Prelims
a) R23C5 = {19/28/37/46}, no 5
b) R3C78 = {89}
c) R7C89 = {89}
d) R8C23 = {89}
e) R9C78 = {18/27/36/45}, no 9
f) 20(3) cage at R1C1 = {389/479/569/578}, no 1,2
g) 9(3) cage at R6C1 = {126/135/234}, no 7,8,9
h) 10(4) cage at R1C8 = {1234}
i) 43(8) cage at R5C7 = {13456789}, no 2
Steps resulting from Prelims.
1a. Naked pair {89} in R3C78, locked for R3 and N3, clean-up: no 1,2 in R2C5
1b. Naked pair {89} in R7C89, locked for R7 and N9, clean-up: no 1 in R9C78
1c. Naked pair {89} in R8C23, locked for R8 and N7
1d. Naked quad {1234} in 10(4) cage at R1C8, locked for N3
2. 45 rule on N1 3 innies R123C3 = 11, no 9 in R12C3
3. 45 rule on N3 1 innie R1C7 = 1 outie R4C9 + 3, R1C7 = {567} -> R4C9 = {234}
4. 45 rule on N4 3 innies R4C23 + R5C3 = 19 = {289/379/469/478/568}, no 1
4a. 2 of {289} must be in R45C3 (R45C3 cannot be {89} which clashes with R8C3) -> no 2 in R4C2
5. 45 rule on N7 2 outies R79C4 = 12 = [39/48/57/75], no 1,2,6, no 3,4 in R9C4
6. 45 rule on N9 2 innies R78C7 = 7 = {16/34}, no 5,7
6a. R9C78 = {27/45} (cannot be {36} which clashes with R78C7)
7. 45 rule on C789 2 outies R89C6 = 1 innie R1C7 + 6
7a. Min R1C7 = 5 -> min R89C6 = 11, no 1
7b. Min R89C6 = 11, max R8C6 = 7 -> min R9C6 = 4
8. 45 rule on N36789 2 outies R56C6 = 1 innie R1C7 + 9
8a. Min R1C7 = 5 -> min R56C6 = 14, no 1,2,3,4
8b. Min R56C6 = 14 -> max R7C56 = 6, no 6,7
9. R1C7 = {567} -> R89C6 (step 7) = 11,12,13, R56C6 (step 8) = 14,15,16 -> R5689C6 = 25,27,29 = {3589/4678},{3789/4689/5679},{5789} (cannot be {3679} because cannot form 11 and 14 separately from this combination, cannot be {4579} = 14+11 = {59}{47} because 20(4) cage at R5C6 = {59}{24} clashes with R89C6 = {47})
9a. {4678} = 14+11 must be {68}{47}, {4689} = 15+11 must be {69}[48], {5679} = 15+11 must be {69}{57} -> no 6 in R89C6
9b. R5689C6 = {3589/4678},{3789/4689/5679} (cannot be {5789} = 16+13 = {79}[58] because 20(4) cage at R5C6 = {79}{13} + R89C6 = [58] clash with R79C4) -> R56C6 = 14,15, R89C6 = 11,12 -> R1C7 = {56}, clean-up: no 4 in R4C9 (step 3)
9c. R78C7 (step 6) = {16/34}
9d. R1C7 + R89C6 = [5+11]/[6+12] = 5[38]/5{47}/6{57} (cannot be 6[39]/6[48] which clash with R78C7), no 9 in R9C6
9e. R178C7 = 5{16}/6{34} (cannot be 5{34} which clashes with R1C7 + R89C6), 6 locked for C7
9f. R56C6 = 14,15 = {59/68/69} (cannot be {78} which clashes with R89C6, while {79} has already been eliminated in step 9b), no 7 in R56C6
9g. R56C6 = 14,15 -> R7C56 = 5,6 = {14/23/15/24}
9h. Hidden killer pair 1,2 in R7C56 and 16(3) cage at R8C4 for N8, R7C56 contains one of 1,2 -> 16(3) cage must contain one of 1,2
9i. 6 in N8 only in 16(3) cage at R8C4 which must contain one of 1,2 = {169/268} -> R9C5 = {89}, R8C45 = {16/26}, 6 locked for R8, clean-up: no 1 in R7C7 (step 6)
9j. 43(8) cage at R5C7 = {13456789}, 9 locked for N6
9k. 43(8) cage at R5C7 = {13456789}, CPE no 1 in R4C7
10. 7 in N3 only in R23C9, locked for C9
10a. 12(3) cage at R8C8 = {156/345} (cannot be {147/246} which clash with R78C7, cannot be {237} which clashes with R4C9, no 2,7
10b. 6 of {156} must be in R9C9 -> no 1 in R9C9
[This elimination in R9C6 proved to be a very important one.]
