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Assassin 76
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Para
Yokozuna
Yokozuna


Joined: 08 Nov 2006
Posts: 384
Location: The Netherlands

PostPosted: Thu Nov 15, 2007 12:00 pm    Post subject: Reply with quote

Hi all

Nice puzzle. Actually did something on the diagonal this time. Although i am not sure that step is really needed, just very clear. I think this puzzle is a rating 1.25, 1.5 is a bit high as it doesn't require much finesse. I think step 18 is the most important step because that single elimination gets things going after step 19 and 20.

Walk-through Assassin 76X

1. R1C12 = {39/48/57}: no 1,2,6

2. 11(3) at R1C6 and R2C8 = {128/137/146/236/245}: no 9

3. R1C89 = {59/68}: no 1,2,3,4,7

4. 20(3) at R3C4 and R8C2 = {389/479/569/578}: no 1,2

5. 21(3) at R6C8 = {489/579/678}: no 1,2,3

6. R89C1 = {15/24}: no 3,6,7,8,9

7. R89C9 = {13} -> locked for C9 and N9

8. 6(3) at R2C2 = {123} -> locked for N1
8a. Naked Triple {123} in R2C2,R3C3 and R9C9 -> locked for D\; 2 locked within R2C2 + R3C3 locked for N1
8b. Clean up: R1C12 = {48/57}: no 9
8c. Killer Pair {58} in R1C12 + R1C89 -> locked for R1

9. 45 on N2: 2 innies: R1C46 = 5 = {14/23}: no 6,7,9
9a. Cage overlap: R1C67 can't equal 5: R2C7: no 6

10. 45 on N5: 2 innies: R5C46 = 7 = {16/25/34}: no 7,8,9

11. 45 on N8: 2 innies: R8C46 = 10 = {19/28/37/46}: no 5

12. 45 on R1234: 2 innies: R4C37 = 16 = {79} -> locked for R4

13. 45 on R1234: 2 outies: R5C37 = 7 = {16/25/34}: no 7,8,9

14. 45 on R89: 2 innies: R8C37 = 12 = [39]/{48/57}: no 1,2,6; R8C3: no 9
14a. 2 in N9 locked for R9
14b. Clean up: R8C1: no 4

15. 45 on C12: 2 outies: R39C3 = 8 = [17/26/35]: R9C3 = {567}
15a. 20(3) at R8C2 = {479/569/578}: {389} blocked by R9C3: no 3
15b. Killer Pair {45} in R89C1 + 20(3) at R8C2 -> locked for N7
15c. Clean up: R8C7: no 7,8

16. 45 on C89: R39C7 = 8 = [17/35]/{26}: no 4,8,9; R3C7: no 5,7

17. 45 on C789: R158C6 = 8 = {1[5]2/134}: no 6,7,8,9; R5C6: no 2; 1 locked for C6
17a. Clean up: R5C4: no 1,5; R8C4: no 1,2,3,4

18. 16(3) at R1C3 = {69}[1]/[781/952/682]/{49}[3]/{67}[3]: {457} blocked by R1C12: R1C4: no 4
18a. Clean up: R1C6: no 1

19. 45 on C123: R158C4 = 14 = [149]/{23}[9]/[167]:[248/347] blocked by R158C6: R8C4: no 6,8
19a. Clean up: R8C6: no 2,4
19b. Naked Pair {13} in R8C69 -> locked for R8
19c. Clean up: R9C1: no 5; R8C7: no 9

20. Killer Triple {789} in R12C3 + R48C3 -> locked for C3
20a. Clean up: R3C3: no 1
20b. 1 in N1 lockd for C2

21. 20(3) at R8C2 = {569/578}: {479} blocked by R9C3: no 4; 5 locked for N7
21a. R89C1 = [24]
21b. Clean up: R1C2: no 8
21c. 1 in N7 locked for R7

22. 2 in R7 locked within 12(3) cage at R7C4 -> 12(3) = {237/246}: no 5,8,9
22a. 5 in R7 locked for N9
22b. R8C37 = 12 = [84]
22c. Clean up: R3C7: no 3; R5C3: no 3

