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Hi folks,

Time for the next Assassin and I still haven't had the chance to take a look at Afmob's A106V2 .

Here's an interesting and unusual cage pattern that will drive you crazy (well, at least it almost drove me crazy!). Like trying to break out of a fortress guarded by a huge moat...

Assassin 107 (A107) (Est. rating: 1.25)




Solution:

6 3 2 5 4 1 9 7 8
8 5 1 7 3 9 6 4 2
4 9 7 8 2 6 3 1 5
3 8 9 4 6 2 1 5 7
7 4 6 1 5 3 2 8 9
2 1 5 9 8 7 4 6 3
9 2 4 6 7 5 8 3 1
1 7 8 3 9 4 5 2 6
5 6 3 2 1 8 7 9 4

SudokuSolver v3.2.0 score: 1.05.


And just so you don't have to wait in suspense, here's a V2:

Assassin 107 V2 (A107 V2) (Est. rating: 1.75)




Solution:

9 6 3 4 2 8 5 1 7
2 4 1 7 5 6 3 8 9
7 8 5 3 1 9 4 2 6
3 2 9 5 7 1 6 4 8
1 5 8 2 6 4 7 9 3
4 7 6 9 8 3 1 5 2
6 1 4 8 3 2 9 7 5
8 9 7 6 4 5 2 3 1
5 3 2 1 9 7 8 6 4

SudokuSolver v3.2.0 score: 1.26.

Happy solving!

Thank you Mike for a relaxing killer! I had to use one small trick (step 3f) to crack it which I think deserves to be of rating 1.0. By the way, this is my shortest wt so far (farewell, A102X ).

Edit: I think my initial rating is too low like Andrew mentioned, so following my wt A107 would be of rating (Hard?) 1.0.

A107 Walkthrough:

1. R456
a) Innies R4 = 6 = R4C5
b) Innies R5 = 5 = R5C5
c) Innies R6 = 8 = R6C5
d) R6C4 = 9 -> R6C3 = 5
e) 13(2) = [94] -> R4C3 = 9, R4C4 = 4
f) Outies N6 = 12(3) = {237} -> R4C6 = 2, R5C6 = 3, R6C6 = 7
g) Cage sums: R4C7 = 1, R5C7 = 2, R6C7 = 4
h) R5C4 = 1 -> R5C3 = 6
i) 11(2) @ R4 = {38} locked for R4+N5

2. C123
a) Innies N7 = 15(3) = {348} locked for C3+N7
b) 8(3) = {125} -> R1C4 = 5, {12} locked for N1
c) R3C3 = 7
d) 14(3) = {356} -> R2C2 = 5, {36} locked for R1+N1
e) 12(3) = 1{29/56} -> 1 locked for N7

3. C789 !
a) Innies+Outies N3: -2 = R1C6 - R3C7 -> R1C6 <> 8,9; R3C7 = (36)
b) 5 locked in 8(3) @ N3 = {125} locked for N3
c) 5 locked in Innies N9 @ C7 = 20(3) = 5{69/78} -> 5 locked for N9
d) 10(3) = 1{27/36} -> 1 locked for N9
e) 4 locked in 15(3) @ N9 = 4{29/38}
f) ! 10(3) = {136} locked for N9 because R234C9 = (1257) blocks {127}
g) Innies N7 = 20(3) = {578} locked for C7+N9
h) 10(3): R7C8 <> 1 because 1{36} blocked by R6C9 = (36)
i) Hidden Single: R3C8 = 1 @ C8, R9C5 = 1 @ R9

4. R789
a) Innies+Outies N7: -2 = R9C4 - R7C3 -> R7C3 <> 3 and R9C4 = (26)
b) Innies+Outies N9: R7C7 = R9C6 = (58)
c) 4,7,8 locked in 30(5) @ R7 = 478{29/56} -> 7 locked for N8
d) 17(4) = 19{25/34} because R9C4 = (26) and R9C6 = (58) block 18{26/35}
-> 9 locked for R8+N8
e) 30(5) = {45678} locked for R7, 6 locked for N8
f) 13(3) = {238} -> R9C4 = 2, {38} locked for N7

5. Rest is singles.

Rating: (Easy) 1.0. I used a combo blocker.

