Prelims
a) R12C1 = {29/38/47/56}, no 1
b) R23C6 = {19/28/37/46}, no 5
c) R3C45 = {39/48/57}, no 1,2,6
d) R45C2 = {19/28/37/46}, no 5
e) R67C8 = {17/26/35}, no 4,8,9
f) R78C4 = {39/48/57}, no 1,2,6
g) R7C56 = {79}
h) R89C9 = {19/28/37/46}, no 5
i) 22(3) cage at R1C2 = {589/679}
j) 11(3) cage at R1C8 = {128/137/146/236/245}, no 9
k) 22(3) cage at R2C9 = {589/679}
l) 23(3) cage at R8C8 = {689}
m) 14(4) cage at R5C8 = {1238/1247/1256/1346/2345}, no 9
n) 28(7) cage at R1C4 = {1234567}, no 8,9
Steps resulting from Prelims
1a. 22(3) cage at R1C2 = {589/679}, 9 locked for N1, clean-up: no 2 in R12C1
1b. 23(3) cage at R8C8 = {689}, locked for N9, clean-up: no 2 in R6C8, no 1,2,4 in R89C9
1c. Naked pair {79} in R7C56, locked for R7 and N8, clean-up: no 1 in R6C8, no 3,5 in R78C4
1d. Naked pair {48} in R78C4, locked for C4 and N8, clean-up: no 4,8 in R3C5
1e. Naked pair {37} in R89C9, locked for C9 and N9, clean-up: no 6 in 22(3) cage at R2C9, no 5 in R6C8
1f. Naked triple {589} in 22(3) cage at R2C9, locked for C9
1g. 3,6 in R7 only in R7C123, locked for N7
2. R12C1 = {38/47} (cannot be {56} which clashes with 22(3) cage at R1C2), no 5,6
2a. Killer pair 7,8 in R12C1 and 22(3) cage, locked for N1
3. 45 rule on R1234 2 outies R5C12 = 9 = {18/27/36}/[54], no 9, no 4 in R5C1, clean-up: no 1 in R4C2
4. 45 rule on R6789 2 innies R67C9 = 3 = {12}, locked for C9 and 14(4) cage at R5C8
4a. R67C9 = {12} = 3 -> R5C89 = 11 = [56/74]
5. 11(3) cage at R1C8 = {146/236/245} (cannot be {128/137} because R1C9 only contains 4,6), no 7,8
5a. 45 rule on C9 3 outies R125C8 = 12 = {147} (only remaining combination, cannot be {156/237} which clash with R67C8, cannot be {246} because R5C8 only contains 5,7, cannot be {345} because 11(3) cage cannot be {34}4) -> R5C8 = 7, R12C8 = {14}, locked for C8 and N3 -> R1C9 = 6, R5C9 = 4, clean-up: no 2,5 in R5C1, no 2 in R5C2 (both step 3), no 3,6,8 in R4C2
6. 4 in N9 only in R78C7, locked for 33(7) cage at R6C5, no 4 in R6C56
6a. 33(7) cage containing 4 must also contain 8, locked for R6
7. 4 in R6 only in R6C123, locked for N4, clean-up: no 6 in R5C2, no 3 in R5C1 (step 3)
8. 45 rule on C12 1 outie R1C3 = 2 innies R67C2 + 1
8a. Max R67C2 = 8, no 8,9 in R67C2
9. Hidden killer pair 4,8 in 19(5) cage at R1C5 and R23C6 for N2, neither can contain both of 4,8 -> R23C9 must contain one of 4,8 = {28/46}, 19(5) cage must contain one of 4,8 in N2 -> no 8 in R12C7
9a. 19(5) cage must contain 1, locked for N2
9b. 1 in R3 only in R3C123, locked for N1
[I spotted a caged X-Wing for 1 first, but these steps give more for R23C6.]
