Prelims
a) R1C45 = {49/58/67}, no 1,2,3
b) R12C9 = {59/68}
c) 12(2) cage at R2C4 = {39/48/57}, no 1,2,6
d) 8(2) cage at R2C5 = {17/26/35}, no 4,8,9
e) 12(2) cage at R5C2 = {39/48/57}, no 1,2,6
f) R5C89 = {18/27/36/45}, no 9
g) R67C5 = {29/38/47/56}, no 1
h) R89C3 = {59/68}
i) R9C45 = {19/28/37/46}, no 5
j) 11(3) cage at R2C3 = {128/137/146/236/245}, no 9
k) 9(3) cage at R2C6 = {126/135/234}, no 7,8,9
l) 26(4) cage at R8C7 = {2789/3689/4589/4679/5678}, no 1
m) 15(5) cage at R4C6 = {12345}
n) 32(5) cage at R2C2 = {26789/35789/45689}, no 1
1. Naked quint {12345} in 15(5) cage at R4C6, locked for N5, clean-up: no 6,7,8,9 in R7C5
2. 45 rule on C12 3 outies R126C3 = 8 = {125/134}, 1 locked for C3 and N1, clean-up: no 3,4,5 in R5C2
2a. 5 of {125} must be in R6C3 -> no 5 in R12C3
3. 45 rule on N69 2(1+1) outies R3C9 + R9C6 = 16 = [79/88/97]
4. 45 rule on N7 1 innie R7C3 = 1 outie R6C1 + 3, no 7,8,9 in R6C1, no 2,3 in R7C3
5. 45 rule on N9 1 outie R9C6 = 2 innies R7C78
5a. Max R7C78 = 9, no 9 in R7C78
6. 45 rule on N6789 3(1+2) outies R3C9 + R6C15 = 21
6a. Max R3C9 = 9 -> min R6C15 = 12, no 1,2 in R6C1, no 6 in R6C5 (because R6C15 cannot be [66]), clean-up: no 4,5 in R7C3 (step 4), no 5 in R7C5
6b. 6 in N5 only in R4C45 + R6C4, CPE no 6 in R4C3
7. R4C2 = 1 (hidden single in N4), R2C3 + R3C2 = 10 = [28/37/46]
7a. R1C3 = 1 (hidden single in C3)
8. 12(3) cage at R7C2 = {237/246/345}, no 8,9
9. 45 rule on C123 2 outies R26C4 = 1 innie R7C3 + 7
9a. Min R7C3 = 6 -> min R26C4 = 13, no 3 in R2C4, clean-up: no 9 in R3C3
10. 45 rule on N1 3 remaining innies R2C3 + R3C13 = 20 = {389/479/569/578}, no 2
11. 2 in R126C3 (step 2) = [125] or 18(3) cage at R4C3 -> 18(3) cage = {279/369/378/468} (cannot be {459/567}, locking-out cages), no 5
12. 15(4) cage at R1C1 = {1239/1248/1257/1347/1356}
12a. 45 rule on C12 2 remaining innies R35C2 = 15 = [69/78/87]
12b. Consider combinations for R35C2
R35C2 = [69] => 9 in 32(5) cage at R2C2 must be in R2C2 + R3C1
or R35C2 = {78} => R2C3 = {23} (step 7) => 15(4) cage cannot be {1239}
-> 15(4) cage = {1248/1257/1347/1356}, no 9
12c. 9 in N1 only in R2C2 + R3C1, locked for 32(5) cage at R2C2, no 9 in R45C1 + R6C2
12d. R2C3 + R3C13 (step 10) contains 9 = {389/479/569}
13. 2 in R126C3 = [125] or 18(3) cage at R4C3 -> 18(3) cage = {279/369/378/468} (step 11)
13a. 9 in N4 only in 18(3) cage or 12(2) cage at R5C2 = [93] -> 18(3) cage = {279/369} (cannot be {378/468}, locking-out cages using both 2 in C3 and 9 in N4), no 4,8
13b. 9 in N4 only in 18(3) cage or 12(2) cage at R5C2 = [93] -> 18(3) cage = {369} can only be {39}6 (locking-out cages), no 6 in R5C3
