED=11.7/1.2/1.2 ........1....23.....4..5.....6....47.78...91.94..7.6.8..71...96.8......461.9..87. ED=11.7/1.2/1.2 ........1....23.....4..5.....6....47.78...91.94..7.6.8..71....6.89.....461.9..87. ED=11.7/1.2/1.2 ........1....23.....1.45.....6....7...8...9.479..8..16.674....81.9...6..84.9..1.7 ED=11.7/1.2/1.2 ........1....23......145.....6....7...8...9.279..8..16.672....81.9...6..82.9..1.7 ED=11.7/1.2/1.2 ...............123.....1.45..6..7....7168...985...9....9..765..16.95.8..7..1.8... ED=11.7/1.2/1.2 ...............123.....1.45..4..6....1789..6..8...7....7.1......96.784..1.864.9.. ED=11.7/1.2/1.2 ........1....23.....1.45.....6.....4..7....8.89..6..17.784...961......7.64.9..1.8 ED=11.7/1.2/1.2 ........1....23......145.....1....6..278..1.996.2...87.89.7..166......9.7.......2 ED=11.7/1.2/1.2 ........1....23......145.....1....6..278..1.996.2...87.89.7..166.....9..7.......2 ED=11.7/1.2/1.2 ...............123.....1.45..4..6....1789..6.8....7....7.1......96.784..1.864.9.. ED=11.7/1.2/1.2 ........1....23.....1.45.....6.....4..7....8.89..6.1.7.784..9.61.....7..64.9...18 ED=11.7/1.2/1.2 ..............1..2....34.56.14.78...73.94....8.9..37...98......17.......4.3.19.8. ED=11.7/1.2/1.2 ........1.....2.3...4...2.5....34.6....26..73...8.7....76.483..42.6....78.3...... ED=11.7/1.2/1.2 ........1.....2.3...4...2.5....34.6....26..73...8.7....76.48...42.6.3..78.3...... ED=11.7/1.2/1.2 ........1...234.....2..5..........67..6.8.1.9..7..928..28...7..17.9..8.66.9...... ED=11.7/1.2/1.2 ........1..2..3....1..45..........67......8.9.78.6.12..69....727.19....882.6..... ED=11.7/1.2/1.2 ........1.....2..3....4..56..472.....28.19....7.8.49...81......4.2.87.1.79....... ED=11.7/1.2/1.2 ..............1..2....3..45..361.7...18.79....6.8.39...87......3.1.86.7.69....... ED=11.7/1.2/1.2 ........1....23.45..25..36......51.4....16.32.1.3...5.1......267....4...89....... ED=11.7/1.2/1.2 ........1.......23....45.....4..6....5748.9..86..795...4..5.....98.6..7.6....8... ED=11.7/1.2/1.2

