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Thanks for A142, Frank! It seems SudokuSolver took a harder path by using Killer triples to eliminate combos and candidates.

A142 Walkthrough

1. C789
a) 3(2) = {12} locked for R1+N3
b) 17(2) = {89} locked for C8+N6
c) 29(4) = {5789}; 5,7 locked for C8+N9; 8,9 locked for C7+N9
d) 13(3) = {346} locked for N9
e) 1,2 locked in 14(4) @ C9 = 12{47/56}
f) Killer pair (67) locked in 15(2) + 14(4) for C9
g) 13(3) = {346} -> R9C8 = 6; {34} locked for C9
h) 14(4) = {1256} -> 5,6 locked for C9+N6
i) 15(2) = {78} locked for N3
j) R3C9 = 9

2. C789
a) 11(2) = {47} locked for C7+N6
b) 18(4) = 39{15/24} because (12) only possible @ R4C8 -> R4C8 <> 3; 3 locked for R3+N3
c) R2C8 = 4, R3C8 = 3, R3C7 = 5 -> R4C8 = 1, R1C8 = 2, R1C7 = 1, R2C7 = 6
d) R7C7 = 2, R7C9 = 1, R4C7 = 3
e) 31(6) = 3469{18/27} -> 9 locked for C6
f) 12(3) = 2{37/46}

3. R123 !
a) 16(4) = 1{249/258/267/456} -> 1 locked for R3+N1
b) Outies R1 = 16(3) = 8{17/35} because R2C9 = (78) -> 8 locked for R2
c) ! Killer pair (13) locked in Outies R1 + 29(6) for R2
d) 31(6) = {234679} -> 2,7 locked for C6

4. N478 !
a) 12(3) = {246} -> 4,6 locked for C6+N8
b) Innies N78 = 15(2) = [69/78/87]
c) Hidden Killer pair (89) in R7C2 for R7 since Innies N78 can only have one of it -> R7C2 = (89)
d) ! Hidden Killer pair (46) in R7C13 for R7
e) Innies N7 = 11(2) = [65/74]
f) 9(3) = 3{15/24} -> R8C4 = (12), R7C4 = 3

5. N12 !
a) 3 locked in 19(4) @ N2 = 3{169/457} since {1378} blocked by R1C9 = (78)
b) Outies R1 = 16(3): R2C5 <> 7 because R2C19 <> 1
c) ! 13(2) = {58} locked for R1+N1 since (49,67) are Killer pairs of 19(4)
d) 11(2) = {47} -> R1C1 = 4, R2C1 = 7
e) 16(4) = {1267} since R3C123 = (126) -> R4C2 = 7; {26} locked for R3+N1
f) R1C6 = 3, R3C6 = 7
g) 19(4) = {1369} -> R2C5 = 1; 9 locked for N2
h) 29(6) = {234569} -> R3C4 = 4; 6 locked for R4
i) R3C5 = 8

6. N5
a) 20(5) = 128{36/45} -> 2 locked for N5
b) R2C6 = 2, R4C6 = 9, R2C4 = 5, R4C4 = 6, R4C3 = 2
c) 27(4) = {3789} -> R6C6 = 8, R6C4 = 7, R7C5 = 9, R6C5 = 3

7. C123
a) Both 10(2) <> 3,8
b) Killer pair (16) locked in R3C3 + 10(2) for C3
c) R7C1 = 6, R7C2 = 8
d) Innie N7 = R7C3 = 5
e) Cage sum: R8C4 = 1
f) 20(4) = {2378} -> R8C2 = 2; 3 locked for C3+N7
g) R8C1 = 9, R9C1 = 1, R9C2 = 4
h) 10(2) @ C2 = {19} locked for C2+N4

8. Rest is singles.

Rating: (Easy) 1.25. I used Hidden Killer pairs.

Thanks Frank!

Not sure when we last, if ever, had two walkthroughs posted the same day that a new Assassin was posted. I would have been first if I hadn't watched TV for a couple of hours this evening.

I've just finished and it's too late to go through Afmob's walkthrough tonight but from the comments my walkthrough should be a bit different.

Edit. I've now gone through Afmob's walkthrough and it's a lot different, even our later breakthrough steps were different.

