Prelims
a) R12C6 = {12}
b) R23C8 = {49/58/67}, no 1,2,3
c) R4C34 = {19/28/37/46}, no 5
d) R56C1 = {19/28/37/46}, no 5
e) R89C5 = {18/27/36/45}, no 9
f) 8(3) cage at R8C1 = {125/134}
g) 41(8) cage at R1C7 = {12356789}, no 4
Steps resulting from Prelims
1a. Naked pair {12} in R12C6, locked for C6 and NR1C4
1b. 8(3) cage at R8C1 = {125/134}, 1 locked for NR7C1, clean-up: no 8 in R8C5
[Since the SS score is 1.25 and Ed has commented that he got a lot of early placements, the puzzle is probably a bit harder than that, so I’ll start using Law of Leftovers (LoL) as soon as possible. A SS score of 1.25 is about the level where I decide whether to use LoL early or avoid using it as long as possible.]
2. LoL for C789 four outies R1C45 + R12C6 must exactly equal four innies R56C78, R12C6 = {12} -> R56C78 must contain both of 1,2, locked for NR5C7, clean-up: no 7,8 in R9C5
3. 45 rule on NR1C7 2 innies R2C8 + R6C9 = 1 outie R5C8 + 4, IOU no 4 in R6C9
[I’d missed 41(8) cage at R1C7 = {12356789}, CPE no 5,6,7,8,9 in R2C8 -> R2C8 = 4.]
3a. R2C8 = 4 (hidden single in NR1C7) -> R3C8 = 9, placed for NR1C4
3b. 41(8) cage at R1C7 = {12356789}, 9 locked for NR1C7, no 9 in R6C9
3c. 4 in C9 only in R789C9, locked for NR7C7
3d. LoL for C789 four outies R1C45 + R12C6 must exactly equal four innies R56C78, no 9 in four outies -> no 9 in R56C7
3e. 34(6) cage at R1C3 must contain 9, CPE no 9 in R2C12 + R45C3, clean-up: no 1 in R4C4
4. 45 rule on NR7C7 1 outie R9C6 = 1 innie R7C7 + 1, no 1,9 in R7C7, no 5 in R9C6
5. LoL for R789 four outies R456C1 + R6C2 must exactly equal four innies R78C56, no 1,2 in R78C56 -> no 1,2 in R456C1 + R6C2, clean-up: no 8,9 in R56C1
5a. 2 in NR4C1 only in R7C234 + R8C34, CPE no 2 in R7C1
6. 45 rule on NR1C1 4(3+1) outies R1C45 + R2C5 + R4C1 = 30
6a. Max R1C45 + R2C5 = 24 -> min R4C1 = 6
6b. Min R1C45 + R2C5 = 21, no 1,2,3 in R1C45 + R2C5
6c. Min R1C45 + R2C5 = 21, max R1C45 = 15 -> min R2C5 = 6
6d. LoL for C789 four outies R1C45 + R12C6 must exactly equal four innies R56C78, no 3 in R1C45 + R12C6 -> no 3 in R56C78
6e. 41(8) cage at R1C7 = {12356789}, 3 locked for NR1C7, no 3 in R6C9
7. 45 rule on NR7C7 4 outies R6789C6 = 26
7a. 45 rule on C6 using R6789C6 = 26, 3 remaining innies R345C6 = 16
7b. 45 rule on C6789 3 innies R2C7 + R3C67 = 15
7c. R2C7 + R3C67 + 22(4) cage at R4C6 = 37, R345C6 = 16 -> R234C7 + R4C8 = 21
[Alternatively can use 45 rule on C789 combined with innies-outies for NR7C7 to get R234C7 + R4C8 = 21; however R345C6 is still useful later.]