This one was certainly tougher than V1. I wonder whether my chain in step 8a can somehow be replaced by an easier move.

A107 V2 Walkthrough:

1. R456
a) Innies R4 = 7 = R4C5
b) Innies R5 = 6 = R5C5
c) Innies R6 = 8 = R6C5
d) R6C4 = 9 -> R6C3 = 6, R4C4 = 5 -> R4C3 = 9
e) 12(2) @ R4 = {48} locked for R4+N6
f) 7(2) @ R4 = {16} -> R4C6 = 1, R4C7 = 6
g) Outies N6 = 8(3) = {134} -> R6C6 = 3, R6C5 = 4
h) Cage sum: R5C7 = 7, R6C7 = 1, R5C4 = 2 -> R5C3 = 8
i) 7(2) @ R6 = {25} locked for R6+N6

2. C1234
a) Innies N1 = 9(3) = 3{15/24} -> 3 locked for C3+N1
b) 7 locked in R789C3 @ C3 for N7
c) Innies+Outies N1: -1 = R1C4 - R3C3 -> R3C3 <> 1,3
d) 3 locked in 8(3) @ C3 = {134}; R1C4 <> 3
e) Innies+Outies N1: -1 = R1C4 - R3C3 -> R3C3 <> 4
f) Innies+Outies N7: -3 = R9C4 - R7C3 -> R9C4 = (14), R7C3 = (47)
g) Naked pair (14) locked in R19C4 for C4
h) 17(3) @ N1 <> 5 because {458} blocked by Killer pair (45) of Innies N1

3. N39
a) Innies N3 = 12(3) = {345} locked for C7+N3
b) Naked triple (289) locked in R789C7 for N9
c) Innies+Outies N3: 4 = R1C6 - R3C7 -> R1C6 = (789)
d) Innies+Outies N9: -2 = R9C6 - R7C7 -> R9C6 = (67), R7C7 <> 2

4. R123
a) Innies R12 = 11(2) <> 1,6,8; R2C1 <> 7
b) 17(3) @ N1: R3C12 <> 4 because R2C1 <> 6,7
c) 3,4,5 locked in 22(5) @ R3 = 345{19/28}

5. R789 !
a) Innies R89 = 9(2): R8C1 <> 1,9
b) Innies N7 = 7{15/24}
c) 17(3) @ N7 <> 4 because (458) blocked by Killer pair (45) of Innies N7
d) 15(3) <> 5{28/46} because they are blocked by Killer pairs (25,45) of Innies N7
e) Outies R89 = 19(4) <> 47{26/35} because R7C3 = (47)
f) ! Outies R89 = 1{279/369/378/459/468/567} because:
- 2 can only be @ 15(3) which must be {29}4 and R7C89 @ 13(3) cannot be {35}
-> 1 locked for R7
g) 24(4) <> 2 because {2589} blocked by Killer pair (25) of 26(5) and {2679}
blocked by R9C6 = (67)
h) 2 locked in 26(5) @ N8 for R7

6. R789
a) 15(3): R7C12 <> 5 because R8C1 <> 1,9
b) 13(3) @ R7: R8C9 <> 6 because Outies R89 have no combo with (34)
c) Innies R89 = 9(2): R8C1 <> 3
d) Innies+Outies N9: -2 = R9C6 - R7C7 -> R7C7+R9C6 = [86/97]
e) 26(5) <> {23678} because 8 in R7C7 forces R9C6 = 6
f) 15(3) <> 2 because {49}2 blocked by Killer pair (49) of 26(5)
g) Innies R89 = 9(2): R8C9 <> 7

7. C4+N8
a) Hidden Killer pair (67) in R78C4 for C4 since only other place possible is R2C4
b) Killer pair (67) locked in R78C4+R9C6 for N8