I would say it took a different path. Afmob's steps 3a and 3c are probably as difficult as my step 16. Therefore this puzzle seems to be one where using killer pairs and combination work isn't necessarily simpler than using killer triples; the two ways probably have a similar level of difficulty. Edit. I'll change my view on that a bit. Afmob told me that SS used several triples, not just the two that I used in step 16, so I'll agree that SS took a harder solving path.

I'll rate A142 at Easy 1.25.

Here is my walkthrough. An easy start with a lot of early placements, then I had to work a bit harder but after step 17 it fell quickly. Edit. I've deleted step 10 which had a typo in a combination . Fortunately that didn't affect later steps.

Prelims

a) R12C1 = {29/38/47/56}, no 1
b) R1C23 = {49/58/67}, no 1,2,3
c) R1C78 = {12}, locked for R1 and N1, clean-up: no 9 in R2C1
d) R12C9 = {69/78}
e) R56C2 = {19/28/37/46}, no 5
f) R56C3 = {19/28/37/46}, no 5
g) R56C7 = {29/38/47/56}, no 1
h) R56C8 = {89}, locked for C8 and N6, clean-up: no 2,3 in R56C7
i) 9(3) cage at R7C3 = {126/135/234}, no 7,8,9
j) R4567C9 = {1238/1247/1256/1346/2345}, no 9
l) 27(4) cage at R6C4 = {3789/4689/5679}, no 1,2, CPE no 9 in R45C5
m) 29(4) cage at R7C8 = {5789}, locked for N9

1. Naked pair {57} in R78C8, locked for C8 and N9
1a. Naked pair {89} in R89C7, locked for C7

2. Hidden killer pair 8,9 in R12C9 and R3C9 for C9 -> R3C9 = {89}
2a. Max R3C78 + R4C8 = 10 must contain one of 1,2 -> R4C8 = {12}
2b. Min R3C9 + R4C8 = 9 -> max R3C78 = 9, no 7 in R3C7

3. 45 rule on N9 2 innies R7C79 = 3 = {12}, locked for R7 and N9
[This made the naked pair in R14C8 unnecessary.]

4. 9(3) cage at R7C3 = {135/234} (cannot be {126} because 1,2 only in R8C4), no 6
4a. 1,2 only in R8C4 -> R8C4 = {12}
4b. 3 locked in R7C34, locked for R7

5. 1,2,5 in C9 locked in R4567C9 = {1256}, locked for C9, 5,6 locked in R456C9, locked for N6, clean-up: no 9 in R12C9

6. R3C9 = 9 (hidden single in C9), R4C7 = 3 (hidden single in N6), R9C8 = 6 (hidden single in N9), R23C8 = [43], clean-up: no 7 in R1C1

7. R3C89 = [39] = 12 -> R3C7 + R4C8 = 6 = [51], R1C78 = [12], R7C79 = [21]
7a. R2C7 = 6 (hidden single in N3), clean-up: no 5 in R1C1
7b. 1 in N5 locked in R5C456, locked for R5 and 20(5) cage at R3C5, clean-up: no 9 in R6C23

8. 31(6) cage at R2C6 = {134689/234679}, no 5, 9 locked for C6

9. R7C7 = 2 -> R78C6 = 10 = [46/64/73]

10. Deleted

11. 45 rule on C1234 4 innies R1569C4 = 26 = {2789/3689/4589/4679/5678}, no 1

12. 45 rule on R789 2 remaining innies R7C15 = 15 = {69/78}
12a. Hidden killer pair 8,9 in R7C15 and R7C2 for R7 -> R7C2 = {89}

13. 45 rule on N7 2 innies R7C13 = 11 = [65/74/83], clean-up: no 6 in R7C5 (step 12)

14. 45 rule on C1 1 outie R9C2 = 1 innie R3C1 + 2, no 4,8 in R3C1, no 1,2,5,7 in R9C2

15. 45 rule on R1 3 outies R2C159 = 16 = {178/358} (cannot be {259} because R2C9 only contains 7,8), no 2,9, 8 locked for R2, clean-up: no 9 in R1C1
15a. 1 of {178} must be in R2C5, 8 of {358} must be in R2C9 -> no 7,8 in R2C5

16. 19(4) cage in N2 = {1369/1459/1567/3457} (cannot be {1378} which clashes with R1C9, cannot be {1468} which clashes with R1C23), no 8
16a. 3 of {3457} must be in R1C456 (R1C456 cannot be {457} which clashes with R1C23), no 3 in R2C5
16b. Killer triple 7,8,9 in R1C23, R1C456 and R1C9, locked for R1, clean-up: no 3 in R2C1

