Prelims
a) 15(2) cage at R1C1 = {69/78}
b) R23C1 = {29/38/47/56}, no 1
c) R23C4 = {29/38/47/56}, no 1
d) R5C78 = {29/38/47/56}, no 1
e) R7C56 = {18/27/36/45}, no 9
f) R78C9 = {39/48/57}, no 1,2,6
g) 20(3) cage at R1C5 = {389/479/569/578}, no 1,2
h) 11(3) cage at R3C8 = {128/137/146/236/245}, no 9
i) 11(3) cage at R4C1 = {128/137/146/236/245}, no 9
j) 20(3) cage at R4C3 = {389/479/569/578}, no 1,2
k) 10(3) cage at R6C1 = {127/136/145/235}, no 8,9
l) 20(3) cage at R8C4 = {389/479/569/578}, no 1,2
m) 10(3) cage at R8C5 = {127/136/145/235}, no 8,9
n) 12(4) cage at R2C3 = {1236/1245}, no 7,8,9
o) 27(4) cage at R8C6 = {3789/4689/5679}, no 1,2
p) 42(8) cage at R2C6 = {12456789}, no 3
1a. 45 rule on C1234 1 outie R6C5 = 4, clean-up: no 5 in R7C6
1b. 45 rule on R6789 2 outies R5C49 = 6 = [15/24/51]
1c. 45 rule on N1 2 outies R14C4 = 7 = {16/25}/[43], no 3,7,8,9 in R1C4
1d. 45 rule on N3 2 outies R1C6 + R4C9 = 7 = {16/25/34}, no 7,8,9
1e. 45 rule on N4 1 innie R6C3 = 1 outie R7C1 + 4, no 1,2,3 in R6C3, no 6,7 in R7C1
1f. 45 rule on N2 4 innies R1C46 + R23C6 = 14 contains 1 = {1238/1247/1256/1346}, no 9
1g. 45 rule on N8 4 innies R789C4 + R8C6 = 26 must contain 9 for N8 = {2789/3689/4589/4679}, no 1
1h. 45 rule on N9 2 outies R68C6 = 1 innie R7C8 + 11
1i. Min R68C6 = 12, no 1,2 in R6C6
1j. Max R68C6 = 17 -> max R7C8 = 6
1k. 45 rule on D\ 1 innie R5C5 = 2 outies R2C3 + R3C2 + 1
1l. Min R2C3 + R3C2 = 3 -> no 1,2 in R5C5
1m. 45 rule on N47 3 innies R679C3 = 20 = {389/479/569/578}, no 1,2
1n. Hidden killer pair for 1,2 in R3C3 + R4C4 and 17(4) cage at R6C6, R3C3 + R4C4 must contain at least one of 1,2 (1,2 cannot both be in R2C3 + R3C2 because min R5C5 = 5), 17(4) cage must contain at least one of 1,2 -> R3C3 + R4C4 and 17(4) cage must each contain one of 1,2
1o. 17(4) cage = {1349/1358/1457/2348/2357/2456} (cannot be {1367} which clashes with 15(2) cage at R1C1)
1p. Hidden killer pair for 17(4) cage and R7C8 in N9, 17(4) contains one of 1,2 in N9 -> R7C8 = {12}
2a. 45 rule on D\ 3 innies R3C3 + R4C4 + R5C5 = 13 = {139/148/157/238/247/256} (cannot be {346} = {34}6 because 12(4) cage at R2C3 cannot contain both of 3,4)
2b. Consider combinations for 15(2) cage at R1C1 = {69/78}
15(2) cage = {69}, locked for D\
or 15(2) cage = {78}, locked for N1 => R23C1 = {29/56} => R2C3 + R3C2 cannot be {26} which clashes with R23C1 => R3C3 + R4C4 + R5C5 cannot be {13}9
-> no 9 in R5C5
2c. Consider again combinations for 15(2) cage at R1C1 = {69/78}
15(2) cage = {69}, locked for N1 and D\ => 12(4) cage = {1245} => R3C3 + R4C4 cannot be {23}
or 15(2) cage = {78}, locked for D\
-> R3C3 + R4C4 + R5C5 = {148/157/247/256} (cannot be {238}), no 3 in R3C3 + R4C4, clean-up: no 4 in R1C4 (step 1c)
2d. 3 on D\ only in 17(4) cage at R6C6 (step 1o) = {1349/1358/2348/2357}, no 6
2e. R23C4 = {29/38/47} (cannot be {56} which clashes with R14C4), no 5,6
2f. 3 in N5 only in R6C46, locked for R6
2g. 21(5) cage at R5C9 = {12459/12468/12567}, CPE no 1,2 in R45C8, clean-up: no 9 in R5C7
2h. R5C78 = [29]/{38/56} (cannot be {47} which clashes with 21(5) cage), no 4,7
2i. Combined half cage R14C4 + R5C4 = {16}2/{16}5/{25}1, 1 locked for C4
2j. Consider placement for R7C8
R7C8 = 1 => 17(4) cage = {2348/2357}
or R7C8 = 2 => R68C6 = 13 (step 1h) => 17(4) cage = {1349/1358} with 9 of {1349} in R6C6
-> no 9 in R7C7 + R8C8 + R9C9
3a. R14C4 (step 1c) = 7 = {16/25}, R3C3 + R4C4 + R5C5 (step 2c) = {148/157/247/256}
3b. Consider combinations for 20(3) cage at R1C5 = {389/569/578}
20(3) cage = {389/569}, 9 locked for N2
or 20(3) cage = {578}, locked for C5 => R5C5 = 6, R4C4 = {25} => R14C4 = [25]
-> R23C4 = {38/47}, no 2,9
3c. 9 in N2 only in 20(3) cage = {389/569}, no 7, 9 locked for C5
3d. R1C46 + R23C6 (step 1f) = {1238/1247/1256} (cannot be {1346} which clashes with R23C4)
3e. 21(5) cage at R5C9 (step 2g) = {12459/12468/12567}
3f. Consider placement for 7 in N6
7 in R4C78 => no 7 in R23C6 => R1C46 + R23C6 = {1238/1256} => 4 in 42(8) cage at R2C6 only in R4C78 = {47}, locked for N6
or 7 in 21(5) cage = {12567}
-> no 4 in R5C9, clean-up: no 2 in R5C4 (step 1b)
[Looks like a key step.]
