Prelims
a) R12C5 = {18/27/36/45}, no 9
b) R9C12 = {39/48/57}, no 1,2,6
c) R9C89 = {39/48/57}, no 1,2,6
d) 11(3) cage at R1C6 = {128/137/146/236/245}, no 9
e) 24(3) cage at R2C8 = {789}
f) 10(3) cage at R4C1 = {127/136/145/235}, no 8,9
g) 19(3) cage at R6C5 = {289/379/469/478/568}, no 1
h) 8(3) cage at R6C7 = {125/134}
i) 8(3) cage at R6C9 = {125/134}
j) 27(4) cage at R7C7 = {3789/4689/5679}, no 1,2
Steps resulting from Prelims
1a. Naked triple {789} in 24(3) cage at R2C8, locked for C8, clean-up: no 3,4,5 in R9C9
1b. 8(3) cage at R6C9 = {125/134}, 1 locked for C9
1c. 34(7) cage at R2C3 must contain 1, CPE no 1 in R2C5, clean-up: no 8 in R1C5
2. 8(3) cage at R6C7 and 8(3) cage at R6C9 both contain 1, locked for R6, N6 and N9
3. 45 rule on C1 2 outies R59C2 = 7 = [25/34/43], clean-up: no 3,4,5 in R9C1
4. 45 rule on C9 2 outies R59C8 = 9 = [45/54/63]
5. 45 rule on C9 3 innies R459C9 = 20 = {389/479/578} (cannot be {569} because 17(3) cage at R4C9 cannot be {56}6), no 2,6
6. 6 in C9 only in 17(3) cage at R1C9, locked for N3
6a. 17(3) cage = {269/368/467}, no 5
6b. Killer triple 7,8,9 in 17(3) cage and R23C8, locked for N3
7. 45 rule on C89 4 innies R1678C8 = 12 = {1236/1245}
7a. 6 of {1236} must be in R8C8 -> no 3 in R8C8
7b. 27(4) cage at R7C7 = {4689/5679} (cannot be {3789} because R8C8 only contains 4,5,6), no 3
8. 45 rule on C1 3 innies R459C1 = 15 = {159/168/258/357} (cannot be {249/267/348} because 10(3) cage at R4C1 cannot be {24}4/{26}2/{34}3, cannot be {456} because R9C1 only contains 7,8,9), no 4 in R45C1
8a. R9C1 = {789} -> no 7 in R45C1
9. 45 rule on R1 2 innies R1C19 = 1 outie R2C5 + 9
9a. Min R2C5 = 2 -> min R1C19 = 11, no 1 in R1C1
10. 45 rule on R123 2 outies R4C28 = 2 innies R3C37 + 10
10a. Max R4C28 = 17 -> max R3C37 = 7, no 7,8,9 in R3C3
10b. Min R3C37 = 3 -> min R4C28 = 13, no 1,2,3 in R4C2
11. 45 rule on R6789 2 outies R5C46 = 9 = {18/27/36/45}, no 9
12. 11(3) cage at R1C6 = {128/137/146/236/245}
12a. Max R1C78 = 9 -> min R1C6 = 2
12b. 6,7 of {137/236} must be in R1C6 -> no 3 in R1C6
[It took me some time to find this step; the first key breakthrough.]
13. R59C8 (step 4) = [45/54/63]
13a. R59C8 = {45} => 4,5 of 8(3) cage at R6C7 must be in R6C7 => R5C8 + R6C7 = {45}
or R5C8 = 6
13b. 17(3) cage at R4C9 = {359/368/467} (cannot be {458} which clashes with R5C8 + R6C7 or doesn’t contain 6, killer locking-out cages?)
