Prelims
a) R2C34 = {29/38/47/56}, no 1
b) R4C12 = {19/28/37/46}, no 5
c) R78C1 = {29/38/47/56}, no 1
d) R89C5 = {39/48/57}, no 1,2,6
e) R89C6 = {69/78}
f) R9C78 = {39/48/57}, no 1,2,6
g) 20(3) cage at R1C2 = {389/479/569/578}, no 1,2
h) 18(5) cage at R6C7 = {12348/12357/12456}, no 9
1a. 45 rule on R12 1 innie R2C9 = 1
1b. 45 rule on R3 2 innies R3C59 = 16 = {79}, locked for R3
1c. 45 rule on N12 3 innies R2C6 + R3C56 = 19 = {289/379/469/478} (cannot be {568} because R3C5 only contains 7,9), no 1,5
1d. 3 of {379} must be in R3C6 -> no 3 in R2C6
1e. 45 rule on N8 5 innies R7C456 + R89C4 = 18 = {12348/12357/12456}, no 9
1f. 45 rule on N9 3 outies R6C67 + R7C6 which ‘see’ each other = 11 = {128/137/146/236/245}, no 9
1g. 45 rule on N78 2 innies R7C16 = 10 = {28/37/46}/[91], no 5, no 1 in R7C1
1h. 45 rule on N4 1 innie R4C3 = 1 outie R7C1 -> no 1,5 in R4C3
1i. 1 in R9 only in R9C1234, CPE no 1 in R8C3
1j. 45 rule on C1234 3 outies R567C5 = 11 = {128/137/146/236/245}, no 9
1k. 35(6) cage at R4C3 = {146789/236789/245789/345689}, CPE no 8,9 in R4C56
1l. 45 rule on C6789 2 outies R34C5 = 1 innie R1C6 + 9
1m. Max R34C5 = 16 -> max R1C6 = 7
2a. 27(6) cage at R2C9 and 18(3) cage at R5C9 total 45
2b. R56C9 ‘see’ all the cells of the 27(6) cage -> 27(6) cage + 18(3) cage must contain at least 8 different numbers so form a 45(9) cage = {123456789}, no 1 in R6C8
2c. 1 in N6 only in R46C7, locked for C7
2d. 45 rule on N6 2 outies R23C9 = 2 innies R46C7
2e. R23C9 = 1{79} -> R46C7 = 1{79}
2f. 18(3) cage = {369/378/459/468/567} (cannot be {279} which clashes with R46C7), no 2
2g. R6C67 + R7C6 (step 1f) = {128/137/146} (cannot be {236/245} because R6C7 only contains 1,7), no 5
2g. Consider placement for 1 in R6C67 + R7C6
2h. R6C6 = 1
or 1 in R6C7 + R7C6 => R8C8 = 1 (hidden single in N9)
-> 1 in R6C6 + R8C8, locked for D\
2i. Disjoint 15(3) cage at R6C6 contains 1 = {159/168}
2j. R6C67 + R7C6 = {128/137/146}
2k. 2,3,4 only in R7C6 -> R7C6 = {234}, R7C1 = {678} (step 1g), R4C3 = {678} (step 1h)
2l. 35(6) cage at R4C3 = {146789/236789/245789/345689}, 9 locked for C4 and N5, clean-up: no 2 in R2C3
2m. 1 in R1 only in R1C56, locked for N2
2n. 13(3) cage at R1C5 = {139/148/157}, no 2,6
2o. 12(3) cage at R1C1 = {237/246/345}, no 8,9
Observations.
3a. 30(7) cage at R7C4 and R89C5 contain 9 cells but only total 42 -> R8C5 must be the same as one of R9C12 and/or R9C5 must be the same as R8C3 -> R7C789 must contain one or both of the numbers in R89C5
The original observation 3a seems still to be valid. However for clarity I’ll now look at things the other way round
New 3a. R8C3 ‘sees’ all the cells in N8 except for R7C6 + R9C56, R9C23 ‘see’ all the cells in N8 except for R7C6 + R8C56
New 3b. BECAUSE 30(7) cage at R7C4 must contain all of 1,2,3,4,5 and R7C6 only contains 2,3,4, the value in R7C6 must be repeated in R8C3 + R9C23.
The pair in R89C6 = 15 cannot be repeated in the 30(7) cage because that would leave the other possible pair totalling 15 which cannot both fit into R89C5 = 12.
Therefore, since R89C6 cannot repeat in the 30(7) cage -> R89C5 must repeat in R8C3 + R9C23 with R9C5 in R8C3 and R7C6 + R8C5 in R9C23.