23. 15(3) at R4C3 = [9]{24}/[7]{26}/[753]: R5C3: no 1
23a. Clean up: R5C7: no 6

24. 1 in C3 locked in 20(4) cage at R6C3 -> 20(4) = 8[219/417]: R6C3 = {24}; R7C3 = 1

25. 20(3) at R8C2 = {569}(last combo) -> locked for N7; 9 locked for C2

26. 15(3) at R6C2 = [5]{37}(last combo): R6C2 = 5; R7C12 = {37} -> locked for R7
26a. R9C3 = 5(hidden); R3C3 = 3; R89C9 = [31]; R23C2 = [21];
26b. R8C6 = 1; R8C4 = 9; R89C2 = [69]; R6C3 = 2; R8C8 = 7; R8C5 = 5
26c. R3C7 = 2; R9C78 = [62]
26d. Clean up: R5C4: no 6; R5C7: no 5

27. 15(3) at R4C7 = [951/753]: R5C6 = 5
27a. R5C4 = 2; R1C46 = [32](45 on C123 and C789)

And the rest is naked and hidden singles.

greetings

Para

ps. The new rank is yokozuna(highest rank in professional Sumo-wrestling) accompanied with the nice moderator colour.

pps. i really hoped you would all be solving my Toroidal Killer Sudoku, but i guess/hope that's for another time.


Last edited by Para on Tue Nov 27, 2007 6:28 pm; edited 1 time in total
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mhparker
Grandmaster
Grandmaster


Joined: 20 Jan 2007
Posts: 345
Location: Germany

PostPosted: Thu Nov 15, 2007 4:39 pm    Post subject: Reply with quote

Hi Para,

Thanks for providing the A76X WT so quickly. I was wondering whether anyone would manage it before A77 arrives on the scene.

Para wrote:
ps. The new rank is yokozuna(highest rank in professional Sumo-wrestling) accompanied with the nice moderator colour.

Congratulations! salut

As someone who's been around here for considerably longer than I have, it's indeed befitting that you will remain one step ahead of me even when I get my next promotion (after making my 250th post). The only slight injustice is that it looks like I will reach Grandmaster status before Ed does, unless Ed surprises us all with a flurry of posts in the next few days. Ed deserves the title more than I do, but I'm apparently more talkative (at least publically on the forum) than he is. Just a thought: maybe I should be promoted to Talkmaster rather than Grandmaster?! Wink

Para wrote:
Actually did something on the diagonal this time.

Whilst creating this puzzle, I made a discovery (probably common knowledge amongst puzzle makers?). As seems to have often been the case recently, Ruud's cage pattern for the A76 turned out to be very complex. In other words, my (admittedly ropey and old) software was struggling to come up with any variants with a unique solution. When it eventually did (typically after 15 - 20 minutes searching), the result was almost invariably either too easy or too boring (or both).

When I decided to try making a Killer-X out of it, something interesting happened. Not only did it take much less time to generate each variant (typically only around 30 seconds), but a high percentage of the generated puzzles suddenly became much harder (and usually much more interesting).

My theory for this observed effect is as follows: If the cage pattern is very complex, the puzzle needs big initial cage constraints (in the form of cages with fixed or very limited combinations, for example) in order to still have a unique solution. These cage constraints serve as obvious "footholds" for the solver when doing the puzzle, often making it relatively straightforward. Because a Killer-X has extra constraints due to the diagonals, it doesn't require so many initial cage constraints as a non-X variant in order to possess a unique solution. Consequently, it can afford to offer fewer footholds, thus often making the puzzle (contrary to expectation) harder. (Note: The A76X was about mid-range in terms of difficulty).

The interesting thing to note here (and the reason why I'm mentioning all this) is that the diagonals can serve a big purpose (in terms of increasing the quality of the puzzle) even if it appears to the solver as though there are no moves making use of the diagonals whatsoever. Even in this case, they can come into play in the endgame in order to provide a unique solution, usually at the stage where (for the solver) the puzzle has long since become trivial. In other words, even if the solver doesn't require the diagonals, the puzzle itself may do!

Para wrote:
pps. i really hoped you would all be solving my Toroidal Killer Sudoku, but i guess/hope that's for another time.