Hidden Text: Show

Code: Select all: ........1.....2.3.....45.....14...63.34.1.7.876....1...43.816.7.7.....846.87..3.. ED=11.6/1.2/1.2 ........1.....2.3.....45.....14...63.34.1.7.876........43.816.7.7.....846.87..31. ED=11.6/1.2/1.2 ........1....23.....4..5.....6.7..48.87...91.9...8.6.7.79...1.44.8.....661.9..78. ED=11.6/1.2/1.2 ........1....23.....4..5.....6.....7.89....1471.9..68...7.894.6.4..6..78.68...19. ED=11.6/1.2/1.2 ........1....23.....4..5.......6..78..7.894.6.68...19..89....144.6.....771.9..68. ED=11.6/1.2/1.2 ........1....23.....4..5.....6.....7.89....1471.9..68...7.8.4.6.68...19.94..6..78 ED=11.6/1.2/1.2 ........1....23.....4..5.....6....78.9....1.481.7..96...8.9..46.69...71.74..6.8.9 ED=11.6/1.2/1.2 ........1....23.....4..5.....6.7..48.87...91.9...8.6.7.7....1.44.8....9661.9..78. ED=11.6/1.2/1.2 ........1.......23....45.....6..7.8..7859....94..68.7..89.54...4.7..6...65.78.9.. ED=11.6/1.2/1.2 ........1.......23....45.....4.67.8..6...8.79.7859.....97.54...48...6...6.587.9.. ED=11.6/1.2/1.2 ........1.......23....45.....6..7.8..7859....94..68.7..8..54...4.79.6...65.78.9.. ED=11.6/1.2/1.2 ........1...234.....21.5.....6.7.28..87....1992......6..87..69..79.....261.....78 ED=11.6/1.2/1.2 ........1...234.....21.5.....6.7.28..87....1992....7.6..87..69..79.....261......8 ED=11.6/1.2/1.2 .......12......345.....3..6..7..2....38.192...9.78.1...728.....31.29....8.9.37... ED=11.6/1.2/1.2 ........1......234.....2.56..7.86....9625....82.7.9....78.9.5..6.2..8...95...76.. ED=11.6/1.2/1.2 ........1......234.....2.5...6..7....28.197..79.68.1...6.8.....21.79....8.9.26..7 ED=11.6/1.2/1.2 ........1......234.....2.5...6.7.....8925....72.6.8.9..6798.5..85...69..9.2..7... ED=11.6/1.2/1.2 .......12......345.....3..6..7..2....38.19...29.78.1...7.8.....31.29....8.9.37.2. ED=11.6/1.2/1.2 ........1......234.....2.56..7.8.....9625....82.7.9.6..7869.5..6.2..8...95...7... ED=11.6/1.2/1.2 .......12......345.....1..6..748.6...86.19...94...7....69..4...4.879....71..68... ED=11.6/1.2/1.2 ........1......234.....2.56..7.86....9625....82.7.9....7869.5..6.2..8...95...7... ED=11.6/1.2/1.2 .......12......345....14.....64.78...78.91...94.86.....89......4.76.9.8.61..7.... ED=11.6/1.2/1.2 .......12......345....14.....64.78...78.91...94..6.....89......4.76.9.8.61..78... ED=11.6/1.2/1.2 .......12.....3..4..3....5...6.37.....85.97..79.86.5...6..8....35.97....8.93.6.7. ED=11.6/1.2/1.2 .......12.....3..4..3....5...6..7....38.597..79.68.5...6.8.....35.79....8.9.36.7. ED=11.6/1.2/1.2 ........1....12......345.....67..48..87....1994....7.6.79.....44.8.7.69.61......8 ED=11.6/1.2/1.2 ........1....12......345.....67..48..87....1994......6.79.....44.8.7.69.61.....78 ED=11.6/1.2/1.2 ........1......234....25.6...2.7.....7...8.96.9856.....69.52...28.6.7...7.598.... ED=11.6/1.2/1.2 ........1......234.....5.6...2.7.....7.2.8.96.9856.....69.52...28.6.7...7.598.... ED=11.6/1.2/1.2 ........1.....2.3.....45.....1....63.34.1.7.876.....4..43.8.6.717.....846.87..3.. ED=11.6/1.2/1.2 ........1.....2.3...3.45....1.....63.4..1.7.86.7....4..867..3..43..8.6.77.1....84 ED=11.6/1.2/1.2 ........1.....2.3...3.45....1.4...63.4..1.7.86.7.......867..3..43..816.77......84 ED=11.6/1.2/1.2 ........1.....2.3...3.45....1.4...63.4..1.7.86.7.......867..3..43..8.6.77.1....84 ED=11.6/1.2/1.2 ........1.....2.3...3.45....1.....63.4..1.7.86.7....4..867..3..43..816.77......84 ED=11.6/1.2/1.2 ........1.....2.3.....45.....14...63.34.1.7.876........43.8.6.717.....846.87..3.. ED=11.6/1.2/1.2 ........1.....2.3.....45.....1....63.34.1.7.876.....4..43.816.7.7.....846.87..3.. ED=11.6/1.2/1.2 ........1.....2.....3....24..5.67....782.3...26.58..3..36......5....8.738.7.356.2 ED=11.6/1.2/1.2 ........1.....2.....3....24..5.672...78..3...26.58..3..3672....5....8.738...356.. ED=11.6/1.2/1.2 ........1.....2.....3....24..5.672...78..3...26.58..3..367.....5....8.738...356.2 ED=11.6/1.2/1.2 ........1.....2.....3....24..5.67....782.3...26.58..3..367.....5....8.738...356.2 ED=11.6/1.2/1.2 ........1.....2.....3....24..5.672...78..3...26.58..3..36.2....5....8.738.7.356.. ED=11.6/1.2/1.2 ........1.....2345.23....6...2.67.....893....97...86...6.72.....8...9...3.7.86.92 ED=11.6/1.2/1.2 ........1......234....25.6...7..8....8629..7595...6....69.5....5.87.96..72..8.... ED=11.6/1.2/1.2 ........1.....2.34..2.1356......6.43....241.5.4.35..2.2.......67....5...8.9.61... ED=11.6/1.2/1.2 ........1.....2.34..2.1356.........6.7...5....89.61.......241.52....6.434..35..2. ED=11.6/1.2/1.2 ........1.....2.34..2.1356.........6.7..35....89.61........41.52....6.434..35..2. ED=11.6/1.2/1.2 ........1.....2.34..2.1356.........6.7..35....89.6.........41.52..1.6.434..35..2. ED=11.6/1.2/1.2 ........1......234....25.....6..2....758.69..82.79.....92.67...6.85.9...7...8..6. ED=11.6/1.2/1.2 ........1......234....25.....6.7..8..785.9...92..86....89..2...2.76.....65.7.89.. ED=11.6/1.2/1.2 ........1......234....25.....627..8..785.9...92..86....89......2.76.....65.7.89.. ED=11.6/1.2/1.2 ........1......234....25.....6.......758.69..82.79.....92.67...6.85.9...7..28..6. ED=11.6/1.2/1.2 ........1......234....25.....6..2....758.69..82.79.....92.67...6.85.....7...8..69 ED=11.6/1.2/1.2 ........1......234....25.....6.......758.69..82.79.....92.67...6.85.....7..28..69 ED=11.6/1.2/1.2 ........1......234....25.....2.67....685.9....7..8..69.96..2...28.7.....7.58.69.. ED=11.6/1.2/1.2 ........1......234....25.....2.67....685.9....7.28..69.96......28.7.....7.58.69.. ED=11.6/1.2/1.2 ........1......234.....2.5...6..7....28.197...9.68.1...678.....21..9....8.9.26..7 ED=11.6/1.2/1.2 ........1......234.....2.5...6.78....9.25....72.6.9.8..67.9.5..8.2..7...95...68.. ED=11.6/1.2/1.2 .......12......345.....3..6..7..2....38.19....9.78.1...728.....31..9....8.9.37.2. ED=11.6/1.2/1.2 ........1......234.....2.56..7.86....9.25....82.7.9.6..78.9.5..6.2..8...95...7... ED=11.6/1.2/1.2 .......12.....3..4..3....5...6.37.....85.97...9.86.5...67.8....35.9.....8.93.6.7. ED=11.6/1.2/1.2 .......12.....3..4..3....5...6..7....38.597...9.68.5...678.....35..9....8.9.36.7. ED=11.6/1.2/1.2 ........1...234.....21.5.....6.7.28..87....199.....7.6.79......2.87..69.61......8 ED=11.6/1.2/1.2 ........1...234.....21.5.....6.7.28..87....199.......6.79......2.87..69.61.....78 ED=11.6/1.2/1.2 ..............1.23.....24.5..6.......1789....82.1.69...7826..9.2.17.....69...8.7. ED=11.6/1.2/1.2 ..............1.23.....24.5.12.67...68.92....7.9..8.6..97......16..8....2.8.197.. ED=11.6/1.2/1.2 ..............1.23.....24.5..6.......1789....82.1.69...7826....2.17.9...69...8.7. ED=11.6/1.2/1.2 ........1.....2.....3....24..526.....7...8.63.8.3.75...375.62..52.87..3.6.8...... ED=11.6/1.2/1.2 ........1.....2.....3....24..5.6.....7...8.63.8.3.75.2.375.62..52.87..3.6.8...... ED=11.6/1.2/1.2 ........1.....2.....3....24..5.6.....7...8.63.8.3.75.2.375.6...52.87..3.6.8.2.... ED=11.6/1.2/1.2 ........1.....2.....3....24..5.......6...7.83.78.365.2.36.58...52.67..3.8.72..... ED=11.6/1.2/1.2 ........1.....2.....3....24..5.2.....6...7.83.78.365...36.582..52.67..3.8.7...... ED=11.6/1.2/1.2 ........1.....2345.36..........67.....389..2.78.2...9..2..3.....9.7.8...6.892.7.. ED=11.6/1.2/1.2 ..............1.23....45..6..7.......18..97..45..8.9...957.4...1.4.58..78..1....5 ED=11.6/1.2/1.2 ........1....23....24....56.4.7.8.9.2.7......98..3..4..78..9...3....4...49.28.7.. ED=11.6/1.2/1.2 ..............1.23....45..6..7.....5.41.57..895.8.4....8..1....17...98..5.4.7.9.. ED=11.6/1.2/1.2 ........1....23....24....56.4.7.8.9.2........98..3..47.78..9...3....4...49.28.7.. ED=11.6/1.2/1.2 ........1.....2.34....56..7..8.......29..18..56..9.1...168.5...2.5.69...9..2....6 ED=11.6/1.2/1.2 ........1.....2.34....56..7..8.....6.52.68...16.9.5....9..2....28...19..6.5.8.1.. ED=11.6/1.2/1.2 ........1.....2345.36..........6......378..2997.2...8..2..3.....8.9.7...6.782.9.. ED=11.6/1.2/1.2 ........1....23.....4..5....1......64.6.7..8.87....4.9.487...1.16.9..7.49......68 ED=11.6/1.2/1.2 ........1....23.....4..5....1.....674.7.6.18.8.....4.9.486.....17.9....49.6....78 ED=11.6/1.2/1.2 ........1....23.....4..5....1.....674.7.6..8.8.....4.9.486...1.17.9....49.6....78 ED=11.6/1.2/1.2 ........1....23.....4..5....1......64.6.7.18.87....4.9.487.....16.9..7.49......68 ED=11.6/1.2/1.2 ........1...234.....2..5.....6....7..89.....271.9...68..7.8.6...68....1992....8.7 ED=11.6/1.2/1.2 ........1....2345...21.4.36.....61.4...31..2.12..453.....65.....7....5..89....... ED=11.6/1.2/1.2 ........1..2..3.....1.45....1......62.6.7..8.87....2.9.287...1..6.9..7.29......68 ED=11.6/1.2/1.2 ........1..2..3.....1.45..........67.27.6.18..8....2.9.96....782.86.....71.9....2 ED=11.6/1.2/1.2 ........1....23.45..21.436.........6.7.4......896..........51.32..34....5...16.24 ED=11.6/1.2/1.2 ........1...234.....2..5.....6...78..9......271.8..6.9..7.9..6..69...1.882.....97 ED=11.6/1.2/1.2 ........1...234.....2..5..........67..7.6829..96...1.8.68......2.9...7..71.8..9.6 ED=11.6/1.2/1.2 ........1....2345...21.4.36.....61.4...31..2.12..453.....6......7....5..895...... ED=11.6/1.2/1.2 ........1...234.....2..5.....6...78..9......271.8..6.9..7.98.6..69...1..82.....97 ED=11.6/1.2/1.2 ........1....23.45..21.436......51.3...34.....5..16.242.......67..4.....8.96..... ED=11.6/1.2/1.2 ........1...234.....2..5.....6.7.28..87...1.99......76.79......2.8...6..61.9..8.7 ED=11.6/1.2/1.2 ........1....2345...21.4.36...2.61.4...31..2..1..453.....6.....7.....5..895...... ED=11.6/1.2/1.2 ........1.....2.3...435..62.......1..7...6...689.........6...2.1...23.545..41.6.3 ED=11.6/1.2/1.2 ........1.....2.3...435..62.......16.7...6....89.........6...2.1...23.545..41.6.3 ED=11.6/1.2/1.2 ........1.....234...3.5.2.........16..4...7..16....4.8..7.45....9.8...7.6..9..... ED=11.6/11.6/2.6

