Prelims
a) R12C5 = {39/48/57}, no 1,2,6
b) R1C67 = {49/58/67}, no 1,2,3
c) R34C2 = {19/28/37/46}, no 5
d) R3C34 = {13}
e) R4C34 = {15/24}
f) R4C56 = {39/48/57}, no 1,2,6
g) R5C34 = {19/28/37/46}, no 5
h) R6C12 = {29/38/47/56}, no 1
i) R6C67 = {29/38/47/56}, no 1
j) R7C12 = {17/26/35}, no 4,8,9
k) R89C1 = {16/25/34}, no 7,8,9
l) R89C2 = {89}
m) R8C78 = {17/26/35}, no 4,8,9
n) 19(3) cage at R1C3 = {289/379/469/478/568}, no 1
Steps resulting from Prelims and Initial Placements
1a. 45 rule on N9 1 innie R9C7 = 9 -> R89C2 = [98], clean-up: no 4 in R1C6, no 1,2 in R34C2, no 2,3 in R6C1, no 2 in R6C6
1b. 45 rule on C1234 1 outie R6C5 = 1, clean-up: no 5 in R4C3, no 9 in R5C3
1c. Naked pair {13} in R3C34, locked for R3, clean-up: no 7 in R4C2
2. 45 rule on R6 2 outies R5C29 = 12 = [39/48]/{57}, no 1,2,6, no 3,4 in R5C9
2a. 45 rule on R56 2 innies R5C18 = 10 = {19/28/37/46}, no 5
2b. 45 rule on R89 2 innies R8C46 = 10 = {28/37/46}, no 1,5
3a. 45 rule on N7 1 outie R9C4 = 1 innie R7C3 + 1, no 7 in R7C3, no 1 in R9C4
3b. Max R7C3 = 6 -> min R78C4 = 11, no 1,2 in R7C4
3c. R3C4 = 1 (hidden single in C4) -> R3C3 = 3, clean-up: no 7 in R5C4, no 4 in R9C4
3d. 45 rule on C12 1 outie R1C3 = 1 innie R5C2 + 2, R5C2 = {3457} -> R1C3 = {5679}
3e. 45 rule on N2 1 outie R2C3 = 1 remaining innie R1C6 + 2 -> R1C6 = {567}, R2C3 = {789}, clean-up: no 4,5 in R1C7
3f. 45 rule on N36 3 innies R156C7 = 14, min R1C7 = 6 -> max R67C7 = 8, no 7,8 in R5C7, no 8 in R6C7, clean-up: no 3 in R6C6
[Ed pointed out that I’d overlooked no 6 in R5C7; I hadn’t looked at the min-max options carefully enough.]
3g. 45 rule on N1 3 remaining innies R2C3 + R3C12 = 18 = {279/459/468/567}
3h. 2 of {279} must be in R3C1, 7,8,9 of {459/468/567} must be in R2C3 -> no 7,8,9 in R3C1
4. R1C3 = R5C2 + 2 (step 3d), R2C3 = R1C6 + 2 (step 3e), R2C3 + R3C12 (step 3g) = {279/459/468/567}
4a. Consider placements for R1C3 = {5679}
R1C3 = 5 => R5C2 = 3 => R46C2 = {46} => R2C3 + R3C12 = {468} (cannot be {279} because 7,9 only in R2C3), no 9 in R2C3
or R1C3 = {67} => R1C67 = [58] (cannot be {67} which clashes with R1C3) => R2C3 = 7
or R1C3 = 9
-> no 9 in R2C3, clean-up: no 7 in R1C6, no 6 in R1C7
[Re-worked after I realised that I’d found an improvement on my original forcing chain starting from R2C3.]
4b. R2C3 + R3C12 = {468/567}, no 2, 6 locked for R3 and N1, clean-up: no 4 in R5C2, no 8 in R5C9 (step 2)
4c. R2C3 = {78} -> no 7 in R3C2, clean-up: no 3 in R4C2
4d. Naked pair {46} in R34C2, locked for C2, clean-up: no 5,7 in R6C1, no 2 in R7C1
4e. 45 rule on N2 2 outies R1C7 + R2C3 = 15 = naked pair {78}, CPE no 7,8 in R1C123 + R2C789, clean-up: no 5 in R5C2, no 7 in R5C9 (step 2)
[I overlooked no 7,8 in R1C4 from the same CPE. Thanks Ed. That would have simplified step 4i.]