17. 45 rule on R1 2 innies R1C19 = 1 outie R2C5 + 10
17a. Max R1C19 = 14 -> no 5 in R2C5
17b. R2C5 = 1, R1C19 = 11 = [38/47], clean-up: no 5 in R2C1, no 8 in R234C6 (step 8)
17c. Naked pair {78} in R2C19, locked for R2
17d. Naked triple {279} in R234C6, locked for C6, clean-up: no 3 in R8C6 (step 9)
17e. Naked pair {46} in R78C6, locked for C6 and N8
17f. 8 in N2 locked in R3C45, locked for R3

18. R5C6 = 1 (hidden single in R5), R8C4 = 1 (hidden single in N8), clean-up: no 4 in R7C3 (step 4), no 7 in R7C1 (step 13), no 8 in R7C5 (step 12)
18a. Naked pair {35} in R7C34, locked for R7 -> R78C8 = [75], R7C5 = 9, R7C12 = [68], R78C6 = [46], R7C3 = 5 (step 13), R7C4 = 3, clean-up: no 2 in R56C2

19. 45 rule on N1 2 innies R2C23 = 1 outie R4C2 + 5
19a. 3 in R2 locked in R2C23 -> R2C23 = {39} (cannot be {23/35} because R2C23 must total more than 5 and no 3 in R4C2), locked for R2, N1 and 29(6) cage at R2C2) -> R1C1 = 4, R2C1 = 7, R12C9 = [78], R2C46 = [52], R1C456 = [963], R1C23 = [58], R34C6 = [79], clean-up: no 2 in R56C3, no 9 in R9C2 (step 14)
19b. R2C23 = {39} = 12 -> R4C2 = 7, clean-up: no 3 in R56C23

20. 29(6) cage at R2C2 = {234569} (only remaining combination) -> R3C4 = 4, R3C5 = 8, R4C34 = {26}, locked for R4 -> R4C9 = 5, R4C1 = 8, R4C5 = 4
20a. R56C1 = {35} (hidden pair in N4), locked for C1
20b. R4C3 = 2 (hidden single in N4), R4C4 = 6

21. 9 in C1 locked in R89C1, locked for N7

22. 20(4) cage at R3C5 = {12458} (only remaining combination) -> R5C45 = [25], R6C456 = [738], R56C9 = [62], R56C7 = [74]

23. Naked triple {349} in R259C2, locked for C2 -> R8C2 = 2, R89C1 = [91], R3C1 = 2, R9C2 = 4 (step 14), R5C2 = 9, R6C2 = 1

and the rest is naked singles

This one drove me nuts! At first I even had to used some chains to progress but then I found a move which cracks this Killer quite early since after the first placement you only need some Killer pairs to finish it.

A142 V2 Walkthrough:

1. C789 !
a) 16(2) = {79} locked for R1+N3
b) 8 locked in R2C78 @ N3 for R2+33(6)
c) Innies N9 = 5(2) = {14/23}
d) Innies N69 = 7(2+1) <> 6,7,9; R4C8 <> 5
e) ! Innies N69 = 7(2+1) <> 4 because {124} blocked by Killer pairs (12,14) of 6(2)
f) Innies+Outies N3: -10 = R4C8 - R2C78 -> R2C78 = 11/12/13 = 8{3/4/5} <> 1,2,6 (step 1b)
g) 6,7,9 in C7 locked in R1389C7 and one of (79) must be in R89C7
h) ! Innies+Outies C89: 22 = R1389C7 - R2C8; R2C8 = (3458) -> R1389C7 = 25/26/27/30 = 679{3/4/5/8} <> 1,2:
- R89C7 <> 6 (IOU @ N3 because of step 1g; R189C7 = {679} implies R3C7 = R2C8)
i) Hidden Single: R3C7 = 6 @ C7
j) 7(2) <> 1

2. C789
a) 1 in N3 locked in R3C89 for R3+12(4)
b) Innies+Outies N3: -10 = R4C8 - R2C78 -> R2C78 <> 3 since R4C8 >= 2
c) 12(4) = {1236}
d) 24(4): R456C9 <> 1,2 since R7C9 <= 4
e) 1 locked in R456C7 @ N6 for C7
f) Innies N9 = 5(2) = {23} locked for R7+N9
g) 2,3 locked in R34C8 @ C8 for 12(4)
h) R3C9 = 1
i) Outie C9 = R9C8 = 5
j) 13(2) <> 8
k) Killer pair (79) locked in R1C8 + 13(2) for C8