8. R789 !
a) ! 15(3): R8C1 <> 4 because it forces Outies R89 = 19(4) = {1378} but at
the same time R8C1 = 4 -> R7C3 = 7
b) Innies N7 = 13(3) = 7{15/24}: R89C3 <> 4 because 10(3) cannot have (24)
c) 4 locked in R7C123 @ N7 for R7
d) 13(3) @ R7: R8C9 <> 3 because R7C89 <> 4
e) Innies R89 = 9(2) = [54/81]
f) 13(3) @ R7 must have 1 xor 4 and R8C9 = (14) -> R7C89 <> 1
g) 1 locked in 15(3) @ R7 for N7 -> 15(3) = 1{59/68}
h) Innies N7 = {247} locked C3+N7 -> R7C3 = 4
i) 10(3) = {127} -> R9C4 = 1

9. R123
a) R1C4 = 4
b) 22(5) = 345{18/29} -> R3C3 = 5, R3C7 = 4; 3 locked for N2
c) 20(4) = {2567} because 18{29/56} blocked by Killer pair (18) of 22(5)
-> R12C5 = {25} locked for C5+N2; R2C46 = {67} locked for R2
d) Innies R12 = 11(2) = {29} locked for R2
e) 16(3) @ N2 = {358} -> R1C6 = 8
f) R3C6 = 9, R8C6 = 5, R8C1 = 8
g) Hidden Single: R2C2 = 4 @ R2 -> R1C12 = 15(2) = {69} locked for R1+N1

10. R789
a) Innies R89 = 9(2) = [81] -> R8C9 = 1

11. Rest is singles.

Rating: (Easy) 1.5. I used some combo analysis and a small contradiction chain.

V1 solving path same as Afmob's..including the "blocking" move.I don't think I'll ever beat him to the tape though !!!

Regards

Gary

Ditto & ditto. Easy 1.0 rating seems generous but correct. For once I agree with Afmob .

No chance to try the A107 V2 at the moment - want to keep onto the archive. With no word from Ruud, seems just a matter of when the forums go down again as well as sudocue.net. Also can't do much solving at the moment as Richard is back onto killer SudokuSolver. Get your orders in quick! Already up to v3 of the next batch of changes so lots of bug hunting and spreadsheets to come .

Already managed to get A106 down to around a score of 1.15. Perfect. But uA94 has gone through the roof! Andrew won't mind that!! The "dependent cages" routine Richard has implemented, that has made such a big difference to A106, found a simpler (but much longer) way to get to the key move in uA94. As usual, 1 win, 1 loss. Sigh....

Cheers
Ed

Yes, this puzzle certainly proved to be less stable than it was intended to be! Unfortunately, I ran out of time to search for other alternatives, after having spent too long unsuccessfully trying to solve another grid that I had hoped would have been the original V1, and which SudokuSolver 3.2.0 thought only deserved a rating of 1.14. It effectively knocked me out of action for more than one week...

_________________
Cheers,
Mike

Congratulations to Afmob, Ed, Gary and anyone else who found Afmob's step 3f.

I think the key to solving this puzzle was interactions between L-shaped cages in R23 or R78 and adjacent 2-cell cages in R4 or R6. That's what Afmob's step 3f was except that it used two L-shaped cages. Having not spotted that one and done a lot of nibbling I eventually found one of those interactions in step 38.

It's interesting to note that Afmob's walkthrough could have been a bit shorter by using a UR after step 3f but we know that Afmob prefers not to use UR. I've given a note in my walkthrough where I could have used the UR.

I'm surprised at that rating. IMHO step 3f belongs in a 1.25 rating, probably an Easy 1.25 since it's the only tricky move in Afmob's walkthrough. If I'd found it, I would definitely have rated it a 1.25.

Here is my rather longer walkthrough. I'll rate it as solid to hard 1.25. Thanks Afmob and Ed for pointing out the minor error in step 21 and to Ed for pointing out that I'd missed a pair of permutations in step 32d. Steps 39 to 41 have been deleted and steps 44 to 47 have been reworked to take account of these extra permutations.

