Prelims
a) R1C34 = {39/48/57}, no 1,2,6
b) R1C56 = {19/28/37/46}, no 5
c) R12C7 = {18/27/36/45}, no 9
d) R34C2 = {29/38/47/56}, no 1
e) R45C9 = {18/27/36/45}, no 9
f) R67C2 = {49/58/67}, no 1,2,3
g) R78C7 = {18/27/36/45}, no 9
h) R8C12 = {19/28/37/46}, no 5
i) R8C56 = {49/58/67}, no 1,2,3
j) R8C89 = {19/28/37/46}, no 5
k) R9C67 = {69/78}
l) R9C89 = {17/26/35}, no 4,8,9
m) 8(3) cage at R2C1 = {125/134}
n) 11(3) cage at R5C6 = {128/137/146/236/245}, no 9
o) 11(3) cage at R9C1 = {128/137/146/236/245}, no 9
1a. 8(3) cage at R2C1 = {125/134}, 1 locked for C1, clean-up: no 9 in R8C2
1b. 45 rule on C1 3 innies R189C1 = 20 = {389/479/569/578}, no 2, clean-up: no 8 in R8C2
2. 45 rule on N2 2 innies R13C4 = 15 = [78/87/96] -> R1C3 = {345}, R4C4 = {345}
3a. 45 rule on N3 3 outies R4C8 + R5C78 = 21 = {489/579/678}, no 1,2,3
3b. 45 rule on C89 2 outies R35C7 = 12 = [39]/{48/57}, no 1,2,6, no 9 in R3C7
4. 45 rule on N8 2 innies R7C4 + R9C6 = 11 = [29/38/47/56]
5. 45 rule on C456789 2 outies R16C3 = 11 = [38/47/56]
6a. 45 rule on R9 2 innies R9C45 = 11 = {29/38/47/56}, no 1
6b. 45 rule on R9 3 outies R7C56 + R8C4 must contain 1 for N8 = 10 = {127/136/145}, no 8,9
6c. 9 in R9 only in R9C45 = {29} or R9C67 = {69} -> R9C45 = {29/38/47} (cannot be {56}, locking-out cages), no 5,6
6d. Hidden killer pair 1,5 in 11(3) cage at R9C1 and R9C89 for R9, neither can contain both of 1,5 -> 11(3) cage at R9C1 = {128/137/146/245}, R9C89 = {17/35}, no 2,6
[Alternatively R9C89 = {26} clashes with R9C45 = {29} or R9C67 = {69}.]
6e. R8C89 = {19/28/46} (cannot be {37} which clashes with R9C89), no 3,7
6f. 11(3) cage at R9C1 = {128/146/245} (cannot be {137} which clashes with R9C89), no 3,7
6g. 8 of {128} must be in R9C1 -> no 8 in R9C23
7a. 45 rule on C789 3 innies R469C7 = 15
7b. Min R69C7 = 7 -> max R4C7 = 8
7c. 45 rule on N9 2 outies R6C89 = 1 innie R9C7
7d. Max R6C89 = 9, no 9 in R6C89
7e. 9 in N6 only in R4C8 + R5C78, locked for 33(6) cage at R3C7
7f. R4C8 + R5C78 (step 3a) contains 9 = {489/579}, no 6
[Oops! At this stage I missed R35C7 cannot be {48/57} which clash with R4C8 + R5C78 = {489/579}. To express this another way, the 12+9 in R4C8 + R5C78 cannot be the same 12 that are in R35C7 because of the overlap at R5C7.
If I’d eliminated the other 9s from N6 earlier, as wellbeback and Ed did, I might have then spotted it when I found step 3b. It would have simplified things, including the time I spent trying to find a nice way to do step 11.]
7g. R45C9 = {18/27/36} (cannot be {45} which clash with R4C8 + R5C78), no 4,5
7h. 45 rule on N3 3 innies R3C789 = 12 = {138/156/237/246} (cannot be {147/345} which clash with R4C8 + R5C78)
7i. R12C7 = {18/27/45} (cannot be {36} which clashes with R3C789), no 3,6
8. 45 rule on R12 2 outies R3C56 = 1 innie R2C1 + 4
8a. Max R2C1 = 5 -> max R3C56 = 9, no 9 in R3C56
8b. 9 in R3 only in R3C23, locked for N1
8c. R189C1 (step 1b) = {389/479/569/578}
8d. 9 of {389/479/569} must be in R8C1 -> no 3,4,6 in R8C1, clean-up: no 4,6,7 in R8C2
8e. 4 of {479} must be in R9C1 -> no 4 in R1C1
9. 11(3) cage at R9C1 (step 6f) = {128/146/245}
9a. Consider combinations for R189C1 (step 1b) = {389/479/569/578}
R189C1 = {389/479/569} => R8C1 = 9, R8C2 = 1 => 11(3) cage = {245}
or R189C1 = {578} => R9C1 = {58} => 11(3) cage = {128/245}
-> 11(3) cage = {128/245}, no 6, 2 locked for R9 and N7, clean-up: no 8 in R8C1
9b. 6 in R9 only in R9C67 = {69}, 9 locked for R9, clean-up: no 3,4 in R7C4 (step 4)
10. R469C7 = 15 (step 7a) = {159/168/249/267/456} (cannot be {258/348/357} because R9C7 only contains 6,9), no 3
10a. 6 of {168/267/456} must be in R9C7 -> no 6 in R46C7
10b. 6 in C7 only in R789C7, locked for N9, clean-up: no 4 in R8C89
10c. Killer triple 7,8,9 in R8C1, R8C56 and R8C89, locked for R8, clean-up: no 1,2 in R7C7
[The best way I could find for this step, even though it’s not a pure forcing chain.]