by **mith** » Wed Mar 16, 2022 4:39 pm

For further clarity on the SE rating difference, I reverted Loki to gsf minlex (still 11.9), added singles to both puzzles without morphing, and compared the solve paths.

Mapping between puzzles: digits 6<->8; rows 4<->8, 5<->7, 6<->9; columns 2<->3

Code: Select all: ........1.....2.......3..458.6.......71.8....23..67..8.827.61..61..23...7.381.6.. ED=11.9/11.8/3.4 3.4, Hidden Pair: Cells R4C4,R5C4: 2,3 in block: r4c4<>1,4,5,9, r5c4<>4,5,9 9.2, Contradiction Forcing Chain (w/48 nodes): R4C6.9 on ==> R5C1.5 both on & off: r4c6<>9 10.2, Region Forcing Chains (w/48 nodes): 1 in row ==> R6C8.5 off: r6c8<>5 9.6, Contradiction Forcing Chain (w/139 nodes): R4C6.5 on ==> R1C8.6 both on & off: r4c6<>5 10.4, Contradiction Forcing Chain (w/78 nodes): R1C1.4 on ==> R6C4.4 both on & off: r1c1<>4 10.4, Contradiction Forcing Chain (w/76 nodes): R1C1.5 on ==> R6C4.5 both on & off: r1c1<>5 10.6, Contradiction Forcing Chain (w/140 nodes): R2C4.4 on ==> R5C6.5 both on & off: r2c4<>4 10.4, Contradiction Forcing Chain (w/77 nodes): R1C2.4 on ==> R8C4.4 both on & off: r1c2<>4 11.0, Contradiction Forcing Chain (w/126 nodes): R4C9.4 on ==> R6C3.4 both on & off: r4c9<>4 11.1, Contradiction Forcing Chain (w/139 nodes): R4C8.5 on ==> R6C3.5 both on & off: r4c8<>5 11.1, Contradiction Forcing Chain (w/148 nodes): R1C3.5 on ==> R9C6.5 both on & off: r1c3<>5 10.4, Contradiction Forcing Chain (w/78 nodes): R2C5.5 on ==> R8C4.5 both on & off: r2c5<>5 11.1, Contradiction Forcing Chain (w/158 nodes): R5C9.4 on ==> R6C3.4 both on & off: r5c9<>4 2.6, Pointing: Cells R4C7,R5C7,R6C7: 4 in block and column: r8c7<>4 11.2, Contradiction Forcing Chain (w/195 nodes): R2C4.9 on ==> R9C6.5 both on & off: r2c4<>9 11.2, Contradiction Forcing Chain (w/218 nodes): R4C7.2 on ==> R6C3.9 both on & off: r4c7<>2 11.5, Contradiction Forcing Chain (w/128 nodes): R1C2.9 on ==> R9C6.9 both on & off: r1c2<>9 11.8, Contradiction Forcing Chain (w/303 nodes): R1C8.3 on ==> R7C8.5 both on & off: r1c8<>3 11.8, Contradiction Forcing Chain (w/325 nodes): R2C1.3 on ==> R6C3.9 both on & off: r2c1<>3 1.2, Hidden Single: R1C1: 3 in block 1: r1c1=3 11.9, Contradiction Forcing Chain (w/418 nodes): R2C3.7 on ==> R4C2.9 both on & off: r2c3<>7 11.9, Contradiction Forcing Chain (w/446 nodes): R1C3.9 on ==> R6C3.5 both on & off: r1c3<>9

Code: Select all: ........1.....2.......3..458.1.23....267.81..73.61.8...17.6....68.......2.3.87..6 ED=11.8/11.8/3.4 3.4, Hidden Pair: Cells R7C4,R8C4: 2,3 in block: r7c4<>4,5,9, r8c4<>1,4,5,9 9.2, Contradiction Forcing Chain (w/48 nodes): R8C6.9 on ==> R7C1.5 both on & off: r8c6<>9 10.2, Region Forcing Chains (w/48 nodes): 1 in row ==> R9C8.5 off: r9c8<>5 9.6, Contradiction Forcing Chain (w/139 nodes): R8C6.5 on ==> R1C8.8 both on & off: r8c6<>5 10.4, Contradiction Forcing Chain (w/78 nodes): R1C1.4 on ==> R9C2.4 both on & off: r1c1<>4 10.4, Contradiction Forcing Chain (w/76 nodes): R1C1.5 on ==> R9C2.5 both on & off: r1c1<>5 10.6, Contradiction Forcing Chain (w/140 nodes): R2C4.4 on ==> R7C6.5 both on & off: r2c4<>4 10.4, Contradiction Forcing Chain (w/77 nodes): R1C3.4 on ==> R4C4.4 both on & off: r1c3<>4 11.0, Contradiction Forcing Chain (w/126 nodes): R8C9.4 on ==> R9C2.4 both on & off: r8c9<>4 11.1, Contradiction Forcing Chain (w/139 nodes): R8C8.5 on ==> R9C2.5 both on & off: r8c8<>5 11.1, Contradiction Forcing Chain (w/143 nodes): R1C2.5 on ==> R9C4.5 both on & off: r1c2<>5 10.4, Contradiction Forcing Chain (w/78 nodes): R2C5.5 on ==> R4C4.5 both on & off: r2c5<>5 11.1, Contradiction Forcing Chain (w/158 nodes): R4C7.4 on ==> R9C2.4 both on & off: r4c7<>4 2.6, Pointing: Cells R4C9,R5C9,R6C9: 4 in block and column: r7c9<>4 11.2, Contradiction Forcing Chain (w/195 nodes): R2C4.9 on ==> R6C6.5 both on & off: r2c4<>9 11.2, Contradiction Forcing Chain (w/218 nodes): R8C7.2 on ==> R4C2.5 both on & off: r8c7<>2 11.5, Contradiction Forcing Chain (w/128 nodes): R1C3.9 on ==> R6C6.9 both on & off: r1c3<>9 11.8, Contradiction Forcing Chain (w/303 nodes): R1C8.3 on ==> R5C8.5 both on & off: r1c8<>3 11.8, Contradiction Forcing Chain (w/320 nodes): R1C8.2 on ==> R5C5.5 both on & off: r1c8<>2 2.6, Pointing: Cells R1C7,R3C7: 2 in block and column: r7c7<>2 11.1, Contradiction Forcing Chain (w/149 nodes): R2C1.3 on ==> R9C2.9 both on & off: r2c1<>3 1.2, Hidden Single: R1C1: 3 in block 1: r1c1=3

There are some minor differences before the 11.8 steps:

Steps 5-6 give the same eliminations at the same complexity, using a different column for the contradiction.
Step 11 gives the same elimination at different complexity (148 vs. 143 nodes) and with a different contradiction.
Step 13 gives different eliminations at the same complexity, but step 14 provides the other elimination via pointing so the pencilmark grids return to the same state.
Step 16 gives the same elimination at the same complexity, using a different row for the contradiction.
Step 18, the first 11.8 step, is identical.
Finally at Step 19 they diverge. In the 11.9, step 19 eliminates the same digit as step 18 in a different cell, directly giving the hidden single on that digit. In the 11.8, step 19 eliminates a different digit from the same cell, resulting in pointing on the second digit, followed by an 11.1 eliminating the same candidate as the second 11.8 in the first puzzle.

