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[mith]

Starting a new thread for puzzles which are not in T&E(2,singles), which is where my current searching is focused. I am continuing to rate these puzzles by SER - however, the SER is not particularly meaningful for these puzzles. Rather, all known puzzles so far have potential trivalue oddagon deductions, of varying usefulness complexity (the simple single-guardian case, which immediately places a digit, vs. up to 10 guardian cells spread across all four boxes of the chromatic pattern).

I am still running some scripts on the latest update, but I will go ahead and link those puzzles here.

Update as of 2022-11-06

The database "expanded_te3.db" currently holds 847778 depth 3 expanded forms*, with clue counts ranging from 24c-40c.

expanded_te3_20221106
unix format
dos format

Note: The IDs for these puzzles go up to 847781 - there are three skipped IDs (155299-155301) due to a duplication issue in the first update, this will be retained for subsequent updates in order to preserve consistency in IDs.

From this database, the min-expand** and max-expand*** puzzles have been determined.

min_expands_20221106 (158276 puzzles)
unix format
dos format

max_expands_20221106 (48071 puzzles)
unix format
dos format

There are a total of 44251 solution grids represented in the database (most have exactly one max-expand representative, however not all "trees" converge to a single "parent" puzzle).

* expanded form - these are puzzles which may or may not be minimal, but which do not have any singles available; all minimals of such puzzles will have the same T&E depth WRT singles.
** min-expand - these are expanded forms which cannot have clues removed while remaining expanded forms; such puzzles again may or may not be minimal.
*** max-expand - these are expanded forms which cannot have clues added while remaining depth 3

[mith]

The following link contains trivalue oddagon and guardian count results for the latest update (20221106).

Trivalue Oddagons

There are three sets of files:

min_expand_20221106_singles_trivalue_oddagons*
This run determined, for each min-expand puzzle, all possible trivalue oddagon patterns (digit triples and cells).
The base file includes global counts for the puzzle (number of trivalue oddagons and the range of guardian_candidate_counts) along with the trivalue oddagon(s) with the minimum number of guardian candidates.
The _full file contains all valid trivalue oddagons, one per line.

min_expand_20221106_basics_special_only*
This run determined, for each min-expand puzzle, only the "special" trivalue oddagons meeting the criteria of the triple digits only appearing at most once and only in one box + the cells appearing in the four boxes which do not share a band/stack with that one box.
The output is in the same format.

max_expand_20221106_basics_special_only*
This run determined, for each max-expand puzzle, only the "special" trivalue oddagons meeting the criteria of the triple digits only appearing at most once and only in one box + the cells appearing in the four boxes which do not share a band/stack with that one box.
The output is in the same format.

Each file is provided in unix format (no file extension) and dos format (.csv).

Note that these results only give the possible choices of trivalue oddagon, and do not say anything about the number or complexity of deductions stemming from those choices (the count of guardian candidates simply gives the number of "branches" which would need to converge on the same elimination) nor how productive those deductions are in terms of solving the puzzle. See Denis' classification of these puzzles for a complete rating up to choice of resolution rules.

To do: Analysis of "special" trivalue oddagons vs. minimal guardian count choices - is it ever productive to consider non-special choices for additional eliminations?

[mith]

[Placeholder for minimals]

There are currently 3729054 minimals from the 847778 expanded forms, ranging from 21c-32c.

[denis_berthier]

.
Hi mith,
Congrats for this new leap forward, from 63,137 to 158,276 min-expands in T&E(3).
(As I explained before, in the analyses I'm making, I'm mainly interested in the min-expands; but having a large collection of minimals is also interesting per se.)

all known puzzles so far have potential trivalue oddagon deductions, of varying usefulness (the simple single-guardian case, which immediately places a digit, vs. up to 10 guardian cells spread across all four boxes of the chromatic pattern)

I think you're making a shortcut:
- of course, the number of guardians (and their distribution among the cells) is related to how easy it is to use the pattern for some elimination (the obvious case being 1 guardian or n guardians in a single cell), but
- number of guardians and (potential) usefulness are two very different things; this can easily be understood when you consider the general observation that, in any context, the elimination of a candidate has unpredictable effects;
- (potential) usefulness largely depends on which kind of rules one is ready to use, as shown by my results in the tridagon thread: http://forum.enjoysudoku.com/the-tridagon-rule-t39859.html
.

[mith]

Yes, that was a bit imprecise.

What I should have said is "varying complexity" (of the trivalue oddagon deduction itself), which of course depends on the number of guardians (but also on the complexity of the guardian => elimination for each guardian individually). Usefulness, of course, is completely unrelated to how complex a step is.

Currently running the TO finder script on the min_expands. Output looks like this:

Code: Select all

solution_minlex;tree;ID;valid_count;max_guardian_candidates;[for each valid pattern with min_guardian_candidate count]guardian_cells;guardian_candidates;digits;cells;[/for]
1.3.56....571.9...69.37......1.93..75.96.73.....51.......96..........4....5...86.;1;3;52;26;1;1;n248;b1p249+b2p159+b4p159+b5p159;
...4.678....18.2.6....72.14.15...6......41.....9..........68...7..2.48....271.46.;2;2;46;26;1;1;n359;b2p267+b3p357+b8p159+b9p159;
...4.6...4...89...68.37..4..68.479..73.......9.48.36..3.6....52....3..9.87....3..;3;105;18;22;1;1;n125;b1p159+b2p249+b4p168+b5p168;
...4.6...4...89...68.37..4..68.479..73.96....9.48.36..3......52....3..9.87....3..;3;7;15;20;1;1;n125;b1p159+b2p249+b4p168+b5p168;
...4.....4...89...68.37..4..68.479..73.96.4..9.48.36..3......52....3..9.87....3..;3;396;41;25;2;2;n125;b1p159+b2p249+b4p168+b5p168;

To a separate file, I am printing the full list of valid cell/triple choices, e.g.:

Code: Select all

1.3.56....571.9...69.37......1.93..75.96.73.....51.......96..........4....5...86.;1;3
1;1;n248;b1p249+b2p159+b4p159+b5p159
5;10;n248;b1p249+b2p159+b7p348+b8p348
5;11;n248;b1p249+b2p159+b7p357+b8p357
9;24;n267;b1p267+b3p267+b7p267+b9p168
10;25;n239;b1p348+b3p168+b7p267+b9p159
8;22;n359;b2p267+b3p159+b8p168+b9p159
6;11;n248;b4p159+b5p159+b7p348+b8p357
6;12;n248;b4p159+b5p159+b7p357+b8p348
10;26;n289;b4p267+b6p159+b7p267+b9p357
10;25;n289;b4p267+b6p267+b7p267+b9p357
10;23;n249;b4p267+b6p267+b7p348+b9p249
10;19;n246;b4p159+b6p159+b7p357+b9p348
10;25;n289;b4p267+b6p267+b7p357+b9p267
10;25;n289;b4p267+b6p168+b7p267+b9p267
10;25;n279;b4p267+b6p357+b7p267+b9p159
10;24;n279;b4p267+b6p357+b7p348+b9p159
10;25;n289;b4p168+b6p267+b7p267+b9p267
9;23;n258;b4p249+b6p159+b7p159+b9p267
8;23;n456;b4p249+b6p159+b7p159+b9p348
9;22;n258;b4p249+b6p267+b7p159+b9p267
8;23;n456;b4p249+b6p267+b7p159+b9p348
10;24;n128;b4p357+b6p159+b7p267+b9p267
10;23;n128;b4p357+b6p159+b7p348+b9p267
10;23;n128;b4p357+b6p267+b7p267+b9p267
10;22;n128;b4p357+b6p267+b7p348+b9p267
10;25;n289;b4p168+b6p168+b7p357+b9p267
9;22;n258;b4p249+b6p168+b7p249+b9p267
10;22;n128;b4p357+b6p168+b7p168+b9p267
10;23;n128;b4p357+b6p168+b7p357+b9p267
10;23;n249;b4p168+b6p168+b7p348+b9p249
9;23;n245;b4p249+b6p168+b7p159+b9p249
9;22;n258;b4p249+b6p168+b7p159+b9p357
10;22;n124;b4p357+b6p168+b7p267+b9p249
10;23;n128;b4p357+b6p168+b7p267+b9p357
10;21;n124;b4p357+b6p168+b7p348+b9p249
10;22;n128;b4p357+b6p168+b7p348+b9p357
10;26;n289;b4p168+b6p159+b7p357+b9p357
10;25;n289;b4p168+b6p267+b7p357+b9p357
9;24;n245;b4p249+b6p159+b7p249+b9p249
9;23;n258;b4p249+b6p159+b7p249+b9p357
9;22;n258;b4p249+b6p267+b7p249+b9p357
10;22;n124;b4p357+b6p159+b7p168+b9p249
10;23;n128;b4p357+b6p159+b7p168+b9p357
10;23;n124;b4p357+b6p159+b7p357+b9p249
10;24;n128;b4p357+b6p159+b7p357+b9p357
10;21;n124;b4p357+b6p267+b7p168+b9p249
10;22;n128;b4p357+b6p267+b7p168+b9p357
10;22;n124;b4p357+b6p267+b7p357+b9p249
10;23;n128;b4p357+b6p267+b7p357+b9p357
9;18;n127;b5p168+b6p357+b8p357+b9p159
9;21;n235;b5p357+b6p249+b8p357+b9p159
8;21;n358;b5p357+b6p249+b8p348+b9p357