Prelims

a) R4C12 = {29/38/47/56}, no 1
b) R4C34 = {49/58/67}, no 1,2,3
c) R4C67 = {12}, locked for R4, clean-up: no 9 in R4C12
d) R4C89 = {39/48/57}
e) R5C12 = {29/38/47/56}, no 1
f) R5C34 = {16/25/34}, no 7,8,9
g) R5C67 = {14/23}
h) R5C89 = {89}, locked for R5 and N6, clean-up: no 3,4 in R4C89, no 2,3 in R5C12
i) R6C12 = {12}, locked for R6 and N4, clean-up: no 5,6 in R5C4
j) R6C34 = {59/68}
k) R6C67 = {47/56}/[83], no 9, no 3 in R6C6
l) R6C89 = {36/45}, no 7
m) 8(3) cage at R1C3 = {125/134}, CPE no 1 in R1C12
n) 21(3) cage in N1 = {489/579/678}, no 1,2,3
o) 19(3) cage in N3 = {289/379/469/478/568}, no 1
p) 8(3) cage in N3 = {125/134}, 1 locked for N3
q) 10(3) cage in N9 = {127/136/145/235}, no 8,9
r) 20(3) cage at R8C7 = {389/479/569/578}, no 1,2

1. Naked pair {57} in R4C89, locked for R4 and N6, clean-up: no 4,6 in R4C12, no 6,8 in R4C34, no 4,6 in R6C6, no 4 in R6C89

2. Naked pair {38} in R4C12, locked for R4 and N4, clean-up: no 4 in R5C4, no 6 in R6C4

3. Naked pair {36} in R6C89, locked for R6 and N6, clean-up: no 2 in R5C6, no 8 in R6C4, no 5,8 in R6C6

4. Naked pair [74] in R6C67, clean-up: no 1 in R5C6

5. Naked pair {12} in R45C7, locked for C7

6. R46C5 = [68] (hidden singles in R4 and R6), R5C5 = 5 (cage sum), R6C34 = [59], R4C34 = [94], R5C6 = 3, R5C7 = 2, R5C4 = 1, R5C3 = 6, R4C67 = [21]

7. 8(3) cage at R1C3 = {125/134}
7a. 1 locked in R12C3, locked for C3 and N1
7b. 5 of {125} must be in R1C4 -> no 2 in R1C4
7c. 3 of {134} must be in R1C4 -> no 3 in R12C3

8. 45 rule on N1 3 innies R123C3 = 10 = {127} (only remaining combination) -> R3C3 = 7, R12C3 = {12}, locked for C3 and N1, R1C4 = 5 (step 7)

9. Naked triple {348} in R789C3, locked for N7

10. 18(3) cage in N7 = {279/567}, no 1, 7 locked for N7

11. 45 rule on N3 1 innie R3C7 = 1 outie R1C6 + 2, no 8,9 in R1C6, no 5,9 in R3C7

12. 45 rule on N7 1 innie R7C3 = 1 outie R9C4 + 2, no 3 in R7C3, no 3,7,8 in R9C4

13. 45 rule on N9 1 innie R7C7 = 1 outie R9C6, no 3,7 in R7C7, no 4 in R9C6

14. 45 rule on R3 2 outies R2C19 = 10 = [64/82/91], no 4,5 in R2C1, no 3,5 in R2C9

15. 45 rule on R7 2 outies R8C19 = 7 = {16/25}, no 9 in R8C1, no 3,4,7 in R8C9

16. 14(3) cage in N1 = {356} (only remaining combination) -> R2C2 = 5, R1C12 = {36}, locked for R1 and N1, clean-up: no 4 in R2C9 (step 14), no 8 in R3C7 (step 11)
16a. 4 in N1 locked in R3C12, locked for R3

17. 8(3) cage in N3 = {125} (only remaining combination), no 3, locked for N3

18. Naked pair {12} in R2C39, locked for R2

19. 5 in C7 locked in R789C7, locked for N9, clean-up: no 2 in R8C1 (step 15)
19a. 45 rule on N9 3 innies R789C7 = 20 = {569/578}, no 3
19b. 3 in C7 locked in R23C7, locked for N3

20. 10(3) cage in N9 = {127/136}, no 4, 1 locked for N9
[I could have used Afmob’s combo blocker here if I’d spotted it.]