I can't speak for others, but I will definitely be attempting it myself. I just need to pick a day where I've had a good sleep the night before! I also assumed that, because you labelled it "Toroidal Killer Sudoku #1", it's the first of a new series, like my Mavericks. For the Mavericks, I'm thinking more in terms of a monthly (at most 3-weekly) cycle, rather than a weekly one, in order to keep them interesting and in order to ensure that the focus of the forum remains on the Assassins. I assume you're thinking along similar lines for your toroidal killers? If so, we still have a bit of time left to solve the first one...
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Cheers,
Mike
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Afmob
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Joined: 22 Sep 2007
Posts: 103
Location: MV, Germany

PostPosted: Thu Nov 15, 2007 6:04 pm    Post subject: Reply with quote

Congratulations Para!

By the way, here's my walkthrough for Mike's A76 variant. I pretty much used the same "breakthrough" move like Para.

A76X Walkthrough:

1. C12
a) 6(3) = {123} locked for N1
b) 12(2) = {48/57}
c) Outies = 8(2) = [17/26/35]
d) 20(3) <> 3 because R9C3 = (567)

2. R1234
a) Killer pair (58) locked in 12(2) + 14(2)
b) Innies = 16(2) = {79} locked for R4

3. N258
a) Innies N2 = 5(2) = {14/23}
b) Innies N5 = 7(2) = {16/25/34}
c) 7,9 locked in 25(4) @ N5 = 79{18/36/45}
d) Innies N8 = 10(2) -> no 5

4. C789
a) 4(2) = {13} locked for C9+N9
b) Outies C89 = 8(2) = [17/26/35/62]
c) Outies C789 = 8(3) = 1{25/34} -> 1 locked for C6
d) Outies C789: R5C6 <> 2 since it's the only place where 5 is possible

5. R6789+D\
a) Naked triple (123) locked in R2C2+R3C3+R9C9 for D\
b) 2 locked in D\ for N1 -> R3C2 <> 2
c) Innies N8 = [64/73/82/91]
d) Innies R89 = 22(4) - Innies N8 = 10(2) -> R8C37 = 12(2) = {39/48/57}; R8C3 <> 9
e) 2 locked in 15(3) @ R9 for R9
f) 6(2): R8C1 <> 4
g) Outies R6789 = 45(9) - 31(5) - Innies N5 = 7(2) -> R5C37 = 7(2) <> 7,8,9
h) Killer pair (45) locked in 6(2) + 20(3) for N7
i) R8C37 = 12(2): R8C7 <> 7,8

6. N258
a) Innies N5 = 7(2): R5C4 <> 1,5
b) 15(3) @ R4C3 <> 1 because 9 only possible @ R4C3 and R5C4 <> 1,5
-> 15(3) = {249/267/357}
c) 15(3) @ R4C3: R5C3 <> 3 because 5 only possible there
d) Clean up: R5C7 <> 4,6 (step 5g) -> 15(3) @ R4C7 = (159/249/357}
e) Killer pair (25) locked in both 15(3) for R5
f) Outies C123 = 14(3): R8C4 <> 6 because R25C4 would be <= 7
g) Innies N8: R8C6 <> 4

7. N258 !
a) ! 16(3) <> {178/457} since it's blocked by Killer pairs (45,78) of 12(2)
b) 16(3) must have 1,2 xor 3 and it's only possible @ R1C4 -> R1C4 = (123)
c) Innies N2 = 5(2): R1C6 <> 1
d) Outies C789 = 8(3): R8C6 <> 2 because R1C6 <> 1,5
e) Innies N8 = 10(2): R8C4 <> 8

8. R789
a) Naked pair (13) locked in R8C69 for R8
b) 6(2): R9C1 <> 5
c) 20(4): R67C3 <> 5,6,7,8,9 because R8C34 >= 15
d) R8C37 = 12(2): R8C7 <> 9
e) 3 locked in R7C123 for R7

9. C123 !
a) ! 7,8,9 locked in 16(3)+R48C3 for C3
b) Outies C12 = 8(2) = [26/35]
c) 1 locked in R23C2 for C2
d) 20(3) <> 4 because R9C3 = (56)
e) Hidden Single: R9C1 = 4 @ N7 -> R8C1 = 2
f) 20(3) = {569} because R8C3 = (78) blocks {578}; {569} locked for N7, 9 locked for C2