10. 45 rule on C1 1 outie R3C2 = 1 innie R9C1, no 3,6 in R3C2, no 7,8,9 in R9C1
11. 14(3) cage at R8C2 = {149/158/248/257}
11a. 22(3) cage at R1C2 = {589/679}
11b. Killer triple 7,8,9 in 22(3) cage, R45C2 and 14(3) cage, locked for C2
11c. Hidden killer triple 7,8,9 in 22(3) cage, R45C2 and 14(3) cage, R45C2 and 14(3) cage each contain one of 7,8,9 -> 22(3) cage must contain one of 7,8,9 in C2 -> R1C3 = {789}
12. 17(3) cage at R6C1 = {179/269/359/368/467} (cannot be {278/458} which clash with R12C1)
12a. 1 of {179} must be in R7C1 -> no 1 in R68C1
13. 17(4) cage at R3C1 = {1259/1268/1358/1457/2456} (cannot be {1349/1367/2348/2357} which clash with R12C1)
13a. Hidden killer triple 1,4,6 in R3C12, R3C3 and R3C6 for R3 -> R3C12 must contain at least one of 1,4,6
13b. 17(4) cage = {1268/1358/1457/2456} (cannot be {1259} which must have 1 in R5C1), no 9
14. 45 rule on R89 5(4+1) innies R8C1467 + R9C6 = 20
14a. R8C467 + R9C6 cannot total 11 (because R8C4 and R8C7 cannot both contain 4) -> no 9 in R8C1
15. R6C9 = 9 (hidden single in C1), R78C1 = 8 = [17/35/62], clean-up: no 1 in R5C2, no 8 in R5C1 (step 3)
15a. 17(4) cage at R3C1 (step 13b) = {1268/1457/2456} (cannot be {1358} = [3581] because no 1,4,6 in R3C12, step 13a), no 3
16. 33(7) cage at R6C5 = {1245678} (only remaining combination), no 3, 7 locked for R6 and N5
16a. 7 in C4 only in R123C4, locked for N2, clean-up: no 5 in R3C4
17. 4 in N4 only in R6C23, locked for 21(5) cage at R6C2, no 4 in R6C4 + R7C23
17a. 21(5) cage = {12468/13458} -> R7C3 = 8, R78C4 = [48]
17b. Naked triple {125} in R7C789, locked for R7 and N9 -> R8C4 = 4, clean-up: no 7 in R8C1 (step 15)
18. Hidden killer pair 7,8 in R12C1 and R4C1, R12C1 contains one of 7,8 -> R4C1 = {78}
18a. Killer pair 7,8 in R4C1 and R45C2, locked for N4
19. 17(4) cage at R3C1 (step 15a) = {1268/1457} (cannot be {2456} because R4C1 only contains 7,8)
19a. Hidden killer triple 1,4,6 in R3C12, R3C3 and R3C6 for R3, R3C12 contains one of 1,4 (because R5C1 = {16} contains the other one of 1,4,6 for 17(4) cage) -> R3C3 = {146}, R3C6 = {46}, clean-up: no 2,8 in R2C6
20. Naked pair {46} in R23C6, locked for C6 and N2
21. 33(7) cage at R6C5 (step 16) = {1245678} -> R6C567 = {678}, locked for R6 -> R6C8 = 3, R7C8 = 5
21a. 33(7) cage = {1245678}, 5 locked for C6 and N8
22. 8 in N2 only in 19(5) cage at R1C5 = {12358} (only remaining combination), no 7,9
23. R3C7 = 7 (hidden single in C7), clean-up: no 5 in R3C5
23a. Naked pair {39} in R3C45, locked for R3 and N2
24. 2 in C8 only in R34C8, locked for 27(5) cage at R3C7, no 2 in R4C67
25. 9 in R1 only in R1C23, locked for 22(3) cage at R1C2, no 9 in R2C2
25a. R2C9 = 9 (hidden single in R2)
26. 3,6 in N8 only in 23(5) cage at R8C3 = {13469/23567} -> R89C3 = [57/75/94]
26a. 45 rule on N7 (using R7C12 = {36} = 9) 3 remaining innies R8C1 + R89C3 = 14 -> R8C1 = 2, R89C3 = {57}, locked for C3 and N7, R1C3 = 9, R7C1 = 6 (cage sum), R7C2 = 3, R5C1 = 1, R9C1 = 4, R5C2 = 8, R4C12 = [72], R3C12 = [54], R12C2 = [76], R34C9 = [85], R3C3 = 1
27. Naked pair {36} in R4C34, locked for R4 and 28(7) cage at R1C4 -> R2C3 = 2, R4C5 = 4
28. R6C23 = [54], R6C4 = 1 (cage sum), R67C9 = [21], R7C7 = 2, R89C6 = {15}, locked for C6 and N8
29. R5C6 = 3 (hidden single in C6)
and the rest is naked singles.
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