13c. Killer pair 2,3 in R26C3 and 18(3) cage, locked for C3, clean-up: no 9 in R2C4
13d. 8 in N4 only in R45C1 + R56C2, CPE no 8 in R2C2
13e. 8 in N5 only in R4C45 + R6C5, CPE no 8 in R3C5
14. 6 in C3 only in R789C3, locked for N7
14a. 12(3) cage at R7C2 (step 8) = {237/345}, 3 locked for C2 and N7
15. Hidden killer pair 6,8 in R37C3 and R89C3 for C3, R89C3 contains both or neither of 6,8 -> R37C3 must contain both or neither of 6,8, there's no 6 in R3C3 -> no 8 in R7C3, clean-up: no 5 in R6C1 (step 4)
16. 45 rule on C1 3 remaining outies R126C2 = 17 = {269/278/458} (cannot be {467} which clashes with R35C2)
16a. 9 of {269} must be in R2C2 -> no 6 in R2C2
16b. 7 of {278} must be in R2C2 -> no 7 in R16C2
17. R2C3 + R3C2 (step 7) = [28/37/46]
17a. R2C3 + R3C13 (step 12d) = {479/569} (cannot be {389} = [938] because R2C3 + R3C2 = [46] clashes with R2C4 + R3C3 = [48]), no 3,8, clean-up: no 4 in R2C4
17b. 15(4) cage at R1C1 (step 12b) = {1248/1347/1356} (cannot be {1257} which clashes with R2C3 + R3C13)
17c. R2C3 + R3C2 = [28/37] (cannot be [46] which clashes with 15(4) cage), clean-up: no 9 in R5C2 (step 12a), no 3 in R6C3
[Cracked. The rest is fairly straightforward.]
18. Naked pair {78} in R35C2, locked for C2
18a. 12(3) cage at R7C2 = {345} (only remaining combination), locked for C2 and N7 -> R2C2 = 9, clean-up: no 5 in R1C9, no 9 in R89C5
18b. Naked pair {68} in R89C3, locked forN7, clean-up: no 3 in R6C1 (step 4)
19. 9 in N4 only in R45C3, locked for C3 -> R7C3 = 7, R6C1 = 4 (step 4), R6C3 = 5, R5C2 = 7, R3C2 = 8, R2C3 = 2 (step 7), R1C2 = 6, R6C2 = 2, R3C3 = 4, R2C4 = 8
19a. R1C23 = [61] = 7 -> R12C1 = 8 = {35}, locked for C1 -> R3C1 = 7, R3C9 = 9, R1C9 = 8, R2C9 = 6, R9C6 = 7 (step 3)
19b. R1C45 = {49} (only remaining combination), locked for R1 and N2
19c. R2C5 = 7 (hidden single in N2), R3C4 = 1
[Routine clean-ups omitted from here]
20. 45 rule on R1 2 remaining outies R2C18 = 9 = [54], R1C1 = 3
20a. R2C67 = {13} = 4 -> R3C8 = 5
20b. Naked pair {27} in R1C78, locked for R1 and N3 -> R1C6 = 5, R3C7 = 3, R2C67 = [31]
21. R6C6 = 1, R45C6 = {24}, locked for C6 and N5, R3C56 = [26]
22. R3C567 = [263] = 11 -> R4C45 = 16 = [79], R1C45 = [94], R45C3 = [39], R6C45 = [68], R7C5 = 3
23. 13(3) cage at R6C8 = {139} (only remaining combination) -> R6C89 = [93], R7C8 = 1
24. R3C9 = 9 -> R4C89 = 8 = [62], R5C8 = 8, R5C9 = 1
25. R789C9 = {457} = 16 -> R8C8 = 3, R9C8 = 2, R9C4 = 4, R9C5 = 6
and the rest is naked singles.
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