Prelims
a) R12C2 = {16/25/34}, no 7,8,9
b) R12C8 = {14/23}
c) R67C1 = {18/27/36/45}, no 9
d) R67C3 = {12}
e) R67C4 = {14/23}
f) R67C6 = {59/68}
g) R67C7 = {18/27/36/45}, no 9
h) R67C9 = {29/38/47/56}, no 1
i) 19(3) cage at R1C3 = {289/379/469/478/568}, no 1
j) 20(3) cage at R2C4 = {389/479/569/578}, no 1,2
k) 10(3) cage at R3C2 = {127/136/145/235}, no 8,9
l) 19(3) cage at R3C7 = {289/379/469/478/568}, no 1
m) 20(3) cage at R6C2 = {389/479/569/578}, no 1,2
n) 20(3) cage at R6C8 = {389/479/569/578}, no 1,2
o) 9(3) cage at R8C1 = {126/135/234}, no 7,8,9
p) 19(3) cage at R8C3 = {289/379/469/478/568}, no 1
q) 11(3) cage at R8C6 = {128/137/146/236/245}, no 9
r) 21(3) cage at R9C4 = {489/579/678}, no 1,2,3
s) 26(4) cage at R4C2 = {2789/3689/4589/4679/5678}, no 1
t) 14(4) cage at R4C8 = {1238/1247/1256/1346/2345}, no 9
1a. Naked pair {12} in R67C3, locked for C3
1b. 10(3) cage at R3C2 = {136/145/235} (cannot be {127} because 1,2 only in R3C2), no 7
1c. -> R3C2 = {12}
1d. Combined half cage R7C3 + 9(3) cage at R8C1 = 10,11 = {1234/1235}, no 6, 1,2,3 locked for N7, clean-up: no 6,7,8 in R6C1
1e. 45 rule on C12 1 outie R5C3 = 1 innie R3C2 + 4, R3C2 = {12} -> R5C3 = {56}
1f. 45 rule on C12 3 outies R345C3 = 14 = {356} (only remaining combination), locked for C3
1g. 45 rule on C123 2 outies R18C4 = 10 = {28/37/46}, no 5,9
1h. 45 rule on C1234 2 innies R59C4 = 10 = [19/28/37/46/64], no 5 -> R5C4 = {12346}
1i. 5 in C4 only in 20(3) cage at R2C4 = {569/578}, no 3,4
1j. 1 in C4 only in R59C4 = [19] or R67C4 = {14} -> R59C4 = [19/28/37] (cannot be {46}, blocking cages), no 4,6
[That would also have eliminated {479} from 20(3) cage if that hadn’t been eliminated in step 1i.
Before I saw the step this way I was looking at the interactions between 21(3) cage at R9C4 and R5C46; they eliminated R59C4 = [64] but not [46].]
1k. 45 rule on C6789 2 innies R59C6 = 14 = {59/68}
1l. Naked quad {5689} in R5679C6, locked for C6
1m. 14(3) cage at R2C6 = {347} (only remaining combination), locked for C6
1n. Max R1C6 = 2 -> min R12C7 = 13, no 1,2,3 in R12C7
1o. 45 rule on C89 1 innie R3C8 = 1 outie R5C7 + 6 -> R3C8 = {789}, R5C7 = {123}
1p. 26(4) cage at R4C2 = {3689/4589/4679/5678} (cannot be {2789} because R5C3 only contains 5,6), no 2
1q. 45 rule on C1 1 innie R5C1 = 1 outie R9C2 + 4 -> min R5C1 = 5
1r. 3 in N7 only in 9(3) cage at R8C1 = {135/234}
1s. 45 rule on C1 3 innies R589C1 = 13 = {139/157/238/247/346} (cannot be {148/256} which aren’t consistent with 9(3) cage) -> R5C1 = {6789}, clean-up: no 1 in R9C2
1t. 1 in C2 only in R123C2, locked for N1
2a. R18C4 (step 1g) = {28/37/46}
2b. 45 rule on N7 4(3+1) outies R6C123 + R8C4 = 15
2c. R6C123 + R8C4 = 15 cannot be {124}8 because R18C4 = [28] clashes with R67C4 = [32] -> no 8 in R8C4, clean-up: no 2 in R1C4
2d. Hidden killer pair 1,2 in 17(3) cage at R1C5 and R1C6 for N2, R1C6 = {12} -> 17(3) cage must contain one of 1,2 = {179/269/278}, no 3,4,5
2e. 5 in N2 only in R23C4 -> no 5 in R4C4
2f. 14(3) cage at R6C5 = {149/158/248/347/356} (cannot be {167/239/257} which clash with 17(3) cage)
2g. 45 rule on C5 3 innies R459C5 = 14 = {149/158/248/347/356} (cannot be {167/239/257} which clash with 17(3) cage)
2h. 19(3) cage at R8C3 = {289/379/469/478}
2i. 3 of {379} must be in R8C4, {478} cannot be [478/874] which clashes with 21(3) cage at R9C4 -> no 7 in R8C4, clean-up: no 3 in R1C4
2j. 3 in N2 only in R23C6 -> no 3 in R4C6
2k. 20(3) cage at R6C2 = {479/569/578} (cannot be {389} which clashes with 19(3) cage), no 3
2l. {479} cannot be 4{79} which clashes with 19(3) cage -> no 4 in R6C2
3a. 45 rule on C9 1 innie R5C9 = 1 outie R9C8, max R5C9 = 8 -> max R9C8 = 8
3b. 14(4) cage at R4C8 = {1238/1247/1256/1346/2345}
3c. Consider placement for 6 in C8
6 in R459C8 => 6 in R45C8 + R5C9 => 14(4) cage = {1256/1346}
or 6 in 20(3) cage at R6C8 = {569}, locked for C8 => R3C8 = {78} => one of 7,8 in R459C8 => one of 7,8 in R45C8 + R5C9 => 14(4) cage = {1238/1247}
-> 14(4) cage = {1238/1247/1256/1346}, 1 locked for N6, clean-up: no 8 in R7C7
[Alternatively 45 rule on N3 and 45 rule on N9 eliminate 1 from R4C9 and R6C7.]
