Prelims
a) R1C34 = {59/68}
b) R1C67 = {49/58/67}, no 1,2,3
c) R2C23 = {49/58/67}, no 1,2,3
d) R2C78 = {39/48/57}, no 1,2,6
e) R34C1 = {17/26/35}, no 4,8,9
f) R34C4 = {39/48/57}, no 1,2,6
g) R34C6 = {29/38/47/56}, no 1
h) R34C9 = {69/78}
i) R5C34 = {18/27/36/45}, no 9
j) R5C67 = {59/68}
k) R67C1 = {69/78}
l) R67C4 = {18/27/36/45}, no 9
m) R67C6 = {18/27/36/45}, no 9
n) R67C9 = {17/26/35}, no 4,8,9
o) R8C23 = {49/58/67}, no 1,2,3
p) R8C78 = {39/48/57}, no 1,2,6
q) R9C34 = {19/28/37/46}, no 5
r) R9C67= {39/48/57}, no 1,2,6
s) 6(3) cage at R1C8 = {123}
t) 8(3) cage at R6C7 = {125/134}
u) 6(3) cage at R8C1 = {123}
1a. Naked triple {123} in 6(3) cage at R1C8, locked for N3, clean-up: no 9 in R2C78
1b. 13(3) cage at R3C7 must contain at least one of 1,2,3 -> R4C7 = {123}
1c. Killer triple 1,2,3 in R12C9 and R67C9, locked for C9
1d. Min R89C9 = 9 -> max R9C8 = 8
1e. Naked triple {123} in 6(3) cage at R8C1, locked for N7, clean-up: no 7,8,9 in R9C4
1f. Killer triple 1,2,3 in R34C1 and R89C1, locked for C1
1g. R1C67 = {49/67} (cannot be {58} which clashes with R1C34), no 5,8
1h. Killer pair 6,9 in R1C34 and R1C67, locked for R1
1i. R2C23 = {49/67} (cannot be {58} which clashes with R2C78), no 5,8
1j. Killer pair 4,7 in R2C23 and R2C78, locked for R2
2a. 15(3) cage at R1C1 = {159/249/258/348/357} (cannot be {168/267} which clash with R67C1, cannot be {456} which clashes with R2C23), no 6
2b. 1,2,3 only in R1C2 -> R1C2 = {123}
2c. Naked triple {123} in R1C289, locked for R1
2d. 45 rule on R1 3 remaining innies R1C15 = 12 = {48/57}
3a. 45 rule on C12 4 innies R2378C2 = 26 = {2789/3689/4589/4679/5678}, no 1
3b. 45 rule on C1234 2 innies R28C4 = 7 = {16/25}/[34], no 7,8,9, no 3 in R8C4
3c. 45 rule on C6789 2 innies R28C6 = 10 = {19/28}/[37/64], no 5, no 3,6 in R8C6
[Note. From steps 3b and 3c, R2C4 cannot be 3 lower than R2C6 and R8C6 cannot be 3 more than R8C4.]
