## pat's 8.6 in the patterns game

Post puzzles for others to solve here.

### pat's 8.6 in the patterns game

original post
Pat wrote:
999_Springs wrote:

i'd love to try out the puzzles that have been posted,
but if you keep posting them at like 2 AM
then i'll never get a chance

i'm usually on Lunar Standard Time

just posted this one in the Patterns Game
— does this suit you?

.17...2..58.9.....6.9.....1.9.6....7.....76......3..5.2...6.3.......9.8...14....2
[ play ]

Code: Select all
` . 1 7 . . . 2 . . 5 8 . 9 . . . . . 6 . 9 . . . . . 1 . 9 . 6 . . . . 7 . . . . . 7 6 . . . . . . 3 . . 5 . 2 . . . 6 . 3 . . . . . . . 9 . 8 . . . 1 4 . . . . 2`

this is really hard

here's my solution path (spoiler) with a few thoughts and stuff
Hidden Text: Show
1. bunch of easy starting singles

2. hidden pair 58 at r1c9 r3c7

3. 6r1c6 is a single

at this point i got stuck for nearly two hours without finding anything. then i realised that i'd copied down the puzzle wrong and left out 2 numbers. what a waste of time

4. r1c5=/=8: flobalobalobalob r1c5-8-r9c5=8=r9c6=3=r8c4-3-r1c4=3=r1c1=4=r1c5

5. r2c56=/=4: 6-cell xy-chain r2c56-4-r2c7-7-r7c9-5-r3c7-8-r1c9-5-r1c5-4-r2c56

6. r9c8=/=7: a carcul-style horrible long chain. i couldn't find anything neater here so went on an all out attack on a candidate
r9c8 (-7-r9c7-5-r3c7-8-(r3c6)r1c9-5-r1c5-4-r1c1-3-r2c3)
(=6=r9c2=3=r9c6-3-r2(3)c6)
-7-r23c8=7=r2c7-7-r2c5=als:127r2c56-2-(r3c6)r2c3-4-r2c89=4=r3c8-4-(r7c8)r3c6-5-r7c6-1-r7c8-7-r9c8

7. bunch of singles r9c8=r2c9=6 r5c9=3

8. xy-wing r8c7=/=4: 457 r29c7 r8c9

9. grouped m-ring r8c45=/=5: r8c45-5-r8c9-4-r8c123=4=r7c2=5=r7c46-5-r8c45 this was probably unnecessary

at this point i tried to start an attack on the 4 in column 1 and see what happened

10. r8c7, r9c2=/=7
r1c1-4-r1c5-5-r1c9=5=r3c7-7-r8c7,r9c2
||
r8c1-4-r8c9-5-same as above
||
r456c1-4-r1(8)c1-3-r8c1-7-r8c7,r9c2 kinda like an almost w-wing

11. r8c7=/=5
r1c1-4-r1c5-5-r1c9=5=r3c7-5-r8c7
||
r8c1-4-r8c9-5-r8c7
||
r456c1-4-(r4c3,r6c23)als:245r5c23(-25-r4c3-3-r8c3)-2-r6c3-6-als:456r8c39-5-r8c7

12. r8c7=1

13. naked pair 48 in b6

14. locked candidate type 2: 4 in r5 and b4 i spent like over half an hour searching for other chains and stuff going nowhere before finding this... sudoku can be so cruel sometimes

15. r1c9=8
r1c1-4-r1c5-5-r1c9-8
||
r8c1-4-r8c9-5-r1c9-8
||
r5c1=8=r5c4-8-r1c4=8=r1c9

16. bunch of singles

17. r4c1=/=3: r4c1-3-r1(8)c1-4-r8c1-7-r8c(4)5-2-r8c2-3-r1c4=3=r1c1-3-r4c1

18. singles to end. the 1 in c5 was a particularly sneaky single

total solving time: ~5 hours including the idiot mistake at the start

if anyone else wants to try doing this, go for it. if you could find a neater solution path too, that would be great. knowing this forum, that's not likely to happen

se rates it 8.6
Once upon a time I was a teenager who was active on here 2007-2011
ocean and eleven should have paired up to make a sudoku-solving duo called Ocean's Eleven
999_Springs

Posts: 487
Joined: 27 January 2007
Location: In the toilet, flushing down springs, one by one.

### Re: pat's 8.6 in the patterns game

For those of us who are not familiar with SE and its ratings, could you provide some perspective about the meaning of an 8.6?
Marty R.