4a. Consider placement for 6 in N7
R7C1 = 6 => R6C1 = 3
or 20(3) cage at R6C2 = {569}, locked for C2 => R12C2 = {34}, locked for C2
-> no 3 in R45C2
4b. Hidden killer triple 1,2,3 in R12C2, R3C2 and R9C2 for C2, R12C2 contains one of 1,2,3, R3C2 = {12} -> R9C2 = {23}, clean-up: no 8,9 in R5C1 (step 1q)
4c. R67C1 = [18/27/36] (cannot be {45} which clash with 9(3) cage at R8C1), no 4,5
4d. 26(4) cage at R4C2 (step 1p) = {4679/5678} (cannot be {4589} because R6C1 only contains 6,7), 6,7 locked for N4
4e. 45 rule on R1234 3 innies R4C258 = 12 = {129/138/156/237/246} (cannot be {147} which clashes with R4C6, cannot be {345} which clashes with R4C3)
4f. 7,8,9 of {129/138/237} must be in R4C2 -> no 7,8,9 in R4C58
4g. R4C258 = {129/138/156/246} (cannot be {237} = 7{23} which clashes with R4C3 + R5C13 only contain 3,5,6,7) -> no 7 in R4C2
4h. 7 in N4 only in R5C12, locked for R5
5a. 19(3) cage at R8C3 (step 2h) = {289/379/469/478} -> R89C3 + R8C4 = {49}6/{78}4/{79}3/{89}2, 20(3) cage at R6C2 (step 2k) = {479/569/578}
5b. Consider placements for R7C1 = {678}
R7C1 = 6 => 20(3) cage at R6C2 = {479/578}, 7 locked for N7 => R89C3 + R8C4 = {49}6/{89}2
or R7C2 = 7 => R89C3 + R8C4 = {49}6/{89}2
or R7C1 = 8 => R6C1 = 1, R67C3 = [21] => R67C4 = [32] => R89C3 + R8C4 = {49}6
-> R89C3 + R8C4 = {49}6/{89}2, no 7 in R89C3, no 3,4 in R8C4, 9 locked for C3 and N7, clean-up: no 6,7 in R1C4 (step 1g)
5c. 21(3) cage at R9C4 = {579/678} (cannot be {489} which clashes with R9C3), no 4, 7 locked for R9 and N8
5d. Killer pair 2,4 in R18C4 and R67C4, locked for C4, clean-up: no 8 in R9C4 (step 1h)
6a. 45 rule on N1 3(2+1) outies R1C4 + R4C13 = 14, R4C13 cannot total 10 -> R1C4 = 8, R4C13 = 6 = [15], R5C13 = [76], naked pair {47} in R12C3, 4 locked for C3 and N1, R6C13 = [32], R3C3 = 3, R3C2 = 2 (cage sum), R9C2 = 3, 20(3) cage at R2C4 = {569} (only remaining combination), 6,9 locked for C4 -> R89C4 = [27], R18C6 = [21]
[Cracked. Fairly straightforward from here; clean-ups omitted]
6b. R4C1 = 1 -> R123C1 = 22 = {589}, 5,8 locked for C1, 5 locked for N1 -> R7C1 = 6
6c. R4C258 (step 4g) = {246} (only remaining combination) -> R4C2 = 4, R4C58 = {26}, locked for R4
6d. R5C13 = [76] = 13 -> R45C2 = 13 = [49], R6C2 = 8, R4C46 = [97], R23C6 = [34]
6e. R67C4 = [14] (only remaining permutation) -> R5C3 = 3
6f. 45 rule on N5 2 remaining innies R6C56 = 11 = {56}, locked for R6 and N5 -> 17(4) cage at R4C5 = [2348]
6g. R67C6 = [59] (only remaining permutation), R9C6 = 6 -> R9C5 = 8 (cage sum)
6h. 45 rule on R89 3 innies R8C258 = 15 = {357} (only possible combination with 3,5,7 in R8C25), locked for R8
6i. R89C1 = [42], R89C3 = [89], R8C67 = [16] -> R9C7 = 4 (cage sum), R6C7 = 7 -> R7C7 = 2
6j. R1C6 = 2 -> R12C7 = 13 = [58], 19(3) cage at R3C7 = [973]
6k. Naked pair {15} in R9C89, 5 locked for N9, R78C8 = [83] -> R6C8 = 9 (cage sum)
6l. R12C8 = {14} (only remaining combination), locked for N3, 1 locked for C8
and the rest is naked singles.