11. R7C4 + R9C6 (step 4) = [29/56]
11a. Consider combinations for R8C56 = {49/58/67}
R8C56 = {49}, 9 locked for N8 => R9C56 = [69], 6 in C7 only in R78C7 = {36}, 3 locked for N9, R9C89 = {17} => R7C89 = {45} (hidden pair in N9), 5 locked for R7 => R7C4 = 2
or R8C56 = {58/67} => R7C4 + R9C6 = [29] (cannot be [56] which clashes with R8C56)
-> R7C4 + R9C6 = [29] -> R9C7 = 6, clean-up: no 1 in R1C5, no 4 in R8C56, no 3 in R78C7
11b. R35C7 = [39] (hidden pair in C7), clean-up: no 8 in R4C2
11c. R3C7 = 3 -> R3C789 (step 7h) = {138/237}, no 4,5,6
11d. 33(6) cage at R3C7 = {138}{579}/{237}{489}, CPE no 7,8 in R12C8
11e. 45 rule on N9 2 remaining innies R7C89 = 12 = {39/48} (cannot be {57} which clashes with R9C89), no 1,5,7
11f. 45 rule on N9 2 remaining outies R6C89 = 6 = {15/24}
11g. Killer pair 4,5 in R45C8 and R6C89, locked for N6
11h. 4 in N6 only in R45C8 + R6C89, CPE no 4 in R7C8, clean-up: no 8 in R7C9
[With hindsight, step 14 would follow from here.]
11i. R45C9 = {36} (hidden pair in N6), locked for C9, clean-up: no 9 in R7C8, no 5 in R9C8
11j. R7C56 + R8C4 (step 6b) = {136/145}, no 7
11k. 9 in R8 only in R8C12 = [91] or R8C89 = {19}, 1 locked for R8 (locking cages), clean-up: no 8 in R7C7
11l. 1 in N8 only in R7C56, locked for R7
12. R189C1 (step 1b) = {389/479/569/578}
12a. Consider combinations for 11(3) cage at R9C1 (step 9a) = {128/245}
11(3) cage = {128} = 8{12}, 1 locked for N7 => R8C12 = [73] => R189C1 = [578]
or 11(3) cage = {245}, R9C1 = {45} => R189C1 = {479/569/578} -> R189C1 = {479/569/578}, no 3
12b. Killer pair 4,5 in R189C1 and 8(3) cage at R2C1, locked for C1
13. R469C7 = 15 (step 7a), R9C7 = 6 -> R46C7 = 9 = {18/27}
13a. Consider combinations for R3C89 (step 11c) = {18/27}
R3C89 = {18}
or R3C89 = {27}, 7 locked for 33(6) cage at R3C7 => 7 in N6 only in R46C7 = {27} => R12C7 = {18} (hidden pair in C7)
-> 1,8 in R12C7 + R3C89, locked for N3
14. R4C8 + R5C78 (step 3a) = {489/579} -> R45C8 = {48/57}, R6C89 (step 11f) = {15/24}, R7C89 (step 11e) = [39/84]
14a. 18(4) cage at R6C8 = {15}[39]/{24}[39] (cannot be {15}[84] which clashes with R45C8) -> R7C89 = [39], clean-up: no 4 in R6C2, no 1 in R8C89, no 5 in R9C9
[Cracked, at last. The rest is straightforward.]
14b. Naked pair {28} in R8C89, locked for R8, clean-up: no 5 in R8C56
14c. Naked pair {17} in R9C89, locked for R9, 7 locked for N9
14d. 11(3) cage at R9C1 = {245} (only remaining combination), 4,5 locked for N7, 4 locked for R9, clean-up: no 8,9 in R6C2
14e. Naked pair {67} in R8C56, locked for R8, 6 locked for N8 -> R8C123 = [913], clean-up: no 9 in R1C4
14f. R13C4 (step 2) = 15 = {78}, locked for C4 and N2 -> R9C45 = [38], clean-up: no 2,3 in R1C56
14g. R34C4 = 11 = [74], R1C4 = 8 -> R1C3 = 4, R8C4 = 5, R78C7 = [54], clean-up: no 6 in R1C56, no 2 in R3C89 (step 11f), no 7 in R4C2
14h. R1C56 = [91], R7C56 = [14], R256C4 = [619]
14i. Naked pair {18} in R3C89, locked for R3 and N3, 8 locked for 33(6) cage at R3C7, clean-up: no 3 in R4C2
14j. Naked pair {27} in R12C7, locked for C7 and N3
14k. Naked pair {25} in R3C16, locked for R3 -> R3C5 = 4, clean-up: no 6,9 in R4C2
14l. R45C8 = 12 = {57}, locked for C8, 5 locked for N6
14m. 8(3) cage at R2C1 = {125} (only remaining combination), 2,5 locked for C1
14n. 17(3) cage at R5C1 = {368} (only remaining combination), 6 locked for C1, 3 locked for N4
14o. R1C1 = 7, R1C2 = 3 (hidden single in R1) -> R2C23 = 9 = [81], clean-up: no 5 in R6C2
14p. R4C1 = 1 (hidden single in C1) -> R46C7 = [81]
14q. R4C7 = 8 -> R4C56 = 10 = {37}, locked for R4 and N5
14r. Naked pair {56} in R56C5, locked for C5 and N5
14s. R1C3 = 4 -> R6C3 = 7 (step 5), R6C2 = 6, R3C2 = 9 -> R4C2 = 2
and the rest is naked singles.