Posts: 1508
Joined: 23 October 2012
Location: Rochester, New York, USA

### Re: pat's 8.6 in the patterns game

Marty R. wrote:For those of us who are not familiar with SE and its ratings, could you provide some perspective about the meaning of an 8.6?

This means that in SE sequence the hardest step is something like this
Code: Select all
`A    B     C    |D     E     F     |G   H   I   34   1     7    |358   45    6     |2   9   58  5    8     234  |9     127   123   |47  367 36  6    234   9    |23578 2457  23458 |58  347 1   -----------------------------------------------1348 9     2345 |6     12458 12458 |148 123 7   148  2345  2345 |1258  9     7     |6   123 348 1478 267   26   |128   3     1248  |9   5   48  -----------------------------------------------2    457   8    |157   6     15    |3   147 9   347  34567 3456 |12357 1257  9     |14  8   456 9    3567  1    |4     78    358   |57  67  2   `

[] 1r8c4 -> ~1r8c7 -> 4r8c7 -> ~4r2c7 -> 4r2c3 -> ~2r2c3 -> 2r3c2 -> ~2r3c6
[] 1r8c4 -> ~3r8c4 -> 3r9c6 -> ~3r3c6
[] 1r8c4 -> ~1r8c7 -> 1r7c8 -> ~4r7c8 -> 4r3c8 -> ~4r3c6
[] 1r8c4 -> ~1r7c6 -> 5r7c6 -> ~5r3c6
[] 1r8c4 -> ~1r8c7 -> 1r4c7 -> ~8r4c7 -> 8r3c7 -> ~8r3c6

<1r8c4>

This does not exclude a nicer solution using other rules.
champagne
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Joined: 02 August 2007
Location: France Brittany

### Re: pat's 8.6 in the patterns game

<6.5: se's solving methodology is highly flawed at this level
6.5-7.0: x-chains, x-cycles, y-cycles (not xy-chains. i think that's a bug though)
7.1-7.5: standard aic's (no grouping)
7.6-8.1: fish (expressed as nishio)
8.2-8.7: kraken row/column/block/cell (no grouping) (like what champagne is doing but backwards, so you start from a cell and work towards the elimination, but the logic is the same)
8.7-9.5: singles forcing nets
9.3-10.1: forcing nets with lc's and pairs and stuff
10.0-10.9: forcing nets with nested aic's
10.8+: forcing nets with nested... more stuff

from a manual solving perspective:

<6.5: easy
6.5-7.0: the easier end of stuff posted here
7.1-7.5: the harder end of stuff posted here, sometimes there's a one-step solution, sometimes not
7.6-8.1: you may wish to set aside a decent chunk of time to do this. look for various types of fish
8.2-8.7: not necessarily that much harder than above, except don't look for fish, and don't expect one-step solutions
8.7-9.5: expect a long hard chain-based slog. sometimes there's a quick solution if you're lucky, but usually not
9.3-10.1: excruciatingly painful, can still be done, but do you really want to
10.0+: may i suggest throwing toilet paper at a tree as a more productive use of your time
Once upon a time I was a teenager who was active on here 2007-2011
ocean and eleven should have paired up to make a sudoku-solving duo called Ocean's Eleven
999_Springs

Posts: 487
Joined: 27 January 2007
Location: In the toilet, flushing down springs, one by one.

### Re: pat's 8.6 in the patterns game

999_Springs wrote:from a manual solving perspective:
9.3-10.1: excruciatingly painful, can still be done, but do you really want to
10.0+: may i suggest throwing toilet paper at a tree as a more productive use of your time

If you limit yourself to chain nets, this is true.

A significant part of these puzzles can be easily solved using "exotic" patterns as SK loop, Exocets, big rank 0 logic. Most of them see the difficulty significantly reduced applying such rules.

champagne
2017 Supporter

Posts: 7030
Joined: 02 August 2007
Location: France Brittany

### re: Pat # 212 # 56

999_Springs wrote:this is really hard

the 8.6 rating indicates a tough move

but then again, for me even 7.3 is too tough

.38...2..12.9.....4.9.....1.9.1....6.....28......5..7.2...8.7.......6.3...12....5 # Pat # 212 # 54
the real question:
how many tough moves does it take?

i tried to choose puzzles which require multiple tough moves

• on my 7.3,
SuDoku Explainer says:
23 x Forcing Chain
• on my 8.6,
SuDoku Explainer says:
15 x Region Forcing Chains

- . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - . - .
999_Springs wrote:6.5-7.0: x-chains, x-cycles, y-cycles

gsf's solver says the above puzzles
can be solved by X-cycles + Y-cycles
-q{NF}{B2B3}{H2T2H3T3H4T4}{W2W3W4}{XY}-G
( W2W3W4 was never used )
7.3: Show
Code: Select all
`      1  1      N3  [35][89][48]=2      2  1      N3  [63]=2 [29][98]=8      3  1      N1  [19]=7      4  1      N1  [18]=9      5  2      B4  [47][67]^3 [25][26]^4      6  5      X3                 Y1  [55]=9 [85][95]^9 [97]^9      7  2      B2  [14][34]^6      8  3      H2                 T2  [67][87]={19} [67][87]^{3456}      9  5      Y9  [16][54][64][74]^4 [27][28][32][51][91]^6     10  5      Y3  [14][34][76]^5     11  1      F1  [14]=4     12  5      Y1  [41]^5     13  5      Y3  [27][37]^5 [47]=5     14  5      Y5  [34][36]^3 [72][76]^4 [26]^5     15  1      N1  [37]=3     16  1      N2                 F1  [38][97]=6 [27]=4     17  1      N1                 F1  [28]=5     18  2      B1  [73]^4     19  3      H1                 T1  [76]={149} [76]^{356}     20  5      Y19 [46][53][66]^3 [52][72]^5 [52][62][73]^6 [72]=6                    [25][26][43][53][81][82][83][95][96]^7 [23]=7     21  1      N3                 F2  [25][53][11]=6 [26]=3 [32]=5     22  1      N3                 F1  [16][51]=5 [64]=6 [15]=1     23  1      N3  [87][76]=1 [34]=8     24  1      N7                 F3  [36]=7 [62][58]=1 [79][81][96][67]=9 [78]=4     25  1      N1  [82]=8     26  2      B1  [45]^4     27  5      X6                 Y7  [54][95][69][73]=3 [59][45]^3 [69][46][95]^4 [92][59][85][66]=4     28  1      N10                F8  [41]=3 [43]=4 [52][84][91][45]=7 [61][46]=8 [74][83]=5                S`

8.6: Show
Code: Select all
`      1  1      N2  [91][55]=9      2  1      N2  [67]=9 [73]=8      3  3      H5                 T3  [19][37]={58} [18][19][37]={589} [18][19][37]^{3467}      4  1      N3                 F1  [16]=6 [18][79]=9      5  5      Y6  [23][41][51][53]^3 [25][26]^4      6  5      Y2  [36][45]^4      7  5      Y3  [15]^5 [34][35]^8      8  5      Y3  [48][52][83]^3      9  1      N1  [43]=3     10  2      B1  [54]^5     11  5      Y5  [45]^1 [25][63]^2 [85][95]^5     12  5      Y1  [25]=1     13  2      B1  [38]^7     14  5      Y24 [35][45][46]^2 [48][85]=2 [11][84][92]=3 [14][29][34][82][58]^3                    [11][29][35][48][52][53][58][62][63][81]^4 [87]^7     15  1      N4                 F4  [15]=4 [59]=3 [87]=1 [98]=6 [29][63]=6 [58]=1 [81]=7     16  1      N3  [51]=4 [62]=7 [82]=6     17  1      N1  [54]=8     18  1      N3                 F2  [19][95][36]=8 [14][45]=5     19  1      N9                 F4  [38][26]=3 [37][89]=5 [61][47]=8 [97][35][74]=7 [96]=5 [69]=4     20  1      N14                F6  [23][66][34]=2 [28]=7 [32][46][83][27][78]=4 [76][41][64]=1                    [72][53]=5     21  1      N1                 F1  [52]=2                S`

Pat

Posts: 3879
Joined: 18 July 2005

### Re: pat's 8.6 in the patterns game

champagne wrote:
999_Springs wrote:from a manual solving perspective:
9.3-10.1: excruciatingly painful, can still be done, but do you really want to
10.0+: may i suggest throwing toilet paper at a tree as a more productive use of your time

If you limit yourself to chain nets, this is true.