4f. 7,8 in N1 only in R2C123, locked for R2, clean-up: no 4,5 in R1C5
4g. R5C34 = [19]/{28/46} (cannot be [73] which clashes with R5C2), no 3,7
4h. R5C18 (step 2a) = {19/28/46} (cannot be {37} which clashes with R5C2), no 3,7
4i. 19(3) cage at R1C4 = {289/379/478} (cannot be {469} because R2C3 only contains 7,8, cannot be {568} which clashes with R1C6), no 5,6
4j. 4 of {478} must be in R2C4 -> no 4 in R1C4
4k. 6 in N2 only in R12C6, locked for C6, clean-up: no 5 in R6C7, no 4 in R8C4 (step 2b)
4l. R156C7 = 14 (step 3f), min R1C7 = 7 -> max R56C7 = 7, no 6 in R5C7, no 7 in R6C7, clean-up: no 4 in R6C6
5. 15(3) cage at R3C1 = {159/168/249/258/267/348} (cannot be {357} = 5{37} which clashes with R5C2, cannot be {456} which clashes with R89C1)
5a. R3C1 = {456} -> no 4,5,6 in R45C1, clean-up: no 4,6 in R5C8 (step 2a)
5b. 5 in N4 only in R6C23, locked for R6, clean-up: no 6 in R6C7
5c. Max R16C7 = 12 -> min R5C7 = 2
[Clean-ups more limited from here.]
6. 1 in R5 only in R5C18 (step 2a) = {19} or R5C34 = [19] (locking cages) -> 9 in R5C148, locked for R5-> R5C9 = 5, R5C2 = 7 (step 2), R1C3 = 9 (step 3d), clean-up: no 3 in R2C5, no 4 in R6C1, no 1 in R7C1
[Cracked.]
6a. R5C2 = 7, R6C5 = 1 -> R6C34 = 11 = [29/56/83], no 4,6 in R6C3, no 2,4,8 in R6C4
6b. R5C9 = 5 -> R6C89 = 11 = {29/38/47}, no 6
6c. 4 in R6 only in R6C789, locked for N6
6d. R156C7 = 14 (step 3f) = [734/824] -> R6C7 = 4, R6C6 = 7, clean-up: no 5 in R4C56
6e. 3 in R5 only in 13(3) cage at R5C5 = {238/346}
6f. R4C56 = {39} (cannot be {48} which clashes with 13(3) cage), locked for R4 and N5 -> R6C4 = 6, R6C3 = 5 (cage sum), clean-up: no 1,4 in R5C3
6g. 13(3) cage = {238} (only remaining combination) -> R5C56 = {28}, locked for R4 and N5 -> R5C34 = [64], R5C7 = 3, R1C7 = 7, R1C6 = 6, R2C3 = 8 (step 4e), clean-up: no 8 in R6C89
6h. Naked pair {29} in R6C89, locked for R6 and N6 -> R6C12 = [83]
6i. R4C4 = 5 -> R4C3 = 1, R45C1 = [29] -> R3C1 = 4 (cage sum)
6j. R89C1 = {16} (only remaining combination), locked for C1 and N7 -> R12C1 = [57], R7C1 = 3 -> R7C2 = 5
6k. R9C4 = R7C3 + 1 (step 3a) -> R7C3 = 2, R9C4 = 3, R12C4 = [29]
[Alternatively 17(3) cage at R7C3 = {278} (only remaining combination) …]
6l. R8C46 = 10 (step 2b) = [82], R7C4 = 7
6m. R12C5 = [84], R23C6 = [35]
6n. 15(3) cage at R7C7 = {168} (only remaining combination), locked for R7 and N9, R8C7 = 5 -> R8C8 = 3
6o. R5C8 = 1 -> R3C8 + R4C89 = 22 = {679}, R3C8 = 9, R4C89 = {67}, locked for N6
[A long time ago I’d spotted 45 rule on N3 2 innies R1C7 + R3C8 = 1 outie R4C7 + 8, no 8 in R3C8 (IOU) but didn’t use it as it didn’t lead to anything at the time.]
6p. R4C7 = 8 -> R3C7 = 2, R3C9 = 8 -> R12C9 = 9 = [36]
and the rest is naked singles.