3. C123 !
a) 3(2) = {12} locked for C3+N4
b) Innies N7 = 13(2) <> 1
c) 24(4) = 3{489/579/678} because 45{69/78} blocked by Killer pair (45) of 6(2) -> 3 locked for C1+N4
d) Innies+Outies C1: 2 = R9C2 - R3C1 -> R3C1 <> 8,9; R9C2 = (4679)
e) 3 in locked in 15(4) @ N7 = 3{129/147/156/246} <> 8
f) Innies C1 = 15(3) = {159/249/258/267/456} because R3C1 <> 1,6,8
g) ! Killer pair (25) locked in 6(2) + Innies C1 for C1
h) 24(4) = 38{49/67} -> 8 locked for C1

4. R789
a) 8 in N7 locked in Innies N7 = 13(2) = {58} -> R7C1 = 8, R7C3 = 5
b) Innies N89 = 10(2) = [73] -> R7C5 = 7, R7C9 = 3
c) R7C7 = 2
d) 13(3) = {256} -> R7C6 = 6, R8C6 = 5
e) 15(4): R78C2 <> 9 because R89C3 >= 7
f) Hidden Single: R7C4 = 9 @ R7
g) Cage sum: R8C4 = 4

5. C789
a) 6(2) = {15} locked for C7+N6
b) Naked pair (48) locked in R2C78 for R2+N3+33(6)
c) 33(6) = {234789} -> R4C7 = 3; {279} locked for C6

6. R123
a) Outies R1 = 9(3) <> 9; R2C15 <> 2,5 since R2C9 = (25)
b) R2C1 = 1 -> R1C1 = 5
c) 16(4) = {1348} -> R2C5 = 3; {48} locked for R1+N2
d) 9(2) = {36} locked for N1
e) 4,8 locked in 24(4) @ N1 = {4578} -> R4C2 = 5; 7 locked for R3+N1
f) R2C3 = 9, R2C2 = 2, R3C4 = 5, R2C6 = 7, R2C4 = 6, R4C8 = 2, R4C6 = 9
g) 30(6) = {125679} -> R4C3 = 7, R4C4 = 1

7. N45
a) 23(4) = 78{26/35} -> 8 locked for R6+N5; R6C5 = (56)
b) 14(2) = {68} -> R6C2 = 6, R5C2 = 8

8. Rest is singles.

Rating: Hard 1.25 - Easy 1.5. I used an IOU move.

Like its predecessor this one offered some nice IOU moves though you need way more to solve it.

A142 V3 Walkthrough:

1. C789
a) 6(3) = {123} locked for N9
b) Innies+Outies C9: -6 = R9C8 - R3C9 -> R3C9 = (789)
c) 4 locked in 25(4) @ C9 = 47{59/68} -> 7 locked for C9
d) Innies N69 = 13(2+1): R4C78 <> 9 since R7C7 >= 4
e) 9 locked in R456C9 @ N6 for C9
f) R3C9 = 8
g) Outie C9 = R9C2 = 2
h) 6(3) = {123} -> 1,3 locked for C9
i) 8(2) @ R1C9 = {26} locked for C9+N3
j) Innies N9 = 11(2): R7C7 <> 5,8

2. C789 !
a) 20(4) = 8{147/156/345} <> 9
b) Innies+Outies N3: -9 = R4C8 - R2C78 -> R2C7 <> 9 (IOU @ C8)
c) Hidden Single: R2C8 = 9 @ N3
d) Innies+Outies N3: R2C7 = R4C8 <> 6
e) ! Hidden Killer triple (268) locked in both 9(2) + R4C7 for N6 since 9(2) can only have one of (268)
-> Both 9(2) <> 4,5; R4C7 = (268)
f) ! 9 locked in Outies C89 @ C7 = 23(4) = 9{158/347} because 69{17/35} unplaceable
since 69 only possible @ R89C7 and R13C7 = 8 clashes with 8(2) @ R1C8 (IOU @ N3)
g) Outies C89 = 23(4): R89C7 <> 5 since R13C7 <> 8,9
h) 5 locked in R123C7 @ C7 for N3
i) 8(2) @ R1C7: R1C7 <> 3
j) Outies C89 = 23(4): R3C7 <> 4,7 since 3 only possible there
k) 20(4): R34C8 <> 1 because R3C7 <> 4,7
l) Innies N69 = 13(2+1) <> 8 because R4C8+R7C7 >= 7