10. R789
a) 1 locked in R7C13 for R7
b) 12(3) = {246} locked for R7+N8
c) 5,9 locked in R7C789 for N9
d) R8C7 = 4 -> R8C3 = 8 (step 9d)
e) 15(3) @ N9 = {267} locked for N9
f) 15(3) @ N7 = {357} because R7C12 = {13/17/37} and only {37} leads to a combination for 15(3)
-> R6C2 = 5, {37} locked for R7
g) R7C3 = 1
h) 20(4) = 18[29/47]
i) Hidden Single: R3C3 = 3 @ C3 -> R3C2 = 1, R2C2 = 2, R9C9 = 1, R8C9 = 3, R8C6 = 1
j) Innies N8 = 10(2) = [91] -> R8C4 = 9
k) R8C2 = 6, R9C2 = 9, R9C3 = 5, R8C8 = 7, R8C5 = 5, R6C3 = 2, R3C7 = 2, R9C7 = 6, R9C8 = 2

11. N23
a) 11(3) @ R2C8 = 2[36/54]
b) Killer pair (56) locked in 11(3) @ R2C8 + 14(2) for N3
c) 11(3) @ N2 = 1{28/37} -> 1 locked for C7
d) Innies N2 = 5(2) = {23} locked for R1+N2
e) 16(3) = 3{49/67} -> R1C4 = 3
f) 11(3) = [218] -> R1C6 = 2, R1C7 = 1, R2C7 = 8
g) 14(2) = {59} locked for R1+N3

12. Rest is clean-up and singles.

Rating: 1.25, nothing too complicated.


Last edited by Afmob on Wed Nov 21, 2007 6:17 am; edited 3 times in total
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Para
Yokozuna
Yokozuna


Joined: 08 Nov 2006
Posts: 384
Location: The Netherlands

PostPosted: Thu Nov 15, 2007 8:33 pm    Post subject: Reply with quote

mhparker wrote:
Whilst creating this puzzle, I made a discovery (probably common knowledge amongst puzzle makers?). As seems to have often been the case recently, Ruud's cage pattern for the A76 turned out to be very complex. In other words, my (admittedly ropey and old) software was struggling to come up with any variants with a unique solution. When it eventually did (typically after 15 - 20 minutes searching), the result was almost invariably either too easy or too boring (or both).

When I decided to try making a Killer-X out of it, something interesting happened. Not only did it take much less time to generate each variant (typically only around 30 seconds), but a high percentage of the generated puzzles suddenly became much harder (and usually much more interesting).

The reason this pattern gives so much problems are the center 3 nonets(258). There are 6 rows in these nonets that have all there cells in one cage. This gives a lot of options for non-unique patterns. Because of this there are a limited set of valid combinations that would give a unique solution of digits for this pattern. Then there is the extra constraint that the cages also have to be able to be uniquely filled. This tends to give either rather easy of extremely difficult puzzles. The brick wall pattern is another example of the uniqueness problem.
When adding the diagonals you can substantially increase the number of unique combinations for the center 3 nonets, thus also increasing the number of unique puzzles and you will have a wider range of difficulty in your puzzles as you have more options to choose from.


mhparker wrote:
Para wrote:
pps. i really hoped you would all be solving my Toroidal Killer Sudoku, but i guess/hope that's for another time.

I can't speak for others, but I will definitely be attempting it myself. I just need to pick a day where I've had a good sleep the night before! I also assumed that, because you labelled it "Toroidal Killer Sudoku #1", it's the first of a new series, like my Mavericks. For the Mavericks, I'm thinking more in terms of a monthly (at most 3-weekly) cycle, rather than a weekly one, in order to keep them interesting and in order to ensure that the focus of the forum remains on the Assassins. I assume you're thinking along similar lines for your toroidal killers? If so, we still have a bit of time left to solve the first one...