13c. 6 of {467} must be in R5C8 -> no 4 in R5C8, clean-up: no 5 in R9C8 (step 4), no 7 in R9C9
14. 7 in N9 only in R789C7, locked for C7
15. Hidden killer triple 7,8,9 in 14(3) cage at R3C7, R4C8 and 17(3) cage at R4C9 for N6, R4C8 = {789}, 17(3) cage contains one of 7,8,9 -> 14(3) cage must contain one of 8,9 = {149/158/239/248} (cannot be {356} which doesn’t contain any of 7,8,9), no 6
16. R5C8 = 6 (hidden single in N6), R9C8 = 3 (step 4), R9C9 = 9, clean-up: no 2 in 17(3) cage at R1C9 (step 6a), no 4 in R5C2 (step 3), no 3 in R5C46 (step 11)
17. R789C7 = {678} (hidden triple in N9), locked for C7
17a. 14(3) cage at R3C7 (step 15) = {149/239}, no 5, 9 locked for N6
17b. 1 of {149} must be in R3C7 -> no 4 in R3C7
18. R5C8 = 6 -> R45C9 = 11 = {38/47}, no 5
18a. 5 in N6 only in R6C789, locked for R6
19. 8(3) cage at R6C9 = {125} (hidden triple in C9)
19a. 4 in N9 only in R78C8, locked for C8
20. 27(4) cage at R7C7 (step 7b) = {4689/5679} -> R8C6 = 9
20a. 27(4) cage = {4689/5679}, 6 locked for N9
20b. Naked pair {78} in R9C17, locked for R9
21. 10(3) cage at R4C1 = {136/235}, 3 locked for N4
21a. 6 of {136} must be in R4C1 -> no 1 in R4C1
22. 17(3) cage at R6C2 = {179/269/278/368/458/467} (cannot be {359} because 3,5 only in R7C2)
22a. 1 of {179} must be in R7C2, 9 of {269} must be in R6C23 (R6C23 cannot be {26} which clashes with 10(3) cage at R4C1) -> no 9 in R7C2
23. 14(3) cage at R3C7 (step 17a) = {149/239}
23a. Consider combinations for R45C9 (step 18) = {38/47}
R45C9 = {38}, locked for N6
or R45C9 = {47}, locked for N6 => 14(3) cage = {239}, 3 locked for C7
-> no 3 in R6C7
[Cracked. The rest is fairly straightforward.]
24. 8(3) cage at R6C7 = {125} (only remaining combination)
24a. Naked triple {125} in R6C789, locked for R6 and N6
24b. 14(3) cage at R3C7 (step 17a) = {149/239}
24c. 1,2 only in R3C7 -> R3C7 = {12}
25. R8C8 = 4 (hidden single in N9)
25a. R8C68 = [94] = 13 -> R78C7 = 14 = {68}, locked for N9 -> R9C7 = 7, R9C1 = 8, R9C2 = 4, R5C2 = 3 (step 3), clean-up: no 8 in R4C9 (step 18)
26. R5C46 = 9 (step 11)
26a. 3 in R6 only in R6C456, locked for N5
26b. 45 rule on N5 3 remaining innies R6C456 = 19 = {379} (only remaining combination), locked for R6 and N5, clean-up: no 2 in R5C45 (step 11)
[I’d seen this 45 much earlier but it’s only useful now.]
27. 17(3) cage at R6C2 (step 22) = {458/467} (cannot be {278} because 2,7 only in R7C2) -> R6C3 = 4, R6C1 = 6, R6C2 = 8, R7C2 = 5
27a. R6C1 = 6 -> R78C1 = 8 = {17} (only remaining combination), locked for C1 and N7
27b. Naked pair {25} in R45C1, locked for C1 and N4
27c. R78C7 = [93] (hidden pair in N7)
27d. R78C7 = [93] = 12 -> R8C24 = 8 = {26}, locked for R8 -> R78C7 = [68]
28. Naked pair {12} in R7C89, locked for R7 and N9 -> R8C9 = 5, R8C5 = 7, clean-up: no 2 in R12C5
28a. R8C5 = 7 -> R67C5 = {39} (only remaining combination, cannot be {48} because 4,8 only in R7C6) -> R6C5 = 9, R7C5 = 3, clean-up: no 6 in R12C5
29. Naked pair {17} in R45C3, locked for C3 and N4, R3C3 = 5 (cage sum)
29a. R4C2 = 9, R23C2 = 9 = {27}, locked for C2 and N1 -> R8C24 = [62], R1C2 = 1, clean-up: no 8 in R2C5
30. 11(3) cage at R1C6 = {236} (only remaining combination, cannot be {245} which clashes with R1C5) -> R1C6 = 6, R1C78 = [32], R1C3 = 8, R1C4 = 7 (cage sum), R6C46 = [37]
30a. R1C9 = 4 -> R23C9 = 13 = {67}, locked for C9 and N3 -> R45C9 = [38], R4C8 = 7, clean-up: no 1 in R5C46 (step 11)
31. R6C4 = 3 -> R57C4 = 12 = [48]
and the rest is naked singles.
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