So the only relevant differences in these puzzles after the 30th digit is placed is that 2 is eliminated from r1c8 and r7c7. However, these eliminations are not found at 11.8 or less in the first puzzle, as you would expect - so either the less complex version of the -2r1c8 chain is pruned at this point, or something else screwy is going on to actually increase the complexity of that chain despite the extra digit.

by **1to9only** » Wed Mar 16, 2022 6:28 pm

mith wrote:For further clarity on the SE rating difference, I reverted Loki to gsf minlex (still 11.9), added singles to both puzzles without morphing, and compared the solve paths.

This is probably related to an SE bug first found in 2010: http://forum.enjoysudoku.com/help-with-sudoku-explainer-t6677.html#p201789

lksudoku (partially) fixed this problem: http://forum.enjoysudoku.com/help-with-sudoku-explainer-t6677-15.html#p204072

I recently mentioned the lksudoku fix here: http://forum.enjoysudoku.com/pgexplainer-a-minimal-sudokuexplainer-in-56-712-bytes-t39049-15.html#p317486

1to9only wrote:The lksudoku fix is only applied to the nested parts of chains in dynamic and multiple chains.
A version of lksudoku fix must also be applied to the final chain when determining the chain to be selected/applied.

I have a version of SE with an improved lksudoku fix, but this also has one issue, and I've not investigated further.

Wrt to the last 2 Loki gsf minlex'ed puzzles, the 'improved' lksudoku fix rates both as ED=11.8/11.8/3.4 with similarities in the solving path:

Code: Select all: ........1.....2.......3..458.6.......71.8....23..67..8.827.61..61..23...7.381.6.. ED=11.9/11.8/3.4