3. R789
a) Innies R789 = 12(3) <> 1 because R7C19 = 11 or R7C59 = 11 clashes with Innies N7 = 11(2) = Innies N9
b) 10(4) = {1234} -> 1 locked for R6+N5; CPE: R45C5 <> 2,3,4
c) Innies R789 = 12(3): R7C1 <> 7,8,9 since R7C59 >= 6
d) Innies N7 = 11(2): R7C3 <> 1,2,3,4

4. R456
a) 1 locked in R5C78 @ N6 locked for R5
b) Both 9(2) @ N6: R5C78 <> 8
c) 8 locked in R6C78 @ N6 locked for R6
d) 10(2) <> 9; R5C2 <> 2
e) 9(2) @ N4 <> 8

5. R123
a) Innies+Outies R1: -3 = R2C5 - R1C19: R2C5 <> 1 since R1C9 <> 1,3
b) Outies R1 = 12(3): R2C1 <> 5 because R2C5 <> 1
c) Outies R1 = 12(3): R2C5 <> 3,8 since R2C9 = (26) and R2C1 <> 7
d) Innies+Outies N1: -9 = R4C2 - R2C23 -> R4C2 <> 7,8,9 because R2C23 <= 15
e) Innies+Outies N1: -9 = R4C2 - R2C23 -> R2C23 <> 1 since R2C23 <> 9

6. C123 !
a) 22(4) <> 56{38/47} since (356,456) are Killer triples of 7(2)
b) 10(2)+9(2) = 19(4) = {2368/2458/2467/3457}
c) ! 22(4): R7C1 <> 6 because {178/259/349}6 blocked by Killer pairs (25,34,78) of combined 19(4) cage
d) Innies+Outies N4: -4 = R7C1 - R4C23 -> R4C3 <> 9 since R7C1 <= 5
e) 9 locked in 22(4) @ N4 for C1; 22(4) = 9{148/157/238/247/256/346}

7. R789+N3 !
a) Innies R789 = 12(3) = 3{27/45} -> 3 locked in R7C15 for R7
b) ! Innies N78 = 12(3+1): R7C1 <> 5 since R7C5 would be 3 but R7C56+R8C6 must be {124}
c) Innies N7 = 11(2) <> 5,6
d) Innies+Outies N78: -1 = R7C7 - R7C15: R7C15 = 3{2/4} -> R7C7 <> 7
e) Innies N9 = 11(2): R7C9 <> 4
f) 25(4) = {4579} -> 4 locked for N6
g) Hidden Single: R3C8 = 4 @ 20(4)

8. C123
a) Innies N47 = 15(2+1): R4C3 <> 7,8 since R7C3 = (789)
b) Hidden Killer pair (78) in 22(4) for N4 since combined 19(4) cage can only have one of it
-> 22(4) <> 6 because 69{25/34} unplaceable
c) 22(4) <> 5 because R7C1 <> 1,7,9
d) 7(2) <> 3,4 since (34) is a Killer pair of 22(4)

9. R123 !
a) Outies R1 = 12(3) = 6{15/24} since R2C19 = (126) and R2C5 <> 9 -> 6 locked for R2; R2C5 = (45)
b) 23(4) = {1589/2489/2579/3569/4568} because R2C5 = (45) and 37{49/58} blocked by Killer pair (37) of 8(2) @ R1C7
c) ! Hidden Killer pair (49) in 11(2) for R1 because in 23(4) 4 of {2489} must be placed in R2C5
-> 11(2) = {29/47}
d) 8 locked in 23(4) @ R1 for N2; 23(4) = 8{159/249/456} <> 3,7
e) Hidden Single: R1C8 = 3 @ R1 -> R1C7 = 5
f) R3C7 = 1 -> R4C8 = 7

10. R456
a) 9(2) @ C8 = {18} -> R5C8 = 1, R6C8 = 8
b) 9(2) @ C7 = {36} locked for C7
c) R2C7 = 7, R4C7 = 2, R7C7 = 4
d) 31(6) @ R2C6 = {234679} because R3C6 = (356) -> 3,4,6 locked for C6

