Prelims
a) R12C9 = {18/27/36/45}, no 9
b) R23C1 = {16/25/34}, no 7,8,9
c) R2C45 = {49/58/67}, no 1,2,3
d) R3C45 = {69/78}
e) R3C89 = {18/27/36/45}, no 9
f) R45C2 = {39/48/57}, no 1,2,6
g) R5C78 = {17/26/35}, no 4,8,9
h) R7C12 = {17/26/35}, no 4,8,9
i) R78C5 = {17/26/35}, no 4,8,9
j) R89C1 = {39/48/57}, no 1,2,6
k) R8C34 = {49/58/67}, no 1,2,3
l) R89C8 = {89}
m) R9C45 = {18/27/36/45}, no 9
n) 8(3) cage at R3C3 = {125/134}
o) 19(3) cage in N5 = {289/379/469/478/568}, no 1
p) 10(3) cage in N7 = {127/136/145/235}, no 8,9
q) 11(3) cage in N9 = {128/137/146/236/245}, no 9
r) 28(4) cage in N1 = {4789/5689}, no 1,2,3
s) 14(4) cage at R1C5 = {1238/1247/1256/1346/2345}, no 9
Steps resulting from Prelims
1a. Naked pair {89} in R89C8, locked for C8 and N9, clean-up: no 1 in R3C9
1b. 8(3) cage at R3C3 = {125/134}, 1 locked for C3
1c. 28(4) cage in N1 = {4789/5689}, 8,9 locked for N1
1d. 9 in C9 only in R456C9, locked for N6
2. R2C45 = {49/58} (cannot be {67} which clashes with R3C45), no 6,7
2a. Killer pair 8,9 in R2C45 and R3C45, locked for N2
2b. 14(4) cage at R1C5 = {1238/1247/1256/1346} (cannot be {2345} which clashes with R2C45)
3. 45 rule on N3 2 innies R23C7 = 11 = [29]/{38/47/56}, no 1, no 2 in R3C7
3a. 14(4) cage at R1C5 (step 2b) = {1238/1247/1256/1346}, 1 locked for N2
4. 45 rule on N7 2 innies R78C3 = 15 = {69/78}, clean-up: no 8,9 in R8C4
4a. Hidden killer pair 8,9 in R78C3 and R89C1 for N7, R78C3 contains one of 8,9 -> R89C1 must contain one of 8,9 -> R89C1 = {39/48} (cannot be {57} which doesn’t contain one of 8,9), no 5,7
4b. R23C1 = {16/25} (cannot be {34} which clashes with R89C1), no 3,4
5. 45 rule on N1 3 innies R2C3 + R3C23 = 10 = {136/235} (cannot be {127/145} which clash with R23C1), no 4,7
6. 4,7 in N1 only in 28(4) cage = {4789} (only remaining combination), no 5,6
7. 45 rule on N4 2 innies R4C1 + R6C3 = 1 outies R3C3 + 12
7a. Min R4C1 + R6C3 = 13, no 1,2,3
8. 10(3) cage in N7 = {127/145/235} (cannot be {136} which clashes with R7C12), no 6
9. 45 rule on N9 2(1+1) outies R6C9 + R9C6 = 11 = {29/38/47/56}, no 1 in R6C9 + R9C6
10. 45 rule on R6789 1 innie R6C5 = 1 outie R5C1 + 1, no 9 in R5C1
11. 45 rule on N5 3 innies R4C6 + R6C46 = 11 = {128/137/146/236/245}, no 9
12. 45 rule on N4 4 innies R4C1 + R456C3 = 20 = {1289/1469/1478/1568/2567} (cannot be {1379} because no 4 in R3C3, cannot be {2369/2378/2468} because 8(3) cage at R3C3 cannot contain both of 2,3 or both of 2,4, cannot be {2459/3458/3467} which clash with R45C2), no 3 in R45C3
12a. 6,7,8,9 only in R4C1 + R6C3 -> no 4,5 in R4C1 + R6C3
13. 6 in N1 only in R2C13 + R3C23, CPE no 6 in R4C1
14. 45 rule on N1 2 outies R1C4 + R4C1 = 1 innie R3C3 + 8
14a. Max R3C3 = 5 -> max R1C4 + R4C1 = 13, min R4C1 = 7 -> max R1C4 = 6
[Ed wrote “This one is technically easier than the last couple. No tricks or really advanced moves needed.” Even so I’ve used a short forcing chain which cracks the puzzle; after that it’s straightforward.]
