Prelims
a) R12C5 = {49/58/67}, no 1,2,3
b) R23C7 = {39/48/57}, no 1,2,6
c) R34C9 = {29/38/47/56}, no 1
d) R45C5 = {49/58/67}, no 1,2,3
e) R56C9 = {69/78}
f) R6C12 = {89}
g) R7C56 = {16/25/34}, no 7,8,9
h) R89C1 = {13}
i) 26(4) cage at R7C1 = {2789/3689/4589/4679/5678}, no 1
j) 23(6) cage at R4C2 = {123458/123467}, no 9
Steps resulting from Prelims and Initial Placement
1a. Naked pair {89} in R6C12, locked for R6 and N4, clean-up: no 6,7 in R5C9
1b. Naked pair {13} in R89C1, locked for C1 and N7
1c. 45 rule on N7 1 outie R9C4 = 9
2. 45 rule on N5 1 innie R4C4 = 1 outie R4C7 + 5 -> R4C4 = {678}, R4C7 = {123}
2a. 23(6) cage at R4C2 = {123458/123467}, 2,4 locked for N4
2b. 45 rule on N4 2 innies R45C1 = 1 outie R4C4 + 5, IOU no 5 in R5C1
3a. 45 rule on N89 2 outies R6C78 = 6 = {15/24}
3b. 45 rule on N689 2 innies R4C79 = 8 = [17/26/35], R4C7 = {123} -> R4C9 = {567}, R3C9 = {456}
3c. Killer pair 6,7 in R34C9 and R6C9, locked for C9, 7 also locked for N6
3d. 45 rule on N3 2(1+1) outies R1C6 + R4C9 = 14 = [77/86/95], R4C9 = {567} -> R1C6 = {789}
3e. 45 rule on N2 3 innies R1C46 + R2C4 = 15 which can only contain one of 7,8,9 -> no 7,8 in R12C4
3f. 15(4) cage at R5C4 cannot be 7{125}/7{134}/8{124} which clash with R6C78 -> no 7,8 in R5C4
4. 15(4) cage at R5C4 = {1257/1347/1356/2346}
4a. 45 rule on N5 3 innies R4C46 + R5C6 = 17 = {179/269/278/368} (cannot be {359/458/467} which clash with 15(4) cage), no 4,5 in R45C6
4b. 6 of {269/368} must be in R4C4 (R4C46 + R5C6 cannot be 8{36} because 12(3) cage at R4C6 cannot be {36}3), no 6 in R45C6
4c. R45C5 = {49/58} (cannot be {67} which clashes with R4C46 + R6C5), no 6,7
5. 26(5) cage at R2C1 = {14579/14678/23579/23678/24569/24578/34568} (cannot be {13589/13679} because 1,3 only in R3C2, cannot be {12689/23489} because must contain two of 5,6,7 for R45C1)
5a. 26(5) cage only contains two of 5,6,7 which must be in R45C1 -> no 5,6,7 in R2C1 + R3C12
6. 16(3) cage at R4C8 = {169/349/358} (cannot be {259} which clash with R6C78, cannot be {268} which clashes with R56C9), no 2
7. 45 rule on R9 2 outies R8C13 = 1 innie R9C9 + 5
7a. Max R8C13 = 11 -> no 8 in max R9C9
7b. Min R8C13 = 6 -> no 2 in R8C3
8. 45 rule on R1 2 innies R1C45 = 11 = [29/38/47]/{56}, no 1 in R1C4, no 4 in R1C5, clean-up: no 9 in R2C5
8a. 15(3) cage at R1C1 = {159/168/258/267/357/456} (cannot be {249/348} which clash with 26(5) cage at R5C1)
8b. R1C45 = [29/38/47] (cannot be {56} which clashes with 15(3) cage), no 5,6, clean-up: no 7,8 in R2C5
8c. R1C46 + R2C4 = 15 (step 3e) = {249/258/267/348/357} (cannot be {159/168} because R1C4 only contains 2,3,4, cannot be {456} because R1C6 only contains 7,8,9), no 1 in R2C4
8d. R1C45 = 11 -> R1C46 cannot be 11 (CCC) -> no 4 in R2C4
8e. R1C46 + R2C4 = {249/258/267/348} (cannot be {357} = [375] which clashes with R1C45 + R2C5 = [385])
8f. 4 of {348} must be in R1C4 -> no 3 in R1C4, clean-up: no 8 in R1C5, no 5 in R2C5
9. Consider placement for 8 in R1
9a. 8 in 15(3) cage at R1C1, locked for N1 => 26(5) cage at R2C1 (step 5) = {14579/23579/24569} => R4C1 = 5
or 8 in 19(4) cage at R1C7 which cannot contain both of 8,9 => no 9 in R1C6, no 5 in R4C9 (step 3d)
-> no 5 in R4C9, clean-up: no 9 in R1C6 (step 3d), no 6 in R3C9, no 3 in R4C7 (step 3b), no 8 in R4C4 (step 2)
[The first breakthrough, thanks to the elimination of 8 from R1C5 in step 8f.]
