Thanks wellbeback for spotting my typos
Prelims
a) R1C67 = {13}
b) R56C5 = {16/25/34}, no 7,8,9
c) R56C7 = {17/26/35}, no 4,8,9
d) R89C1 = {19/28/37/46}, no 5
e) R9C23 = {19/28/37/46}, no 5
f) R9C45 = {15/24}
g) 21(3) cage at R1C1 = {489/579/678}, no 1,2,3
h) 11(3) cage at R1C8 = {128/137/146/236/245}, no 9
i) 21(3) cage at R2C6 = {489/579/678}, no 1,2,3
j) 20(3) cage at R7C8 = {389/479/569/578}, no 1,2
k) 26(4) cage at R2C9 = {2789/3689/4589/4679/5678}, no 1
1. Naked pair {13} in R1C67, locked for R1
1a. 13(3) cage at R1C4 = {148/157/238/247/256/346} (cannot be {139} which clashes with R1C6), no 9
1b. Min R1C45 = 6 -> max R2C5 = 7
1c. 11(3) cage at R1C8 = {128/146/236/245} (cannot be {137} which clashes with R1C7), no 7
1d. Min R1C89 = 6 -> max R2C8 = 5
1e. 9 in R1 only in R1C123, locked for N1
2. 45 rule on N7 2 innies R7C13 = 10 = {19/28/37/46}, no 5
3. 45 rule on C123 2 innies R67C3 = 10 = {19/28/37/46}, no 5
3a. 45 rule on C123 2 outies R67C4 = 12 = {39/48/57}, no 1,2,6
4. 45 rule on C89 2 outies R47C7 = 13 = {49/58/67}, no 1,2,3
5. 45 rule on C6789 2 outies R4C5 + R5C4 = 13 = {49/58/57}, no 1,2,3
5a. 45 rule on C6789 5 innies R34567C6 = 20 = {12458/12467/23456} (cannot be {12359/12368/13457} which clash with R1C6), no 9, 2,4 locked for C6, 4 also locked for 33(7) cage at R3C6, no 4 in R4C5 + R5C4, clean-up: no 9 in R4C5 + R5C4
[With hindsight 33(7) cage at R3C6 contains 1, locked for C6 -> R1C67 = [31] would have been simpler.]
5b. R34567C6 = {12458/12467} (cannot be {23456} which clashes with R4C5 + R5C4), 1 locked for C6 -> R1C67 = [31], clean-up: no 7 in R56C7
5c. 11(3) cage at R1C8 (step 1c) = {236/245}, no 8, 2 locked for N3
5d. 45 rule on N3 2(1+1) remaining outies R2C6 + R4C9 = 14 = {59/68} (cannot be [77] because R2C6 + R4C9 ‘see’ all 7s in N3)
6. 45 rule on R1 3 outies R2C158 = 1 innie R1C3 + 4
6a. Min R2C158 = 7 -> no 2 in R1C3
7. 9 in N5 only in R46C4, locked for C4, clean-up: no 3 in R6C4 (step 3a)
7a. 9 in R4C4 or R67C4 = [93] -> no 3 in R4C4 (locking-out cages)
7b. 3 in N5 only in R56C5 = {34}, locked for C5 and N5, clean-up: no 8 in R7C4, no 2 in R9C4
7c. 9 in N2 only in R2C6 + R3C5, CPE no 9 in R3C7
7d. 13(3) cage at R1C4 (step 1a) = {148/157/247/256}
7e. 4 of {148} must be in R1C4 -> no 8 in R1C4
8. 1 in N9 only in R789C9, locked for 32(7) cage at R5C9, no 1 in R5C9 + R6C89
8a. 1 in N6 only in 14(3) cage at R4C7 = {149/158/167}, no 2,3
9. Hidden killer pair 2,3 in R56C7 and 24(4) cage at R8C6 for C7, R56C7 contains one of 2,3 -> 24(4) cage must contain one of 2,3 = {2589/2679/3489/3579/3678} (cannot be {4569/4578} which don’t contain 2 or 3)
9a. Hidden killer pair 2,3 in R56C7 and 32(7) cage at R5C9 for N6, R56C7 contains one of 2,3 -> 32(7) cage contains one of 2,3 in R5C9 + R6C78, 32(7) cage contains both of 2,3 -> it must contain one of 2,3 in R789C9
9b. Killer pair 2,3 in 24(4) cage and R789C9, locked for N9
10. 14(3) cage at R4C7 (step 8a) = {149/158/167}, 32(7) cage at R5C9 = {1234589/1234679/1235678}, R47C7 (step 4) = {49/58/67}
10a. Consider placements for 4 in N6
4 in 14(3) cage = {149}, locked for N6
or 4 in R5C9 + R6C89 => 32(7) cage = {1234589/1234679}, 14(3) cage = {158/167} => R47C7 = {58/67} => 9 in 32(7) cage must be in R5C9 + R6C89 + R789C9, CPE no 9 in R4C9
-> no 9 in R4C9, clean-up: no 5 in R2C6 (step 5d)
11. 45 rule on C789 4 remaining innies R2389C7 = 23 = {2489/2678/3479/3578} (cannot be {2579/3569/4568} which clash with R56C7)
