Prelims
a) R1C45 = {12}
b) R1C89 = {69/78}
c) R2C45 = {18/27/36/45}, no 9
d) R3C12 = {16/25/34}, no 7,8,9
e) R45C6 = {15/24}
f) R5C23 = {49/58/67}, no 1,2,3
g) R67C2 = {39/48/57}, no 1,2,6
h) R7C34 = {69/78}
i) R78C9 = {49/58/67}, no 1,2,3
j) R8C67 = {39/48/57}, no 1,2,6
k) 19(3) cage at R1C6 = {289/379/469/478/568}, no 1
l) 21(3) cage at R2C2 = {489/579/678}, no 1,2,3
m) 20(3) cage at R4C7 = {389/479/569/578}, no 1,2
n) 23(6) cage at R7C8 = {123458/123467}, no 9
1a. Naked pair {12} in R1C45, locked for R1 and N2, clean-up: no 7,8 in R2C45
1b. 17(4) cage at R1C1 must contain at least one of 1,2 -> R2C1 = {12}
1c. 45 rule on R1 2 outies R2C16 = 9 = [18/27]
1d. 45 rule on R12 1 outie R3C3 = 8 -> R2C23 = 13 = {49/67}, no 5, clean-up: no 5 in R5C2, no 7 in R7C4
1e. Killer pair 4,6 in R2C23 and R2C45, locked for R2
1f. Hidden killer pair 1,2 in R2C1 and R3C12 for N1, R2C1 = {12} -> R3C12 must contain one of 1,2 = {16/25}, no 3,4
1g. 8 in N2 only in R12C6, locked for C6 and 19(3) cage at R1C6, no 8 in R1C7 clean-up: no 4 in R8C7
1h. 19(3) cage contains 8 = {478/568}, no 3,9
1i. 45 rule on C6789 1 innie R3C6 = 9, clean-up: no 3 in R8C7
1j. 45 rule on N2 2(1+1) outies R1C7 + R4C5 = 11 = {47/56}
1k. 45 rule on R89 2 outies R7C89 = 8 = [17/26/35], clean-up: no 4,5,9 in R8C9
1l. 9 in N9 only in R78C7, locked for C7
1m. 45 rule on N9 2 innies R78C7 = 1 outie R9C6 + 9
1n. R78C7 contains 9, no 8 in R9C6 -> no 8 in R78C7, clean-up: no 4 in R8C6
1o. 20(3) cage at R4C7 = {389/479/569/578}
1p. 9 of {389/479/569} must be in R5C8 -> no 3,4,6 in R5C8
1q. 3 in R1 only in 17(4) cage at R1C1 = {1349/2357} (cannot be {1367} = {367}1 which clashes with R1C89), no 6
1r. Killer pair 7,9 in 17(4) cage and R1C89, locked for R1, clean-up: no 4 in R4C5
1s. 19(3) cage = {478/568} = [847]/{56}8, no 4 in R1C6
1t. Consider placement for 4 in N3
R1C7 = 4 => R12C6 = [87]
or 4 in R3C789, locked for R3 => 4 in N2 only in R2C45 = {45}, 5 locked for N2
-> no 5 in R1C6
1u. 19(3) cage = [658/847], no 6 in R1C7, clean-up: no 5 in R4C5
1v. R8C67 = [39/57] (cannot be [75] which clashes with 19(3) cage)
2a. 15(4) cage at R3C9 = {1239/1248/1347/2346} (cannot be {1257/1356} which clash with R78C9), no 5
2b. 45 rule on C9 3 innies R129C9 = 17 = {179/269/359/458/467} (cannot be {278/368} which clash with R78C9)
2c. 4 of {458} only in R9C9 -> no 8 in R9C9
2d. 45 rule on R123 2 outies R4C58 = 1 innie R3C9 + 9
2e. Min R4C58 = 10, max R4C5 = 7 -> min R4C8 = 3
2f. 8,9 in N5 only in R4C4 + R6C45, locked for 34(7) cage at R4C4, no 8,9 in R6C3 + R7C5
2g. Hidden killer pair 8,9 in R1C89 and 14(3) cage at R2C7 for N3, R1C89 contains one of 8,9 -> 14(3) cage must contain one of 8,9 = {158/239}, no 7
2h. 45 rule on N47 2 outies R78C4 = 1 innie R6C3 + 10
2i. Min R78C4 = 11 -> no 1 in R8C4
2j. 45 rule on N36 3(1+2) innies R1C7 + R6C78 = 12, min R1C7 = 4 -> max R6C78 = 8, no 8,9 in R6C78
2k. 45 rule on N3 4 innies R1C7 + R3C789 = 16 must contain 4 for N3 = {1456/2347} = 4{237}/5{146} (cannot be 4{156} which clashes with R3C12), no 5 in R3C78
2l. 14(3) cage at R3C7 = {149/167/239/257/347} (cannot be {158} because 5,8 only in R4C8, cannot be {248} because R3C78 cannot contain both of 2,4, cannot be {356} because R3C78 cannot contain both of 3,6), no 8
2m. 6 of {167} must be in R3C78 (cannot be {17}6) -> no 6 in R4C8
2n. 4 of {347} must be in R4C8 (cannot be {47}3), 9 of {239} must be in R4C8 -> no 3 in R4C8
2o. R3C789 = {14}6/{16}4/{23}7/{27}3/{37}2 (14(3) cage cannot be {46}4), no 1 in R3C9