It's going to take about a day to run - taking somewhere between 1/2 and 3/4 of a second per puzzle.

[denis_berthier]

Output looks like this:

Code: Select all

solution_minlex;tree;ID;valid_count;max_guardian_candidates;[for each valid pattern with min_guardian_candidate count]guardian_cells;guardian_candidates;digits;cells;[/for]
...4.....4...89...68.37..4..68.479..73.96.4..9.48.36..3......52....3..9.87....3..;3;396;41;25;2;2;n125;b1p159+b2p249+b4p168+b5p168;

tree = 3
ID = 396
valid_count = 41 (= "valid cell/triple count")
max_guardian_candidates = 25 (= the max nb of guardians in a valid cell/triple choice ??? - better named max_guardians_count
Why not a "min_guardians_count" slot?
The rest matches the stated pattern: "[for each valid pattern with min_guardian_candidate count]guardian_cells;guardian_candidates;digits;cells;[/for]" only if
min_guardians_count = 1
guardian_cells = 2, better named guardian_cells_count
guardian_candidates = 2, better named guardians_count

To a separate file, I am printing the full list of valid cell/triple choices, e.g.:

Code: Select all

1.3.56....571.9...69.37......1.93..75.96.73.....51.......96..........4....5...86.;1;3
1;1;n248;b1p249+b2p159+b4p159+b5p159
5;10;n248;b1p249+b2p159+b7p348+b8p348
5;11;n248;b1p249+b2p159+b7p357+b8p357
9;24;n267;b1p267+b3p267+b7p267+b9p168
10;25;n239;b1p348+b3p168+b7p267+b9p159
8;22;n359;b2p267+b3p159+b8p168+b9p159
6;11;n248;b4p159+b5p159+b7p348+b8p357
6;12;n248;b4p159+b5p159+b7p357+b8p348
10;26;n289;b4p267+b6p159+b7p267+b9p357
10;25;n289;b4p267+b6p267+b7p267+b9p357
10;23;n249;b4p267+b6p267+b7p348+b9p249
10;19;n246;b4p159+b6p159+b7p357+b9p348
10;25;n289;b4p267+b6p267+b7p357+b9p267
10;25;n289;b4p267+b6p168+b7p267+b9p267
10;25;n279;b4p267+b6p357+b7p267+b9p159
10;24;n279;b4p267+b6p357+b7p348+b9p159
10;25;n289;b4p168+b6p267+b7p267+b9p267
9;23;n258;b4p249+b6p159+b7p159+b9p267
8;23;n456;b4p249+b6p159+b7p159+b9p348
9;22;n258;b4p249+b6p267+b7p159+b9p267
8;23;n456;b4p249+b6p267+b7p159+b9p348
10;24;n128;b4p357+b6p159+b7p267+b9p267
10;23;n128;b4p357+b6p159+b7p348+b9p267
10;23;n128;b4p357+b6p267+b7p267+b9p267
10;22;n128;b4p357+b6p267+b7p348+b9p267
10;25;n289;b4p168+b6p168+b7p357+b9p267
9;22;n258;b4p249+b6p168+b7p249+b9p267
10;22;n128;b4p357+b6p168+b7p168+b9p267
10;23;n128;b4p357+b6p168+b7p357+b9p267
10;23;n249;b4p168+b6p168+b7p348+b9p249
9;23;n245;b4p249+b6p168+b7p159+b9p249
9;22;n258;b4p249+b6p168+b7p159+b9p357
10;22;n124;b4p357+b6p168+b7p267+b9p249
10;23;n128;b4p357+b6p168+b7p267+b9p357
10;21;n124;b4p357+b6p168+b7p348+b9p249
10;22;n128;b4p357+b6p168+b7p348+b9p357
10;26;n289;b4p168+b6p159+b7p357+b9p357
10;25;n289;b4p168+b6p267+b7p357+b9p357
9;24;n245;b4p249+b6p159+b7p249+b9p249
9;23;n258;b4p249+b6p159+b7p249+b9p357
9;22;n258;b4p249+b6p267+b7p249+b9p357
10;22;n124;b4p357+b6p159+b7p168+b9p249
10;23;n128;b4p357+b6p159+b7p168+b9p357
10;23;n124;b4p357+b6p159+b7p357+b9p249
10;24;n128;b4p357+b6p159+b7p357+b9p357
10;21;n124;b4p357+b6p267+b7p168+b9p249
10;22;n128;b4p357+b6p267+b7p168+b9p357
10;22;n124;b4p357+b6p267+b7p357+b9p249
10;23;n128;b4p357+b6p267+b7p357+b9p357
9;18;n127;b5p168+b6p357+b8p357+b9p159
9;21;n235;b5p357+b6p249+b8p357+b9p159
8;21;n358;b5p357+b6p249+b8p348+b9p357

It'd probably be better to write the first line as: 1.3.56....571.9...69.37......1.93..75.96.73.....51.......96..........4....5...86.;1;3;52
Some redundancy may be important to be understood.
What do the 1st 2 numbers on each mine mean?

To be completely clear: these calculations are done without using any resolution rule before them, right?

[mith]

max_guardian_candidates = 25 (= the max nb of guardians in a valid cell/triple choice ??? - better named max_guardians_count

Correct. (Re: Names, these are not included in the files themselves - I'll add them after the script is done - just keeping them compact for the examples.)