21. R9C5 = 1 (hidden single in R9)

22. 1 in R8 locked in R8C19 = {16} (step 15), locked for R8

23. 16(3) cage at R1C6 = {169/178/349} (cannot be {367} because R1C6 only contains 1,4)
23a. 3,6 of {169/349} must be in R2C7 -> no 9 in R2C7

24. 3,6,7 in R3 locked in R3C34567 = {13679/23678}
24a. 1 of {13679} must be in R3C6 -> no 9 in R3C6

25. 17(4) cage in N8 = {1259/1349/1358/1457}
25a. 2 of {1259} must be in R8C4 -> no 2 in R8C5
25b. 5 of {1358} must be in R8C6 -> no 8 in R8C6
25c. 7 of {1457} must be in R8C4 -> no 7 in R8C5

26. 4 in N9 locked in 15(3) cage = {249/348}, no 6,7

27. 18(3) cage in N7 (step 10) = {279/567}
27a. 5 of {567} must be in R9C1 -> no 6 in R9C1

28. 5 in R8 locked in R8C67, CPE no 5 in R9C6, clean-up: no 5 in R7C7 (step 13)

29. 20(3) cage at R8C7 = {569/578}
29a. 8 of {578} must be in R9C6 -> no 8 in R89C7

30. 8 in R1 locked in R1C789, locked for N3

31. 16(3) cage at R1C6 (step 23) = {169/178/349}
31a. 8 of {178} must be in R1C7 -> no 7 in R1C7

32. 1 in R7 locked in R7C1289
32a. 45 rule on R7 4 innies R7C1289 = 15 = {1239/1257/1356}
32b. {1239} must be {29}{13}
32c. {1257} must be [51]{27} (other permutations not consistent with the 12(3) and 10(3) cages)
32d. {1356} must be [51]{36}/[56]{13}
32e. -> no 1,6 in R7C1 (no eliminations from R7C2)
[Thanks Ed for pointing out the extra permutations in step 32d. Ed also pointed out that I could have got the eliminations from R7C1 from the permutations of the 12(3) cage.]

33. 45 rule on R9 4 outies R8C2378 = 22 = {2389/2479/2578} (cannot be {3478} because 3,4,8 only in R8C38), 2 locked for R8
33a. 4 of {2479} must be in R8C3 -> no 4 in R8C8

34. 4 in N9 locked in R9C89, locked for R9

35. 4 in N3 locked in 19(3) cage = {469/478}
35a. 6 of {469} must be in R2C8 -> no 9 in R2C8

36. Killer pair 8,9 in R1C7 and R1C89, locked for R1

37. 1 in N2 locked in R13C6
37a. 45 rule on N2 4 remaining innies R1C6 + R3C456 = 17 = {1268/1349}
37b. 2 of {1268} must be in R3C5 -> no 2 in R3C4
37c. 9 of {1349} must be in R3C5 -> no 3 in R3C5
37d. 2 in N2 locked in R13C5, locked for C5

38. 12(3) cage in N7 = {129/156}
38a. {129} must be [921] (R7C12 cannot be [29] because R78C1 clashes with R6C1), no 2 in R7C1, no 9 in R7C2

39 to 41. Deleted

42. 10(3) cage in N9 (step 20) = {127/136}
42a. 1 must be in R78C9 (R78C9 cannot be {36} which clashes with R6C9) -> no 1 in R7C8
[A similar interaction between R6C89 and 10(3) cage in N9 could be used to eliminate 6 from R7C89 but that would involve using a UR, which I don’t use. Fortunately the finish is quick without using it.]