I don't know for sure how often i will post these puzzles. But it is mostly that i am not really formiliar yet with all the special techniques for this type. I think the toroidal shapes have more impact than i can find right now. But the idea now is to increase the difficulty by each puzzle and so finding more techniques so i can increase the difficulty a bit more. There's a few things i know that might help me make these puzzles more difficult but you'll find about that when solving.

greetings

Para
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azpaull
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Joined: 27 Aug 2007
Posts: 14
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PostPosted: Thu Nov 15, 2007 10:39 pm    Post subject: Reply with quote

I enjoyed this one, and actually finished on Tuesday - which is very unusual for me. I'll have to give it a 1.0 - to paraphrase Groucho, any Assassin that will allow me to solve it....
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mhparker
Grandmaster
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Joined: 20 Jan 2007
Posts: 345
Location: Germany

PostPosted: Fri Nov 16, 2007 11:02 am    Post subject: Reply with quote

Para wrote:
The reason this pattern gives so much problems are the center 3 nonets(258). There are 6 rows in these nonets that have all there cells in one cage. This gives a lot of options for non-unique patterns.

You're right. For example, if I remove the diagonal constraints from the A76X, it not only doesn't have a unique solution any more, but 15 of the cells within the resulting deadly pattern belong to the 18 cells in n258 that you mention.

I also looked at some of the other grids I generated based on this cage pattern. The easiest one had c123 outies = 6(3) = {123} and c789 outies = 24(3) = {789}. Of course, these fixed cage combinations significantly reduce the number of valid arrangements for the digits in n258 and were possibly critical in causing the puzzle to have a unique solution. But they also made the puzzle trivial. That's the point I was trying to make, and which accounts for the "uniqueness problem" you mention where a high proportion of the generated puzzles become (too) easy.

Para wrote:
This tends to give either rather easy of extremely difficult puzzles.

In practice, none of the non-X variants I generated were extremely difficult this time. But (because it took so long to generate each one, by which time the laptop was running very hot) I only generated 9 of these, which means that the results are maybe not statistically significant.

BTW, with the A73 (where I didn't publish anything), the situation was really extreme. Around 60% of the variants I privately generated were extremely difficult (read: practically unsolvable!), around one third were trivial, and only 5 - 10% of the attempts resulted in something in the 1.25 - 1.75 rating range.

Para wrote:
When adding the diagonals you can substantially increase the number of unique combinations for the center 3 nonets, thus also increasing the number of unique puzzles and you will have a wider range of difficulty in your puzzles as you have more options to choose from.

In practice, it appeared that having diagonals not only increased the number available to choose from, but also made the difficulty spread more even. For now it's somewhat of an unconfirmed observation due to not having enough non-X variants available to say definitely that this is the case. Maybe I need another 20 or so grids to see a definite trend. But at an estimated 20 minutes for each puzzle, that exercise would tie me (and my computer!) up for not much less than a whole working day!
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Cheers,
Mike
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Andrew
Grandmaster
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Joined: 11 Aug 2006
Posts: 300
Location: Lethbridge, Alberta

PostPosted: Tue Nov 20, 2007 11:22 pm    Post subject: Reply with quote

Mike estimated A76X as 1.5 but both Para and Afmob rated it as 1.25. I can see where both the estimate and the rating come from. If one spots the two key moves quickly then it's definitely 1.25. Maybe both Para and Afmob spotted those two moves quickly. I didn't!

Perhaps Mike's estimate took account of the difficulty in spotting the key moves. Both key moves are based on the 16(3) cage at R1C3; Para's steps 18 and 20, Afmob's steps 7a and 9a.

I spent several hours struggling and nibbling, see my comment after step 19, before I revisited that 16(3) cage and found the {457} elimination. Then because that gave key eliminations in R1C46 I missed the killer triple in C3 so my later stage was a lot longer.

It is, in fact, possible to solve the puzzle using only one of the key eliminations. I did that without spotting the killer triple. Similarly I think it would be possible to solve it using the killer triple, but missing the {457} elimination, since the killer triple is present for all combinations in 16(3) except for {178} which clashes with the killer triple as well as with R1C12.


Here is my walkthrough for A76X, without the killer triple in C3.

This is a Killer-X. I've included all the eliminations on the diagonals because it's so easy to overlook them if you are doing your own eliminations.