I'll do a more thorough write-up of this when I get the code on github, but to clarify what "valid" means:

• Iterate through all possible 12-cell trivalue oddagon patterns. Each of four boxes forming a rectangle (9 choices) contains three row- and column-spanning cells such that there is an odd (3 vs. 1) split in diagonal parities (8 choices - 2 for which parity has the majority * 4 for which box has the minority parity). After the previous choices there are 3 choices for cells per box (corresponding to which of p123 is included, for example), for a total of 5832 cell patterns.
• Iterate through all possible triples of digits (84 choices).
• For each box, the digits in the triple must be naively placeable in the pattern cells for that box - that is, there is a choice of digit <-> cell mapping such that each of the three cells contains the mapped digit as a candidate (this eliminates triples where a cell in the pattern doesn't have any of the triple as a candidate, but also cases where two of the digits are only placeable in the same single cell).
• For each box, the other cells in the box must not contain only candidates from the triple. I actually just noticed a minor bug here - it was only checking for cells that contained all three digits from the triple, rather than any subset. I've restarted the run (not a huge loss of progress, my computer restarted last night anyway so it was only about 25% complete), even though it won't matter a great deal. This is going to filter out a few more of the extremely bad choices - of the first five puzzles posted in the example above, the first has dropped to 51 valid and the fifth to 35, with the others unchanged.
• For each triple passing the previous filters, determine any fixed digits in each box (that is, of the valid placements, digits which only end up in one pattern cell), and rule out any pattern with the same fixed digit in the same row or column of two different boxes. (I'm no longer printing any stats on this, but IIRC this eliminated around 25% of choices.)
• If it passes all the filters, increment valid_count, add it to the list for printing in the "verbose" file, and check guardian (candidate) counts against current min and max. (Any matching the current minimum is also added to that list for potential printing in the "compact" file - this is reset whenever the minimum is lowered.)

Why not a "min_guardians_count" slot?

It's redundant - guardian_candidates will always match min_guardians_count (for that puzzle) because only the triple/cell choices that are min are printed. (But I've added it back in for the restart.)

What do the 1st 2 numbers on each mine mean?

The format of these lines matches the format from the compact version: guardian_cells;guardian_candidates;digits;cells

To be completely clear: these calculations are done without using any resolution rule before them, right?

Correct, this is from the min-expand (singles expanded) with candidates only eliminated directly by a given digit. I will probably run this with basics (subsets+whips[1]) later.

[mith]

As another example, here is the earliest puzzle with multiple minimal guardian candidate choices:

Code: Select all

1..4.6....5.....3..89....4.2...1...47....4.12......36....24..7..7..634....67.1..3;861;29033;39;7;27;6;7;n589;b5p357+b6p249+b8p348+b9p168;6;7;n589;b5p357+b6p249+b8p348+b9p357;

Assuming the minimal count will always correspond to the single choice of triple and boxes (where the digits in the triple only occur at most once in the grid and all in the box not sharing a row/column with the trivalue oddagon boxes), these can only occur when at least one of the boxes has two possible diagonal choices. I will separately do a run which only allows for this choice of triple/boxes for comparison purposes (the current run will capture all diagonal choices in the "verbose" file, but the "compact" file will only include multiple choices if the candidate count is the same for each).

[mith]

Another thing to filter is “degenerate” patterns - degenerate here in the sense of containing a smaller chromatic pattern.

This will again only matter for the non-minimal choices (the minimal choice is - apparently - always non-degenerate in a more specific sense of containing all three candidates in every cell of the pattern, which makes it non-degenerate in the above sense too), but in some cases a pattern is a “valid” trivalue oddagon cell/triple choice but has candidates missing in such a way as to contain a bivalue oddagon in some subset of cells.

Currently considering how best to efficiently look for these cases.

[mith]

Here's an interesting example (the current "worst case" for guardian candidate count in the best TO choice):

Code: Select all

....56.8....18....6..3.7.1...5.3....3.....6.7.........53....4.171....9.2....1..5.;10451;367518;203;15;33;
7;15;n249;b1p249+b2p168+b4p168+b5p159;
7;15;n249;b1p249+b2p168+b4p159+b5p357;
7;15;n249;b1p249+b2p168+b4p267+b5p357;

Not only are there three "best" choices because of the lack of digits in b45, there are different *parity* choices.

This particular example collapses after basics (leaving 5;7;n249;b1p249+b2p168+b4p159+b5p357; as the only best choice), but it's entirely possible we'll find one that doesn't.

[denis_berthier]

Here's an interesting example (the current "worst case" for guardian candidate count in the best TO choice):

Code: Select all

....56.8....18....6..3.7.1...5.3....3.....6.7.........53....4.171....9.2....1..5.

Not only are there three "best" choices because of the lack of digits in b45, there are different *parity* choices.
This particular example collapses after basics

I think what happens before Singles or whips[1] doesn't mean much.

In spite of a relatively low SER (10.2), this puzzle is hard; it requires Trid-OR7-ctr-whip[9].

The start is easy:

Code: Select all

Resolution state after Singles and whips[1]:
   +----------------------+----------------------+----------------------+
   ! 1249   2479   123479 ! 249    5      6      ! 237    8      349    !
   ! 249    5      23479  ! 1      8      249    ! 237    234679 3469   !
   ! 6      8      249    ! 3      249    7      ! 25     1      459    !
   +----------------------+----------------------+----------------------+
   ! 12489  24679  5      ! 24679  3      1249   ! 128    249    489    !
   ! 3      249    1249   ! 24589  249    124589 ! 6      249    7      !
   ! 12489  24679  124679 ! 24679  24679  1249   ! 12358  2349   34589  !
   +----------------------+----------------------+----------------------+
   ! 5      3      2689   ! 26789  2679   289    ! 4      67     1      !
   ! 7      1      468    ! 4568   46     3458   ! 9      36     2      !
   ! 249    2469   2469   ! 24679  1      2349   ! 378    5      368    !
   +----------------------+----------------------+----------------------+
213 candidates.

hidden-pairs-in-a-row: r5{n5 n8}{c4 c6} ==> r5c6≠9, r5c6≠4, r5c6≠2, r5c6≠1, r5c4≠9, r5c4≠4, r5c4≠2
singles ==> r5c3=1, r1c1=1
hidden-pairs-in-a-row: r4{n6 n7}{c2 c4} ==> r4c4≠9, r4c4≠4, r4c4≠2, r4c2≠9, r4c2≠4, r4c2≠2
   +----------------------+----------------------+----------------------+
   ! 1      2479   23479  ! 249    5      6      ! 237    8      349    !
   ! 249    5      23479  ! 1      8      249    ! 237    234679 3469   !
   ! 6      8      249    ! 3      249    7      ! 25     1      459    !
   +----------------------+----------------------+----------------------+
   ! 2489   67     5      ! 67     3      1249   ! 128    249    489    !
   ! 3      249    1      ! 58     249    58     ! 6      249    7      !
   ! 2489   24679  24679  ! 24679  24679  1249   ! 12358  2349   34589  !
   +----------------------+----------------------+----------------------+
   ! 5      3      2689   ! 26789  2679   289    ! 4      67     1      !
   ! 7      1      468    ! 4568   46     3458   ! 9      36     2      !
   ! 249    2469   2469   ! 24679  1      2349   ! 378    5      368    !
   +----------------------+----------------------+----------------------+

OR7-anti-tridagon[12] for digits 2, 4 and 9 in blocks:
        b1, with cells: r1c2, r2c1, r3c3
        b2, with cells: r1c4, r2c6, r3c5
        b4, with cells: r5c2, r4c1, r6c3
        b5, with cells: r5c5, r4c6, r6c4
with 7 guardians: n7r1c2 n8r4c1 n1r4c6 n6r6c3 n7r6c3 n6r6c4 n7r6c4

biv-chain[3]: r7c8{n6 n7} - c5n7{r7 r6} - r4c4{n7 n6} ==> r7c4≠6
biv-chain[3]: r8c5{n4 n6} - r8c8{n6 n3} - b8n3{r8c6 r9c6} ==> r9c6≠4

Here is our first tridagon rule, based on the above OR7 relation:
Trid-OR7-ctr-whip[9]: r6n5{c9 c7} - c7n1{r6 r4} - b6n8{r4c7 r4c9} - r9c9{n8 n6} - r7c8{n6 n7} - c5n7{r7 r6} - r4n7{c4 c2} - c2n6{r4 r6} - OR7{{n7r6c4 n6r6c4 n7r6c3 n6r6c3 n1r4c6 n8r4c1 n7r1c2 | .}} ==> r6c9≠3