10a. Min R2C6 = 6 -> max R23C7 = 15 -> 4 of {2489/3479} must be in R23C7 -> no 4 in R89C7
12. 45 rule on N689 3(1+2) innies R4C9 + R7C46 = 15, R4C9 = {568} -> R7C46 = 7,9,10 = [34/36/72/37/46] (cannot be {16/18/28} because 1,2,6,8 only in R7C6, cannot be {45}/[52] which clash with R9C45) -> R7C4 = {347}, R7C6 = {2467}, clean-up: no 7 in R6C4 (step 3a)
12a. 16(3) cage at R7C5 = {169/178/268/358} (cannot be {259/457} which clash with R9C45, cannot be {349} because 3,4 only in R8C4, cannot be {367} which clashes with R7C46), no 4
12b. Consider placement for 3 in N8
R7C4 = 3 => R7C46 = [34/36/37]
or R8C4 = 3 => 16(3) cage = {358} => R9C45 = [42] => R7C35 = [72]
-> R7C46 = [34/36/72/37], no 4 in R7C4, clean-up: no 8 in R6C4 (step 3a)
12c. Combined cage R7C46 + R9C45 = [34]{15}/[36]{15/24}/[37]{15/24} (cannot be [72]{15} which clashes with 16(3) cage -> R7C4 = 3, R7C6 = {467}, R6C4 = 9 (step 3a), clean-up: no 1,7 in R67C3 (step 3), no 1,7,9 in R7C1 (step 2)
12d. 16(3) cage = {169/178/268}, no 5
13. 45 rule on N8 3 remaining innies R789C6 = 20 = {479/569/578}
13a. 6 of {569} must be in R7C6 -> no 6 in R89C6
13b. R89C6 = {58/59/79} -> 24(4) cage at R8C6 (step 9) = {2589/2679/3579} (cannot be {3678} because R89C6 cannot contain both of 7,8)
13c. 7,9 of {2679/3579} must be in R89C6 -> no 7 in R89C7
13d. R89C7 = {26/28/29/35} -> R2389C7 (step 11) = {2489/2678/3578} (cannot be {3479} because R89C7 cannot contain two of 3,9), 8 locked for C7, clean-up: no 5 in R47C7 (step 4)
13e. 3,5 of {3578} must be in R89C7 -> no 5 in R23C7
13f. 14(3) cage at R4C7 (step 8a) = {149/167} (cannot be {158} because no 1,5,8 in R4C7), no 5,8
13g. 21(3) cage at R2C6 = {489/678}, CPE no 8 in R2C9
13h. 8 in N6 only in R45C9 + R6C89, CPE no 8 in R789C9
14. 26(4) cage at R2C9 = {3689/4589/5678} (cannot be {4679} which clashes with 11(3) cage at R1C8 + 21(3) cage at R2C6)
14a. 5 of {4589/5678} must be in R2C9 + R3C89 (R2C9 + R3C89 cannot be {489/678} which clash with 11(3) cage at R1C8 + 21(3) cage at R2C6) -> no 5 in R4C9, clean-up: no 9 in R2C6 (step 5d)
14b. R2C6 + R4C9 (step 5d) = {68}, CPE no 6,8 in R2C9 + R4C6
14c. 9 in C6 only in R89C6, locked for N8 and 24(4) cage at R8C6, no 9 in R89C7
14d. R789C6 (step 13) contains 9 = {479/569}, no 8
14e. 4,6 only in R7C6 -> R7C6 = {46}
15. 14(3) cage at R4C7 (step 13e) = {149/167}, R47C7 (step 4) = {49/67}, 32(7) cage at R5C9 = {1234589/1234679/1235678}
15a. Consider combinations for R47C7
R47C7 = {49} => 32(7) cage = {1234589/1234679}
or R47C7 = {67} => 14(3) cage = {167}, locked for N6 => R4C9 = 8 => 32(7) cage = {1234679}
-> 32(7) cage = {1234589/1234679}
16. 45 rule on N3689 1 remaining outie R2C6 = 1 remaining innie R7C6 + 2 -> R27C6 = [64/86], 6 locked for C6
17. R7C13 (step 2) = {28} (only remaining combination, cannot be {46} which clashes with R7C6), locked for R7 and N7, clean-up: no 4,6 in R6C3 (step 3)
17a. Naked pair {28} in R67C3, locked for C3
17b. 32(7) cage at R1C3 = {1234589/1234679/1235678}, 2 locked for N1
18. 16(3) cage at R7C5 (step 12d) = {178/268}
18a. 6 of {268} must be in R7C5 -> no 6 in R8C45
18b. 6 in N8 only in R7C56, locked for R7, clean-up: no 7 in R4C7 (step 4)
18c. 14(3) cage at R4C7 (step 13e) = {149/167}
18d. 6 of {167} must be in R4C7 -> no 6 in R45C8
19. 20(3) cage at R7C8 = {569/578} (cannot be {479} which clashes with R7C7), 5 locked for C8 and N9
