Prelims
a) R12C1 = {17/26/35}, no 4,8,9
b) R1C23 = {18/27/36/45}, no 9
c) R1C89 = {49/58/67}, no 1,2,3
d) R2C89 = {18/27/36/45}, no 9
e) R67C1 = {17/26/35}, no 4,8,9
f) R7C23 = {18/27/36/45}, no 9
g) R8C23 = {29/38/47/56}, no 1
h) R9C78 = {18/27/36/45}, no 9
1. 45 rule on C89 2 outies R39C7 = 17 = [98] -> R9C8 = 1, clean-up: no 4 in R1C89, no 8 in R2C9
1a. R3C7 = 9 -> R3C89 = 7 = {25/34}/[61], no 7,8, no 6 in R3C9
1b. 45 rule on N3 2 innies R12C7 = {16/25/34}, no 7
1c. Hidden killer pair 7,8 in R1C89 and R2C89 for N3, R1C89 contains one of 7,8 -> R2C89 must contain one of 7,8 = {27}[81], no 3,4,5,6
[Alternatively killer triple 4,5,6 in R12C7, R1C89 and R3C89 …]
1d. 45 rule on N23 1 innie R3C4 = 1 outie R4C5, no 9 in R3C4 -> no 9 in R4C5
1e. 9 in R1 only in R1C456, locked for N2
1f. 30(7) cage at R1C4 contains 9 = {1234569}, no 7,8
1g. 8 in R1 only in R1C23 = {18} or R1C89 = {58} -> R1C23 = {18/27/36} (cannot be {45}, locking-out cages), no 4,5
1h. 4 in R1 only in R1C4567, locked for 30(7) cage, clean-up: no 3 in R1C7
1i. 9 in N1 only in R2C23, CPE no 9 in R4C3
1j. 45 rule on R12 3 innies R2C235 = 21 and must contain 4 for R2 = {489}, 8 locked for R2, clean-up: no 1 in R2C9
1k. Naked pair {27} in R2C89, locked for R2 and N3, clean-up: no 1,6 in R1C1, no 6 in R1C89
1l. Naked pair {58} in R1C89, locked for R1, 5 locked for N3, clean-up: no 1 in R1C23, no 3 in R2C1
1m. 2 in 30(7) cage only in R1C456, locked for R1 and N2, clean-up: no 7 in R1C23, no 6 in R2C1, no 2 in R4C5
1n. Naked pair {36} in R1C23, locked for R1 and N1, R1C1 = 7 -> R2C1 = 1
1o. 5 in R2 only in R2C46, locked for N2, clean-up: no 5 in R4C5
1p. Naked quint {24589} in R2C23 + R3C123, CPE no 2,4,5,8 in R4C3
1q. 1 in N7 only in R7C23 = {18}, locked for R7, 8 locked for N7, clean-up: no 3 in R8C23
1r. 18(3) cage at R6C8 = {369/378/459/468/567} (cannot be {279} which clashes with R2C8), no 2
2. 45 rule on N7 2 innies R7C1 + R9C3 = 8 = {26/35}
2a. R8C23 = {29/47} (cannot be {56} which clashes with R7C1 + R9C3), no 5,6
2b. 17(3) cage at R8C1 = {269/359/467}
2c. 7 of {467} must be in R9C2 -> no 4 in R9C2
2d. 17(3) cage at R8C4 = {269/278/359/368/458/467} (cannot be {179} because no 1,7,9 in R9C3), no 1
3. 15(3) cage at R2C3 = {159/249/348/456} (cannot be {168/267/357} because 1,3,6,7 only in R3C4, cannot be {258} because 2,5 only in R3C3), no 7, clean-up: no 7 in R4C5 (step 1d)
3a. 3 of {348} must be in R3C4 -> no 8 in R3C4, clean-up: no 8 in R4C5 (step 1d)
[Fairly heavy combination analysis; I’ve tried to make it as clear as possible.]
