Prelims
a) R12C1 = {19/28/37/46}, no 5
b) R1C23 = {49/58/67}, no 1,2,3
c) R12C4 = {19/28/37/46}, no 5
d) R12C7 = {19/28/37/46}, no 5
e) R1C89 = {49/58/67}, no 1,2,3
f) R23C2 = {13}
g) R23C6 = {49/58/67}, no 1,2,3
h) R4C23 = {19/28/37/46}, no 5
i) R4C45 = {19/28/37/46}, no 5
j) R45C6 = {39/48/57}, no 1,2,6
k) R45C7 = {39/48/57}, no 1,2,6
l) R56C3 = {16/25/34}, no 7,8,9
m) R56C4 = {19/28/37/46}, no 5
n) R56C9 = {39/48/57}, no 1,2,6
o) R6C56 = {17/26/35}, no 4,8,9
p) R6C78 = {14/23}
q) R7C23 = {59/68}
r) R78C4 = {18/27/36/45}, no 9
s) R89C6 = {17/26/35}, no 4,8,9
t) R89C9 = {13}
u) R9C12 = {39/48/57}, no 1,2,6
v) R9C78 = {29/38/47/56}, no 1
1a. 45 rule on N5 1 innie R5C5 = 5, clean-up: no 7 in R45C6, no 7 in R4C7, no 2 in R6C3, no 3 in R6C56, no 7 in R6C9
1b. Killer pair 1,2 in R6C56 and R6C78, locked for R6, clean-up: no 6 in R5C3, no 8,9 in R5C4
1c. R23C6 = {58/67} (cannot be {49} which clashes with R45C6), no 4,9
2a. Naked pair {13} in R23C2, locked for C2 and N1, clean-up: no 7,9 in R12C1, no 7,9 in R4C3, no 9 in R9C1
2b. Naked pair {13} in R89C9, locked for C9 and N9, clean-up: no 9 in R56C9, no 8 in R9C78
3a. Killer triple 3,4,5 in R45C7, R56C9 and R6C78, locked for N6
3b. 5 in N6 only in R45C7 = [57] or R56C9 = [75] (locking cages), 7 locked for R5 and N6, clean-up: no 3 in R6C4
[Alternatively I could have used the 45 in step 8a with 6 in N6 only in R4C89 + R5C8 so no 7 in them.]
4. 45 rule on C6 3 innies R167C6 = 12 = {129/147/246} (cannot be {138/345} which clash with R45C6, cannot be {156/237} which clash with R89C6), no 3,5,8
5. R7C23 = {59/68}, R9C12 = [39]/{48/57} -> combined cage R7C23 + R9C12 = {59}{48}/{68}[39]/{68}{57}, 8 locked for N7
6. 45 rule on N1 3 innies R2C3 + R3C13 = 18 = {279/459/567} (cannot be {468} which clashes with R1C23), no 8
7. 45 rule on N1 2 outies R3C45 = 1 innie R3C1 + 2, IOU no 2 in R3C45
[Now it gets harder.]
8a. 45 rule on N6 3 innies R4C89 + R5C8 = 16 = {169/268}
8b. Consider combinations for R4C89 + R5C8
R4C89 + R5C8 = {169} => R4C23 and R4C45 both cannot be {19} which clashes with R4C89 (ALS block)
or R4C89 + R5C8 = {268}, locked for N6, R6C78 = {14}, locked for R6, 4 locked for N6 => R56C9 = [75], R6C56 = {26}, 6 locked for R6, R56C3 = [43] => R4C23 = [91] (cannot be {28} which clashes with R4C89 + R5C8, ALS block), locked for R4
-> R4C45 = {28/37/46}, no 1,9
[Note. I could have taken the R4C89 + R5C8 = {268} path through to a contradiction but that’s not my solving style, I’ll look for an alternative method to make that elimination.]