Now an easy part:

Code: Select all

t-whip[5]: r8c8{n6 n3} - r6n3{c8 c7} - c7n1{r6 r4} - c7n8{r4 r9} - b9n7{r9c7 .} ==> r7c8≠6
singles ==> r7c8=7, r9c4=7

At least one candidate of a previous Trid-OR7-relation has just been eliminated.
There remains a Trid-OR6-relation between candidates: n7r1c2 n8r4c1 n1r4c6 n6r6c3 n7r6c3 n6r6c4
   +-------------------+-------------------+-------------------+
   ! 1     2479  23479 ! 249   5     6     ! 237   8     349   !
   ! 249   5     23479 ! 1     8     249   ! 237   23469 3469  !
   ! 6     8     249   ! 3     249   7     ! 25    1     459   !
   +-------------------+-------------------+-------------------+
   ! 2489  67    5     ! 67    3     1249  ! 128   249   489   !
   ! 3     249   1     ! 58    249   58    ! 6     249   7     !
   ! 2489  24679 24679 ! 2469  24679 1249  ! 12358 2349  4589  !
   +-------------------+-------------------+-------------------+
   ! 5     3     2689  ! 289   269   289   ! 4     7     1     !
   ! 7     1     468   ! 4568  46    3458  ! 9     36    2     !
   ! 249   2469  2469  ! 7     1     239   ! 38    5     368   !
   +-------------------+-------------------+-------------------+
naked-single ==> r4c4=6

At least one candidate of a previous Trid-OR6-relation has just been eliminated.
There remains a Trid-OR5-relation between candidates: n7r1c2 n8r4c1 n1r4c6 n6r6c3 n7r6c3
   +-------------------+-------------------+-------------------+
   ! 1     2479  23479 ! 249   5     6     ! 237   8     349   !
   ! 249   5     23479 ! 1     8     249   ! 237   23469 3469  !
   ! 6     8     249   ! 3     249   7     ! 25    1     459   !
   +-------------------+-------------------+-------------------+
   ! 2489  67    5     ! 6     3     1249  ! 128   249   489   !
   ! 3     249   1     ! 58    249   58    ! 6     249   7     !
   ! 2489  24679 24679 ! 249   2479  1249  ! 12358 2349  4589  !
   +-------------------+-------------------+-------------------+
   ! 5     3     2689  ! 289   269   289   ! 4     7     1     !
   ! 7     1     468   ! 458   46    3458  ! 9     36    2     !
   ! 249   2469  2469  ! 7     1     239   ! 38    5     368   !
   +-------------------+-------------------+-------------------+
naked-single ==> r4c2=7

At least one candidate of a previous Trid-OR5-relation has just been eliminated.
There remains a Trid-OR4-relation between candidates: n7r1c2 n8r4c1 n1r4c6 n6r6c3
   +-------------------+-------------------+-------------------+
   ! 1     2479  23479 ! 249   5     6     ! 237   8     349   !
   ! 249   5     23479 ! 1     8     249   ! 237   23469 3469  !
   ! 6     8     249   ! 3     249   7     ! 25    1     459   !
   +-------------------+-------------------+-------------------+
   ! 2489  7     5     ! 6     3     1249  ! 128   249   489   !
   ! 3     249   1     ! 58    249   58    ! 6     249   7     !
   ! 2489  24679 2469  ! 249   2479  1249  ! 12358 2349  4589  !
   +-------------------+-------------------+-------------------+
   ! 5     3     2689  ! 289   269   289   ! 4     7     1     !
   ! 7     1     468   ! 458   46    3458  ! 9     36    2     !
   ! 249   2469  2469  ! 7     1     239   ! 38    5     368   !
   +-------------------+-------------------+-------------------+

At least one candidate of a previous Trid-OR4-relation has just been eliminated.
There remains a Trid-OR3-relation between candidates: n8r4c1 n1r4c6 n6r6c3
   +-------------------+-------------------+-------------------+
   ! 1     249   23479 ! 249   5     6     ! 237   8     349   !
   ! 249   5     23479 ! 1     8     249   ! 237   23469 3469  !
   ! 6     8     249   ! 3     249   7     ! 25    1     459   !
   +-------------------+-------------------+-------------------+
   ! 2489  7     5     ! 6     3     1249  ! 128   249   489   !
   ! 3     249   1     ! 58    249   58    ! 6     249   7     !
   ! 2489  2469  2469  ! 249   2479  1249  ! 12358 2349  4589  !
   +-------------------+-------------------+-------------------+
   ! 5     3     2689  ! 289   269   289   ! 4     7     1     !
   ! 7     1     468   ! 458   46    3458  ! 9     36    2     !
   ! 249   2469  2469  ! 7     1     239   ! 38    5     368   !
   +-------------------+-------------------+-------------------+

hidden-single-in-a-row ==> r6c5=7
whip[1]: r9n4{c3 .} ==> r8c3≠4
hidden-pairs-in-a-block: b1{n3 n7}{r1c3 r2c3} ==> r2c3≠9, r2c3≠4, r2c3≠2, r1c3≠9, r1c3≠4, r1c3≠2

And more tridagons:
Trid-OR3-whip[6]: r8c8{n6 n3} - r6n3{c8 c7} - c7n1{r6 r4} - OR3{{n1r4c6 n6r6c3 | n8r4c1}} - b6n8{r4c7 r6c9} - r6n5{c9 .} ==> r8c3≠6
naked-single ==> r8c3=8
Trid-OR3-whip[7]: r9n6{c3 c9} - r8c8{n6 n3} - r6n3{c8 c7} - r6n1{c7 c6} - OR3{{n1r4c6 n6r6c3 | n8r4c1}} - b6n8{r4c7 r6c9} - r6n5{c9 .} ==> r7c3≠6
singles ==> r7c5=6, r8c5=4, r8c4=5, r5c4=8, r5c6=5, r8c6=3, r8c8=6, r2c9=6, r7c6=8
z-chain[3]: b5n4{r6c6 r4c6} - r2n4{c6 c8} - r5n4{c8 .} ==> r6c1≠4
whip[5]: c9n5{r6 r3} - r3n4{c9 c3} - r3n9{c3 c5} - r1n9{c4 c2} - r5n9{c2 .} ==> r6c9≠9
whip[6]: c5n9{r5 r3} - c4n9{r1 r7} - r7n2{c4 c3} - r3n2{c3 c7} - c7n5{r3 r6} - r6n1{c7 .} ==> r6c6≠9
whip[5]: c8n9{r6 r2} - c6n9{r2 r9} - c1n9{r9 r6} - c2n9{r5 r1} - c4n9{r1 .} ==> r4c9≠9
whip[1]: b6n9{r6c8 .} ==> r2c8≠9
Trid-OR3-whip[3]: b5n4{r6c6 r4c6} - OR3{{n1r4c6 n6r6c3 | n8r4c1}} - r4c9{n8 .} ==> r6c3≠4

The end is easy:

