Prelims
a) R1C45 = {69/78}
b) R4C56 = {49/58/67}, no 1,2,3
c) R5C56 = {29/38/47/56}, no 1
d) R56C7 = {17/26/35}, no 4,8,9
e) R67C9 = {18/27/36/45}, no 9
f) R78C8 = {18/27/36/45}, no 9
g) R89C6 = {39/48/57}, no 1,2,6
h) 21(3) cage at R5C8 = {489/579/678}, no 1,2,3
i) 11(3) cage at R6C6 = {128/137/146/236/245}, no 9
j) 14(4) cage at R7C3 = {1238/1247/1256/1346/2345}, no 9
1a. 45 rule on C123 5 outies R25678C3 = 16 = {12346}, locked for C4, clean-up: no 9 in R1C5
1b. 45 rule on N1 1 outie R2C4 = 1 innie R3C3 + 3 -> R2C4 = {46}, R3C3 = {13}
1c. 45 rule on R6789 1 innie R6C8 = 2 outies R5C27 + 5 -> R6C8 = {89}, R5C28 = {12/13}, 1 locked for R5, clean-up: no 1,2,3 in R6C7
[Ed and wellbeback both extended this with what is effectively the short forcing chain
21(3) cage at R5C8 = {489/579/678}
Consider combinations for R5C27 = {12/13}
R5C27 = {12} => R6C7 = {67}, R6C8 = 8 => 21(3) cage = {489} (cannot be {678} which clashes with R6C7)
or R5C27 = {13} => R6C7 = {57}, R6C8 = 9 => 21(3) cage = {489} (cannot be {579} which clashes with R6C7)
-> 21(3) cage = {489}, locked for N6
It didn’t matter that I didn’t spot that, I got that result in step 4g without trying; I don’t usually look for early forcing chains.]
1d. 1 in N5 only in R6C456, locked for R6, clean-up: no 8 in R7C9
1e. 45 rule on R1234 4(1+3) innies R3C3 + R4C123 = 12
1f. R3C3 = {13} -> R4C123 = 9,11, no 9
1g. 45 rule on R1234 3 outies R5C134 = 18 = {279/369/378/459/468/567}
1h. 2 on {279} must be in R5C4 -> no 2 in R5C13
[Note. 18(4) cage at R3C3 and R5C134 = 18 share R5C34 -> R34C3 = R5C1 may be useful later.]
2a. 45 rule on N1236 2(1+1) outies R4C4 + R7C9 = 1 innie R3C3 + 13
2b. R3C3 = {13} -> R4C4 + R7C9 = 14,16 = [77/86/95/97] -> R4C4 = {789}, R7C9 = {567}, R6C9 = {234}
2c. 45 rule on N9 1 outie R6C9 = 1 innie R7C7 = {234}
2d. 32(7) cage at R1C6 = {1234589/1234679/1235678}
2e. R1C45 = {78} (cannot be [96] which clashes with 32(7) cage), locked for R1 and N2
2f. 32(7) cage must contain at least one of 7,8, only in R4C4 -> 32(7) cage = {1234589/1234679}, R4C4 = {78}
2g. Naked pair {78} in R14C4, locked for C4
2h. R4C4 + R7C9 = [77/86], R7C9 = {67} -> R6C9 = {23}, clean-up: no 4 in R7C7 (step 1j)
2i. R4C4 + R7C9 = [77/86] = 14 -> R3C3 = 1, R2C4 = 4 (step 1b)
2j. Hidden killer triple 1,2,3 in R4C789, R5C7 and R6C9 for N6, R5C7 = {123}, R6C9 = {23} -> R4C789 must contain one of 1,2,3
2k. Hidden killer triple 1,2,3 in R4C123 and R4C789 for R4, R4C789 contains one of 1,2,3 -> R4C123 must contain two of 1,2,3
2l. Killer triple 1,2,3 in R4C123 and R5C2 for N4
2m. R3C3 = 1 -> R4C123 (step 1e) = 11 = {128/137/236} (cannot be {146/235} which only contain one of 1,2,3), no 4,5
2n. 12(3) cage at R4C1, R4C123 = 11 -> R5C1 = R4C3 + 1, no 2 in R4C3, no 5,6 in R5C1
2o. R4C123 = {128/137/236}
2p. 8 of {128} must be in R4C3 -> no 8 in R4C12
2q. 4 in N1 only in R3C12, locked for R3
2r. 15(3) cage at R2C2 contains 4 = {249/348/456}, no 7