Prelims

a) R1C12 = {39/48/57}, no 1,2,6
b) R1C89 = {59/68}
c) R89C1 = {15/24}
d) R89C9 = {13}, locked for C9 and N9
e) 6(3) cage in N1 = {123}, locked for N1, clean-up: no 9 in R1C12
f) R3C456 = {389/479/569/578}, no 1,2
g) 11(3) cage at R1C6 = {128/137/146/236/245}, no 9
h) 11(3) cage at R2C8 = {128/137/146/236/245}, no 9
i) 21(3) cage at R6C8 = {489/579/678}, no 1,2,3
j) 20(3) cage in N7 = {389/479/569/578}, no 1,2

1. Naked triple {123} in R2C2 + R3C3 + R9C9, locked for D\
1a. 2 locked in R2C2 + R3C3 for D\ -> no 2 in R3C2

2. Killer pair 5,8 in R1C12 and R1C89, locked for R1

3. 45 rule on N2 2 innies R1C46 = 5 = {14/23}

4. 45 rule on R1234 2 innies R4C37 = 16 = {79}, locked for R4

5. 45 rule on N5 2 innies R5C46 = 7 = {16/25/34}, no 7,8,9

6. 45 rule on R1234 4 outies R5C3467 = 14, R5C46 = 7 -> R5C37 = 7 = {16/25/34}, no 7,8,9

7. 45 rule on N8 2 innies R8C46 = 10 = {19/28/37/46}, no 5

8. 45 rule on R89 4 innies R8C3467 = 22 -> R8C37 = 12 = [39]/{48/57}, no 1,2,6, no 9 in R8C3

9. 2 in N9 locked in R9C78, locked for R9, clean-up: no 4 in R8C1
9a. 15(3) cage in N9 = 2{49/58/67}

10. 45 rule on C12 2 outies R39C3 = 8 = [17/26/35]

11. 45 rule on C89 2 outies R39C7 = 8 = [17/26/35/62], no 4,8,9, no 5,7 in R3C7

12. 45 rule on C12 4 innies R2389C2 = 18, max R23C2 = 5 -> min R89C2 = 13, no 3

13. 20(3) cage in N7 = {479/569/578}
13a. Killer pair 4,5 in R89C1 and 20(3) cage, locked for N7, clean-up: no 7,8 in R8C7 (step 8)
13b. 3 in N7 locked in R7C123 + R8C3
13c. 45 rule on N7 4 innies R7C123 + R8C3 = 19 = {1369/1378/2368}

14. 45 rule on N3 3 outies R1C6 + R4C89 = 9, max R4C89 = 8, no 8

15. 45 rule on R6789 2 innies R6C19 = 1 outie R5C5
15a. Max R5C5 = 9 -> max R6C19 = 9, no 9, no 8 in R6C1
15b. R6C19 cannot make 4 -> no 4 in R5C5

16. 7,9 in N5 locked in 25(4) cage = 79{18/36/45}, no 2

17. 45 rule on C789 3 outies R158C6 = 8 = 1{25/34}, 1 locked for C6, clean-up: no 1 in R5C4 (step 5), no 1,2,3,4 in R8C4 (step 7)
17a. 5 of {125} must be in R5C6 -> no 2 in R5C6, clean-up: no 5 in R5C4 (step 5)

18. 45 rule on C123 3 outies R158C4 = 14 = {149/167/239/248/347}
18a. 7 of {167} must be in R8C4 -> no 6 in R8C4, clean-up: no 4 in R8C6 (step 7)
18b. R158C4 = [149/167/239/329/428/437] (cannot be [248/347] which donít give valid permutations for R158C6]

19. 16(3) cage at R1C3 = {169/259/268/349/367} (cannot be {178/358/457} which clash with R1C12; all 3 cells of the 16(3) cage see R1C12)
19a. 3 of {349} must be in R1C4 -> no 4 in R1C4, clean-up: no 1 in R1C6 (step 3)
[This move was actually available after step 3! Initially I saw the {178/358} clashes but missed the {457} clash which is less obvious.]