43. 1 in N9 locked in R78C9, locked for C9 -> R23C9 = [25], R3C8 = 1, R12C3 = [21], R4C12 = [57]

44. R1C6 = 1, R3C5 = 2 (hidden singles in N2), R3C46 = {68} (step 37a), locked for R3 and N2 -> R3C7 = 3

Rest of step 44 deleted, steps 45 to 47 reworked.

45. R2C1 = 8 (hidden single in R2), R4C12 = [38], R1C12 = [63], R8C1 = 1, R6C12 = [21], R8C9 = 6, R6C89 = [63], R7C9 = 1, R7C8 = 3 (step 20), clean-up: no 9 in 19(3) cage in N3 (step 35), no 6 in R9C6 (step 13), no 8 in 15(3) cage in N9 (step 26)

46. Naked triple {478} in 19(3) cage in N3, locked for N3 -> R12C7 = [96], R7C7 = 8, R9C6 = 8 (step 13), R3C46 = [86], R7C3 = 4, R89C3 = [83], R9C4 = 2 (cage sum)

47. Naked pair {59} in R7C16, locked for R7 -> R7C45 = [67]


and the rest is naked singles

Continuing working through my backlog, the next variant I tried was A107 V2 which I found that I hadn't started.

Thanks Mike for a challenging variant!

My solving path is very similar to Afmob's and we used the same breakthrough move but our reasoning for steps was sometimes different so I'm posting my walkthrough.


Here is my walkthrough for A107 V2.

Prelims

a) R4C12 = {14/23}
b) R4C34 = {59/68}
c) R4C67 = {16/25/34}, no 7,8,9
d) R4C89 = {39/48/57}, no 1,2,6
e) R5C12 = {15/24}
f) R5C34 = {19/28/37/46}, no 5
g) R5C67 = {29/38/47/56}, no 1
h) R5C89 = {39/48/57}, no 1,2,6
i) R6C12 = {29/38/47/56}, no 1
j) R6C34 = {69/78}
k) R6C67 = {13}
l) R6C89 = {16/25/34}, no 7,8,9
m) 19(3) cage in N1 = {289/379/469/478/568}, no 1
n) 8(3) cage at R1C3 = {125/134}
o) 21(3) cage at R4C5 = {489/579/678}, no 1,2,3
p) 10(3) cage at R8C3 = {127/136/145/235}, no 8,9

Steps resulting from Prelims
1a. Naked pair {13} in R6C67, locked for R6, clean-up: no 8 in R6C12, no 4,6 on R6C89
1b. Naked pair {25} in R6C89, locked for R6 and N6, clean-up: no 2,5 in R4C6, no 7 in R4C89, no 6,9 in R5C6, no 7 in R5C89, no 6,9 in R6C12
1c. Naked pair {47} in R6C12, locked for R6 and N4, clean-up: no 1 in R4C12, no 2 in R5C12, no 3,6 in R5C4, no 8 in R6C34
1d. Naked pair {23} in R4C12, locked for R4 and N4, clean-up: no 4 in R4C67, no 9 in R4C89, no 7,8 in R5C4
1e. Naked pair {16} in R4C67, locked for R4, clean-up: no 8 in R4C34
1f. Naked pair {15} in R5C12, locked for R5 and N4 -> R4C34 = [95], R6C34 = [69], R5C3 = 8, R5C4 = 2, R6C5 = 8, clean-up: no 3 in R5C6, no 3,6,9 in R5C7, no 4 in R5C89
1g. Naked pair {48} in R4C89, locked for R4 and N6 -> R4C5 = 7, R5C67 = [47], R5C5 = 6, R4C67 = [16], R6C67 = [31]
What an incredible start to an Assassin variant! The hard work will come later.