19a. 5 in C7 only in R56C7 = {35}, locked for C7 and N6
19b. 2 in C7 only in R89C7, locked for N9
20. 32(7) cage at R5C9 (step 15a) = {1234679} (only remaining combination), CPE no 6 in R4C9
20a. R4C9 = 8 -> R2C6 = 6 (step 5d), R7C6 = 4, clean-up: no 2 in R9C5
20b. R2C6 = 6 -> R23C7 = 15 = {78}, locked for C7 and N3 -> R7C7 = 9, naked pair {26} in R89C7, locked for C7 and N9 -> R4C7 = 4
20c. Naked pair {15} in R9C45, locked for R9 and N8, clean-up: no 9 in R8C1, no 9 in R9C23
20d. Naked pair {79} in R89C6, locked for C6 and N8 -> R7C5 = 6
20e. Naked pair {28} in R8C45, locked for R8 -> R89C7 = [62], clean-up: no 4 in R9C1
20f. Naked pair {57} in R78C8, locked for C8 and N9 -> R7C9 = 1, R9C8 = 8
20g. Naked pair {34} in R89C9, locked for C9
21. 33(7) cage at R3C6 = {1245678} -> R5C4 = 6, R4C5 = 7
21a. 6 in N6 only in R6C89, locked for R6
21b. 8 in N5 only in R56C6, locked for C6
21c. R3C5 = 9 (hidden single in C5) -> R2C9 = 9 (hidden single in C9)
21d. R3C5 = 9 -> R234C4 = 14 = {257} (only possible combination, cannot be {158} which clashes with R9C4, cannot be {248} which clashes with R8C4), locked for C4 -> R1C4 = 4, R12C5 = 9 = [81]
22. 15(4) cage at R5C1 = {1239/1248/1257} (cannot be {1347} because R7C1 only contains 2,8), 1 locked for N4
22a. 45 rule on N14 4 innies R5C1 + R6C123 = 15 and shares three cells with 15(4) cage so must have the same combination = {1239/1248/1257}, 2 locked for N4
22b. 32(7) cage at R1C3 contains 1, locked for N1
23. 45 rule on N1 2 outies R45C3 = 1 innie R3C2 + 8
23a. 32(7) cage at R1C3 contains 3, R45C3 cannot be {38} (because no 8 in R45C3) -> no 3 in R3C2
23b. 3 in N1 only in R2C23 + R4C13, locked for 32(7) cage, no 3 in R45C3
24. 21(3) cage at R1C1 = {579/678} (cannot be {489} because 4,8 only in R2C1), no 4, 7 locked for N1
24a. 7 in R1 only in R1C12, locked for N1
25. 3 in R4 only in R4C12, locked for N4
25a. 22(4) cage at R3C2 contains 3 = {3469/3568} (cannot be {3478} because no 4,7,8 in R4C12), no 7
25b. R5C1 + R6C123 (step 22a) = {1248/1257}, no 9
26. Consider combinations for 21(3) cage at R1C1 (step 24) = {579/678}
21(3) cage = {579}, locked for N1 => R1C3 = 6
or 21(3) cage = {678} = {67}8
-> 6 in R1C123, locked for R1 and N1
26a. R1C89 = [25] -> R2C8 = 4 (cage sum), R3C89 = [36]
27. R45C3 = R3C2 + 8 (step 23), R3C2 = {458} -> R45C3 = 12,13,16 = [57/67/94/97] -> R5C4 = {47}
27a. R5C1 + R6C123 (step 25b) = {1248/1257}
27b. Killer pair 4,7 in R5C1 + R6C123 and R5C3, locked for N4
28. 22(4) cage at R3C2 (step 25a) = {3469/3568}
28a. 3,6 only in R4C12 = R4C12 = {36}, locked for R4
[Just spotted a neat step for the final breakthrough …]
29. 32(7) cage at R1C3 = {1234589/1234679/1235678}, R45C3 (step 27) = [57/94/97]
29a. Consider placements for R1C3 = {69}
R1C3 = 6 => 32(7) cage must contain 7 => R5C3 = 7
or R1C3 = 9 => R4C3 = 5 => R5C3 = 7
-> R5C3 = 7, clean-up: no 3 in R9C2
29b. R5C3 = 7 -> 32(7) cage = {1234679/1235678} -> R1C3 = 6, clean-up: no 4 in R9C2
29c. R5C1 + R6C123 (step 25b) = {1248} (only remaining combination), locked for N5
30. R1C12 = {79} -> R2C1 = 5 (cage sum), R2C3 = 3, clean-up: no 7 in R9C2
30a. R9C23 = [64] -> R9C9 = 3, clean-up: no 7 in R8C1
30b. R4C12 = [63], R8C1 = 3 (hidden single in C1) -> R9C1 = 7, R7C2 = 5, R5C2 = 9
30c. R45C3 = [57] = 12 -> R3C2 = 4 (step 23)
and the rest is naked singles.
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