A significant part of these puzzles can be easily solved using "exotic" patterns as SK loop, Exocets, big rank 0 logic. Most of them see the difficulty significantly reduced applying such rules.

ok this is news to me. i've been out of these forums for 3 years so i don't know much about these new methods or anything about how manual players can solve ultra hard puzzles using them. can you point me to some resources please? links that start from the beginning and have a strong emphasis on applying them to manual solving would be great

also did you do this? http://i.imgur.com/0faKbUv.jpg

pat: i have no idea what those weird symbols mean in your output but in sudoku explainer an x-cycle is a closed loop version of a forcing x-chain and a y-cycle is a closed loop version of an xy-chain, which are definitely confined to sub-7 territory. also what's up with the morse code in the middle of your post
Once upon a time I was a teenager who was active on here 2007-2011
ocean and eleven should have paired up to make a sudoku-solving duo called Ocean's Eleven
999_Springs

Posts: 487
Joined: 27 January 2007
Location: In the toilet, flushing down springs, one by one.

### Re: pat's 8.6 in the patterns game

Hi 999_Springs,

a comprehensive answer would require more time, so, just a "flavour" of what can be done

false double exocet

this is a link within a thread dedicated to such powerful rules, but not in the list I mentioned.

Have a look to these puzzles rating at minimum SE diamond 10.3

Most of them should collapse after a short analysis of a visible potential complementary double AAHS
champagne
2017 Supporter

Posts: 7030
Joined: 02 August 2007
Location: France Brittany

999_Springs wrote:in sudoku explainer, an x-cycle is a closed loop version of a forcing x-chain
and a y-cycle is a closed loop version of an xy-chain,
which are definitely confined to sub-7 territory

-q{NF}{B2B3}{H2T2H3T3H4T4}{W2W3W4}{XY}-G
in gsf's command-line [ manual ] —
• level 1 = NF = "singles"
• level 2 = B = b\l, l\b
• level 3 = HT = subsets
• level 4 = W = fish (not used)
• level 5 = XY, quite possibly not matching the SuDoku Explainer definition

this is the 8.6 after "basic" moves [ play ]

Code: Select all
`+-------+-------+---------+| . 1 7 | . . 6 | 2  9 58 || 5 8 . | 9 . . | .  . .  || 6 . 9 | . . . | 58 . 1  |+-------+-------+---------+| . 9 . | 6 . . | .  . 7  || . . . | . 9 7 | 6  . .  || . . . | . 3 . | 9  5 .  |+-------+-------+---------+| 2 . 8 | . 6 . | 3  . 9  || . . . | . . 9 | .  8 .  || 9 . 1 | 4 . . | .  . 2  |+-------+-------+---------+`
at this point, gsf's software offers 6 exclusions by Y-cycles
Code: Select all
`      5  5      Y6  [23][41][51][53]^3 [25][26]^4`
• exclude 3 at [23][41][51][53]
• exclude 4 at [25][26]
does this make sense ?

Pat

Posts: 3879
Joined: 18 July 2005

### re: XY

i've now obtained the detailed report for those 6 exclusions —
Code: Select all
`y-cycle  1  3 20a [23]-[11]>[43]-y-cycle  2  3 20a [41]-[11]>[43]-y-cycle  3  3  3a [51]-[29][59]-y-cycle  4  3 13a [53]-[59]>[83]-y-cycle  5  4 26a [25]-[15]>[83]-[72][78]-  =>  [25]-[15]>[83]-y-cycle  6  4 26a [26]-[15]>[83]-[72][78]-  =>  [26]-[15]>[83]-`

does this make sense ?
Attachments
report.TXT
detailed report
puzzles.txt
2 puzzles: 7.3 and 8.6
XY.bat.TXT
bat for detailed report

Pat

Posts: 3879
Joined: 18 July 2005

### Re: pat's 8.6 in the patterns game

unfortunately i don't get it... sudoku explainer lists all 6 of those eliminations as dynamic fc's, the shortest one being 3r1 --> 3r5c1 off:
Hidden Text: Show
Chain 1: If R1C1 contains the value 3, then R5C1 cannot contain the value 3 (View 1):
(1) If R1C1 contains the value 3, then R5C1 cannot contain the value 3 (the value can occur only once in the column)