Solution Path: Show

Code: Select all: ........1.....2.......3..458.6.......71.8....23..67..8.827.61..61..23...7.381.6.. ED=11.9/11.8/3.4 3.4, Hidden Pair: Cells R4C4,R5C4: 2,3 in block: r4c4<>1,4,5,9, r5c4<>4,5,9 9.2, Contradiction Forcing Chain (w/48 nodes): R4C6.9 on ==> R5C1.5 both on & off: r4c6<>9 10.2, Region Forcing Chains (w/48 nodes): 1 in row ==> R6C8.5 off: r6c8<>5 9.6, Contradiction Forcing Chain (w/139 nodes): R4C6.5 on ==> R1C8.6 both on & off: r4c6<>5 10.4, Contradiction Forcing Chain (w/78 nodes): R1C1.4 on ==> R6C4.4 both on & off: r1c1<>4 10.4, Contradiction Forcing Chain (w/76 nodes): R1C1.5 on ==> R6C4.5 both on & off: r1c1<>5 10.6, Contradiction Forcing Chain (w/140 nodes): R2C4.4 on ==> R5C6.5 both on & off: r2c4<>4 10.4, Contradiction Forcing Chain (w/77 nodes): R1C2.4 on ==> R8C4.4 both on & off: r1c2<>4 11.0, Contradiction Forcing Chain (w/120 nodes): R4C9.4 on ==> R6C3.4 both on & off: r4c9<>4 11.1, Contradiction Forcing Chain (w/137 nodes): R4C8.5 on ==> R6C3.5 both on & off: r4c8<>5 11.1, Contradiction Forcing Chain (w/163 nodes): R1C3.5 on ==> R8C4.5 both on & off: r1c3<>5 10.4, Contradiction Forcing Chain (w/78 nodes): R2C5.5 on ==> R8C4.5 both on & off: r2c5<>5 11.1, Contradiction Forcing Chain (w/177 nodes): R8C7.4 on ==> R6C3.4 both on & off: r8c7<>4 2.6, Pointing: Cells R7C9,R8C9,R9C9: 4 in block and column: r5c9<>4 11.1, Contradiction Forcing Chain (w/180 nodes): R2C4.9 on ==> R6C4.5 both on & off: r2c4<>9 11.1, Contradiction Forcing Chain (w/179 nodes): R4C7.2 on ==> R5C1.4 both on & off: r4c7<>2 11.5, Contradiction Forcing Chain (w/128 nodes): R1C2.9 on ==> R9C6.9 both on & off: r1c2<>9 11.8, Contradiction Forcing Chain (w/303 nodes): R1C8.3 on ==> R7C8.5 both on & off: r1c8<>3 11.8, Contradiction Forcing Chain (w/320 nodes): R2C1.3 on ==> R9C2.9 both on & off: r2c1<>3 1.2, Hidden Single: R1C1: 3 in block 1: r1c1=3 11.8, Contradiction Forcing Chain (w/382 nodes): R2C3.9 on ==> R4C2.5 both on & off: r2c3<>9 11.8, Contradiction Forcing Chain (w/382 nodes): R1C3.9 on ==> R4C2.5 both on & off: r1c3<>9 11.6, Contradiction Forcing Chain (w/132 nodes): R3C6.9 on ==> R8C3.9 both on & off: r3c6<>9 9.2, Region Forcing Chains (w/36 nodes): 1 in block ==> R3C3.8 off: r3c3<>8 11.0, Contradiction Forcing Chain (w/116 nodes): R2C3.7 on ==> R8C4.9 both on & off: r2c3<>7 11.6, Contradiction Forcing Chain (w/180 nodes): R2C5.9 on ==> R5C6.9 both on & off: r2c5<>9 10.6, Contradiction Forcing Chain (w/140 nodes): R2C9.7 on ==> R4C5.5 both on & off: r2c9<>7 10.6, Contradiction Forcing Chain (w/168 nodes): R2C7.7 on ==> R8C3.5 both on & off: r2c7<>7 11.0, Contradiction Forcing Chain (w/109 nodes): R1C7.9 on ==> R9C2.9 both on & off: r1c7<>9 11.7, Contradiction Forcing Chain (w/241 nodes): R7C1.9 on ==> R5C1.5 both on & off: r7c1<>9 10.6, Contradiction Forcing Chain (w/129 nodes): R8C3.4 on ==> R5C1.9 both on & off: r8c3<>4 10.3, Region Forcing Chains (w/54 nodes): 4 in column ==> R7C5.4 off: r7c5<>4 9.1, Contradiction Forcing Chain (w/30 nodes): R2C1.4 on ==> R5C6.4 both on & off: r2c1<>4 9.7, Region Forcing Chains (w/35 nodes): 4 in row ==> R1C8.2 off: r1c8<>2 2.6, Pointing: Cells R1C7,R3C7: 2 in block and column: r5c7<>2 10.0, Contradiction Forcing Chain (w/103 nodes): R5C9.9 on ==> R7C5.9 both on & off: r5c9<>9 10.2, Region Forcing Chains (w/41 nodes): 4 in row ==> R4C7.4 off: r4c7<>4 9.2, Contradiction Forcing Chain (w/48 nodes): R1C3.7 on ==> R7C5.9 both on & off: r1c3<>7 1.2, Hidden Single: R3C3: 7 in block 1: r3c3=7 9.0, Cell Forcing Chains (w/18 nodes): R5C1 ==> R4C2.5 off: r4c2<>5 9.0, Contradiction Forcing Chain (w/17 nodes): R8C4.9 on ==> R6C7.4 both on & off: r8c4<>9 9.0, Cell Forcing Chains (w/18 nodes): R9C2 ==> R9C8.5 off: r9c8<>5 9.0, Contradiction Forcing Chain (w/18 nodes): R9C2.9 on ==> R5C6.4 both on & off: r9c2<>9 1.2, Hidden Single: R8C3: 9 in block 7: r8c3=9 8.5, Cell Forcing Chains (w/18 nodes): R6C4 ==> R4C5.9 off: r4c5<>9 8.3, Cell Forcing Chains (w/10 nodes): R5C1 ==> R6C4.4 off: r6c4<>4 8.3, Cell Forcing Chains (w/12 nodes): R9C6 ==> R6C4.5 off: r6c4<>5 3.0, Naked Pair: Cells R6C4,R6C8: 1,9 in row: r6c7<>9 7.1, Forcing Chain (w/6 nodes): R3C4.9 off: r3c4<>9 2.6, Pointing: Cells R1C4,R1C5,R1C6: 9 in block and row: r1c8<>9 8.3, Cell Forcing Chains (w/12 nodes): R5C1 ==> R7C5.5 off: r7c5<>5 2.0, Direct Hidden Pair: Cells R8C4,R9C6: 4,5 in block: r9c6<>9, r7c5=9 3.0, Naked Pair: Cells R9C2,R9C6: 4,5 in row: r9c9<>4 6.6, Turbot Fish (w/4 nodes): R5C6.4 off: r5c6<>4 2.6, Pointing: Cells R4C5,R4C6: 4 in block and row: r4c2<>4 2.0, Direct Hidden Pair: Cells R5C1,R6C3: 4,5 in block: r5c1<>9, r4c2=9 3.0, Naked Pair: Cells R5C1,R7C1: 4,5 in column: r2c1<>5 6.6, Turbot Fish (w/4 nodes): R5C6.5 off: r5c6<>5 1.2, Hidden Single: R4C5: 5 in block 5: r4c5=5 1.2, Hidden Single: R4C6: 4 in block 5: r4c6=4 1.2, Hidden Single: R6C4: 1 in block 5: r6c4=1 1.2, Hidden Single: R3C6: 1 in block 2: r3c6=1 1.2, Hidden Single: R2C1: 1 in block 1: r2c1=1 1.2, Hidden Single: R3C1: 9 in block 1: r3c1=9 1.2, Hidden Single: R1C6: 8 in block 2: r1c6=8 1.2, Hidden Single: R2C3: 8 in block 1: r2c3=8 1.2, Hidden Single: R1C4: 9 in block 2: r1c4=9 1.2, Hidden Single: R2C4: 5 in block 2: r2c4=5 1.2, Hidden Single: R1C2: 5 in block 1: r1c2=5 1.2, Hidden Single: R3C2: 2 in block 1: r3c2=2 1.2, Hidden Single: R2C2: 6 in block 1: r2c2=6 1.0, Hidden Single: R1C3: 4 in block 1: r1c3=4 1.0, Hidden Single: R9C2: 4 in column 2: r9c2=4 1.0, Hidden Single: R7C1: 5 in block 7: r7c1=5 1.0, Hidden Single: R5C1: 4 in column 1: r5c1=4 1.0, Hidden Single: R6C3: 5 in block 4: r6c3=5 1.2, Hidden Single: R2C5: 4 in block 2: r2c5=4 1.0, Hidden Single: R1C5: 7 in column 5: r1c5=7 1.0, Hidden Single: R3C4: 6 in block 2: r3c4=6 1.0, Hidden Single: R3C7: 8 in row 3: r3c7=8 1.2, Hidden Single: R1C7: 2 in block 3: r1c7=2 1.0, Hidden Single: R1C8: 6 in row 1: r1c8=6 1.2, Hidden Single: R2C8: 7 in block 3: r2c8=7 1.2, Hidden Single: R5C6: 9 in block 5: r5c6=9 1.0, Hidden Single: R9C6: 5 in column 6: r9c6=5 1.0, Hidden Single: R8C4: 4 in block 8: r8c4=4 1.2, Hidden Single: R4C8: 1 in block 6: r4c8=1 1.2, Hidden Single: R6C7: 4 in block 6: r6c7=4 1.0, Hidden Single: R6C8: 9 in row 6: r6c8=9 1.2, Hidden Single: R5C9: 6 in block 6: r5c9=6 1.2, Hidden Single: R7C9: 4 in block 9: r7c9=4 1.0, Hidden Single: R7C8: 3 in row 7: r7c8=3 1.2, Hidden Single: R8C8: 8 in block 9: r8c8=8 1.2, Hidden Single: R8C7: 5 in block 9: r8c7=5 1.0, Hidden Single: R8C9: 7 in row 8: r8c9=7 1.2, Hidden Single: R5C8: 5 in block 6: r5c8=5 1.0, Hidden Single: R9C8: 2 in column 8: r9c8=2 1.0, Hidden Single: R9C9: 9 in block 9: r9c9=9 1.2, Hidden Single: R2C7: 9 in block 3: r2c7=9 1.0, Hidden Single: R2C9: 3 in block 3: r2c9=3 1.0, Hidden Single: R4C9: 2 in column 9: r4c9=2 1.2, Hidden Single: R5C4: 2 in block 5: r5c4=2 1.0, Hidden Single: R4C4: 3 in block 5: r4c4=3 1.0, Hidden Single: R4C7: 7 in row 4: r4c7=7 1.0, Hidden Single: R5C7: 3 in block 6: r5c7=3 ED=11.8/11.8/3.4