4. 35(6) cage at R2C2 = {236789/245789/345689} (cannot be {146789} = {489}{167} which clashes with R2C3), no 1
4a. 15(3) cage at R2C3 (step 3) = {159/249/348/456}, R3C4 = R4C5 (step 1d) -> split 15(3) cage R23C3 + R4C5 = 15 = {159/249/348/456}
[Alternatively 45 rule on N23 3 outies R23C3 + R4C5 = 15 …]
4b. 35(6) cage at R2C2 = {245789/345689} (cannot be {236789} = 9{28}{367} which clashes with split 15(3) cage)
4c. 35(6) cage = {345689} can only be 4{58}{369} (cannot be {489}{356} which clashes with R2C3, cannot be 9{45}{368} which clashes with 15(3) cage at R2C3)
4d. 15(3) cage at R5C1= {159/168/249/258/267/348/357/456}
4e. 35(6) cage = {245789} (cannot be {345689} = 4{58}{369} which clashes with seven of the combinations of 15(3) cage at R5C1 while 4{58}{369} clashes with combined cage 15(3) at R5C1 = {258} + R6C1 = {2356})
4f. 35(6) cage = {245789}, no 3,6 -> R4C3 = 7, clean-up: no 4 in R8C2
5. 15(3) cage at R5C1= {159/249/258/348/456} (cannot be {168} because 15(3) cage = {168} + R7C2 = {18} block all 8s in 35(6) cage at R2C2)
5a. 45 rule on R123 3 remaining outies R4C125 = 15, R4C5 = {1346} -> R4C12 = 9,11,12,14
5b. 15(3) cage = {159/348/456} (cannot be {249/258} because R4C12 cannot be {49/58} = 13), no 2
5c. 45 rule on N7 1 outie R6C1 = R9C3
5d. Consider placement for 5 in C3
R3C3 = 5 => 5 in 35(6) cage at R2C2 must be in R4C12, locked for N4
or 5 in R56C3, locked for N4
or 5 in R9C3 => R6C1 = 5
-> 15(3) cage = {348} (only remaining combination), locked for N4, clean-up: no 5 in R7C1, no 3 in R9C3 (step 2)
5e. 35(6) cage at R2C2 = {245789}, 4,8 locked for N1 -> R2C3 = 9, R3C34 = 5 = [24/51], clean-up: no 3,6 in R4C5 (step 1d)
5f. 9 in N4 only in R4C12, locked for R4
5g. R7C3 = 8 (hidden single in C3)
5h. R1C3 = 3 (hidden single in C3) -> R1C2 = 6
5i. R8C3 = 4 (hidden single in C3) -> R8C2 = 7
5j. Naked quad {1249} in R1C456 + R3C4, locked for N2 -> R2C5 = 8, R2C2 = 4
5k. Naked pair {38} in R56C2, locked for C2 and N4 -> R5C1 = 4
5l. R3C1 = 8 (hidden single in N1)
6. 17(3) cage at R8C4 (step 2d) = {269/278/359/368/458} (cannot be {467} because 4,7 only in R9C4)
6a. 17(3) cage = {269/278/359/368} (cannot be {458} = [854] which clashes with R3C34), no 4
6b. 5 of {359} must be in R9C3 -> no 5 in R89C4
7. 15(3) cage at R4C4 = {159/168/258/267/348/357/456} (cannot be {249} which clashes with R13C4, ALS block)
7a. Deleted; this step has been replaced by step 8g.
[This one took me a long time to spot. I wonder whether steps 8b and 8e are “human” steps.]
