Prelims
a) R1C67 = {69/78}
b) R1C89 = {49/58/67}, no 1,2,3
c) R2C12 = {19/28/37/46}, no 5
d) R1C67 = {18/27/36/45}, no 9
e) R1C89 = {19/28/37/46}, no 5
f) R45C8 = {14/23}
g) R56C2 = {18/27/36/45}, no 9
h) R8C12 = {18/27/36/45}, no 9
i) R8C34 = {19/28/37/46}, no 5
j) R8C89 = {69/78}
k) R9C12 = {49/58/67}, no 1,2,3
l) R9C34 = {18/27/36/45}, no 9
m) 10(3) cage at R2C4 = {127/136/145/235}, no 8,9
n) 11(3) cage at R4C4 = {128/137/146/236/245}, no 9
o) 22(3) cage at R5C6 = {589/679}
p) 26(4) cage at R6C3 = {2789/3689/4589/4679/5678}, no 1
q) 28(7) cage at R3C8 = {1234567}, no 8,9
1. R1C89 = {49/58} (cannot be {67} which clashes with R1C67), no 6,7
1a. Killer pair 8,9 in R1C67 and R1C89, locked for R1
2. 45 rule on NR1C4 2 innies R2C4 + R3C6 = 6 = {15/24}
2a. R2C67 = {18/27/36} (cannot be {45} which clashes with R2C4 + R3C6), no 4,5
3. 45 rule on NR1C4 2 outies R3C45 = 1 innie R3C6 + 4, IOU no 4 in R3C45
4. 45 rule on NR7C4 2 innies R7C4 + R8C6 = 12 = {39/48/57}, no 1,2,6
5. 45 rule on NR7C4 2 outies R7C56 = 1 innie R7C4 + 1, IOU no 1 in R7C56
5a. 13(3) cage at R7C5 = {238/247/256/346}, no 9, clean-up: no 3 in R7C4 (step 3)
6. 45 rule on R8 3 innies R8C567 = 11 = {128/137/146/236/245}, no 9
6a. 8 of {128} must be in R8C6 -> no 8 in R8C57
7. 45 rule on C89 3 innies R6C8 + R9C89 = 19 = {289/379/469/478/568}, no 1
7a. 2,3,5 of {289/379/568} must be in R9C89 (R9C89 cannot be {89/79/68} which clash with R8C89), no 2,3,5 in R6C8
7b. 45 rule on C89 4 outies R6789C7 = 14 = {1238/1247/1256/1346} (cannot be {2345} which clashes with R6C8 + R9C89 or, if preferred, 33(7) cage must contain1), no 9, 1 locked for C7, clean-up: no 8 in R2C6
8. 45 rule on C1234 1 outie R3C5 = 1 innie R1C4, no 4 in R1C4
9. 45 rule on C6789 1 innie R9C6 = 1 outie R7C5 + 1, no 1,2 in R9C6
[Just spotted 5 in R2 only in R2C345]
10. 45 rule on R2 3 innies R2C345 = 16 = {259/358/457}, no 1,6, clean-up: no 5 in R3C6 (step 2)
11. 10(3) cage at R2C4 = {127/145/235} (cannot be {136} because R2C4 only contains 2,4,5), no 6, clean-up: no 6 in R1C4 (step 8)
12. 45 rule on R12 3 outies R3C3 + R4C23 = 1 innie R2C4 + 18
12a. Min R2C4 = 2 -> min R3C3 + R4C23 = 20, no 1,2 in R3C3 + R4C23
13. Law of Leftovers (LoL) for C123 two outies R34C4 must exactly equal two innies R89C3, no 9 in R34C4 -> no 9 in R8C3, clean-up: no 1 in R8C4
14. LoL for C789 two outies R67C6 must exactly equal two innies R12C7, no 4,5 in R12C7 -> no 4,5 in R67C6
15. 45 rule on R1 3 innies R1C123 = 1 outie R2C5 + 2
15a. Min R1C123 = 6 -> min R2C5 = 4
15b. R2C345 (step 10) = {259/358/457}
15c. 3 of {358} must be in R2C3, no 8 in R2C3
16. 3 in NR1C4 only in 15(3) cage at R1C4 = {348/357} or in R2C67 = {36} -> 15(3) cage = {159/249/258/348/357} (cannot be {168/267/456}, locking-out cages), no 6
16a. 15(3) cage = {159/249/348/357} (cannot be {258} which clashes with R2C4 + R3C6)
[With hindsight, 15(3) cage at R1C4 = {159/249/348/357} (cannot be {168/267} which clash with R1C67, cannot be {258/456} which clash with R2C4 + R3C6), no 6 is simpler.]