Chain 2: If R1C4 contains the value 3, then R5C1 cannot contain the value 3 (View 2):
(1) If R1C4 contains the value 3, then R3C6 cannot contain the value 3 (the value can occur only once in the block)
(2) If R1C4 contains the value 3 (initial assumption), then R2C6 cannot contain the value 3 (the value can occur only once in the block)
(3) If R2C6 does not contain the value 3 and R3C6 does not contain the value 3 (1), then R9C6 must contain the value 3 (only remaining possible position in the column)
(4) If R9C6 contains the value 3, then R9C6 cannot contain the value 8 (the cell can contain only one value)
(5) If R9C6 does not contain the value 8, then R9C5 must contain the value 8 (only remaining possible position in the block)
(6) If R9C5 contains the value 8, then R1C5 cannot contain the value 8 (the value can occur only once in the column)
(7) If R1C4 contains the value 3 (initial assumption), then R1C4 cannot contain the value 8 (the cell can contain only one value)
(8) If R1C4 does not contain the value 8 and R1C5 does not contain the value 8 (6), then R1C9 must contain the value 8 (only remaining possible position in the row)
(9) If R1C9 contains the value 8, then R6C9 cannot contain the value 8 (the value can occur only once in the column)
(10) If R6C9 does not contain the value 8, then R6C9 must contain the value 4 (only remaining possible value in the cell)
(11) If R6C9 contains the value 4, then R5C9 cannot contain the value 4 (the value can occur only once in the block)
(12) If R1C9 contains the value 8 (8), then R5C9 cannot contain the value 8 (the value can occur only once in the column)
(13) If R5C9 does not contain the value 8 and R5C9 does not contain the value 4 (11), then R5C9 must contain the value 3 (only remaining possible value in the cell)
(14) If R5C9 contains the value 3, then R5C1 cannot contain the value 3 (the value can occur only once in the row)

and i can't construct that at all out of gsf's output, and it also doesn't even involve r2c9 anywhere.

certainly none of these eliminations can be done with an xy-chain or loop. in particular, the xy-chain in my solution isn't there until you kill 8r1c5 - which gsf does not do

Once upon a time I was a teenager who was active on here 2007-2011
ocean and eleven should have paired up to make a sudoku-solving duo called Ocean's Eleven
999_Springs

Posts: 487
Joined: 27 January 2007
Location: In the toilet, flushing down springs, one by one.

### re: XY in gsf's software

well gsf may be busy

so i changed the reporting to "-v3" (to get greater detail)
-v3: Show
Code: Select all
`                    y-edge      3 22  [11]>[28]                    y-edge      3 18  [11]>[43]                    y-edge      3  8  [11]=[59]                    y-edge      4  9  [15]>[27]                    y-edge      4 19  [15]>[52]                    y-edge      4  7  [15]>[69]                    y-edge      4 11  [15]>[78]                    y-edge      4 21  [15]>[83]                    y-edge      3 11  [59]>[83]                    y-edge      3  6  [59]=[96]                    y-cycle  1  3 20a [23]-[11]>[43]-                    y-cycle  2  3 20a [41]-[11]>[43]-                    y-cycle  3  3  3a [51]-[29][59]-                    y-cycle  4  3 13a [53]-[59]>[83]-                    y-cycle  5  4 26a [25]-[15]>[83]-[72][78]-  =>  [25]-[15]>[83]-                    y-cycle  6  4 26a [26]-[15]>[83]-[72][78]-  =>  [26]-[15]>[83]-      5  5      Y6  [23][41][51][53]^3 [25][26]^4`

then changed the reporting to "-v4" (to get even greater detail)
-v4: Show
Code: Select all
`                    x-hinge     1  6  [41]-[78]                    x-hinge     1  6  [45]-[78]                    x-hinge     1  6  [46]-[78]                    x-hinge     3  7  [32]-[84]                    x-hinge     3  7  [52]-[84]                    x-hinge     4  3  [23]-[78]                    x-hinge     4  3  [25]-[78]                    x-hinge     4  3  [26]-[78]                    x-hinge     5  7  [52]-[89]                    x-hinge     5  9  [82]-[37]                    x-hinge     5  9  [83]-[37]                    x-hinge     5  9  [84]-[37]                    x-hinge     5  9  [85]-[37]                    x-hinge     6  7  [89]-[62]                    x-hinge     7  7  [84]-[62]                    x-hinge     7  7  [85]-[62]                    x-hinge     7  7  [87]-[62]                    x-hinge     8  6  [41]-[19]                    x-hinge     8  6  [45]-[19]                    x-hinge     8  6  [46]-[19]                    y-edge      3 22  [11]>[28]                    y-edge      3 18  [11]>[43]                    y-edge      3  8  [11]=[59]                    y-edge      4  9  [15]>[27]                    y-edge      4 19  [15]>[52]                    y-edge      4  7  [15]>[69]                    y-edge      4 11  [15]>[78]                    y-edge      4 21  [15]>[83]                    y-edge      3 11  [59]>[83]                    y-edge      3  6  [59]=[96]                    y-cycle  1  3 20a [23]-[11]>[43]-                    y-cycle  2  3 20a [41]-[11]>[43]-                    y-cycle  3  3  3a [51]-[29][59]-                    y-cycle  4  3 13a [53]-[59]>[83]-                    y-cycle  5  4 26a [25]-[15]>[83]-[72][78]-  =>  [25]-[15]>[83]-                    y-cycle  6  4 26a [26]-[15]>[83]-[72][78]-  =>  [26]-[15]>[83]-      5  5      Y6  [23][41][51][53]^3 [25][26]^4`