8. 17(3) cage at R8C4 (step 6a) = {269/278/359/368}
8a. 45 rule on N8 3 innies R789C6 = 1 outie R9C3 + 7
8b. Consider placement for 8 in N8
R8C4 = 8 => 17(3) cage = {278/368}, no 5
or R8C6 = 8 -> R789C6 must total at least 13 (cannot be {138} because 1,8 only in R8C6) => R9C3 must be 6
-> no 5 in R9C3, clean-up: no 3 in R7C1 (step 2), no 5 in R6C1
8c. Naked pair {26} in R67C1, locked for C1
8d. Naked pair {26} in R7C1 + R9C3, 2 locked for N7
[Taking step 8b further …]
8e. R789C6 + R9C3 cannot be {238}6 which clashes with R89C4 = {29/38} -> no 8 in R8C6
8f. R8C4 = 8 (hidden single in N8) -> R9C34 = 9 = [27/63]
8g. 15(3) cage at R4C4 (step 7) = {159/267/456} (cannot be {357} which clashes with R9C4), no 3
9. 45 rule on N6789 2 innies R45C7 = 1 outie R6C1 + 3
9a. R6C1 = {26} -> R45C7 = 5,9 = {23}/[27/45] (cannot be {14} which clashes with R1C7, cannot be {36} which clashes with R2C7), no 1,6, no 5 in R4C7
10. 45 rule on N8 4 remaining innies R9C4 + R789C6 = 16 = {1267/1357/2347/2356} (cannot be {1249/1456} because R9C4 only contains 3,7), no 9
10a. 25(6) cage at R6C7 = {124567} (only remaining combination), no 3
10b. R9C4 + R789C6 = {1267/2347} (cannot be {1357} = 3{157} because R678C7 = {246} which clashes with R12C7, cannot be {2356} = 3{256} because R678C6 = {147} clashes with R1C7), no 5, 2,7 locked for N8, 2 locked for C6 and 25(6) cage at R6C7
10c. 2 in C7 only in R45C7, locked for N6
10d. R45C7 = R6C1 + 3 (step 9), R6C1 = {26} -> R45C7 = 5,9 contains 2 = {23}/[27], no 4,5
10e. R9C4 + R789C6 = {1267/2347} -> R789C6 = {126/247}
10f. 1 of {126} must be in R8C6 -> no 6 in R8C6
10g. R789C6 = {126/247} -> R678C7 = {457/156}
10h. 5 of {457} must be in R8C7, 1 of {156} must be in R6C7 -> no 5,6 in R6C7
10i. 5 in C7 only on R78C7, locked for N9
10j. R2C8 = 2 (hidden single in C8) -> R2C9 = 7
10k. 7 in N9 only in R7C78, locked for R7
10l. 2 in N9 only in 20(4) cage at R6C9 = {2369/2459/2468} (cannot be {1289} because 1,8 only in R6C9), no 1
10m. 5,9 in N8 only in 21(4) cage at R7C4 = {1569/3459}
10n. 1 of {1569} must be in R8C5 -> no 6 in R8C5
10o. 6 in R8 only in R8C789, locked for N9
11. R789C6 = {126/247} -> R678C7 = {457/156} (step 10g), 21(4) cage at R7C4 (step 10m) = {1569/3459}
11a. Consider placements for 5 in R8
R8C1 = 5 => R9C1 = 3 (hidden single in N7)
or R8C5 = 5 => 21(4) cage = {3459}, 3 locked for N8
or R8C7 = 5 => R678C7 = {457} => R789C6 = {126} => R8C5 => 21(4) cage = {3459}, 3 locked for N8
-> R9C4 = 7, R9C3 = 2 (cage sum), R67C1 = [26], R3C23 = [25], R3C4 = 1 (cage sum)
[Cracked, at last.]
11b. 15(3) cage at R4C4 (step 8g) = {456} (only remaining combination), locked for C4 and N5 -> R127C4 = [239], R1C7 = 1 (hidden single in R1), R2C7 = 6, R8C7 = 5
11c. Naked pair {47} in R67C7, 7 locked for C7, 4 locked for 25(6) cage at R6C7 -> R789C6 = [216], R8C5 = 3, R8C1 = 9, R9C12 = [35], R4C1 = 5, R8C89 = [62]
11d. Naked pair {34} in R37C9, locked for C9
12. R5C5 = 2 (hidden single in C5), R5C7 = 3, R56C2 = [83]
12a. R79C5 = [54], R4C5 = 1, R16C5 = [97], R5C6 = 9
12b. R5C9 = 1 (hidden single in N6), R5C8 = 7 (hidden single in R5) -> R4C89 = 14 = [86]
12b. R689C9 = [529] = 16 -> R7C9 = 4 (cage sum)
and the rest is naked singles.