7d. 45 rule on NR1C1 + NR1C4 4 innies R234C7 + R4C8 = 2(1+1) outies R2C5 + R4C1 + 3
7e. R234C7 + R4C8 = 21 -> R2C5 + R4C1 = 18 -> R2C5 = 9, placed for NR2C4, R4C1 = 9, placed for NR4C1, clean-up: no 1 in R4C3
7f. R1C2 = 9 (hidden single in NR1C1) -> R12C1 = 8 = {17/26/35}, no 4,8
7g. R5C9 = 9 (hidden single in NR1C7)
7h. R7C6 = 9 (hidden single in R7) , clean-up: no 8 in R7C7 (step 4)
7i. R8C7 = 9 (hidden single in R8)
8. R1C45 + R2C5 + R4C1 = 30 (step 6), R2C5 = 9, R4C1 = 9 -> R1C45 = 12 = {48/57}, no 6
8a. 4 in R1 only in R1C345, locked for 34(6) cage at R1C3, no 4 in R3C3
8b. 34(6) cage at R1C3 = {145789/245689/345679}, CPE no 5 in R1C1, clean-up: no 3 in R2C1 (step 7f)
8c. LoL for C789 four outies R1C45 + R12C6 must exactly equal four innies R56C78, R1C45 + R12C6 = {1248/1257} -> R56C78 = {1248/1257}, no 6
8d. 41(8) cage at R1C7 = {12356789}, 6 locked for NR1C7, no 6 in R6C9
9. 45 rule on R123 2 remaining outies R4C9 + R5C8 = 6 = {15}, locked for 41(8) cage at R1C7
9a. Naked pair {15} in R4C9 + R5C8, CPE no 5 in R4C8
9b. R46C9 = {15} (hidden pair in NR1C7), locked for C9
9c. 5 in R1 only in R1C345, locked for 34(6) cage at R1C3, no 5 in R23C3
9d. Hidden killer pair 4,5 in R1C3 and R1C45 for R1, R1C45 contains one of 4,5 -> R1C3 = {45}
10. 3,6 in NR5C7 only in R68C6 + R78C5, R78C5 cannot contain both of 3,6 (which would clash with R89C5, CCC) -> R68C6 must contain at least one of 3,6
10a. 25(4) cage at R6C6 contains 9 = {2689/3589/3679} (cannot be {4579} which doesn’t contain 3 or 6), no 4
10b. 5 of {3589} must be in R7C7 (R68C6 cannot be {58} which clashes with R56C78), no 5 in R68C6
11. 5 in C6 only in R345C6, locked for NR3C6
11a. R345C6 (step 7a) = 16 = {358/457}, no 6
12. 45 rule on C6789 3 outies R2C4 + R3C45 = 12 cannot contain both of 1,2 (because no 9)
12a. 27(6) cage at R2C4 = {134568/234567} (cannot be {123678/124578} because R2C4 + R3C45 cannot contain both of 1,2), 4 locked for R3
12b. R1C3 = 4 (hidden single in NR1C1), clean-up: no 8 in R1C45 (step 8), no 6 in R4C4
12c. Naked pair {57} in R1C45, locked for R1, 34(6) cage at R1C3 and NR1C4, clean-up: no 1 in R2C1 (step 7f)
12d. 8 in R1 only in R1C789, locked for NR1C7
12e. And, for the last time, LoL for C789 four outies R1C45 + R12C6 must exactly equal four innies R56C78, R1C45 + R12C6 = {1257} -> R56C78 = {1257}, locked for NR5C7, clean-up: no 2,4 in R9C5
12f. 41(8) cage at R1C7 = {12356789}, 7 locked for C9
12g. R1C3 = 4, R1C45 = {57} = 12, R2C5 = 9 -> R23C3 = 9 = {18/36}, no 2
13. 25(4) cage at R6C6 (step 10a) = {2689/3589/3679}
13a. R68C6 = {368} -> no 3,6 in R7C7, clean-up: no 4,7 in R9C6 (step 4)
13b. R345C6 = {457} (hidden triple in C6), locked for NR3C6
14. 22(4) cage at R4C6 = {3478/4567} (cannot be {3568} because R45C6 only contain 4,5,7), 7 locked for C6
14a. 6 of {4567} must be in R4C8 -> no 6 in R4C7
14b. 4 of {4567} must be in R4C7, 4 of {3478} must be in R5C6 (R4C678 cannot be {478} which clashes with R4C34) -> no 4 in R4C6
14c. R2C7 + R3C67 = 15 (step 7b) = {348/456}, 4 locked for R3
14d. 6 of {456} must be in R2C7 -> no 6 in R3C7
15. 4 in R6 only in R6C12, locked for NR4C1, clean-up: no 6 in R6C1
16. 19(4) cage at R4C5 = {1378/2368} (cannot be {1567} because 5,7 only in R5C3), no 5
16a. 7 of {1378} must be in R5C3 -> no 1 in R5C3
16b. 5 in C3 only in C789C3, CPE no 5 in R7C12
17. 20(4) cage at R7C8 contains 4 = {2468/3458/3467} (cannot be {1478} because 1,7 only in R7C8), no 1
17a. 5,7 of {3458/3467} must be in R7C8 -> no 3 in R7C8
17b. 1 in R7 only in R7C234, locked for NR4C1
18. 24(4) cage at R2C2 contains 9 = {2589/2679/3579} (cannot be {1689} which clashes with R23C3), no 1
19. 1 in R123 only in NR1C1 + NR1C4 + NR2C4 -> NR2C4 must contain 1 in 27(6) cage at R2C4 (step 12a) = {134568}, no 2,7, 1 locked for NR2C4
[Step 12 would have been a bit simpler if I’d spotted this back then.]
