Jigsaw nonets identified by their upper-left cells, for example NR3C5
Prelims
a) R12C9 = {18/27/36/45}, no 9
b) R34C1 = {59/68}
c) R3C34 = {19/28/37/46}, no 5
d) R67C9 = {19/28/37/46}, no 5
e) R7C67 = {89}
f) R89C1 = {59/68}
g 9(3) cage at R1C1 = {126/135/234}, no 7,8,9
h) 8(3) cage at R5C4 = {125/134}
i) 11(3) cage at R6C2 = {128/137/146/236/245}, no 9
j) 26(4) cage at R1C4 = {2789/3689/4589/4679/5678}, no 1
Steps resulting from Prelims
1a. 8(3) cage at R5C4 = {125/134}, 1 locked for R5 and NR3C5
1b. Naked pair {89} in R7C67, locked for R7, clean-up: no 1,2 in R6C1
1c. Naked pair {89} in R7C67, CPE no 8,9 in R69C7
1d. Naked quad {5689} in R3489C1, locked for C1
2a. 45 rule on R1234 1 outie R5C3 = 1 innie R4C9 + 6 -> R4C9 = {123}, R5C3 = {789}
2b. 18(3) cage at R5C1 = {279/378} (cannot be {189/369/459/468/567} because 5,6,8,9 only in R5C2) -> R56C1 = {27/37}, 7 locked for C1, R5C2 = {89}
2c. 45 rule on R6789 1 outie R5C7 = 1 innie R6C1 = {237}, R56C1 = {27/37} -> R5C17 = {27/37}, 7 locked for R5, clean-up: no 1 in R4C9
2d. Naked pair {89} in R5C23, locked for R5
2e. Killer triple 2,3 in R5C17 and 8(3) cage at R5C4, locked for R5
2f. 13(3) cage at R4C9 = {256/346}, CPE no 6 in R3C9 + R67C8
2g. 6 in NR3C5 only in R3467C5, locked for C5
3a. Naked pair {89} in R5C23, CPE no 8,9 in R4C2 + R68C3
3b. 8,9 in NR5C2 only in R568C2, locked for C2
3c. Hidden killer pair 8,9 in R34C1 and R23C3 for NR1C1, R34C1 contain one of 8,9 -> R23C3 must contain one of 8,9
3d. Killer pair 8,9 in R23C3 and R5C3, locked for C3
4a. Hidden killer pair 8,9 in R689C4 + R8C5 and R89C1 for NR6C4, R89C1 contain one of 8,9 -> R689C4 + R8C5 can only contain one of 8,9
4b. 25(4) cage at R7C3 = {3679/4579/4678} (cannot be {1789/2689/3589} which contain both of 8,9), no 1,2
4c. 25(4) cage at R7C3 = {3679/4579/4678}, CPE no 7 in R9C3
4d. Killer pair 8,9 in 25(4) cage and R89C1, locked for NR6C4
4e. 45 rule on R89 2 innies R89C4 = 1 outie R7C8 + 11
4f. Max R89C4 = 16 -> max R7C8 = 5
4g. Hidden killer pair 8,9 in R1234C4 and R89C4 for C4, R89C4 contain one of 8,9 -> R1234C4 must contain one of 8,9
5. 45 rule on R12 2 innies R12C6 = 1 outie R3C2 + 2, max R3C2 = 7 -> max R12C6 = 9, no 9 in R12C6
[Time to start using Law of Leftovers (LoL). For those not familiar with LoL, the outies and innies must contain exactly the same numbers.]
6. LoL for R1234 2 outies R5C18 = 2 innies R34C5, no 8,9 in R5C18 -> no 8,9 in R34C5
6a. R5C3 + R6C5 = {89} (hidden pair in NR3C5)
7. LoL for R6789 2 outies R5C29 = 2 innies R67C5, no 2,3,7 in R5C29 -> no 2,3,7 in R67C5
7a. Min R67C5 = 12 -> max R6C34 = 7, no 7 in R6C34
7b. 7 in NR3C5 only in R34C5 + R5C7, CPE no 7 in R4C7
8. LoL for C1234 2 outies R18C5 = 2 innies R5C34, no 7 in R5C34 -> no 7 in 2 R18C5
8a. R5C3 = {89} -> R1C5 = {89}
8b. Naked pair {89} in R16C5, locked for C5
8c. R1234C4 must contain one of 8,9 (step 4g), killer pair 8,9 in R1234C4 and R1C5, locked for NR1C2
9. 45 rule on R789 2 outies R6C29 = 1 innie R7C5 + 7, min R7C5 = 4 -> min R6C29 = 11, no 1 in R6C2
10. 45 rule on R123 2 outies R4C18 = 1 innie R3C5 + 11
10a. Max R4C18 = 17 -> max R3C5 = 6
10b. Min R3C5 = 2 -> min R4C18 = 13 -> min R4C8 = 4
11. 45 rule on R1234 4 innies R4C2349 = 14 = {1238/1247/1256/1346/2345}, no 9
[This step could be considered to be a forcing chain or combined cages.]
