Prelims
a) R23C1 = {19/28/37/46}, no 5
b) R3C56 = {29/38/47/56}, no 1
c) R4C78 = {39/48/57}, no 1,2,6
d) R45C9 = {19/28/37/46}, no 5
e) R5C23 = {15/24}
f) R78C1 = {12}
g) R8C78 = {19/28/37/46}, no 5
h) 19(3) cage at R6C8 = {289/379/469/478/568}, no 1
i) 19(3) cage at R7C4 = {289/379/469/478/568}, no 1
j) 16(5) cage at R1C8 = {12346}
k) 35(5) cage at R5C5 = {56789}
Steps resulting from Prelims
1a. Naked pair {12} in R78C1, locked for C1 and N7, clean-up: no 8,9 in R23C1
1b. Naked quint {56789} in 35(5) cage at R5C5, locked for D/
1c. Min R9C23 = 7 -> max R9C4 = 7
2. Naked quad {1234} in R1C9 + R2C8 + R3C7 + R4C6, CPE no 1,2,3,4 in R1C8 -> R1C8 = 6, clean-up: no 4 in R8C7
3. 45 rule on C1 2 innies R19C1 = 14 = {59}/[86]
3a. 18(3) cage at R4C1 = {378/468} (cannot be {369/567} which clash with R19C1, cannot be {459} which clashes with R5C23), no 5,9, 8 locked for C1 and N4, clean-up: no 6 in R9C1
3b. Naked pair {59} in R19C1, CPE no 5,9 in R5C5 + R9C9 using diagonals
4. 45 rule on N4 2 outies R37C2 = 5 = [14/23]
4a. 13(3) cage at R3C2 = {139/157/247/256} (cannot be {346} because R3C2 only contains 1,2)
4b. R3C2 = {12} -> no 1,2 in R4C23
4c. Killer triple 3,4,5 in 13(3) cage, 18(3) cage at R4C1 and R5C23, locked for N4
4d. 13(3) cage at R6C2 = {139/247} (cannot be {346} because 3,4 only in R7C2), no 6
4e. 2 in N4 only in R5C23 = {24} or 13(3) cage at R6C2 = {247} -> 13(3) cage at R3C2 = {139/157/256} (cannot be {247} which clashes with R5C23 + 13(3) cage at R6C2, blocking cages), no 4
4f. 9 in N4 only in 13(3) cage at R3C2 = {139} or 13(3) cage at R6C2 = {139} (locking cages), CPE no 1 in R5C2, clean-up: no 5 in R5C3
4g. Min R9C23 = {35} = 8 (R9C23 cannot be {34} which clashes with R7C2) -> max R9C4 = 6
5. 45 rule on C6789 2 outies R34C5 = 7 = [34/43/52/61], clean-up: R3C6 = {5678}
6. 45 rule on R6789 1 outie R5C5 = 1 innie R6C1 + 4, R5C5 = {78}, R6C1 = {34}
6a. Killer pair 3,4 in R23C1 and R6C1, locked for C1
7. 45 rule on C1234 2 innies R23C4 = 1 outie R5C5 + 1
7a. Max R5C5 = 8 -> max R23C4 = 9, no 9 in R23C4
8. 25(4) cage at R1C9 = {1789/3589/4579}, no 2, 9 locked for N3
8a. R1C9 = {134} -> no 1,3,4 in R2C9 + R3C89
8b. 2 on D/ only in R2C8 + R3C7 + R4C6, locked for 16(4) cage at R1C8, no 2 in R4C5, clean-up: no 5 in R3C5 (step 5), no 6 in R3C6
9. 45 rule on N23 1 innie R1C4 = 2 outies R4C56 + 1
9a. Min R4C56 = 3 -> min R1C4 = 4
9b. Max R4C56 = 7 -> max R1C4 = 8
10. R5C5 + R6C1 = [73/84] (step 6)
10a. 18(3) cage at R4C1 (step 3a) = {378/468} = [783/864], no 6 in R4C1, no 7 in R5C1
10b. 18(3) cage + R5C5 = [7837/8648], 8 locked for R5, clean-up: no 2 in R4C9
[Afmob got a similar result with 45 rule on R6789 3(2+1) outies R45C1 + R5C5 = 22 …]
10c. 2 in R4 only in R4C46, locked for N5
11. 45 rule on R1234 2 outies R5C49 = 1 innie R4C1 + 2
11a. R4C1 = {78} -> R5C49 = 9,10
11b. Hidden killer pair 3,9 in R5C49 and 15(3) cage at R5C6 for R5, neither can contain both of 3,9 -> R5C49 and 15(3) must each contain one of 3,9
11c. R5C49 = 9,10 = {36/37/19}, no 2,4,5, 15(3) cage = {159/249/357}, no 6, clean-up: no 6,8 in R4C9
11d. R4C1 + R5C49 = [736]/8{37}/8{19} (cannot be [763] because R45C9 = [73] clashes with R4C1), no 6 in R5C4
[These combinations can be reduced further using a forcing chain, but I’ll leave that for now and hope to find something simpler.]