8c. 1 in N5 only in R56C4 = [19] or R6C56 = {17} -> R56C4 = [19/28]/{46} (cannot be {37}, locking-out cages), no 3,7
9a. Consider combinations for R23C6 = {58/67}
R23C6 = {58}
or R23C6 = {67}, locked for C6 and N2 => R89C6 = {35}, 3 locked for C6 => R45C6 = {48}, R6C4 = 9 (hidden single in N5), R5C4 = 1 => R12C4 = {28}
-> 8 in R12C4 + R23C6, locked for N2
9b. Consider combinations for R78C4 = {18/27/36/45}
R78C4 = {18/27}, locked for C4 => 8 in N2 only in R23C6 = {58}
or R78C4 = {36/45} => R89C6 = {17/26} (cannot be {35} which clashes with R78C4) => 5 in C6 only in R23C6 = {58}
-> R23C6 = {58}, locked for C6 and N2, clean-up: no 2 in R12C4, no 4 in R45C6, no 3 in R89C6
[I hadn’t expected R78C4 to crack this puzzle, mostly easier from here.]
9c. Naked pair {39} in R45C6, locked for N5, 9 locked for C6, clean-up: no 7 in R4C45, no 1 in R5C4
9d. R6C56 = {17} (hidden pair in N5), locked for R6, clean-up: no 4 in R6C78
9e. Naked pair {23} in R6C78, locked for N6, 3 locked for R6, clean-up: no 9 in R45C7, no 4 in R5C3
9f. R4C89 + R5C8 = {169} (hidden triple in N6), 1 locked for C8
9g. 9 in R6 only in R6C12, locked for N4 and 21(4) cage at R6C1, clean-up: no 1 in R4C3
9h. R167C6 (step 4) = {147} (cannot be {246} because R6C6 only contains 1,7), locked for C6
9i. Naked pair {26} in R89C6, locked N8, clean-up: no 3 in R78C4
9j. Killer pair 4,8 in R56C4 and R78C4, locked for C4, clean-up: no 6 in R12C4, no 2,6 in R4C5
10a. R2C3 + R3C13 (step 6) = {279/459/567}
10b. 20(4) cage at R2C3 = {1469/2369/3467} (cannot be {1379} = {79}{13} which clashes with R3C2, cannot be {2459/2567} = {25}[94]/{25}{67} because R2C3 + R3C13 only contains one of 2,5), no 5
10c. Killer pair 1,3 in R3C2 and 20(4) cage, locked for R3
10d. Killer pair 1,3 in R12C4 and 20(4) cage, locked for N2
10e. 20(4) cage = {1469/2369/3467}, CPE no 6 in R3C1
10f. 1 in N3 only in R12C7 = {19}, 9 locked for C7 and N3, clean-up: no 4 in R1C89, no 2 in R9C8
10g. R2C8 = 3 (hidden single in N3) -> R23C2 = [13], R6C78 = [32], R12C7 = [19], R2C4 = 7 -> R1C4 = 3, R1C6 = 4, clean-up: no 9 in R1C23
10h. R1C15 = [29] (hidden pair in R1) -> R2C1 = 8, R2C5 = 2 (hidden single in N2), R23C6 = [58], clean-up: no 5 in R1C23, no 4 in R9C2
10i. Naked pair {67} in R1C23, locked for R1 and N1 -> R23C3 = [49], R3C1 = 5, R2C9 = 6, R4C89 = [19], R3C9 = 7 (cage sum), R3C78 = [24], R5C8 = 6, R45C6 = [39], R5C7 = 7 (hidden single in N6) -> R4C7 = 5, clean-up: no 6,7 in R4C2, no 3 in R5C3, no 5 in R7C2, no 7 in R9C2, no 9 in R9C8
10j. Naked pair {48} in R56C9, locked for C9 -> R1C89 = [85]
11a. 8 in N9 only in 16(4) cage at R7C5 = [31]{48}, clean-up: no 8 in R78C4
11b. R9C7 = 6 (hidden single in N9) -> R9C8 = 5, R9C4 = 9, R9C2 = 8 -> R9C1 = 4
11c. R6C4 = 8 (hidden single in C4)
and the rest is naked singles.