3d. 45 rule on C123 3 outies R159C4 = 17, max R19C4 = 15 -> min R5C4 = 2, clean-up: no 8 in R5C3
3e. 45 rule on C789 3 innies R159C7 = 24 = {789}, locked for C7, clean-up: no 7,9 in R1C6, no 4,5 in R2C8, no 8,9 in R5C6, no 3,4,5 in R8C8, no 7,8,9 in R9C6
3f. 45 rule on C789 3 outies R159C6 = 15 = {456} (only remaining combination), no 3, clean-up: no 7 in R34C6, no 3 in R67C6, no 9 in R9C7
3g. Killer pair 2,8 in R34C6 and R67C6, locked for C6
3h. R3C7 = 6 (hidden single in C7), clean-up: no 2 in R4C1, no 9 in R4C9
3i. R3C7 = 6 -> R3C8 + R4C7 = 7 = [43/52]
3j. 1 in C7 only in R67C7, locked for 8(3) cage at R6C7
3k. 45 rule on C89 4 innies R2378C8 = 24 = {2589/4578} (cannot be {3489/3579} because R78C8 = [39] clashes with R8C78 = [39], CCC), no 3, 5,8 locked for C8
3l. 8(3) cage at R6C7 = {125/134}
3m. 4 of {134} must be in R7C8 -> no 4 in R67C7
4a. 45 rule on R9 2 outies R8C19 = 1 innie R9C5
4b. Min R8C19 = 5 -> min R9C5 = 5
4c. Max R8C19 = 9, no 9 in R8C9
4d. 45 rule on N1 2 innies R1C3 + R3C1 = 1 outie R4C3 + 5, IOU no 5 in R3C1, clean-up: no 3 in R4C1
5a. Consider placement for 6 in N9
6 in R789C9, locked for C9 => R34C9 = {78}, 7 locked for C9
or 6 in R9C8 => R89C9 = 11 = {47}, 7 locked for C9
-> 7 in R3489C9, locked for C9, clean-up: no 1 in R67C9
5b. 1 in C9 only in R12C9, locked for N3
5c. 17(3) cage at R8C9 = {269/359/458/467} (cannot be {179/368} which clash with R34C9, cannot be {278} which clashes with R9C7), no 1
5d. R7C7 = 1 (hidden single in N9), clean-up: no 8 in R6C4, no 8 in R6C6
5e. 2 in C7 only in R46C7, locked for N6, clean-up: no 6 in R7C9
5f. 6 in N9 only in 17(3) cage = {269/467}, no 3,5,8
5g. 4 of {467} must be in R89C9 (R89C9 cannot be {67} which clashes with R34C9), no 4 in R9C8
5h. Consider combinations for 17(3) cage
17(3) cage = {269} = [629]
or 17(3) cage = {467} = [476/674], 6 locked for C9
or 17(3) cage = {467} = [467/764], 7 locked for C9 => R34C9 = [96]
-> 6 in R489C9, clean-up: no 2 in R7C9
5i. Naked pair {35} in R67C9, locked for C9
5j. R1C8 = 3 (hidden single in N3)
5k. 15(3) cage at R1C1 (step 2a) = {159/249/258}, no 7
5l. Killer pair 8,9 in R12C1 and R67C1, locked for C1
5m. Hidden killer pair 6,7 in 20(4) cage at R4C8 = {1469/1478} and R4C9 for N6, 20(4) cage contains one of 6,7 -> R4C9 = {67}, clean-up: no 7 in R3C9
5n. 3 in N1 only in R3C123, locked for R3, clean-up: no 9 in R4C4, no 8 in R4C6
5o. 5 in N6 only in R6C79, locked for R6, clean-up no 4 in R7C4
5p. 8 in N6 only in R5C79, locked for R5, clean-up: no 1 in R5C3
6a. 5 in R9 only in R9C56, locked for N8, clean-up: no 2 in R2C4 (step 3b), no 4 in R6C4
6b. 5 in R8 only in R8C23 = {58} or R8C78 = [57] -> R8C23 = {49/58} (cannot be {67}, locking-out cages), R8C78 = [39/57] (cannot be [48], locking-out cages)
[Don’t think I’ve ever used double locking-out cages like that before.
Fairly straightforward from here, routine clean-ups omitted.]
6c. Killer pair 5,9 in R8C23 and R8C67, locked for R8, clean-up: no 1 in R2C6 (step 3c)
6d. R9C7 = 8 (hidden single in N9) -> R9C6 = 4, R15C7 = [79], R15C6 = [65], R2C78 = [48], R3C8 = 5, R34C9 = [96], R89C9 = [47], R79C8 = [26], R8C78 = [39], R67C9 = [35], R9C34 = [91], R9C5 = 5, R7C5 = 9 (hidden single in N8)
6e. R78C6 = [87], R34C6 = [29], R2C456 = [513] -> R1C5 = 8 (cage sum), R34C4 = [48] (only remaining permutation), R7C4 = 3 (hidden single in N8) -> R6C4 = 6, R8C4 = 2, R5C34 = [27]
6f. 4 in N7 only in R7C23, locked for 17(3) cage at R6C3 = {467} (only possible combination) -> R6C3 = 7, R2C123 = [976]
6g. R34C1 = [35] (only remaining permutation)
and the rest is naked singles.
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