20. 7 in NR2C4 only in R4C234 + R5C23, 7 in NR3C6 only in R45C6
20a. Grouped X-Wing for 7 in R4C234 + R5C23 and R45C6, no other 7 in R4C45, clean-up: no 3 in R6C1
20b. 7 in NR5C7 only in R6C78, locked for C7, clean-up: no 3 in R5C1
[Alternatively Grouped Swordfish for 7 in R4C234 + R5C23, R45C6 and R5C7 + R6C78, no other 7 in R456, then 7 in R6C78, locked for NR5C7 …]
20c. R56C1 = [64], placed for NR4C1, clean-up: no 2 in R12C1 (step 7f)
20d. Killer pair 1,3 in R12C1 and R23C3, locked for NR1C1
20e. 6 in R1 only in R1C789, locked for NR1C7
21. R4C34 = {28/37}/[64], 22(4) cage at R4C6 (step 14) = {3478/4567} -> R4C78 = {38}/[46]
21a. Combined cage R4C3478 = {28}[46]/{37}[46]/[64]{38}, 4,6 locked for R4
21b. 19(4) cage at R4C5 (step 16) = {1378} (only remaining cage) -> R5C3 = 7, R4C5 + R5C45 = {138}, locked for NR4C5
21c. R4C6 = 7 (hidden single in C6), clean-up: no 3 in R4C34
21d. Combined cage R4C3478 = {28}[46]//[64]{38}, 8 locked for R4
21e. 19(4) cage = {1378}, 8 locked for R5
21f. Naked triple {269} in R6C345, locked for R6
21g. R6C26 = {38} (hidden pair in R6)
21h. R5C7 = 2 (hidden single in NR5C7), clean-up: no 3 in R9C6 (step 4)
22. 8(3) cage at R8C1 = {125} (only remaining combination, cannot be {134} = {13}4 which clashes with R1C1), locked for NR8C1, clean-up: no 4 in R8C5
22a. 3 in C6 only in R68C6, locked for NR5C7 -> R89C5 = [63], 3 placed for NR7C1
22b. R456C5 = [182], R5C4 = 3, R4C9 = 5, R5C8 = 1, R6C9 = 1, R3C5 = 5, R35C6 = [45], R5C2 = 4, placed for NR2C4, clean-up: no 6 in R4C3
22c. Naked pair {28} in R4C34, locked for R4 and NR2C4 -> R4C2 = 3, R6C2 = 8, placed for NR4C1, R6C3 = 6 (cage sum)
22d. R1C1 = 3 (hidden single in C1), R2C1 = 5 (cage sum)
22e. R68C6 = [38], R9C6 = 6, R7C7 = 5 (step 4), placed for NR7C7
23. 36(7) cage at R7C1 = {1245789} (only remaining combination), no 3
24. R9C7 = 1 (hidden single in NR7C7)
and the rest is naked singles, without using the nonets.