16b. 4 of {249/348} must be in R1C5 -> no 2 in R1C5, no 4 in R2C5
17. R1C67 = {69/78}, R1C89 = {49/58} -> combined cage R1C6789 = {69}{58}/{78}{49}
17a. 15(3) cage at R1C4 (step 16a) = {159/348/357} (cannot be {249} which clashes with combined cage R1C6789), no 2, clean-up: no 2 in R3C5 (step 8)
18. 2 in R1 only in R1C123, locked for NR1C1, clean-up: no 8 in R2C12
18a. 36(7) cage = {1236789/1245789/2345679} must contain 7
19. 15(3) cage at R1C4 (step 17a) = {159/348/357}, R1C4 = R3C5 (step 8) -> R123C5 = {159/348/357}
19a. 17(3) cage at R4C5 = {179/269/278/368/467} (cannot be {359/458} which clash with R123C5), no 5
19b. 45 rule on C6789 3 outies R789C5 = 13 = {148/157/238/247/256/346} (cannot be {139} which clashes with R123C5), no 9
20. 2 in R1 only in R1C123
20a. Hidden killer pair 1,3 in R1C123 and 15(3) cage at R1C4 for R1, 15(3) cage contains one of 1,3 -> R1C123 must contain one of 1,3
20b. R1C123 = R2C5 + 2 (step 15)
20c. R2C5 = {5789} -> R1C123 = 7,9,10,11 = {124/126/127} (cannot be {234/235} which clash with 15(3) cage = {35}7/{34}8), cannot be {236} because R1C123+R2C5 = {236}9 clashes with R2C12, killer combo clash, other combinations don’t contain 2 or one of 1,3), no 3,5, 1 locked for R1 and NR1C1, clean-up: no 9 in R2C12, no 1 in R3C5 (step 8)
20d. 1 in NR1C4 only in R23C6, locked for C6
[I omitted to say that step 20c eliminates 9 from R2C5; however the next step does that.]
21. 15(3) cage at R1C4 (step 17a) = {348/357}, no 9, 3 locked for NR1C4, clean-up: no 6 in R2C67
21a. LoL for C789 two outies R67C6 must exactly equal two innies R12C7, no 3 in R12C7 -> no 3 in R7C6
22. R1C67 = {69} (hidden pair in NR1C4), locked for R1, clean-up: no 4 in R1C89
22a. Naked pair {58} in R1C89, locked for R1 and NR1C8, clean-up: no 2 in R2C89, no 5 in R3C5 (step 8)
22b. Caged X-Wing for 9 in R1C67 and 22(3) cage at R5C6, no other 9 in C67, clean-up: no 8 in R7C5 (step 9)
23. 5 in NR1C4 only in R2C45, locked for R2
23a. R2C345 (step 10) = {259/358} (cannot be {457} which clashes with R2C12), no 4,7, clean-up: no 2 in R3C6 (step 2)
24. Killer pair 4,7 in R1C123 and R2C12, locked for NR1C1
24a. 5,8 in NR1C1 only in R3C12 + R4C1, locked for 37(7) cage at R3C1, no 5,8 in R56C1 + R7C12
24b. 37(7) cage contains 5 so must contain 3 = {1345789/2345689}
25. 10(3) cage at R2C4 (step 11) = {127/235}, 2 locked for C4, clean-up: no 8 in R8C3, no 7 in R9C3
25a. R3C5 = {37} -> no 3,7 in R3C4
26. 1,2 in R1 only in 36(7) cage at R1C1 (step 18a) = {1236789/1245789}, 8 locked for NR3C3, clean-up: no 1 in R6C2
26a. 36(7) cage = {1236789/1245789}, CPE no 9 in R6C4
27. 1 in NR3C3 only in R34C4 + R5C123, CPE no 1 in R5C4
28. LoL for R789 three outies R6C129 must exactly equal three innies R7C567, no 9 in R7C567 -> no 9 in R6C1
[At this stage I spotted some interesting forcing chains starting from the combinations of R45C8, using caged X-wings for C89, in the case for R45C8 = {14} with one caged X-wing leading to a second caged X-wing within the same path of the forcing chain.