9b. Naked pair {67} in R4C49, locked for R4 -> R4C1 = 5, clean-up: no 8 in R5C5
9c. Naked pair {67} in R46C9, 6 locked for N6
9d. 3 in N6 only in 16(3) cage at R4C8 (step 6) = {349/358}, no 1
9e. R4C46 + R5C6 (step 4a) = {179/269/368} (cannot be {278} = 7{28} because 12(3) cage at R4C6 cannot be {28}2)
9f. 7 of {179} must be in R4C4 -> no 7 in R5C6
9g. R1C46 + R2C4 (step 8e) = {258/267/348}
9h. 2 of {258/267} must be in R1C4 -> no 2 in R2C4
9i. 19(4) cage at R1C6 = {1378/1468/1567/2368} (cannot be {1279} which clashes with R1C5, cannot be {2458/2467} which clash with R1C4, cannot be {1459/3457} which clash with R3C9, cannot be {1369/2359} because R1C6 only contains 7,8), no 9
9j. 15(3) cage at R1C1 (step 8a) = {159/357/456} (cannot be {168/258/267} which clash with 19(4) cage), no 2,8, 5 locked for R1 and N1
9k. 7,9 of {159} must be in R1C1 -> no 7,9 in R1C23
9l. R1C45 = 11 (step 8b) = [29/47], R12C5 = [76/94] -> R1C45 + R2C5 = [294/476], 4 locked for N2
9m. Killer triple 3,4,5 in 19(4) cage at R1C6, R23C7 and R3C9, locked for N3
10. R1C6 + R4C9 (step 3d) = 14 = [77/86]
10a. R1C46 + R2C4 (step 9g) = {258/348} (cannot be {267} = [276] which clashes with R4C49 = [67], no 6,7
[The second breakthrough.]
10b. R1C6 = 8 -> R4C9 = 6, R3C9 = 5, R4C7 = 2 (step 3b), R4C4 = 7, R6C9 = 7 -> R5C9 = 8, clean-up: no 7 in R23C7, no 4 in R6C78 (step 3a)
10c. R4C7 = 2 -> R45C6 = 10 = {19}, locked for C6 and N5, clean-up: no 4 in R45C5, no 6 in R7C5
10d. R45C5 = [85], clean-up: no 2 in R7C6
10e. Naked pair {15} in R6C78, locked for R6 and N6
10f. 30(7) cage at R6C7 must contain 2, locked for N8, clean-up: no 5 in R7C6
10g. R7C56 = [16] (cannot be {34} because 30(7) cage at R6C7 must contain both of 3,4 and R7C56 ‘sees’ all its cells except for R68C7 and R6C7 only contains 1,5)
[I’d been aware of this type of ‘sees all except’ step for a while, realising that R678C7 had to contain one of 1,2,3, but this was the first time I found it useful.]
10h. 30(7) cage at R6C7 must contain 1 in R68C7, locked for C7
11. 8 in C4 only in 30(7) cage at R6C7 = {1234578}, no 6,9, no 8 in R78C7
12. R5C1 = 7 (hidden single in N4)
12a. 15(3) cage at R1C1 (step 9j) = {159/456}, no 3
12b. 3 in R1 only in R1C789, locked for N3, clean-up: no 9 in R23C7
12c. Naked pair {48} in R23C7, locked for C7 and N3
12d. R5C7 = 9 (hidden single in C7) -> naked pair {34} in R45C8, locked for C8, R45C6 = [91]
12e. 30(7) cage at R6C7 must contain 4, locked for N8
12f. 22(4) cage at R9C5 = {3568} (only possible combination, cannot be {1678/2578} because 1,2,8 only in R9C8) = [3568], R89C1 = [31]
12g. 7 in N8 only in R8C56, locked for R8 and 30(6) cage at R6C7
12h. Naked pair {15} in R68C7 -> R17C7 = [73]
12i. R1C67 = [87] = 15 -> R1C89 = 4 = [13], R1C5 = 9 -> R2C5 = 4, R1C4 = 2 (step 8b), R23C7 = [84], R6C78 = [15]
12j. Naked pair {37} in R23C6, locked for C6 and N2 -> R3C45 = [16]
12k. Naked pair {48} in R78C4, locked for N8 -> R8C56 = [72]
12l. Naked pair {29} in R2C29, locked for R2 -> R2C8 = 6
12m. R8C8 = 9 -> R89C9 = 5 = [14]
12n. Naked pair {27} in R9C23, locked for N7, R9C4 = 9 -> R8C3 = 6 (cage sum)
and the rest is naked singles.