12. 13(3) cage at R4C9 = {256/346}, 8(3) cage at R5C4 = {125/134} -> combined cage R4C9 + R5C45689 = 2{134}{56}/3{125}{46} -> no 2,3 in R4C5
[The hardest part for me was spotting useful 45s on the jigsaw nonets; this one and the next one crack this puzzle.]
13. 45 rule on NR5C9+NR6C6 2 outies R5C7 + R8C5 = 1 innie R5C9 + 5
13a. Min R5C9 = 4 -> min R5C7 + R8C5 = 9, max R8C5 = 5 -> R5C7 = 7 -> R6C1 = 7 (step 2c), clean-up: no 3 in R7C9
13b. R5C9 = R8C5 + 2, R5C9 = {456} -> R8C5 = {234}
13c. LoL for R1234 (step 6) R5C18 = R34C5 -> R3C5 = {23}
13d. LoL for C1234 (step 8) R18C5 = R5C34 -> R5C4 = {234}
14. 45 rule on NR1C1+NR1C2 2 outies R2C5 + R5C3 = 1 innie R5C1 + 10
14a. R6C1 = 7 -> R5C12 = 11 = [29/38] -> R5C13 = [28/39]
14b. R5C3 6 more than R5C1 -> R2C5 = 4, clean-up: no 5 in R12C9
14c. R9C5 = 7 (hidden single in C5)
14d. 4 in NR3C5 only in 8(3) cage at R5C4 = {134}, 3,4 locked for R5, 3 locked for NR3C5 -> R3C5 = 2, R5C1 = 2, R5C2 = 9, R5C3 = 8, R5C58 = [13], R6C5= 9 -> R1C5 = 8, clean-up: no 1 in R2C9, no 1 in R7C9
14e. R8C5 = 3 -> R5C9 = 5 (step 13b)
14f. R5C89 = [65] -> R4C9 = 2 (cage sum), clean-up: no 7 in R12C9, no 8 in R6C9
14g. R12C9 = [18] (cannot be {36} which clashes with R67C9)
14h. LoL for C1234 (step 8) R18C5 = R5C34, R18C5 = [83] -> R5C34 = [83], R5C6 = 4, clean-up: no 7 in R3C3
14i. R89C5 = [37] = 10 -> R89C6 = 9 = {18}, locked for NR6C6 -> R7C67 = [98]
14j. R12C5 = [84] = 12 -> R12C4 = 14 = {59}, locked for C4, 5 locked for NR1C2, clean-up: no 1 in R3C3
14k. 9(3) cage at R1C1 = {234} (only remaining combination), locked for R1, 2 locked for NR1C2
14l. 25(4) cage at R7C3 = {4678} (only remaining combination), 7 locked for C4, 8 locked for NR6C4, clean-up: no 3 in R3C3, no 6 in R89C1
14m. Naked pair {59} in R89C1, locked for C1, 5 locked for NR6C4
14n. Naked pair {68} in R34C1, 6 locked for NR1C1, clean-up: no 4 in R3C4
14o. R6C4 = 2 (hidden single in C4), R6C5 = 9 -> R6C3 + R7C5 = 8 = [35], clean-up: no 7 in R7C9
14p. Naked pair {46} in R67C9, locked for C9 and NR5C9
14q. Naked pair {46} in R7C39, locked for R7 -> R7C4 = 7
14r. R7C12 = [12] -> R6C2 = 8 (cage sum)
14s. R2C1 = 3 -> 9(3) cage at R1C1 = [432]
14t. R2C1 = 3 -> 21(4) cage at R2C1 = {3567} (only remaining combination) -> R2C2 = 6, R2C3 + R3C2 = {57}, locked for N1, R4C2 = 1, R4C345 = [746], R34C1 = [68], R3C4 = 1 -> R3C3 = 9
and the rest is naked singles, using rows, columns and jigsaw nonets.