But then I spotted the much simpler …, which had been available since step 22b, but I’d been doing steps which were more obvious to me at the time]
29. 9 in NR1C8 only in R2C89 = {19}, locked for R2 and NR1C8 -> R2C3 = 3, placed for NR1C1, clean-up: no 7 in R2C12, no 8 in R2C7, no 4 in R5C8, no 7 in R8C4, no 6 in R9C4
29a. Naked pair {27} in R2C67, locked for R2 and NR1C4 -> R1C45 = [34], 4 placed for NR1C4, R2C45 = [58], R3C6 = 1, R3C4 = 2, placed for NR3C3, R3C5 = 3 (cage sum), placed for NR3C5, clean-up: no 7 in R6C2, no 7 in R8C3, no 4,6 in R9C3, no 4,5 in R9C6 (step 9)
30. R3C12 + R4C1 = {589} (hidden triple in NR1C1), locked for 37(7) cage at R3C1, no 9 in R5C1 + R7C12
31. Naked triple {127} in R1C123, locked for 36(7) cage at R1C1, no 7 in R3C3 + R4C23
31a. 36(7) cage at R1C1 (step 26a) = {1236789} (only remaining combination), no 4,5
31b. Naked triple {689} in R3C3 + R4C23, locked for NR3C3, clean-up: no 3 in R6C2
32. 11(3) cage at R4C4 = {146} (only remaining combination) -> R5C4 = 6, placed for NR3C5, R4C4 + R5C3 = {14}, locked for NR3C3, clean-up: no 5,8 in R6C2, no 4 in R8C3, no 7 in R9C6 (step 9)
32a. 3 in NR3C3 only in R5C12, locked for R5, clean-up: no 2 in R4C8
33. 17(3) cage at R4C5 (step 19a) = {179} (only remaining combination), locked for C5 and NR3C5, clean-up: no 8 in R9C6 (step 9)
34. 14(3) cage at R8C5 = {356} (only remaining combination) -> R9C6 = 3, R89C5 = {56}, locked for C5 and NR7C4 -> R7C5 = 2, placed for NR3C5, clean-up: no 4 in R8C4, no 4 in R9C4
34a. Killer pair 8,9 in R8C4 and R8C89, locked for R8, clean-up: no 1 in R8C12
35. R7C5 = 2 -> R78C6 = 11 = [74], 7 placed for NR4C7, 4 placed for NR7C4, R2C67 = [27], clean-up: no 5 in R8C12
35a. Killer pair 6,7 in R8C12 and R8C89, locked for R8 -> R89C5 = [56], clean-up: no 7 in R9C12
36. Naked pair {58} in R45C6, locked for C6 and NR3C5 -> R6C4 = 4, R4C4 = 1, R5C3 = 4
37. R9C4 = 7 (hidden single in C4), R9C3 = 2, R8C3 = 1, R8C4 = 9, clean-up: no 6 in R8C89
37a. Naked pair {78} in R8C89, locked for R8 and NR6C9, clean-up: no 2 in R8C12
37b. Naked pair {36} in R8C12, locked for R8 and NR6C1 -> R8C7 = 2, placed for NR6C9, R6C2 = 2, R5C2 = 7, R5C1 = 3
38. Naked pair {14} in R7C12, locked for R7 and NR6C1 -> R6C1 = 7, clean-up: no 9 in R9C12
38a. Naked pair {58} in R9C12, locked for R9 and NR6C1 -> R7C3 = 9
39. 22(3) cage at R5C6 = {589} (only remaining combination) -> R6C6 = 9, R5C67 = {58}, locked for R5
39a. 17(3) cage at R4C5 = [791]
40. Naked triple {356} in R6C9 + R7C89, locked for 28(7) cage at R3C8, no 3,6 in R3C89 + R4C9
40a. Naked pair {47} in R3C89, locked for NR1C8 -> R3C7 = 6, R4C8 = 3, R5C8 = 2
41. R3C67 = [16] = 7 -> R4C67 = 9 = [54]
and the rest is naked singles, without using the nonets.