10c. R9C78 = {27} (hidden pair in N9), locked for R9, clean-up: no 5 in R7C4 (step 5)
10d. 1 in N9 only in R8C789, locked for R8
10e. Naked pair {26} in R8C45, 2 locked for R8 and N8
10f. R8C45 = {26} -> R9C5 = 8 (cage sum), clean-up: no 2 in R3C5
10g. R9C4 = 9 (hidden single in N8) -> R7C4 = 3 (step 5)
10h. R8C6 = 7 (hidden single in N8)
10i. R9C4 = 9 -> R9C23 = 6 = {15}, locked for R9 and N7, R9C6 = 4, R7C7 = 6, R1C7 = 5
10j. R7C56 = {15} -> 20(4) cage at R5C6 = {1568}, 6,8 locked for C6 and N5
10k. Naked pair {67} in R23C9, 6 locked for C9, R9C19 = [63] -> R78C1 = 7 = [43], R4C9 = 2, R8C7 = 1
10l. Naked quad {3589} in R5C7 + R6C789, 3,5,8 locked for N6, 5 also locked for R6
10m. 9(3) cage at R6C1 = {126/234}, 2 locked for R6 and N4
10n. 8 in R4 only in R4C123, locked for N4
10o. 7 in R6 only in R6C45, locked for N5
10p. 2 in C6 only in R123C6, locked for N2
11. 17(4) cage at R1C5 contains 5 = {1259/2357} (cannot be {1457/2456} because 4,6,7 only in R1C5), no 4,6, 2 locked for N2
11a. 7 of {2357} must be in R1C5 -> no 3 in R1C5
12. 13(3) cage at R2C4 = {148/157}, no 6, 1 locked for C4
12a. 8 of {148} must be in R2C4 -> no 4 in R2C4
12b. Killer pair 4,7 in 13(3) cage and R6C4, locked for C4
13. 45 rule on N2 2 innies R1C4 + R3C6 = 1 remaining outie R4C4 + 10
13a. Max R1C4 + R3C6 = 13 -> R4C4 = 1, R1C4 + R3C6 = 11 = [65/83]
13b. 1 in R6 only in 9(3) cage at R6C1 (step 10m) = {126}, 6 locked for R6 and N4, 1 also locked for N4 -> R56C6 = [68]
13c. Naked triple {359} in R6C789, 3,9 locked for R6 and N6 -> R5C7 = 8
13d. Naked pair {47} in R6C45, 4 locked for N5
14. 17(3) cage at R4C1 = {359/458} -> R5C2 = {34}, R45C1 = {58/59}, 5 locked for C1 and N4
14a. 7 in C1 only in R123C1, locked for N1
15. Hidden killer pair 1,2 in R23C1 and R6C1 for C1, R6C1 = {12} -> R23C1 must contain one of 1,2, no 1,2 in R3C2 because 14(3) cage cannot contain both of 1,2
15a. R123C3 (step 2) = 11 = {146/236/245} (cannot be {128} which clashes with R23C1), no 8
16. 45 rule on N4 1 innie R5C3 = 1 outie R3C3 + 2 -> R3C3 = {125}, R5C3 = {347}
16a. 17(3) cage at R3C3 = {179/278/458} (cannot be {359} which clashes with R4C56, ALS block), no 3
16b. 3 in N4 only in R5C23, locked for R5
17. 14(3) cage at R2C1 must contain 1 or 2 (step 15) = {158/167/239/257} (cannot be {248} which clash with R123C3, cannot be {149} which clashes with 17(3) cage at R4C1), no 4
17a. 7,8 of {158/167} must be in R2C1 (14(3) cage cannot be [176] which clashes with R3C9) -> no 1 in R2C1
17b. R123C3 (step 15a) = {146/236/245}
17c. 1 of {146} must be in R3C3 -> no 1 in R12C3
17d. 1 in N1 only in R3C13, locked for R3, clean-up: no 9 in R2C5
17e. 9 in N2 only in 17(4) cage at R1C5 (step 11) = {1259}, no 3,7
18a. R3C13 = {12} (hidden pair in R3), 2 locked for N1, clean-up: no 7 in R5C3 (step 16)
18b. Naked pair {34} in R5C23, 4 locked for R5 and N4
18c. R15C9 = [41]
18d. R123C3 = {146/236} (cannot be {245} because 4,5 only in R2C3), no 5, 6 locked for C3 and N1
18e. Naked pair {12} in R36C3, locked for C3 -> R7C23 = [27]
18f. R4C2 = 7 (hidden single in N4)
18g. Naked pair {35} in R3C26, locked for R3, clean-up: no 7 in R2C5
18h. 13(3) cage at R2C4 (step 12) = {148/157}
18i. 5 of {157} must be in R2C4 -> no 7 in R2C4
18j. 7 in N2 only in R3C45, locked for R3 -> R23C9 = [76], clean-up: no 4 in R2C5
18k. R1C1 = 7 (hidden single in N1) -> R12C2 = 13 = [85/94]
19. R1C4 + R3C6 (step 13a) = [65/83]
19a. Consider placements for R3C2 = {35}
R3C2 = 3 => R1C34 = [68]
or R3C2 = 5 => R3C6 = 3
-> R1C4 + R3C6 = [83], R2C5 = 6 -> R3C5 = 4
19b. R13C2 = [95], R2C1 = 8 -> R3C1 = 1 (cage sum)
and the rest is naked singles.