2. 45 rule on R12 2 innies R2C19 = 11 = {29/38/47/56}, no 1

3. 45 rule on R89 2 innies R8C19 = 9 = {18/27/36/45}, no 9

4. 45 rule on N1 1 innie R3C3 = 1 outie R1C4 + 1, no 1,3,7 in R3C3
4a. 45 rule on N1 3 innies R123C3 = 9 = {135/234}, 3 locked for C3, N1 and 8(3) cage at R1C3, no 3 in R1C4, clean-up: no 8 in R2C9 (step 2), no 4 in R3C3
4b. R3C3 = {25} -> no 2,5 in R12C3
4c. 8(3) cage at R1C3 = {134}, CPE no 4 in R1C12

5. 45 rule on N3 3 innies R123C7 = 12 = {345} (only remaining combination), locked for C7 and N3, clean-up: no 6,7,8 in R2C1 (step 2)
5a. 45 rule on N3 1 outie R1C6 = 1 innie R3C7 + 4, no 2,5,6 in R1C6

6. Naked triple {289} in R789C7, locked for N9, clean-up: no 1,7 in R8C1 (step 3)
6a. 45 rule on N9 1 innie R7C7 = 1 outie R9C6 + 2, no 2 in R7C7, no 2,5,8,9 in R9C6

7. 45 rule on N7 1 innie R7C3 = 1 outie R9C4 + 3, no 1,2,5 in R7C3, no 3,6,7 in R9C4
7a. R7C3 + R9C4 = [41/74], CPE no 4 in R7C45 + R89C3 + R9C12

8. 45 rule on N7 3 innies R789C3 = 13 = {157/247}, 7 locked for N7
8a. 10(3) cage at R8C3 = {127/145}, CPE no 1 in R9C12

9. Naked pair {14} in R19C4, locked for C4

10. 17(3) cage in N7 = {269/359/368} (cannot be {458} which clashes with R789C3), no 1,4

11. 17(3) cage in N1 = {179/269/278/467} (cannot be {458} which clashes with R123C3), no 5, clean-up: no 6 in R2C9 (step 2)
11a. 4 of {467} must be in R2C1 -> no 4 in R3C12

12. 3,4,5 in R3 only in 22(5) cage at R3C3 = {13459/23458}, no 6,7
12a. 1 of {13459} must be in R3C5 -> no 9 in R3C5

13. Hidden killer pair 6,7 in R2C4 and R78C4 for C4, R78C4 cannot be {67} which clashes with R9C6 -> R2C4 = {67}, R78C4 must contain one of 6,7
13a. Killer pair 6,7 in R78C4 and R9C6, locked for N8

14. 6 in N2 only in R2C46, locked for R2
14a. 20(4) cage = {1568/2369/2468/2567/3467} (cannot be {1469} which clashes with R1C4)
14b. 6,7,8 of {1568/2567} must be in R2C46 -> no 5 in R2C6

15. 19(3) cage in N1 = {289/469/478/568}
15a. 7 of {478} must be in R1C1 (R12C2 cannot be [74] which clashes with R6C2) -> no 7 in R12C2

16. 15(3) cage in N7 = {159/168/249/348} (cannot be {258/456} which clash with R789C3)
16a. 5 of {159} must be in R8C1 -> no 5 in R7C12

17. 13(3) cage at R7C8 = {157/346}
17a. 45 rule on R7 4 innies R7C1289 = 19 = {1279/1369/1378/1459/1468/1567} (cannot be {2467/3457} which clash with R7C3, cannot be {2359/2458} because {35/45} in R7C89 aren’t consistent with combinations for 13(3) cage at R7C8, cannot be {2368} because no {258} combination in 15(3) cage in N7), 1 locked for R7
17b. No combination for R7C1289 contains both of 3,4 -> R7C89 cannot be {34} -> no 6 in R8C9, clean-up: no 3 in R8C1 (step 3)

18. Hidden killer pair 1,4 in 24(4) cage and R9C4 for N8, R9C4 = {14} -> 24(4) cage must contain one of 1,4 -> {1689/3489/4569/4578} (only combinations containing one of 1,4), no 2
18a. 2 in N8 only in R7C56, locked for R7