i'm still in the dark

"-v5" pours out tons of extra stuff ("y-trip")
which may explain what's really happening
but not something a human can follow

yes we'll have to wait for gsf

Pat

Posts: 3879
Joined: 18 July 2005

### Re: pat's 8.6 in the patterns game - gwhip solution

Full resolution path using Subsets and g-whips. Longest chain = 6

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`(solve ".17...2..58.9.....6.9.....1.9.6....7.....76......3..5.2...6.3.......9.8...14....2")*****  SudoRules 16.2 based on CSP-Rules 1.2, config: gW-S   *****24 givens, 221 candidates, 1451 csp-links and 1451 links. Initial density = 1.49singles ==> r9c1 = 9, r6c7 = 9, r5c5 = 9, r7c3 = 8hidden-pairs-in-a-block: b3{r1c9 r3c7}{n5 n8} ==> r3c7 <> 7, r3c7 <> 4, r1c9 <> 9, r1c9 <> 6, r1c9 <> 4, r1c9 <> 3singles ==> r1c8 = 9, r7c9 = 9, r1c6 = 6biv-chain[4]: b8n8{r9c5 r9c6} - b8n3{r9c6 r8c4} - r1n3{c4 c1} - r1n4{c1 c5} ==> r1c5 <> 8biv-chain[4]: r1c5{n4 n5} - c9n5{r1 r8} - r9c7{n5 n7} - r2c7{n7 n4} ==> r2c6 <> 4, r2c5 <> 4whip[4]: r1n3{c1 c4} - r1n8{c4 c9} - r5c9{n8 n4} - r6c9{n4 .} ==> r5c1 <> 3whip[5]: r6c9{n4 n8} - r1c9{n8 n5} - r1c5{n5 n4} - r3n4{c5 c8} - r7n4{c8 .} ==> r6c2 <> 4whip[5]: r6c9{n4 n8} - r1c9{n8 n5} - r1c5{n5 n4} - r3n4{c5 c2} - r7n4{c2 .} ==> r5c8 <> 4whip[4]: c3n6{r6 r8} - c9n6{r8 r2} - c9n3{r2 r5} - r5n4{c9 .} ==> r6c3 <> 4whip[5]: r6c9{n4 n8} - r1c9{n8 n5} - r1c5{n5 n4} - r3n4{c5 c2} - r7n4{c2 .} ==> r4c8 <> 4whip[3]: r2c7{n7 n4} - c8n4{r2 r7} - b9n1{r7c8 .} ==> r8c7 <> 7whip[4]: c7n1{r8 r4} - c8n1{r4 r7} - b9n4{r7c8 r8c9} - b6n4{r6c9 .} ==> r8c7 <> 5whip[5]: r5n4{c1 c9} - r6c9{n4 n8} - r4c7{n8 n1} - r8c7{n1 n4} - r2n4{c7 .} ==> r4c3 <> 4whip[6]: b9n7{r9c8 r7c8} - c8n4{r7 r3} - r2n4{c7 c3} - r1c1{n4 n3} - r8c1{n3 n4} - b9n4{r8c9 .} ==> r9c2 <> 7whip[6]: r5n4{c1 c9} - r6c9{n4 n8} - r4c7{n8 n1} - b4n1{r4c1 r5c1} - r5n8{c1 c4} - r1n8{c4 .} ==> r6c1 <> 4g-whip[6]: r6c9{n4 n8} - r1n8{c9 c4} - r1n3{c4 c1} - c1n4{r1 r456} - r5n4{c2 c1} - r5n8{c1 .} ==> r8c9 <> 4hidden-pairs-in-a-block: b9{r7c8 r8c7}{n1 n4} ==> r7c8 <> 7whip[1]: b9n7{r9c8 .