Code: Select all

z-chain[4]: r7n2{c4 c3} - b1n2{r3c3 r2c1} - r2n9{c1 c6} - b2n4{r2c6 .} ==> r1c4≠2
biv-chain[3]: r1c4{n4 n9} - b3n9{r1c9 r3c9} - r3n4{c9 c3} ==> r1c2≠4
biv-chain[3]: b1n4{r3c3 r2c1} - r2n9{c1 c6} - b2n2{r2c6 r3c5} ==> r3c3≠2
biv-chain[3]: b1n2{r1c2 r2c1} - r2n9{c1 c6} - r9c6{n9 n2} ==> r9c2≠2
biv-chain[3]: r5c5{n9 n2} - c4n2{r6 r7} - b8n9{r7c4 r9c6} ==> r4c6≠9
biv-chain[3]: r4n9{c8 c1} - r2n9{c1 c6} - c5n9{r3 r5} ==> r5c8≠9
z-chain[3]: r3n2{c7 c5} - r5n2{c5 c2} - r1n2{c2 .} ==> r4c7≠2, r6c7≠2, r2c8≠2
biv-chain[4]: c4n2{r6 r7} - b8n9{r7c4 r9c6} - r2n9{c6 c1} - b1n2{r2c1 r1c2} ==> r6c2≠2
biv-chain[4]: r5c8{n4 n2} - c5n2{r5 r3} - r3c7{n2 n5} - b6n5{r6c7 r6c9} ==> r6c9≠4
z-chain[4]: c2n2{r1 r5} - r5n4{c2 c8} - r2n4{c8 c6} - r2n9{c6 .} ==> r2c1≠2
hidden-single-in-a-block ==> r1c2=2
naked-pairs-in-a-row: r1{c3 c7}{n3 n7} ==> r1c9≠3
singles ==> r9c9=3, r9c7=8, r4c7=1

At least one candidate of a previous Trid-OR3-relation has just been eliminated.
There remains a Trid-OR2-relation between candidates: n8r4c1 n6r6c3
   +----------------+----------------+----------------+
   ! 1    2    37   ! 49   5    6    ! 37   8    49   !
   ! 49   5    37   ! 1    8    249  ! 237  34   6    !
   ! 6    8    49   ! 3    29   7    ! 25   1    459  !
   +----------------+----------------+----------------+
   ! 2489 7    5    ! 6    3    24   ! 1    249  48   !
   ! 3    49   1    ! 8    29   5    ! 6    24   7    !
   ! 289  469  269  ! 249  7    124  ! 35   2349 58   !
   +----------------+----------------+----------------+
   ! 5    3    29   ! 29   6    8    ! 4    7    1    !
   ! 7    1    8    ! 5    4    3    ! 9    6    2    !
   ! 249  469  2469 ! 7    1    29   ! 8    5    3    !
   +----------------+----------------+----------------+

hidden-single-in-a-block ==> r6c6=1
finned-x-wing-in-columns: n4{c6 c8}{r2 r4} ==> r4c9≠4
stte

Note: in addition to this solution in W6+OR7W9, there's also a solution in W10+OR3W6. Hard also.
.

[mith]

Finished the min-expand run starting from only the singles expansion; now running starting after basics using only the "criterion" choice for triple and boxes.

I noticed in testing (before I added the basics) that there was at least one puzzle that had an entirely empty box as part of the TO, meaning three diagonal choices - this was only after singles of course, and it collapsed after basics, but I'm curious if any like this exist after basics as well.

The current script is running a bit faster than the previous, but it's still going to take overnight.

[mith]

There are indeed at least two puzzles (run isn't quite finished) with an empty box after basics:

Code: Select all

..3.56....571.9.3..6.37..5.....1..956.....8.27.....4.33.........7....3.1.96.....7;12346;296061;3
5;11;n248;b1p249+b2p159+b7p267+b8p159
5;6;n248;b1p249+b2p159+b7p267+b8p267
5;10;n248;b1p249+b2p159+b7p267+b8p348

1.3....89.57.8.....68..2....8.....4.......3...34....123....14.....92...3....43..1;15082;482867;3
9;16;n567;b5p168+b6p357+b8p267+b9p357
9;18;n567;b5p249+b6p357+b8p267+b9p357
9;13;n567;b5p357+b6p357+b8p267+b9p357

The first has a relatively tame TOFC from the 6 guardian candidate choice which reduces it to a couple skyscrapers, but the second looks completely disgusting. Curious what your solver will make of this one, Denis.

[denis_berthier]

Code: Select all

..3.56....571.9.3..6.37..5.....1..956.....8.27.....4.33.........7....3.1.96.....7;12346;296061;3

Trivial:
3 different anti-tridagons (computed after W2+S3):

Code: Select all

Resolution state after Singles and whips[1]:
   +----------------------+----------------------+----------------------+
   ! 12489  1248   3      ! 248    5      6      ! 1279   2478   489    !
   ! 248    5      7      ! 1      248    9      ! 26     3      468    !
   ! 12489  6      1248   ! 3      7      248    ! 129    5      489    !
   +----------------------+----------------------+----------------------+
   ! 248    2348   248    ! 24678  1      23478  ! 67     9      5      !
   ! 6      134    1459   ! 4579   349    3457   ! 8      17     2      !
   ! 7      128    12589  ! 25689  2689   258    ! 4      16     3      !
   +----------------------+----------------------+----------------------+
   ! 3      1248   1248   ! 245678 2468   124578 ! 2569   2468   4689   !
   ! 2458   7      248    ! 245689 24689  2458   ! 3      2468   1      !
   ! 12458  9      6      ! 2458   2348   123458 ! 25     248    7      !
   +----------------------+----------------------+----------------------+

202 candidates.

hidden-pairs-in-a-column: c3{n5 n9}{r5 r6} ==> r6c3≠8, r6c3≠2, r6c3≠1, r5c3≠4, r5c3≠1
whip[1]: b4n1{r6c2 .} ==> r1c2≠1, r7c2≠1
   +----------------------+----------------------+----------------------+
   ! 12489  248    3      ! 248    5      6      ! 1279   2478   489    !
   ! 248    5      7      ! 1      248    9      ! 26     3      468    !
   ! 12489  6      1248   ! 3      7      248    ! 129    5      489    !
   +----------------------+----------------------+----------------------+
   ! 248    2348   248    ! 24678  1      23478  ! 67     9      5      !
   ! 6      134    59     ! 4579   349    3457   ! 8      17     2      !
   ! 7      128    59     ! 25689  2689   258    ! 4      16     3      !
   +----------------------+----------------------+----------------------+
   ! 3      248    1248   ! 245678 2468   124578 ! 2569   2468   4689   !
   ! 2458   7      248    ! 245689 24689  2458   ! 3      2468   1      !
   ! 12458  9      6      ! 2458   2348   123458 ! 25     248    7      !
   +----------------------+----------------------+----------------------+

OR11-anti-tridagon[12] for digits 2, 4 and 8 in blocks:
        b1, with cells: r1c2, r2c1, r3c3
        b2, with cells: r1c4, r2c5, r3c6
        b7, with cells: r7c2, r9c1, r8c3
        b8, with cells: r7c4, r9c6, r8c5
with 11 guardians: n1r3c3 n5r7c4 n6r7c4 n7r7c4 n6r8c5 n9r8c5 n1r9c1 n5r9c1 n1r9c6 n3r9c6 n5r9c6

OR6-anti-tridagon[12] for digits 2, 4 and 8 in blocks:
        b1, with cells: r1c2, r2c1, r3c3
        b2, with cells: r1c4, r2c5, r3c6
        b7, with cells: r7c2, r9c1, r8c3
        b8, with cells: r7c5, r9c4, r8c6
with 6 guardians: n1r3c3 n6r7c5 n5r8c6 n1r9c1 n5r9c1 n5r9c4

OR10-anti-tridagon[12] for digits 2, 4 and 8 in blocks:
        b1, with cells: r1c2, r2c1, r3c3
        b2, with cells: r1c4, r2c5, r3c6
        b7, with cells: r7c2, r9c1, r8c3
        b8, with cells: r7c6, r9c5, r8c4
with 10 guardians: n1r3c3 n1r7c6 n5r7c6 n7r7c6 n5r8c4 n6r8c4 n9r8c4 n1r9c1 n5r9c1 n3r9c5

biv-chain[3]: r2c7{n2 n6} - r4c7{n6 n7} - b3n7{r1c7 r1c8} ==> r1c8≠2
+S2 until:

Code: Select all

At least one candidate of a previous Trid-OR6-relation has just been eliminated.
There remains a Trid-OR5-relation between candidates: n5r7c6 n7r7c6 n6r8c4 n9r8c4 n3r9c5
   +-------------------+-------------------+-------------------+
   ! 12489 248   3     ! 248   5     6     ! 1279  478   489   !
   ! 248   5     7     ! 1     248   9     ! 26    3     468   !
   ! 12489 6     248   ! 3     7     248   ! 129   5     489   !
   +-------------------+-------------------+-------------------+
   ! 248   2348  248   ! 24678 1     23478 ! 67    9     5     !
   ! 6     134   59    ! 4579  349   3457  ! 8     17    2     !
   ! 7     128   59    ! 25689 2689  258   ! 4     16    3     !
   +-------------------+-------------------+-------------------+
   ! 3     248   1     ! 57    2468  57    ! 69    2468  4689  !
   ! 5     7     248   ! 24689 24689 248   ! 3     2468  1     !
   ! 248   9     6     ! 248   2348  1     ! 5     248   7     !
   +-------------------+-------------------+-------------------+

hidden-single-in-a-block ==> r9c5=3
ORk-relation with only one candidate => r7c5=6
stte

[denis_berthier]

Code: Select all

1.3....89.57.8.....68..2....8.....4.......3...34....123....14.....92...3....43..1;15082;482867;3

The first has a relatively tame TOFC from the 6 guardian candidate choice which reduces it to a couple skyscrapers, but the second looks completely disgusting. Curious what your solver will make of this one, Denis.

SER = 11.7
Harder, but solved in W8+OR6W9

Code: Select all

Resolution state after Singles and whips[1]:
   +-------------------------+-------------------------+-------------------------+
   ! 1       24      3       ! 4567    567     4567    ! 2567    8       9       !
   ! 249     5       7       ! 1346    8       469     ! 126     236     46      !
   ! 49      6       8       ! 13457   13579   2       ! 157     357     457     !
   +-------------------------+-------------------------+-------------------------+
   ! 2567    8       12569   ! 123567  135679  5679    ! 5679    4       567     !
   ! 2567    1279    12569   ! 1245678 15679   456789  ! 3       5679    5678    !
   ! 567     3       4       ! 5678    5679    56789   ! 56789   1       2       !
   +-------------------------+-------------------------+-------------------------+
   ! 3       279     2569    ! 5678    567     1       ! 4       25679   5678    !
   ! 45678   147     156     ! 9       2       5678    ! 5678    567     3       !
   ! 25678   279     2569    ! 5678    4       3       ! 256789  25679   1       !
   +-------------------------+-------------------------+-------------------------+
220 candidates.

finned-x-wing-in-rows: n8{r7 r5}{c9 c4} ==> r6c4≠8

More anti-tridagons than I can count with one hand:

Code: Select all

   +-------------------------+-------------------------+-------------------------+
   ! 1       24      3       ! 4567    567     4567    ! 2567    8       9       !
   ! 249     5       7       ! 1346    8       469     ! 126     236     46      !
   ! 49      6       8       ! 13457   13579   2       ! 157     357     457     !
   +-------------------------+-------------------------+-------------------------+
   ! 2567    8       12569   ! 123567  135679  5679    ! 5679    4       567     !
   ! 2567    1279    12569   ! 1245678 15679   456789  ! 3       5679    5678    !
   ! 567     3       4       ! 567     5679    56789   ! 56789   1       2       !
   +-------------------------+-------------------------+-------------------------+
   ! 3       279     2569    ! 5678    567     1       ! 4       25679   5678    !
   ! 45678   147     156     ! 9       2       5678    ! 5678    567     3       !
   ! 25678   279     2569    ! 5678    4       3       ! 256789  25679   1       !
   +-------------------------+-------------------------+-------------------------+

OR16-anti-tridagon[12] for digits 5, 6 and 7 in blocks:
        b5, with cells: r4c4, r5c5, r6c6
        b6, with cells: r4c9, r5c8, r6c7
        b8, with cells: r9c4, r7c5, r8c6
        b9, with cells: r9c8, r7c9, r8c7
with 16 guardians: n1r4c4 n2r4c4 n3r4c4 n1r5c5 n9r5c5 n9r5c8 n8r6c6 n9r6c6 n8r6c7 n9r6c7 n8r7c9 n8r8c6 n8r8c7 n8r9c4 n2r9c8 n9r9c8

OR18-anti-tridagon[12] for digits 5, 6 and 7 in blocks:
        b5, with cells: r4c5, r5c4, r6c6
        b6, with cells: r4c9, r5c8, r6c7
        b8, with cells: r7c5, r9c4, r8c6
        b9, with cells: r7c9, r9c7, r8c8
with 18 guardians: n1r4c5 n3r4c5 n9r4c5 n1r5c4 n2r5c4 n4r5c4 n8r5c4 n9r5c8 n8r6c6 n9r6c6 n8r6c7 n9r6c7 n8r7c9 n8r8c6 n8r9c4 n2r9c7 n8r9c7 n9r9c7

OR15-anti-tridagon[12] for digits 5, 6 and 7 in blocks:
        b5, with cells: r4c5, r5c6, r6c4
        b6, with cells: r4c9, r5c8, r6c7
        b8, with cells: r7c5, r8c6, r9c4
        b9, with cells: r7c9, r8c7, r9c8
with 15 guardians: n1r4c5 n3r4c5 n9r4c5 n4r5c6 n8r5c6 n9r5c6 n9r5c8 n8r6c7 n9r6c7 n8r7c9 n8r8c6 n8r8c7 n8r9c4 n2r9c8 n9r9c8

OR12-anti-tridagon[12] for digits 5, 6 and 7 in blocks:
        b5, with cells: r4c6, r5c5, r6c4
        b6, with cells: r4c9, r5c8, r6c7
        b8, with cells: r8c6, r7c5, r9c4
        b9, with cells: r8c8, r7c9, r9c7
with 12 guardians: n9r4c6 n1r5c5 n9r5c5 n9r5c8 n8r6c7 n9r6c7 n8r7c9 n8r8c6 n8r9c4 n2r9c7 n8r9c7 n9r9c7

OR16-anti-tridagon[12] for digits 5, 6 and 7 in blocks:
        b5, with cells: r4c4, r5c6, r6c5
        b6, with cells: r4c9, r5c8, r6c7
        b8, with cells: r9c4, r8c6, r7c5
        b9, with cells: r9c7, r8c8, r7c9
with 16 guardians: n1r4c4 n2r4c4 n3r4c4 n4r5c6 n8r5c6 n9r5c6 n9r5c8 n9r6c5 n8r6c7 n9r6c7 n8r7c9 n8r8c6 n8r9c4 n2r9c7 n8r9c7 n9r9c7

OR15-anti-tridagon[12] for digits 5, 6 and 7 in blocks:
        b5, with cells: r4c6, r5c4, r6c5
        b6, with cells: r4c9, r5c8, r6c7
        b8, with cells: r8c6, r9c4, r7c5
        b9, with cells: r8c7, r9c8, r7c9
with 15 guardians: n9r4c6 n1r5c4 n2r5c4 n4r5c4 n8r5c4 n9r5c8 n9r6c5 n8r6c7 n9r6c7 n8r7c9 n8r8c6 n8r8c7 n8r9c4 n2r9c8 n9r9c8

Then a few easy steps and some cleaning:

Code: Select all

biv-chain[3]: r2n9{c6 c1} - r3c1{n9 n4} - b3n4{r3c9 r2c9} ==> r2c6≠4
z-chain[3]: r8n4{c1 c2} - c2n1{r8 r5} - c2n7{r5 .} ==> r8c1≠7
z-chain[4]: b7n7{r9c2 r9c1} - c1n8{r9 r8} - r8n4{c1 c2} - c2n1{r8 .} ==> r5c2≠7
whip[1]: c2n7{r9 .} ==> r9c1≠7
whip[4]: r2n1{c4 c7} - b3n6{r2c7 r1c7} - b3n2{r1c7 r2c8} - r2n3{c8 .} ==> r2c4≠6
z-chain[5]: c5n3{r3 r4} - c5n1{r4 r5} - c2n1{r5 r8} - c2n4{r8 r1} - r3c1{n4 .} ==> r3c5≠9
hidden-single-in-a-block ==> r2c6=9

At least one candidate of a previous Trid-OR12-relation has just been eliminated.
There remains a Trid-OR11-relation between candidates: n1r5c5 n9r5c5 n9r5c8 n8r6c7 n9r6c7 n8r7c9 n8r8c6 n8r9c4 n2r9c7 n8r9c7 n9r9c7
   +-------------------------+-------------------------+-------------------------+
   ! 1       24      3       ! 4567    567     4567    ! 2567    8       9       !
   ! 249     5       7       ! 134     8       69      ! 126     236     46      !
   ! 49      6       8       ! 13457   1357    2       ! 157     357     457     !
   +-------------------------+-------------------------+-------------------------+
   ! 2567    8       12569   ! 123567  135679  567     ! 5679    4       567     !
   ! 2567    129     12569   ! 1245678 15679   45678   ! 3       5679    5678    !
   ! 567     3       4       ! 567     5679    5678    ! 56789   1       2       !
   +-------------------------+-------------------------+-------------------------+
   ! 3       279     2569    ! 5678    567     1       ! 4       25679   5678    !
   ! 4568    147     156     ! 9       2       5678    ! 5678    567     3       !
   ! 2568    279     2569    ! 5678    4       3       ! 256789  25679   1       !
   +-------------------------+-------------------------+-------------------------+

hidden-single-in-a-block ==> r3c1=9
whip[1]: r2n6{c9 .} ==> r1c7≠6
z-chain[6]: c5n3{r3 r4} - c5n1{r4 r5} - c2n1{r5 r8} - c2n4{r8 r1} - r1n2{c2 c7} - r1n5{c7 .} ==> r3c5≠5
z-chain[6]: c5n3{r3 r4} - c5n1{r4 r5} - c2n1{r5 r8} - c2n4{r8 r1} - r1n2{c2 c7} - r1n7{c7 .} ==> r3c5≠7
t-whip[4]: c4n2{r5 r4} - r4n3{c4 c5} - r3c5{n3 n1} - b5n1{r4c5 .} ==> r5c4≠4, r5c4≠5, r5c4≠6, r5c4≠7, r5c4≠8
hidden-single-in-a-block ==> r5c6=4
hidden-single-in-a-block ==> r6c6=8

At least one candidate of a previous Trid-OR10-relation has just been eliminated.
There remains a Trid-OR9-relation between candidates: n1r5c5 n9r5c5 n9r5c8 n9r6c7 n8r7c9 n8r9c4 n2r9c7 n8r9c7 n9r9c7
   +----------------------+----------------------+----------------------+
   ! 1      24     3      ! 4567   567    567    ! 257    8      9      !
   ! 24     5      7      ! 134    8      9      ! 126    236    46     !
   ! 9      6      8      ! 13457  13     2      ! 157    357    457    !
   +----------------------+----------------------+----------------------+
   ! 2567   8      12569  ! 123567 135679 567    ! 5679   4      567    !
   ! 2567   129    12569  ! 12     15679  4      ! 3      5679   5678   !
   ! 567    3      4      ! 567    5679   5678   ! 5679   1      2      !
   +----------------------+----------------------+----------------------+
   ! 3      279    2569   ! 5678   567    1      ! 4      25679  5678   !
   ! 4568   147    156    ! 9      2      567    ! 5678   567    3      !
   ! 2568   279    2569   ! 5678   4      3      ! 256789 25679  1      !
   +----------------------+----------------------+----------------------+

hidden-single-in-a-block ==> r5c9=8

At least one candidate of a previous Trid-OR9-relation has just been eliminated.
There remains a Trid-OR8-relation between candidates: n1r5c5 n9r5c5 n9r5c8 n9r6c7 n8r9c4 n2r9c7 n8r9c7 n9r9c7
   +----------------------+----------------------+----------------------+
   ! 1      24     3      ! 4567   567    567    ! 257    8      9      !
   ! 24     5      7      ! 134    8      9      ! 126    236    46     !
   ! 9      6      8      ! 13457  13     2      ! 157    357    457    !
   +----------------------+----------------------+----------------------+
   ! 2567   8      12569  ! 123567 135679 567    ! 5679   4      567    !
   ! 2567   129    12569  ! 12     15679  4      ! 3      5679   5678   !
   ! 567    3      4      ! 567    5679   8      ! 5679   1      2      !
   +----------------------+----------------------+----------------------+
   ! 3      279    2569   ! 5678   567    1      ! 4      25679  567    !
   ! 4568   147    156    ! 9      2      567    ! 5678   567    3      !
   ! 2568   279    2569   ! 5678   4      3      ! 256789 25679  1      !
   +----------------------+----------------------+----------------------+

hidden-single-in-a-row ==> r7c4=8

At least one candidate of a previous Trid-OR8-relation has just been eliminated.
There remains a Trid-OR7-relation between candidates: n1r5c5 n9r5c5 n9r5c8 n9r6c7 n2r9c7 n8r9c7 n9r9c7
   +----------------------+----------------------+----------------------+
   ! 1      24     3      ! 4567   567    567    ! 257    8      9      !
   ! 24     5      7      ! 134    8      9      ! 126    236    46     !
   ! 9      6      8      ! 13457  13     2      ! 157    357    457    !
   +----------------------+----------------------+----------------------+
   ! 2567   8      12569  ! 123567 135679 567    ! 5679   4      567    !
   ! 2567   129    12569  ! 12     15679  4      ! 3      5679   8      !
   ! 567    3      4      ! 567    5679   8      ! 5679   1      2      !
   +----------------------+----------------------+----------------------+
   ! 3      279    2569   ! 5678   567    1      ! 4      25679  567    !
   ! 4568   147    156    ! 9      2      567    ! 5678   567    3      !
   ! 2568   279    2569   ! 567    4      3      ! 256789 25679  1      !
   +----------------------+----------------------+----------------------+

biv-chain[4]: r5c4{n2 n1} - c2n1{r5 r8} - r8n4{c2 c1} - r2c1{n4 n2} ==> r5c1≠2
whip[4]: r4n3{c4 c5} - r3c5{n3 n1} - b5n1{r4c5 r5c4} - c4n2{r5 .} ==> r4c4≠5
whip[4]: r4n3{c4 c5} - r3c5{n3 n1} - b5n1{r4c5 r5c4} - c4n2{r5 .} ==> r4c4≠6
whip[4]: r4n3{c4 c5} - r3c5{n3 n1} - b5n1{r4c5 r5c4} - c4n2{r5 .} ==> r4c4≠7
t-whip[6]: r4n1{c5 c3} - c2n1{r5 r8} - c2n4{r8 r1} - b1n2{r1c2 r2c1} - r4n2{c1 c4} - r5c4{n2 .} ==> r5c5≠1