12. 45 rule on C9 1 outie R3C8 = 1 innie R9C9 + 6, R3C8 = {789}, R9C9 = {123}
13. 45 rule on R9 2 innies R9C15 = 1 outie R8C6 + 10
13a. Max R9C15 = 17 -> max R8C6 = 7
13b. Min R9C15 = 11, no 1 in R9C5
14. 45 rule on C1234 3 outies R125C5 = 18 = {189/279/378/567} (cannot be {369/459} because R5C5 only contains 7,8, cannot be {468} which clashes with R34C5), no 4
15. 45 rule on N2 3 innies R1C46 + R2C6 = 15 = {159/168/249/258/267/348/456} (cannot be {357} which clashes with R3C56)
15a. 7 of {267} must be in R1C4 -> no 7 in R12C6
16. 45 rule on N2 2 outies R12C7 = 1 innie R1C4 + 3, IOU no 3 in R2C7
17. 45 rule on R123 4 outies R4C456 + R5C4 = 1 innie R3C2 + 12
17a. Max R3C2 = 2 -> max R4C456 + R5C4 = 14, no 9 in R45C4, clean-up: no 1 in R5C9 (step 11c), no 9 in R4C9
[I was slow to spot the next two steps even though I’d used the 45 earlier; it’s more powerful now.]
18. 9 in R4 only in 13(3) cage at R3C2 = {139} or R4C78 = {39} -> 3 locked for R4 (locking cages), clean-up: no 4 in R3C5 (step 5), no 7 in R3C6, no 7 in R5C9
18a. 16(5) cage at R1C8 = {12346}, 3 locked for N3
19. R1C4 = R4C56 + 1 (step 9)
19a. R4C56 = [12/42] (cannot be {14} = 5 because no 6 in R1C4) -> R4C6 = 2
19b. R4C56 = [12/42] = 3,6 -> R1C4 = {47}
19c. 2 in N3 only in R12C7, locked for C7, clean-up: no 8 in R8C8
19d. Naked triple {134} in R1C9 + R2C8 + R3C7, locked for N3
[Cracked. The rest is straightforward.]
20. R1C46 + R2C6 (step 15) = {348/456} (cannot be {159/168} because no 1,5,6,8,9 in R1C4) -> R1C4 = 4, R12C6 = [38/56/83]
20a. 19(4) cage at R1C5 = {1279} (hidden quad in N2), 9 locked for C5
20b. R125C5 (step 14) contains 9 = {189/279}
20c. R5C5 = {78} -> no 7 in R12C5
20d. 19(4) = {1279}, 7 locked for C4, clean-up: no 3 in R5C9 (step 11d), no 7 in R4C9
21. R1C9 = 1, R4C9 = 4 -> R5C9 = 6, R4C5 = 1, R3C5 = 6 (step 5) -> R3C6 = 5, R5C4 = 3, R45C1 = [78], R6C1 = 3 (cage sum), R23C1 = [64], clean-up: no 7 in R3C8 (step 12), no 5,8 in R4C78
21a. R3C7 = 3, R2C8 = 4, R4C78 = [93], clean-up: no 6,7 in R8C7, no 1,7 in R8C8
21b. Naked pair {56} in R4C23, locked for R4 and N4 -> R4C4 = 8, placed for D\
21c. R4C23 = {56} -> R3C2 = 2 (cage sum), R5C2 = 4 -> R5C3 = 2, R7C2 = 3
21d. R5C5 = 7, placed for both diagonals
21e. Naked pair {29} in R12C5, locked for C5 and N2
21f. Naked pair {17} in R23C4, locked for C4
22. R1C9 = 1 -> 25(4) cage at R1C9 = {1789}, locked for N3, 7 also locked for C9
22a. Naked pair {25} in R12C7, locked for C7 -> 15(3) cage at R5C6 = [915], R8C7 = 8 -> R8C8 = 2
23. R6C7 = 7 -> R67C6 = 7 = [61], 6 placed for D/
and the rest is naked singles, without using the diagonals.