2p. 1 in R3 in R3C12 = {16} or R3C789 = {146} (locking cages), 6 locked for R3
3a. R129C9 (step 2b) = {179/269/359/458/467}, 14(3) cage at R2C7 (step 2g) = {158/239}, 15(4) cage at R3C9 (step 2a) = {1239/1248/1347/2346}
3b. Consider combinations for R1C89 = {69/78}
R1C89 = {69}, locked for N3 => 14(3) cage = {158} => R129C9 = {179/359} (cannot be {269/467} which don’t contain 1,5,8 for R2C9, cannot be {458} which doesn’t contain 6,9 for R1C9)
or R1C89 = {78}, locked for N3 => 14(3) cage = {239}, killer pair 7,8 in R1C9 and R78C9, locked for C9, 15(4) cage = {2346} (cannot be {1239} which clashes with R2C9), locked for C9, R78C9 = [58], R129C9 = {179}
-> R129C9 = {179/359}, no 2,4,6,8, 9 locked for C9 and N3, clean-up: no 7 in R1C8
3c. Killer pair 5,7 in R129C9 and R78C9, locked for C9
3d. R1C7 + R6C78 = 12 (step 2j)
3e. Consider placements for 4 in C9
R3C9 = 4 => R1C7 = 5, R3C78 = 7 = {16}, R4C8 = 7 (cage sum), 20(3) cage at R4C8 = {389} (cannot be {569} = {56}9 which clashes with R1C7)
or 4 in R456C9, locked for N6
-> R4C8 = {579}, 20(3) cage = {389/569/578}, no 4
3f. Consider placements for R1C7 = {45}
R1C7 = 4 => R3C789 (step 2k) = {237}, R3C78 = {27} (cannot be {37} because no 4 in R4C8), R4C8 = 5 (cage sum) => 20(3) cage = {389}
or R1C7 = 5 => 20(3) cage = {389/578} (cannot be {569} = {56}9)
-> 20(3) cage = {389/578}, no 6, 8 locked for N6
3g. R8C9 = 8 (hidden single in C9) -> R7C9 = 5, R7C8 = 3 (step 1k), clean-up: no 7,9 in R6C2
3h. R129C9 = {179} (only remaining combination), no 3, 1 locked for C9
3i. 23(6) cage at R7C8 = {123467} (only remaining combination), no 5
3j. Deleted
4a. 1 in N6 only in R6C78, locked for R6 and 20(5) cage at R6C6, no 1 in R7C67, clean-up: no 1 in R9C6 (step 1m)
4b. R1C7 + R6C78 = 12 (step 2j) with 1 in R6C78 = 4{17}/5{16}
4c. 20(5) cage = {12467} (only remaining combination, cannot be {13456} because 3,5 only in R6C6), no 3,5,9
4d. R8C7 = 9 (hidden single in N9) -> R8C6 = 3
4e. R45C6 = {15} (hidden pair in C6), locked for N5
4f. 34(7) cage at R4C4 contains 8,9 (step 2f) = {1234789/1235689} -> R7C5 = 1, R1C45 = [12]
4g. Hidden killer pair 8,9 in R7C4 and 18(3) cage at R8C5 for N8, 18(3) cage without 1 cannot contain both of 8,9 -> R7C4 = {89}, 18(3) cage must contain one of 8,9, clean-up: no 9 in R7C3
4h. 14(3) cage at R8C2 = {167/257}, no 4, 7 locked for R8
4i. 23(6) cage at R7C8 (step 3i) = {123467}, 7 locked for R9
4j. 18(3) cage = {459/468}, no 2, 4 locked for N8
5a. R6C6 = 4 (hidden single in C6), clean-up: no 8 in R7C2
[Cracked. The rest is straightforward.]
5b. 2 in C6 only in R79C6, locked for N8
5c. 34(7) cage at R4C4 (step 4f) = {1235689} (only remaining combination), no 7, R6C3 = 5, clean-up: no 8 in R5C2, no 7 in R7C2
5d. R4C5 = 7 (hidden single in N5), R3C6 = 9 -> R3C45 = 9 = {45}, locked for R3 and N2, naked pair {36} in R2C45, locked for R2, 6 locked for N2, R12C6 = [87], R1C7 = 4 (cage sum), R1C8 = 6 -> R1C9 = 9, R29C9 = [17], R2C1 = 2
5e. R1C7 = 4 -> R6C78 (step 4b) = 8 = {17}, 7 locked for R6, N6 and 20(5) cage at R6C6, no 7 in R7C67
5f. Naked pair {26} in R7C67, locked for R7, R7C3 = 7 -> R7C4 = 8, clean-up: no 6 in R5C2
5g. 20(3) cage at R4C7 (step 3f) = {389} (only remaining combination) -> R5C8 = 9, 3 locked for C4 and N6, R4C8 = 5, R45C6 = [15], clean-up: no 4 in R5C23
5h. R5C23 = [76], R1C123 = [753]
5i. Naked pair {38} in R5C57, locked for R5 -> R5C49 = [24]
5j. R5C1 = 1 -> R67C1 = 13 = [94], R7C2 = 9 -> R6C2 = 3
5k. R89C1 = [53] (hidden pair in C1) = 8 -> R9C23 = 10 = [82]
and the rest is naked singles.