At least one candidate of a previous Trid-OR7-relation has just been eliminated.
There remains a Trid-OR6-relation between candidates: n9r5c5 n9r5c8 n9r6c7 n2r9c7 n8r9c7 n9r9c7
   +----------------------+----------------------+----------------------+
   ! 1      24     3      ! 4567   567    567    ! 257    8      9      !
   ! 24     5      7      ! 134    8      9      ! 126    236    46     !
   ! 9      6      8      ! 13457  13     2      ! 157    357    457    !
   +----------------------+----------------------+----------------------+
   ! 2567   8      12569  ! 123    135679 567    ! 5679   4      567    !
   ! 567    129    12569  ! 12     5679   4      ! 3      5679   8      !
   ! 567    3      4      ! 567    5679   8      ! 5679   1      2      !
   +----------------------+----------------------+----------------------+
   ! 3      279    2569   ! 8      567    1      ! 4      25679  567    !
   ! 4568   147    156    ! 9      2      567    ! 5678   567    3      !
   ! 2568   279    2569   ! 567    4      3      ! 256789 25679  1      !
   +----------------------+----------------------+----------------------+

hidden-pairs-in-a-column: c5{n1 n3}{r3 r4} ==> r4c5≠9, r4c5≠7, r4c5≠6, r4c5≠5

At last, a few tridagon rules:
Trid-OR6-whip[9]: r4n9{c3 c7} - b9n9{r9c7 r7c8} - c2n9{r7 r5} - c2n1{r5 r8} - c2n4{r8 r1} - r1n2{c2 c7} - OR6{{n2r9c7 n9r9c7 n9r6c7 n9r5c8 n9r5c5 | n8r9c7}} - c1n8{r9 r8} - r8n4{c1 .} ==> r9c3≠9
Trid-OR6-whip[9]: r4n9{c3 c7} - r9n9{c7 c8} - c2n9{r9 r5} - c2n1{r5 r8} - c2n4{r8 r1} - r1n2{c2 c7} - OR6{{n2r9c7 n9r9c7 n9r6c7 n9r5c8 n9r5c5 | n8r9c7}} - c1n8{r9 r8} - r8n4{c1 .} ==> r7c3≠9
whip[1]: b7n9{r9c2 .} ==> r5c2≠9
naked-pairs-in-a-row: r5{c2 c4}{n1 n2} ==> r5c3≠2, r5c3≠1
Trid-OR6-whip[7]: b1n2{r1c2 r2c1} - b4n2{r4c1 r4c3} - r4n9{c3 c7} - r6n9{c7 c5} - OR6{{n9r6c7 n9r9c7 n2r9c7 n9r5c8 n9r5c5 | n8r9c7}} - c1n8{r9 r8} - c1n4{r8 .} ==> r9c2≠2
Trid-OR6-whip[8]: b1n2{r1c2 r2c1} - c8n2{r2 r9} - c3n2{r9 r4} - r4n9{c3 c7} - r6n9{c7 c5} - OR6{{n9r6c7 n9r9c7 n2r9c7 n9r5c8 n9r5c5 | n8r9c7}} - c1n8{r9 r8} - c1n4{r8 .} ==> r7c2≠2
naked-pairs-in-a-block: b7{r7c2 r9c2}{n7 n9} ==> r8c2≠7
biv-chain[5]: r2c9{n6 n4} - b1n4{r2c1 r1c2} - r8c2{n4 n1} - r5n1{c2 c4} - r2n1{c4 c7} ==> r2c7≠6
z-chain[5]: r3c5{n1 n3} - r2c4{n3 n4} - r2c1{n4 n2} - c2n2{r1 r5} - r5n1{c2 .} ==> r3c4≠1
Trid-OR6-whip[8]: r2n2{c8 c1} - c2n2{r1 r5} - b4n1{r5c2 r4c3} - r4n9{c3 c7} - r6n9{c7 c5} - OR6{{n9r6c7 n9r9c7 n2r9c7 n9r5c8 n9r5c5 | n8r9c7}} - c1n8{r9 r8} - c1n4{r8 .} ==> r1c7≠2

Easy end:

Code: Select all

hidden-single-in-a-row ==> r1c2=2
naked-single ==> r2c1=4
naked-single ==> r2c9=6
naked-single ==> r5c2=1
naked-single ==> r5c4=2
naked-single ==> r8c2=4
hidden-single-in-a-column ==> r8c3=1
hidden-single-in-a-block ==> r3c9=4
hidden-single-in-a-block ==> r1c4=4
naked-pairs-in-a-column: c4{r2 r4}{n1 n3} ==> r3c4≠3
t-whip[2]: c9n7{r4 r7} - r8n7{c8 .} ==> r4c6≠7
whip[3]: b3n7{r3c7 r3c8} - r8n7{c8 c6} - c4n7{r9 .} ==> r6c7≠7
t-whip[4]: r4c6{n6 n5} - c9n5{r4 r7} - b8n5{r7c5 r9c4} - c4n6{r9 .} ==> r6c5≠6, r5c5≠6
z-chain[3]: r1n6{c5 c6} - b5n6{r4c6 r6c4} - b5n7{r6c4 .} ==> r1c5≠7
finned-x-wing-in-rows: n7{r1 r8}{c6 c7} ==> r9c7≠7
z-chain[3]: r1n7{c7 c6} - r8n7{c6 c8} - b3n7{r3c8 .} ==> r4c7≠7
biv-chain[4]: r7n2{c8 c3} - r4n2{c3 c1} - r4n7{c1 c9} - c9n5{r4 r7} ==> r7c8≠5
biv-chain[4]: r9n8{c7 c1} - c1n2{r9 r4} - r4n7{c1 c9} - c9n5{r4 r7} ==> r9c7≠5
t-whip[4]: r4c6{n6 n5} - c9n5{r4 r7} - c5n5{r7 r1} - c5n6{r1 .} ==> r8c6≠6
biv-chain[3]: r8c6{n7 n5} - r4c6{n5 n6} - c4n6{r6 r9} ==> r9c4≠7
whip[4]: r7c9{n5 n7} - r7c2{n7 n9} - c8n9{r7 r5} - b6n7{r5c8 .} ==> r9c8≠5
z-chain[4]: r9n5{c3 c4} - c4n6{r9 r6} - r4c6{n6 n5} - c9n5{r4 .} ==> r7c3≠5
z-chain[3]: c1n8{r9 r8} - b7n5{r8c1 r9c3} - r9c4{n5 .} ==> r9c1≠6
whip[3]: r9c4{n6 n5} - b7n5{r9c3 r8c1} - r8n6{c1 .} ==> r9c8≠6
whip[3]: r9c4{n6 n5} - b7n5{r9c3 r8c1} - r8n6{c1 .} ==> r9c7≠6
biv-chain[2]: b5n6{r4c6 r6c4} - r9n6{c4 c3} ==> r4c3≠6
biv-chain[4]: b4n2{r4c1 r4c3} - r7c3{n2 n6} - r9n6{c3 c4} - b5n6{r6c4 r4c6} ==> r4c1≠6
z-chain[5]: r1c5{n5 n6} - b8n6{r7c5 r9c4} - c4n5{r9 r3} - c8n5{r3 r8} - r7n5{c9 .} ==> r5c5≠5
t-whip[3]: b5n5{r6c5 r4c6} - c9n5{r4 r7} - r8n5{c7 .} ==> r6c1≠5
t-whip[3]: r6c1{n7 n6} - r5n6{c3 c8} - b6n7{r5c8 .} ==> r4c1≠7
hidden-single-in-a-row ==> r4c9=7
naked-single ==> r7c9=5
biv-chain[3]: r1c5{n5 n6} - r7c5{n6 n7} - r8c6{n7 n5} ==> r1c6≠5
biv-chain[3]: r4n6{c7 c6} - r1c6{n6 n7} - r1c7{n7 n5} ==> r4c7≠5
finned-x-wing-in-rows: n5{r8 r4}{c6 c1} ==> r5c1≠5
naked-pairs-in-a-block: b4{r5c1 r6c1}{n6 n7} ==> r5c3≠6
whip[1]: c3n6{r9 .} ==> r8c1≠6
whip[1]: r8n6{c8 .} ==> r7c8≠6
biv-chain[3]: r4c6{n5 n6} - r4c7{n6 n9} - r6n9{c7 c5} ==> r6c5≠5
stte