Prelims
a) R23C3 = {16/25/34}, no 7,8,9
b) R23C4 = {69/78}
c) R23C6 = {59/68}
d) R78C4 = {13}
e) R78C6 = {16/25/34}, no 7,8,9
f) R78C7 = {39/48/57}, no 1,2,6
g) 20(3) cage at R4C5 = {389/479/569/578}, no 1,2
h) 21(3) cage at R6C9 = {489/579/678}, no 1,2,3
i) 21(3) cage at R7C1 = {489/579/678}, no 1,2,3
j) 21(6) cage at R8C1 = {123456}, no 7,8,9
Steps resulting from Prelims
1a. Naked pair {13} in R78C4, locked for C4 and NR6C4
1b. 21(6) cage at R8C1 = {123456}, locked for NR7C2
1c. 1 in C5 only in R123C5, locked for NR1C4
2. 21(3) cage at R7C1 = {489/579/678}
2a. 4,5,6 only in R7C1 -> R7C1 = {456}
3. 45 rule on NR1C6 3 innies R1C6 + R3C78 = 11 = {128/137/146/236/245}, no 9
3a. 45 rule on NR1C6 1 outie R3C9 = 1 innie R1C6 + 4, R1C6 = {12345}, R3C9 = {56789}
[I haven’t been using Law of Leftovers for TJKs with low SSscores. However in this case it’s very useful. I think this puzzle would be very hard to solve without using LoL.
I later noticed Ed’s comment in the TJK42 thread “Just to note again that the SS score should usually be low for zero killers since there aren't as many cages to get non-essential solving steps from.”, which implies that TJK42 is harder than its SSscore.]
4. Law of Leftovers (LoL) for C123 three outies R239C4 must exactly equal three innies R456C3, R239C4 only contain 6,7,8,9 -> R456C3 = {6789}
5. 34(6) cage at R4C3 = {245689} (only remaining combination) -> R456C3 = {689}, locked for C3 and NR4C3, R456C4 = {245}, locked for C4, clean-up: no 1 in R23C3
[With hindsight I realise that I’d forgotten to do LoL for C123 in the opposite direction after eliminating 7 from the 34(6) cage.]
[When posting my walkthrough, I checked through the other messages in the TJK 42 thread and found that I’d missed making full use of LoL for C123 and C789.
LoL for C123 R456C3 = R239C4 -> R234569C4 = 34, 45 rule on C4 -> R1C4 = 7, placed for NR1C4
LoL for C789 R456C7 = R178C6 -> R145678C6 = 24, 45 rule on C6 -> R9C6 = 7, placed for NR6C4
R5C5 = 7 (hidden single in C5).]
6. R7C3 = 7, placed for NR7C2, R7C12 = 14 = [59/68], clean-up: no 5 in R8C7
6a. Killer pair 8,9 in R23C4 and R9C4, locked for C4
7. LoL for R789 three outies R6C456 must exactly equal three innies R7C189, no 1,3 in R6C456 -> no 1,3 in R7C8, no 7 in R7C189 -> no 7 in R6C56
8. LoL for R123 three outies R4C456 must exactly equal three innies R3C129, no 1 in R4C456 -> no 1 in R3C12
9. 24(6) cage at R4C6 = {123459/123468/123567}
9a. Killer pair 6,9 in R23C6 and 24(6) cage, locked for C6, clean-up: no 1 in R78C6
10. R7C4 = 1 (hidden single in R7), R8C4 = 3, clean-up: no 4 in R7C6, no 9 in R7C7
10a. 21(6) cage at R8C1 = {123456}, 3 locked for R9
11. Grouped X-Wing for 3 in 24(6) cage at R4C6 and R7C67, no other 3 in C67
12. 24(6) cage at R4C6 = {123459/123468/123567}
12a. Consider placement for 3 in R7C67
3 in R7C6 => R78C6 = [34], R78C7 = {57}, locked for C7 => 24(6) cage = {123459/123468} (cannot be {123567} which clashes with R23C6)
or 3 in R7C7 => R78C6 = {25}, locked for C6, R23C6 = {68}, locked for C6 => 24(6) cage = {123459}
-> 24(6) cage = {123459/123468}, no 7
13. R5C5 = 7 (hidden single in NR4C3), R46C5 = 13 = {49/58}, no 3,6
14. R1C4 = 7 (hidden single in NR1C4), clean-up: no 8 in R23C4
14a. Naked pair {69} in R23C4, locked for C4 and NR1C1 -> R9C4 = 8, R7C2 = 9, R7C1 = 5 (cage sum), placed for NR3C1, clean-up: no 2 in R8C6, no 7 in R8C7
15. R78C6 = [25] (cannot be [34] which clashes with R78C7), both placed for NR7C6, clean-up: no 9 in R23C6
15a. Naked pair {68} in R23C6, locked for C6 and NR1C4
15b. Caged X-Wing for 6 in R23C4 and R23C6, no other 6 in R23
[Cracked. The rest is fairly straightforward.]
16. R7C7 = 3 (hidden single in R7), R8C7 = 9, placed for NR7C6
16a. R9C5 = 9 (hidden single in R9), placed for NR6C4, R6C6 = 4, placed for NR6C4, R1C6 = 1, placed for NR1C6, R3C9 = 5 (step 3a), placed for NR3C9, R45C6 = [93], R9C6 = 7, clean-up: no 2 in R2C3
16b. Naked triple {146}, locked for R9 and NR7C6
16c. Naked triple {235} in R9C123, locked for NR7C2
17. R3C5 = 1 (hidden single in R3), R8C3 = 1 (hidden single in C3), R8C5 = 2 (hidden single in R8), placed for NR6C4, R6C4 = 5, R6C5 = 8, R4C5 = 5 (cage sum), R7C5 = 6
17a. Naked pair {12} in R46C7, locked for C7 and NR4C3 -> R5C4 = 4, R4C4 = 2, R456C7 = [152]
17b. Naked pair {48} in R7C89, locked for NR3C9
17c. 4 in C3 only in R123C3, locked for NR1C1
18. R3C9 = 5 -> R3C78 = [73/82]
18a. 7,8 in C7 only in R123C7, locked for NR1C6
19. 21(3) cage at R6C9 = {489/678}
19a. 6,9 only in R6C9 -> R6C9 = {69}
19b. Naked pair {69} in R6C39, locked for R6
20. 2 in NR3C9 only in R5C89, locked for R5
20a. 2 in NR3C1 only in R3C12, locked for R3 -> R3C8 = 3, placed for NR1C6, R3C7 = 7 (step 18), R3C3 = 4, R2C3 = 3, placed for NR1C1, R12C5 = [34], R2C7 = 8, R23C6 = [68], R23C4 = [96], R2C9 = 2, R2C8 = 5, R3C12 = [92]
21. R3C1 = 9 -> R24C1 = 5 = [14], R8C1 = 6, R5C1 = 8, R1C123 = [285], R9C123 = [352], R2C2 = 7, R6C1 = 7, R6C8 = 1, R456C2 = [613], R4C8 = 7, R78C8 = [48], R78C9 = [87], R6C9 = 6 (cage sum)
and the rest is naked singles, without using the nonets.
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. So, if you don't mind h3lix, I'd like to christen it as a part of the TJK series. I'll post my #43 on the 1st of July. Just to note again that the SS score should usually be low for zero killers since there aren't as many cages to get non-essential solving steps from.