11. R789
a) 11(3) = {245} -> 2,5 locked for C6+N8
b) R7C5 = 3, R7C1 = 2, R6C6 = 1
c) Innie N7 = R7C3 = 9
d) 18(3) = {189} -> 1,8 locked for C4+N8
e) 10(4) = {1234} -> 2,4 locked for R6+N5
f) Hidden Single: R5C3 = 2 @ N4 -> R6C3 = 7, R2C6 = 4 @ C6
g) R1C3 = 4 -> R1C2 = 7

12. N256
a) 23(4) = {1589} -> R2C5 = 5, R1C4 = 9, R1C6 = 8, R1C5 = 1
b) 31(6) @ R2C2 = {235678} -> R3C4 = 7; R4C34 = {56} locked for R4
c) 10(2) = {46} -> R6C2 = 6, R5C2 = 4

13. Rest is singles.
Rating: Hard 1.5. I used Hidden Killer subsets, IOU's and combo analysis.


On a side note, aren't 4 Killers (JFF2, A142 V1-V3) a week more than enough? I don't see a point in posting a fifth Killer especially when the other ones only have 1 or 2 walkthroughs. By the way Frank, did you try to solve V3 and V4?

This V4 is really one of the most difficult I have ever solved : 2 contradiction moves, 2 forcing chains ! Not able to rate it !
Here is my wt :

WALKTHROUGH A 142 V4




1.Innies for n9 r7c79 totals 12 : no 126
2.Innies for n7 : r7c13 totals 10 : no 5
3.Innies for r789 : r7c159 totals 14
4.Outies for c789 : r23478c6 totals 31
a) Max r234c6=24 → Min r78c6= 7 → Max r7c7=7 (cage 14(2))
b) r7c79 = [75] / [57] / [48] / [39]
5.IO for n8 : r7c3+r7c7=r7c5+8
a) r7c3 <>8 since r7c7<>r7c5 (see a)), no 1,2 for r7c3 (cage (20(3))
b) r7c13=[19] / {37} / {46}
6.a) Since r7c1={13467} and r7c9={5789} we deduce from step 3 that r7c159 = [18]5 / {36}5 / [72]5 / {16}7 / {34}7 / [15]8 / [42]8 / {14}9 / [32]9
b) Using steps 1,2 : removed [365] (r7c3=7=r7c7) / [72]5 (r7c1=r7c7=7) /[347] (r7c3=r7c9=7) /[42]8 (r7c1=r7c7=4) /[149] (r7c3=r7c9=9) /[329] (r7c1=r7c7=3)
c) We deduce r7c13579 = [19875] / [64375] / [46357] / [46139] / [64157]
[19657] / [19548] and r4c3 = {469}
7. IO for c1 : r3c1=r9c2
8.a) Forcing chain : from step 7, digit r7c3 is locked for c1 in one of cages 15(2) and 15(4).
* if r4c3 =4, r4c1=6, and 4 is locked in cage 15(4) : impossible since combination {234}6 blocks one of both cages 10(2) in n4
*If r4c3=6, r4c1=4, and 6 can't be locked in cage 15(4) since combination {236}4 blocks one of both cages 10(2) in n4 → 6 locked in cage 15(2)
*If r4c3=9, r4c1=1, and 9 can't be locked in cage 15(4) since combination {239}1 blocks one of both cages 10(2) in n4 → 9 locked in cage 15(2)

→ 6 or 9 locked in cage 15(2), 15(2)={69} locked for c1 and n1, r4c3 <>4

b) Combinations [64375] and [64157] removed from step 6.c)