} ==> r9c5 <> 7biv-chain[3]: r9c5{n8 n5} - c7n5{r9 r3} - c7n8{r3 r4} ==> r4c5 <> 8whip[5]: c7n1{r4 r8} - b8n1{r8c5 r7c4} - r6n1{c4 c1} - c1n7{r6 r8} - b8n7{r8c4 .} ==> r4c6 <> 1whip[5]: c7n1{r4 r8} - c4n1{r8 r7} - r6n1{c4 c1} - c1n7{r6 r8} - b8n7{r8c4 .} ==> r4c5 <> 1biv-chain[3]: c5n1{r2 r8} - r8c7{n1 n4} - r2c7{n4 n7} ==> r2c5 <> 7whip[1]: b2n7{r3c5 .} ==> r3c8 <> 7hidden-pairs-in-a-column: c8{n6 n7}{r2 r9} ==> r2c8 <> 3whip[3]: r3c8{n3 n4} - r2n4{c7 c3} - r1c1{n4 .} ==> r3c2 <> 3whip[3]: b1n3{r1c1 r2c3} - b4n3{r5c3 r5c2} - c9n3{r5 .} ==> r8c1 <> 3biv-chain[3]: r1n4{c5 c1} - r8c1{n4 n7} - c5n7{r8 r3} ==> r3c5 <> 4biv-chain[4]: b8n3{r9c6 r8c4} - r8n2{c4 c5} - c5n7{r8 r3} - c5n8{r3 r9} ==> r9c6 <> 8hidden-single-in-a-block ==> r9c5 = 8biv-chain[4]: r9c6{n3 n5} - b9n5{r9c7 r8c9} - c9n6{r8 r2} - b3n3{r2c9 r3c8} ==> r3c6 <> 3whip[4]: b8n2{r8c5 r8c4} - b8n3{r8c4 r9c6} - r2c6{n3 n1} - r2c5{n1 .} ==> r3c5 <> 2whip[5]: c7n5{r3 r9} - c9n5{r8 r1} - r1n8{c9 c4} - c4n3{r1 r8} - r9c6{n3 .} ==> r3c4 <> 5whip[5]: b8n2{r8c5 r8c4} - c4n7{r8 r3} - c4n3{r3 r1} - r1c1{n3 n4} - r8c1{n4 .} ==> r8c5 <> 7hidden-single-in-a-column ==> r3c5 = 7biv-chain[4]: b9n1{r8c7 r7c8} - r7c6{n1 n5} - r3n5{c6 c7} - c7n8{r3 r4} ==> r4c7 <> 1singles ==> r8c7 = 1, r7c8 = 4, r3c8 = 3, r5c9 = 3, r2c5 = 1whip[1]: c2n3{r9 .} ==> r8c3 <> 3whip[1]: r5n4{c1 .} ==> r4c1 <> 4whip[1]: b3n4{r2c7 .} ==> r2c3 <> 4biv-chain[3]: b1n4{r3c2 r1c1} - r1n3{c1 c4} - r8n3{c4 c2} ==> r8c2 <> 4biv-chain[3]: b7n3{r8c2 r9c2} - r9n6{c2 c8} - r8c9{n6 n5} ==> r8c2 <> 5biv-chain[3]: b6n8{r4c7 r6c9} - r1n8{c9 c4} - r5n8{c4 c1} ==> r4c1 <> 8biv-chain[3]: c4n3{r8 r1} - r1c1{n3 n4} - r8c1{n4 n7} ==> r8c4 <> 7singles ==> r7c4 = 7, r7c2 = 5, r7c6 = 1naked-pairs-in-a-column: c2{r3 r5}{n2 n4} ==> r6c2 <> 2x-wing-in-rows: n5{r3 r9}{c6 c7} ==> r4c6 <> 5whip[2]: r2n2{c6 c3} - r6n2{c3 .} ==> r4c6 <> 2naked-pairs-in-a-row: r4{c6 c7}{n4 n8} ==> r4c5 <> 4singles to the end317546298582913476649278531193654827425897613876132954258761349764329185931485762`

Considering only Subsets and whips, the max. length is 7.
Allowing braids and/or g-braids doesn't change the resolution paths.
denis_berthier
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