15. Consider placement for 1 in N1
R2C1 = 1 => R3C1 = 6 => R3C45 = {78}, locked for R3
or 1 in R3C123 => no 1 in R3C8 => no 8 in R3C9
-> no 8 in R3C9, clean-up: no 1 in R3C8
16. Hidden killer pair 8,9 in R3C45 and R3C7 for R3, R3C45 must contain one of 8,9 -> R3C7 = {89}, clean-up: R2C7 = {23} (step 3)
17. Combined cage R23C45 = 13(2) + 15(2) = 28(4) = {4789/5689}
17a. 14(4) cage at R1C5 (step 2b) = {1247/1256} (cannot be {1346} which clashes with R23C45), no 3 -> R2C7 = 2, R1C56 + R2C6 = {147/156}, R3C7 = 9 (step 3), clean-up: no 7 in R12C9, no 5 in R3C1, no 6 in R3C45, no 7 in R3C89, no 6 in R5C8
18. Naked pair {78} in R3C45, locked for R3 and N2, clean-up: no 5 in R2C45
18a. Naked pair {49} in R2C45, locked for R2 and N2, clean-up: no 5 in R1C9
18b. R1C4 + R3C6 = {23} (hidden pair in N2)
19. 4 in R3 only in R3C89 = {45}, locked for R3 and N3
19a. 6 in R3 only in R3C12, locked for N1, clean-up: no 1 in R3C1
19b. Killer pair 3,5 in R2C3 and 8(3) cage at R3C3, locked for C3
20. 11(3) cage in N9 = {146/236} (cannot be {137} which clashes with R12C9, cannot be {245} which clashes with R3C9), no 5,7, 6 locked for C9 and N9, clean-up: no 3 in R12C9, no 5 in R9C6 (step 9)
20a. Naked pair {18} in R12C9, locked for C9 and N3, clean-up: no 4 in 11(3) cage in N9, no 3 in R9C6 (step 9)
20b. Naked triple {236} in 11(3) cage, locked for C9 and N9, clean-up: no 8,9 in R9C6 (step 9)
21. 14(3) cage at R6C9 = {149} (only remaining combination) -> R6C9 = 9, R7C78 = {14}, locked for R7 and N9, R9C6 = 2 (step 9), R3C6 = 3, R1C4 = 2, clean-up: no 7 in R7C12, no 6 in R7C5, no 6,7 in R8C5, no 7 in R9C45
21a. Naked pair {57} in R89C7, locked for C7, clean-up: no 1,3 in R5C8
21b. 7 in N3 only in R12C8, locked for C8, clean-up: no 1 in R5C7
22. R2C3 + R3C23 (step 5) = {136} (only remaining combination, cannot be {235} because 3,5 only in R2C3) -> R2C3 = 3, R3C23 = [61], R23C1 = [52], R4C1 = 7 (cage sum), clean-up: no 5 in R45C2, no 3 in R7C2
22a. R5C9 = 7 (hidden single in C9)
22b. R45C3 = {25} (hidden pair in C3), locked for N4
23. R3C67 = [39] = 12 -> R4C67 = 9 = {18} (cannot be {45} which clashes with R4C9), locked for R4, clean-up: no 4 in R5C2
24. 9 in N4 only in R45C2 = {39}, locked for C2 and N4
25. 13(3) cage in N4 = {148} (only remaining combination), locked for N4 -> R6C3 = 6, clean-up: no 9 in R78C3 (step 4), no 4,7 in R8C4
25a. Naked pair {78} in R78C3, locked for C3 and N7 -> R9C3 = 4, R1C3 = 9, clean-up: no 5 in R9C45
25b. R7C1 = 6 (hidden single in C1), R7C2 = 2, R7C9 = 3, R9C9 = 6, R8C9 = 2, clean-up: no 5 in R8C5, no 3 in R9C45
25c. 1 in C1 only in R56C1, locked for N4
26. Naked pair {18} in R9C45, locked for R9 and N8 -> R8C5 = 3, R7C5 = 5, R8C4 = 6, R8C3 = 7, R89C1 = [93], R8C6 = 4
26a. 4 in N6 only in 14(3) cage = {347} (only remaining combination) -> R4C89 = [34], R3C89 = [45], R45C2 = [93], R5C7 = 6, R5C8 = 2, R7C78 = [41], R6C8 = 5, R45C3 = [25]
27. R4C45 = [56] -> R5C4 = 4 (cage sum), R2C45 = [94]
27a. Naked pair {89} in R5C56, locked for R5 and N5 -> R4C6 = 1
and the rest is naked singles.