3g. 21(5) cage = {12567} (only remaining combination), 5,6,7 locked for N7, 6,7 locked for R6, clean-up: no 1,2 in R1C6 (step 1d)
3h. Naked pair {15} in R5C46, locked for R5
3i. Killer pair 1,5 in R14C4 and R5C4, locked for C4
3j. 7 in N5 only in R45C56, locked for 42(8) cage, no 7 in R23C6
3k. 7 in N2 only in R23C4 = {47}, locked for C4, 4 locked for N2, clean-up: no 3 in R4C9 (step 1d)
3l. 3 in N6 only in R5C78 = {38}, locked for R5, 8 locked for N6
3m. 9 in N6 only in R4C78, locked for R4 and 42(8) cage
3n. Killer pair 6,7 in 15(2) cage at R1C1 and R5C5, locked for D\, clean-up: no 1 in R1C4
3o. 12(4) cage at R2C3 = {1245} (cannot be {1236} because R3C3 + R4C4 only can only contain one of 1,2, step 1n), 4 locked for N1, clean-up: no 7 in R23C1
3p. R6C46 = {39} (hidden pair in N6), 9 locked for R6 -> R6C3 = 8 (hidden single in R6)
3q. 8 in N5 only in R4C56, locked for 42(8) cage, no 8 in R23C6
3r. R1C46 + R23C6 = {1256} (only remaining combination), 5,6 locked for N2, clean-up: no 4 in R4C9 (step 1d)
3s. R4C78 = {49} (hidden pair in N6), 4 locked for R4
3t. Naked triple {389} in 20(3) cage, 3,8 locked for C5, clean-up: no 1,6 in R7C6
3u. R4C6 = 8 (hidden single in N5), placed for D/, clean-up: no 1 in R7C5
3v. 8 in C4 only in 20(3) cage at R8C4 = {389} (cannot be {578} because 5,7 only in R9C3) -> R89C4 = {389}, R9C3 = {39}, CPE no 3 in R9C6
4a. R6C3 = 8 -> R7C1 = 4 (step 1e), R6C12 = 6 = {15}, locked for R6 and N4, clean-up: no 5 in R7C5, no 8 in R8C9
4b. Naked triple {267} in R4C789, 2 locked for N6 and 21(5) cage at R5C9 -> R45C9 = [15], R7C8 = 1, R5C4 = 1, R1C6 = 6 (step 1d), clean-up: no 9 in R2C2, no 7 in R78C9
4c. 12(4) cage at R2C3 (step 3o) = {1245}, 1 locked for N1
4d. 45 rule on N7 2 remaining innies R79C3 = {39}, locked for C3 and N7
4e. Naked pair {39} in R6C4 + R7C3, locked for D/
4f. R6C4 + R7C3 = {39}, R5C4 = 1, R6C35 = [84] -> R7C4 = 6 (cage sum), clean-up: no 3 in R7C6
4g. 4 on D/ only in R1C9 + R2C8 + R3C7, locked for N3
4h. 17(4) cage at R6C6 (step 2d) = {2348} (only remaining combination) -> R6C6 = 3, R7C7 + R8C8 + R9C9 = {248}, locked for D\, 4,8 locked for N9, clean-up: no 7 in 15(2) cage at R1C1
4i. R6C4 = 9, R79C3 = [39], R78C9 = [93], R89C3 = [83]
4j. R5C5 = 7 (hidden single on D\), placed for D/
4k. R5C1236 = [6942], R4C3 = 7
4l. R4C4 = 5, R1C34 = [52] -> R1C2 = 7 (cage sum)
4m. R23C1 = {38} (hidden pair in N1), 3 locked for C1
4n. R1C6 = 6, R1C7 = 1 (hidden single in R1) -> R2C7 = 9 (cage sum)
4o. Naked pair {38} in R15C8, locked for C8
4p. Naked pair {38} in R2C15, locked for R2
4q. R1C8 = 3 (hidden single in N3)
4r. R3C9 = 8 (hidden single in N3), R4C9 = 1 -> R3C8 = 2 (cage sum)
4s. R8C2 + R9C1 = {12} (hidden pair on D/), locked for N7
and the rest is naked singles, without using the diagonals.