2s. 7 in N1 only in R2C13, locked for R2
2t. 32(7) cage = {1234589/1234679} -> R1C7 = 4
2u. 32(7) cage = {1234589/1234679}, 2,3,9 locked for N2
3a. 45 rule on N5 5 innies R45C4 + R6C456 = 21 contains 1 for N5 = {12378/12567/13458/13467} (cannot be {12369/12459} because R4C4 only contains 7,8, cannot be {12468} which clashes with R4C56), no 9
3b. 9 in N5 only in R4C56 = {49} or R5C56 = {29} -> R5C56 = {29/38/56} (cannot be {47}, locking-out cages), no 4,7
4a. R4C4 + R7C9 (step 2h) = [77/86], R6C9 = R7C7 (step 2c)
4b. Consider permutations for R4C4 + R7C9
R4C4 + R7C9 = [77] => R1C45 = [87], R6C9 = 2, R7C7 = 2 => R67C6 = 9 = {18/36/45} => 7 in C6 only in R89C6 = {57}, 5 locked for C6 => R67C6 = {18/36}
or R4C4 + R7C9 = [86], R1C45 = [78], R6C9 = 3, R7C7 = 3 => R67C6 = 8 = {17/26} => 8 in C6 only in R89C6 = {48}
-> R89C6 = {48/57}, no 3,9, no 4,5 in R67C6
4c. Consider placements for R3C6 = {56}
R3C6 = 5 => R89C6 = {48}, locked for C6
or R3C6 = 6, R67C6 = {17/18}, killer pair 7,8 in R67C7 and R89C6, locked for C6
-> no 8 in R45C6, clean-up: no 5 in R4C5, no 3 in R5C5
4d. Consider again placements for R4C4 + R7C9
R4C4 + R7C9 = [77] => R1C45 = [87], R6C9 = 2, R7C7 = 2 => R67C6 = 9 = {18/36} => 7 in C6 only in R89C6 = {57}, 5 locked for C6
or R4C4 + R7C9 = [86]
-> R4C56 = {49/67}, no 5,8
4e. R4C123 (step 2o) = {128/137/236}
4f. R4C56 = {49} (cannot be {67} because R4C456 = 8{67} clashes with R4C123), locked for R4 and N5, clean-up: no 2 in R5C56
4g. 4,9 in N6 only in 21(3) cage at R5C8 = {489}, 4 locked for R5, 8 locked for N6
4h. 38(7) cage at R2C8 = {1256789}, no 3, 8,9 locked for N3
4i. 22(5) cage at R1C8 = {13567}, no 2 -> R3C7 = 7, clean-up: no 1 in R5C7
4j. 7 in N6 only in R4C89, locked for R4 -> R4C4 = 8, R1C45 = [78], R4C123 = {236}, locked for N4, 2,6 locked for R4, clean-up: no 3 in R5C6
4k. R6C7 = 6 (hidden single in N6) -> R5C7 = 2, R6C9 = 3 -> R7C9 = 6, R7C7 = 3
4l. 22(5) cage = {13567} -> R1C8 = 3, R3C6 = 6, R5C56 = [65],
4m. R7C7 = 3 -> R67C7 = 8 = {17}, locked for C7
5a. Naked pair {48} in R89C6, 4 locked for C6 and N8 -> R4C56 = [49], R12C6 = [23]
5b. R34C3 + R5C4 = [163] = 10 -> R5C3 = 8 (cage sum)
5c. Naked triple {569} locked for N1, 5 locked for R1
5d. R1C9 = 1 -> R24C7 = [51]
5e. R3C45 = {59}, 9 locked for R3 and N2 -> R2C5 = 1
5f. Naked pair {28} in R3C89, locked for R3 and N3 -> R2C89 = [69], R5C89 = [94], R6C8 = 8, R3C89 = [28], R5C1 = 7, clean-up: no 1,7 in R78C8
5g. R3C12 = {34} = 7 -> R2C2 = 8 (cage sum), R2C13 = [27], R4C12 = [32], R3C12 = [43]
5h. Naked pair {45} in R78C8, locked for N9, 5 locked for C8
5i. 18(3) cage at R7C1 = {189} (only remaining combination), 1,9 locked for N7, 9 locked for C1
5j. R6C23 = {49} (hidden pair in R6) -> R6C4 = 2 (cage sum)
5k. R6C1 = 5 -> R789C2 = {467}, 4 locked for C2 and N7
5l. R78C4 = [16] = 7 -> R78C3 = 7 = {25}
and the rest is naked singles.
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