9.a) Outies for r1 : r2c159 totals 12 → r2c156 = [912] [615] [624] ([642] impossible since r1c1 =r1c9=9)
b) 11(2) at c9 : [92] / [74] / [65]
10.Innies for n69 : 14=r4c78+r7c7
11.Combinations of cage 20(4) at c9 not blocked by combinations of cage 11(2) at c9 (see step 9.b) : r4567c9 = {1289} {1379} {2369} {1478} {2378} {1568} {3458} {3467}
12.Step by contradiction : if r7c9=9, r7c7=3 (step 1) and r4c78 totals 11 (step 10) → 4 locked for n6 in cage 20(4) (r4c78 cannot be {47} and 7(2) cannot be {34} , but there is no valid combinations for 20(4) with both digits 4 and 9 (step 11) → removed [46139] from step 6)c.
13.a) Step by contradiction : if r7c79=[57], no 2 in cage 7(2) at n6 and no 2 in r4c78 that totals 9 (step 10), so 2 locked for c9 and n6 in 20(4). 20(4) = {382}7 (step 11) → 11(2) at c9 =[65] and 8(2) at n3 = {17} → 1 locked for c9 in cage 15(3). But there is no combination : {159} blocked by r7c7=5, and {168} blocked by hidden pair {68} locked for c9 in cages 11(2) and 20(4) → r7c79 <>[57]
b) Combinations [46357] [64157] [19657] removed from step 6)c.
14.From step 6)c. : r7c13579= [19875] or [19548] → r7c13=[19]
15.IO for n4 : r4c23 totals 11 → no 1. Digit 1 is locked for c2 and n4 at r56c2. R56c2={19}
16.Combinations for cage 20(4) at c9 from step 11 and 14 : {156}8 {345}8 {168}5 and {348}5 → cage 11(2) at c9 cannot be {56} → (step 9.a)) r2c156 = [912] or [624]
17.Hidden pair {48} for r1 locked in cage 15(4) : 15(4)={1248}, all locked for n2
18.Forcing chain : r7c7={47} (step 14)
*If r7c7=7, r78c6=7 → r234c6 totals 24 (step 4) → r4c6=8 (step 17)
*If r7c7=4 r7c9=8 (step 1) → 8 locked for n6 and r4 in r4c78

→ 8 locked for r4 in r4c678-> hidden cage 11(2) at r4c23 (step 15) must be {47} or {56}

19.a) Combinations for cage 15(4) at c1 (no 6 9) : 15(4)= {842}1 {752}1 {743}1
b) Removed {752}1 since {75} is a killer pair for r4c23 (step 18)
c)r4c23 cannot be {74}, blocked by combinations {842}1 and {743}1 of 15(4),
step 18-> r4c23={56}, locked for r4 and n4.
20.a. Outies for n1 : r234c4 totals 13. Using step 17, r234c4= {63}4 {56}2 {93}1 {57}1
b. Removed {63}4 : blocks all combinations of cage 20(3) at n78
c.Removed {56}2 since r4c3={56}
d.We deduce from 20. a. that r4c4=1
21.IO for n 5 : r3c5+r7c5=2+r4c6
a) r4c6<>8 since r37c5 cannot be [55] or [28]
b) We deduce that 8 is locked for r4 and n6 at r4c78 (step 18)
22.Innies for n69 : r4c78+r7c7 totals 14 and 8 is locked for r4 and n6 at r4c78, so r7c7
cannot be 7 ->r7c7=4, r7c9=8r7c5=5
23.IO for n5 : r4c6 = 3+r3c5
a) r3c5={3679} → r4c6={69}
b) r4c6<>6 since r4c23={56} → r4c6=9 and r3c5=6
24.Innies for n6 : r4c78 totals 10 and contain 8 → r4c78={28}. We deduce :
a. Cage 7(2) at n6 = {16}
b. Cage 20(4) at c9 = {345}8
c. Cage 11(2) at c9 =[92]
d. Cage 15(2) at c1 =[69]
e. r2c5=1
25.6 is locked for c9 in cage 15(3) and r89c9={167}
a) r9c8<>8 → r89c9={67} r9c8=2 r4c78=[28] r3c9=1
b) Last combination for cage 20(4) at n3 : r3c78=[74], cage 8(2) at n3={35}, cage 8(2) at n1 =[71]
26.a) Hidden singles : r2c78=[86]
b) Last combination for cage 37(6) : r23c6=[75]
c) Last combination for cage 14(3) at n89 : r78c6=[28] r1c6=4 (hidden single)
d) Last combination for cage 20(3) at n78 : 20(3)=[974]
27.a) Digit 6 is locked is locked for n5 at r5c46 → cage 7(2) at c7=[61]
b) Cage 10(2) at c2=[19]
c) Cage 16(2) at c8=[97]
d) r2c4=3 and r3c4=9 (hidden singles)
28.Hidden pair {39} at r89c5 locked for c5 and r5 : r9c46=[61] → r89c9=[67]
29.r6c456 totals 12 : last combination {246} → 6c456=[246]
30.Naked singles : cage 29(5) = [67583] at n25 r1c45=[82] and cage 20(8) at c9=[3458]. R4c1=4
31.Hidden single for n7 : r9c3=4
32.Last combination for 20(4) at n7 : [6374]
33.Rest is singles