19. 20(4) cage in N2 (step 14a) = {2369/2468/2567/3467} (cannot be {1568} which clashes with 22(5) cage at R3C3 which must have 1 or 8 in R3C456), no 1

20. 45 rule on N2 5 innies R1C46 + R3C456 = 25 = {12589/13489/13579/14578} (from combinations for 20(4) cage, step 19)
20a. R1C46 + R3C456 cannot be {12589} which clashes with R3C3
R1C46 + R3C456 cannot be {13579} = [17]+[359] which clashes with 22(5) cage at R3C3 because {13459} must have 5 in R3C3)
-> R1C46 + R3C456 = {13489/14578}, 4,8 locked for N2

21. 20(4) cage in N2 (step 19) = {2369/2567}, 2 locked for N2
21a. 5 of {2567} must be in R2C5 (R2C456 cannot be {267} which clashes with R2C19), no 5 in R1C5
21b. R1C4 + R3C5 = {14} (hidden pair in N2)

22. Hidden grouped X-Wing for 1 in 8(3) cage at R1C3 and 16(3) cage at R1C8 for R12, 8(3) cage at R1C3 contains 1 -> 16(3) cage at R1C8 must contain 1 = {169/178}, no 2, 1 locked for N3

23. 45 rule on R3 4 innies R3C1289 = 23 = {1679/2678}
23a. 1 of {1679} only in R3C12, R3C12 cannot be {19} (because no 7 in R2C1) -> no 9 in R3C12

24. 2 in R7 only in 26(5) cage at R7C3 = {23489/23579/24569/24578} (cannot be {23678} which clashes with 24(4) cage in N8 or 24(4) cage + R9C6 in the case of the {4578} combination)
[Alternatively {23678} can be eliminated by looking at the permutations for 5 innies R7C456 + R9C69 in N8.]

25. R7C1289 (step 17a) = {1369/1378/1468/1567} (cannot be {1459} which clashes with 26(5) cage)
25a. 4,6 of {1468} must be in R7C89 (R7C12 cannot contain both of 4,8 because no 3 in R8C1), no 4 in R7C12
25b. 15(3) cage in N7 (step 16) = {159/168/348} (cannot be {249} because 2,4 only in R8C1), no 2, clean-up: no 7 in R8C9 (step 3)
25c. 9 of {159} must be in R7C1 (R78C1 cannot be [15] which clashes with R5C1) -> no 9 in R7C2

26. 15(3) cage in N7 (step 25b) = {159/168/348}
26a. Cannot be {348}
R8C1 = 4 => R7C12 = {38} => R7C89 (step 25) = {17} clashes with R7C3
26b. 15(3) cage = {159/168}, no 3,4, clean-up: no 5 in R8C9 (step 3)

27. R7C3 = 4 (hidden single in N7), R9C4 = 1 (step 7), R1C4 = 4, R3C5 = 1, R3C3 = 5 (step 4), clean-up: no 9 in R3C89 (step 23)

28. Naked quad {2678} in R3C1289, locked for R3 -> R3C46 = [39], R3C7 = 4, R1C6 = 8 (step 5a), R12C5 = [25], R78C6 = [25]

29. Naked pair {67} in R2C46, locked for R2, clean-up: no 4 in R2C1 (step 2)
29a. Naked pair {29} in R2C19, locked for R2

30. R2C2 = 4 (hidden single in N1), R6C12 = [47]
30a. R2C2 = 4 -> R1C12 = 15 = {69}, locked for R1 and N1, R2C1 = 2, R2C9 = 9, R3C12 = [78], R4C12 = [32], R5C89 = [93]

31. Naked pair {17} in R1C89, locked for R1 and N3 -> R2C8 = 8, R4C89 = [48]

32. R8C19 = 9 (step 3) = [81], R1C89 = [17]

33. R8C9 = 1 -> R7C89 = 12 = [75]

34. R7C4 = 8 (hidden single in C4), R7C7 = 9, R9C6 = 7 (step 6a)

and the rest is naked singles.