4a. 45 rule on N3 2 outies R23C6 = 1 remaining innie R3C9 + 3
[Only used later]
4b. Consider placements for 9 in C6
R2C6 = 9 => R3C5 = 7, R3C6 = 3 (step 1c)
or R89C6 = {69}, locked for C6
-> no 6 in R23C6
4c. 6 in N2 only in R123C4, locked for C4
4d. 45 rule on N1 3 outies R123C4 = 13 contains 6 = {256/346}, no 7,8, clean-up: no 3,4 in R2C3
4e. 1 in N8 only in R7C45 + R89C4, locked for 30(7) cage at R7C4
4f. 1 in R9 only in R9C14, CPE no 1 in R6C4 using D/
4g. R6C67 + R7C6 (step 1f) = {128/137/146}, 1 locked for R6
[I ought to have spotted this sooner]
4h. Disjoint 15(3) cage at R6C6 (step 2i) = {159/168}
4i. Consider combinations for 18(5) cage at R6C7 = {12348/12357/12456}
18(5) cage = {12348}, 8 locked for N9 => R9C78 = {39/57} => 15(3) cage = {168} (cannot be {159} = 1{59} which clashes with R9C78
or 18(5) cage = {12357/12456}, 5 locked for N9
-> 15(3) cage = {168}, 6,8 locked for D\
4j. 9 in N9 only in R78C9 = {29} or R9C78 = {39} -> no 3 in R78C9 (locking-out cages), clean-up: no 8 in R78C9
5a. R6C67 + R7C6 (step 1f) = {128/137/146} = [173/614/812], 18(5) cage at R6C7 = {12348/12357/12456}
5b. Consider combinations containing 9 in N9
R78C9 = {29}, 2 locked for N9 => 2 in 18(5) cage only in R7C6 => R6C67 + R7C6 = [812] => R78C6 = {69}
or R9C78 = {39}, locked for R9, no 3,9 in R8C5 => 9 in N8 only in R89C6 = {69}
-> R89C6 = {69}, locked for C6 and N8, clean-up: no 4 in R7C6, no 6 in R7C1 (step 1g), no 6 in R4C3 (step 1h), no 3 in R89C5
5c. R89C6 = {69} -> no 6,9 in 30(7) cage at R7C4 (step 3b, as clarified above)
5d. 1,6,9 in N7 only in 23(5) cage at R7C2 = {12569/13469}, no 7,8
5e. Disjoint 15(3) cage at R6C6 (step 4i) = {168}, 6 locked for N9, clean-up: no 5 in R78C9
[There's a killer pair in C9 here but it's not needed]
6a. R4C4 = 9 (hidden single on D\), clean-up: no 1 in R4C12
6b. Naked pair {17} in R46C7, 7 locked for C7 and N6, clean-up: no 5 in R9C8
6c. 27(6) cage at R2C9 + 18(3) cage at R5C9 form a 45(9) cage (step 2b) = {123456789} -> R3C9 = 7, R3C5 = 9, clean-up: no 4 in R78C9
6d. Naked pair {29} in R78C9, locked for C9 and N9, clean-up: no 3 in R9C78
6e. 18(5) cage at R6C7 = {12348/12357/12456} -> R7C6 = 2 -> R6C67 = [81] (step 5a), R4C7 = 7, R78C9 = [92], 15(3) disjoint cage = [816], R89C6 = [69], R4C3 = 8, R7C1 = 8 (step 1h), clean-up: no 3 in R2C4, no 2,3 in R4C12
6f. Naked pair {46} in R4C12, locked for R4 and N4
6g. R3C9 = 7 -> R23C4 = 10 (step 4a) = [73] -> R3C78 = 11 = {56}, locked for R3 and N3
6h. R123C4 (step 4d) = {256} (only remaining combination) = {56}2, 5 locked for C4 and N2, clean-up: no 9 in R2C3
6i. R3C123 = [184], 4 placed for D\
6j. 26(5) cage at R3C5 contains 1 for R4, 5 for C6 and 7,9 = [91574], 5 placed for D/ -> R3C78 = [65], 6 placed for D/
6k. Naked pair {37} in R56C4, locked for C4 and N5 -> R56C6 = [26], 2 placed for both diagonals
6l. R9C4 = 1 (hidden single in R9) -> R78C4 = [48]
6m. 8 in R9 only in R9C78 = {48}, 4 locked for R9 and N9
7a. R9C1 = 3, placed for D/
7b. Naked pair {57} in R8C35, 5 locked for R8 -> R8C7 = 3, R7C7 = 5, placed for D\
7c. R1C1 = 7, R2C2 = 3 -> R2C1 = 2 (cage sum)
7d. Naked pair {59} in R56C1, locked for N4, 9 locked for C1 -> R8C12 = [49], 9 placed for D/
7e. Naked pair {48} in R1C9 + R2C8, locked for N3
7f. 5 in N6 only in 18(3) cage at R5C9 = {459} (only remaining combination) = [594]
and the rest is naked singles, without using the diagonals.