Code: Select all: ........1.....2.......3..458.1.23....267.81..73.61.8...17.6....68.......2.3.87..6 ED=11.8/11.8/3.4

Solution Path: Show

Code: Select all: ........1.....2.......3..458.1.23....267.81..73.61.8...17.6....68.......2.3.87..6 ED=11.8/11.8/3.4 3.4, Hidden Pair: Cells R7C4,R8C4: 2,3 in block: r7c4<>4,5,9, r8c4<>1,4,5,9 9.2, Contradiction Forcing Chain (w/48 nodes): R8C6.9 on ==> R7C1.5 both on & off: r8c6<>9 10.2, Region Forcing Chains (w/48 nodes): 1 in row ==> R9C8.5 off: r9c8<>5 9.6, Contradiction Forcing Chain (w/139 nodes): R8C6.5 on ==> R1C8.8 both on & off: r8c6<>5 10.4, Contradiction Forcing Chain (w/78 nodes): R1C1.4 on ==> R9C4.4 both on & off: r1c1<>4 10.4, Contradiction Forcing Chain (w/76 nodes): R1C1.5 on ==> R9C4.5 both on & off: r1c1<>5 10.6, Contradiction Forcing Chain (w/140 nodes): R2C4.4 on ==> R7C6.5 both on & off: r2c4<>4 10.4, Contradiction Forcing Chain (w/77 nodes): R1C3.4 on ==> R4C4.4 both on & off: r1c3<>4 11.0, Contradiction Forcing Chain (w/120 nodes): R8C9.4 on ==> R9C2.4 both on & off: r8c9<>4 11.1, Contradiction Forcing Chain (w/137 nodes): R8C8.5 on ==> R9C2.5 both on & off: r8c8<>5 11.1, Contradiction Forcing Chain (w/163 nodes): R1C2.5 on ==> R9C4.5 both on & off: r1c2<>5 10.4, Contradiction Forcing Chain (w/78 nodes): R2C5.5 on ==> R4C4.5 both on & off: r2c5<>5 11.1, Contradiction Forcing Chain (w/177 nodes): R4C7.4 on ==> R8C3.4 both on & off: r4c7<>4 2.6, Pointing: Cells R4C9,R5C9,R6C9: 4 in block and column: r7c9<>4 11.1, Contradiction Forcing Chain (w/180 nodes): R2C4.9 on ==> R9C4.5 both on & off: r2c4<>9 11.1, Contradiction Forcing Chain (w/179 nodes): R8C7.2 on ==> R7C1.4 both on & off: r8c7<>2 11.5, Contradiction Forcing Chain (w/128 nodes): R1C3.9 on ==> R6C6.9 both on & off: r1c3<>9 11.8, Contradiction Forcing Chain (w/303 nodes): R1C8.3 on ==> R5C8.5 both on & off: r1c8<>3 11.8, Contradiction Forcing Chain (w/325 nodes): R2C1.3 on ==> R9C2.9 both on & off: r2c1<>3 1.2, Hidden Single: R1C1: 3 in block 1: r1c1=3 11.8, Contradiction Forcing Chain (w/380 nodes): R2C2.9 on ==> R4C2.5 both on & off: r2c2<>9 11.8, Contradiction Forcing Chain (w/380 nodes): R1C2.9 on ==> R4C2.5 both on & off: r1c2<>9 11.5, Contradiction Forcing Chain (w/127 nodes): R3C6.9 on ==> R4C2.9 both on & off: r3c6<>9 9.2, Region Forcing Chains (w/36 nodes): 1 in block ==> R3C2.6 off: r3c2<>6 11.0, Contradiction Forcing Chain (w/116 nodes): R2C2.7 on ==> R9C4.9 both on & off: r2c2<>7 11.6, Contradiction Forcing Chain (w/180 nodes): R2C5.9 on ==> R7C6.9 both on & off: r2c5<>9 10.6, Contradiction Forcing Chain (w/140 nodes): R2C9.7 on ==> R8C5.5 both on & off: r2c9<>7 10.6, Contradiction Forcing Chain (w/168 nodes): R2C7.7 on ==> R4C2.5 both on & off: r2c7<>7 11.0, Contradiction Forcing Chain (w/109 nodes): R1C7.9 on ==> R6C3.9 both on & off: r1c7<>9 11.7, Contradiction Forcing Chain (w/241 nodes): R5C1.9 on ==> R7C1.5 both on & off: r5c1<>9 10.6, Contradiction Forcing Chain (w/129 nodes): R4C2.4 on ==> R7C1.9 both on & off: r4c2<>4 10.3, Region Forcing Chains (w/54 nodes): 4 in column ==> R5C5.4 off: r5c5<>4 9.1, Contradiction Forcing Chain (w/30 nodes): R2C1.4 on ==> R7C6.4 both on & off: r2c1<>4 9.7, Region Forcing Chains (w/35 nodes): 4 in row ==> R1C8.2 off: r1c8<>2 2.6, Pointing: Cells R1C7,R3C7: 2 in block and column: r7c7<>2 10.0, Contradiction Forcing Chain (w/103 nodes): R7C9.9 on ==> R5C5.9 both on & off: r7c9<>9 10.2, Region Forcing Chains (w/41 nodes): 4 in row ==> R8C7.4 off: r8c7<>4 9.2, Contradiction Forcing Chain (w/48 nodes): R1C2.7 on ==> R5C5.9 both on & off: r1c2<>7 1.2, Hidden Single: R3C2: 7 in block 1: r3c2=7 9.0, Cell Forcing Chains (w/18 nodes): R6C3 ==> R8C3.5 off: r8c3<>5 9.0, Contradiction Forcing Chain (w/17 nodes): R4C4.9 on ==> R9C7.4 both on & off: r4c4<>9 9.0, Cell Forcing Chains (w/18 nodes): R6C3 ==> R6C8.5 off: r6c8<>5 9.0, Contradiction Forcing Chain (w/18 nodes): R4C2.9 off ==> R7C6.4 both on & off: r4c2<>5, r4c2=9 8.5, Cell Forcing Chains (w/18 nodes): R9C4 ==> R8C5.9 off: r8c5<>9 8.3, Cell Forcing Chains (w/10 nodes): R7C1 ==> R9C4.4 off: r9c4<>4 8.3, Cell Forcing Chains (w/12 nodes): R6C6 ==> R9C4.5 off: r9c4<>5 3.0, Naked Pair: Cells R9C4,R9C8: 1,9 in row: r9c7<>9 7.1, Forcing Chain (w/6 nodes): R3C4.9 off: r3c4<>9 2.6, Pointing: Cells R1C4,R1C5,R1C6: 9 in block and row: r1c8<>9 8.3, Cell Forcing Chains (w/12 nodes): R7C1 ==> R5C5.5 off: r5c5<>5 2.0, Direct Hidden Pair: Cells R4C4,R6C6: 4,5 in block: r6c6<>9, r5c5=9 3.0, Naked Pair: Cells R6C3,R6C6: 4,5 in row: r6c9<>4 6.6, Turbot Fish (w/4 nodes): R7C6.4 off: r7c6<>4 2.6, Pointing: Cells R8C5,R8C6: 4 in block and row: r8c3<>4 2.0, Direct Hidden Pair: Cells R7C1,R9C2: 4,5 in block: r7c1<>9, r8c3=9 3.0, Naked Pair: Cells R5C1,R7C1: 4,5 in column: r2c1<>5 6.6, Turbot Fish (w/4 nodes): R7C6.5 off: r7c6<>5 1.2, Hidden Single: R8C5: 5 in block 8: r8c5=5 1.2, Hidden Single: R8C6: 4 in block 8: r8c6=4 1.2, Hidden Single: R4C4: 4 in block 5: r4c4=4 1.0, Hidden Single: R6C6: 5 in block 5: r6c6=5 1.2, Hidden Single: R5C1: 5 in block 4: r5c1=5 1.0, Hidden Single: R6C3: 4 in block 4: r6c3=4 1.2, Hidden Single: R5C9: 4 in block 6: r5c9=4 1.0, Hidden Single: R5C8: 3 in row 5: r5c8=3 1.2, Hidden Single: R9C2: 5 in block 7: r9c2=5 1.0, Hidden Single: R7C1: 4 in block 7: r7c1=4 1.2, Hidden Single: R9C4: 1 in block 8: r9c4=1 1.2, Hidden Single: R3C6: 1 in block 2: r3c6=1 1.2, Hidden Single: R2C1: 1 in block 1: r2c1=1 1.0, Hidden Single: R3C1: 9 in column 1: r3c1=9 1.2, Hidden Single: R1C6: 6 in block 2: r1c6=6 1.0, Hidden Single: R7C6: 9 in column 6: r7c6=9 1.2, Hidden Single: R2C2: 6 in block 1: r2c2=6 1.0, Hidden Single: R1C2: 4 in column 2: r1c2=4 1.2, Hidden Single: R2C5: 4 in block 2: r2c5=4 1.0, Hidden Single: R1C5: 7 in column 5: r1c5=7 1.2, Hidden Single: R1C4: 9 in block 2: r1c4=9 1.2, Hidden Single: R2C4: 5 in block 2: r2c4=5 1.0, Hidden Single: R3C4: 8 in block 2: r3c4=8 1.2, Hidden Single: R1C3: 5 in block 1: r1c3=5 1.2, Hidden Single: R3C3: 2 in block 1: r3c3=2 1.0, Hidden Single: R2C3: 8 in block 1: r2c3=8 1.0, Hidden Single: R3C7: 6 in row 3: r3c7=6 1.2, Hidden Single: R1C7: 2 in block 3: r1c7=2 1.0, Hidden Single: R1C8: 8 in row 1: r1c8=8 1.2, Hidden Single: R2C8: 7 in block 3: r2c8=7 1.2, Hidden Single: R4C8: 6 in block 6: r4c8=6 1.2, Hidden Single: R4C7: 5 in block 6: r4c7=5 1.0, Hidden Single: R4C9: 7 in row 4: r4c9=7 1.2, Hidden Single: R8C8: 1 in block 9: r8c8=1 1.2, Hidden Single: R9C7: 4 in block 9: r9c7=4 1.0, Hidden Single: R9C8: 9 in row 9: r9c8=9 1.2, Hidden Single: R6C9: 9 in block 6: r6c9=9 1.0, Hidden Single: R6C8: 2 in block 6: r6c8=2 1.0, Hidden Single: R7C8: 5 in column 8: r7c8=5 1.2, Hidden Single: R2C7: 9 in block 3: r2c7=9 1.0, Hidden Single: R2C9: 3 in block 3: r2c9=3 1.2, Hidden Single: R8C7: 7 in block 9: r8c7=7 1.0, Hidden Single: R7C7: 3 in column 7: r7c7=3 1.2, Hidden Single: R8C4: 3 in block 8: r8c4=3 1.0, Hidden Single: R7C4: 2 in block 8: r7c4=2 1.0, Hidden Single: R7C9: 8 in row 7: r7c9=8 1.0, Hidden Single: R8C9: 2 in block 9: r8c9=2 ED=11.8/11.8/3.4

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