20. R158C6 (step 17) = 1{25/34}
20a. 1 of {125} must be in R8C6 -> no 2 in R8C6, clean-up: no 8 in R8C4 (step 7)

21. Killer pair 1,3 in R8C69, locked for R8, clean-up: no 9 in R8C7 (step 8), no 5 in R9C1
21a. 3 in R7 locked in R7C123, locked for R7

22. 15(3) cage at R4C3 = {249/267/357} (cannot be {159} because no 1,5,9 in R5C4, cannot be {456} because R4C3 only contains 7,9), no 1, clean-up: no 6 in R5C7 (step 6)
22a. 3 of {357} must be in R5C4 -> no 3 in R5C3, clean-up: no 4 in R5C7 (step 6)
22b. 15(3) cage at R4C7 = {159/249/357}
22c. Taking these cages together R5C3467 = {1256/2345} = 25{16/34}, 2,5 locked for R5

23. Min R8C34 = 15 -> max R67C3 = 5, R6C3 = {1234}, R7C3 = {123}

24. R7C123 + R8C3 (step 13c) = {1378/2368} (cannot be {1369} because R8C3 only contains 7,8), no 9, 8 locked for N7
24a. 20(3) cage (step 13) = {479/569}, 9 locked in R89C2, locked for C2
24b. 7 of {479} must be in R9C3 -> no 7 in R89C2

25. Hidden killer pair 1,2 in R7C123 (step 24) and R7C456 -> R7C456 must contain 1/2
25a. R7C456 = {147/246} (cannot be {129} which contains 1 and 2, cannot be {156} which clashes with R7C123 = {236} because of the hidden killer pair) = 4{17/26}, no 5,8,9, 4 locked for R7 and N8

26. 5,9 in R7 locked in R7C789, locked for N9, clean-up: no 4,8 in 15(3) cage (step 9a) -> R8C7 = 4, R8C3 = 8 (step 8), clean-up: no 3 in R3C7 (step 11)
26a. 15(3) cage in N9 = {267}, locked for N9

27. 21(3) cage at R6C8 = {489/579} (cannot be {678} because 6,7 only in R6C8) = 9{48/57}, no 6
27a. 4,7 only in R6C8 -> R6C8 = {47}
27b. 9 locked in R7C89, locked for R7

28. 9 on D\ locked in R5C5 + R6C6, locked for N5
28a. 9 in C7 locked in R46C7, locked for N6

29. 19(4) cage at R6C7 = {1459/1468/3457} (cannot be {2458/2467} because R8C6 only contains 1,3)
29a. 6,7,9 only in R6C7 -> R6C7 = {679}

30. 8 in N6 locked in 15(3) cage = {168/348} (cannot be {258} because 2,5 only in R6C9), no 2,5,7
30a. 1,3 of {168/348} must be in R5C8 -> R5C8 = {13}
30b. 8 locked in R56C9, locked for C9, clean-up: no 6 in R1C8

31. 2 in C9 locked in R234C9, locked for 18(4) cage -> no 2 in R4C8
31a. 7 in C9 locked in R23C9, locked for N3 and 18(4) cage
31b. 18(4) cage = {2367/2457}, no 1,9
31c. 3 of {2367} must be in R4C8 -> no 6 in R4C8

32. 1 in N6 locked in R5C78, locked for R5, clean-up: no 6 in R5C4 (step 5)

33. R8C6 = 1 (hidden single in C6), R8C4 = 9 (step 7), R89C9 = [31], 1 locked for D\, R89C1 = [24], 4 locked for D/, clean-up: no 8 in R1C2, no 7 in R7C456 (step 25a), no 7 in R9C3 (step 24a)
33a. Naked triple {246} in R7C456, locked for R7 and N8

34. R8C34 = 17 -> R67C3 = 3 = [21], 1 locked for D/, R3C3 = 3, R23C2 = [21]

35. R9C2 = 9 (hidden single in R9)

36. R7C12 = {37} -> R6C2 = 5, R8C2 = 6, locked for D/, R9C3 = 5, R8C8 = 7, locked for D\, R8C5 = 5, R6C8 = 4, R3C7 = 2, locked for D/, R9C78 = [62], clean-up: no 8 in R1C8, no 5 in R7C89 (step 27)
36a. R7C89 = [89], R7C7 = 5, locked for D\, R1C1 = 8, locked for D\, R1C2 = 4, R5C5 = 9, R6C6 = 6, R4C4 = 4, R1C9 = 5, locked for D/, R1C8 = 9, R56C9 = [68], R4C9 = 2, clean-up: no 1 in R1C4 (step 3), no 3 in R5C46 (step 5)

and the rest is naked singles
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