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Prelims

a) R34C4 = {49/58/67}, no 1,2,3
b) R34C6 = {18/27/36/45}, no 9
c) R45C1 = {39/48/57}, no 1,2,6
d) R45C5 = {69/78}
e) R45C9 = {19/28/37/46}, no 5
f) R67C1 = {17/26/35}, no 4,8,9
g) R67C5 = {17/26/35}, no 4,8,9
h) R67C9 = {69/78}
i) R78C2 = {59/68}
j) R78C8 = {15/24}
k) R89C5 = {69/78}
l) R9C34 = {12}
m) R9C67 = {18/27/36/45}, no 9
n) 8(3) cage at R3C1 = {125/134}
o) 11(3) cage at R7C4 = {128/137/146/236/245}, no 9
p) 19(3) cage at R8C1 = {289/379/469/478/568}, no 1

Steps resulting from Prelims
1a. Naked pair {12} in R9C34, locked for R9, clean-up: no 7,8 in R9C67
1b. 8(3) cage at R3C1, CPE no 1 in R12C2

2. Naked quad {6789} in R4589C5, locked for C5, clean-up: no 1,2 in R67C5
2a. Naked pair {35} in R67C5, locked for C5
2b. Naked triple {124} in R123C5, locked for N2, clean-up: no 9 in R4C4, no 5,7,8 in R4C6

3. 45 rule on C1234 1 innie R2C4 = 9, clean-up: no 4 in R4C4

4. 45 rule on C6789 1 innie R2C6 = 5, clean-up: no 8 in R4C4, no 4 in R4C6, no 4 in R9C7

5. 19(3) cage at R8C1 = {379/478} (cannot be {289/469/568} which clash with R78C2), no 2,5,6, 7 locked for N7, clean-up: no 1 in R6C1
5a. Killer pair 8,9 in R78C2 and 19(3) cage, locked for N7

6. 5 in R9 only in R9C789, locked for N9, clean-up: no 1 in R78C8
6a. Naked pair {24} in R78C8, locked for C8 and N9

7. 45 rule on R1234 3 outies R5C159 = 21 = {489/579/678}, no 1,2,3, clean-up: no 9 in R4C1, no 7,8,9 in R4C9

8. 45 rule on C12 2 innies R56C2 = 13 = {49/67} (cannot be {58} which clashes with R78C2)
8a. Killer pair 6,9 in R56C2 and R78C2, locked for C2

9. 45 rule on C12 2 outies R67C3 = 10 = [73/82/91] (cannot be {46} which clashes with R56C2)

10. 45 rule on C89 2 innies R56C8 = 8 = {17/35}
10a. R45C9 = [19/28]/{46} (cannot be [37] which clashes with R56C8), no 3 in R4C9, no 7 in R5C9

11. 45 rule on C89 2 outies R67C7 = 8 = {17}/[26/53], no 4,8,9, no 3,6 in R6C7

12. 45 rule on N9 4 innies R7C79 + R89C7 = 26 = {3689/5678}, no 1, clean-up: no 7 in R6C7 (step 11)
12a. R8C9 = 1 (hidden single in N9) -> R9C89 = 12 = {39/57}, no 6,8, clean-up: no 9 in R5C9
12b. Killer pair 3,5 in R9C67 and R9C89, locked for R9
12c. Killer pair 6,8 in R45C9 and R67C9, locked for C9

13. R5C159 (step 7) = {489/678} (cannot be {579} because R5C9 only contains 4,6,8), no 5, 8 locked for R5, clean-up: no 7 in R4C1

14. 19(3) cage at R8C1 (step 5) = {478} (only remaining combination, cannot be {379} which clashes with R9C89), locked for N7, clean-up: no 6 in R78C2
14a. Naked pair {59} in R78C2, locked for C2 and N7, clean-up: no 4 in R56C2 (step 8), no 3 in R6C1
14b. Naked pair {67} in R56C2, locked for C2 and N4, no 5 in R4C1, no 1,2 in R7C1, no 3 in R7C3 (step 9)

15. R45C1 = [39] (cannot be {48} which clashes with 19(3) cage at R8C1, ALS block), R7C1 = 6 -> R6C1 = 2, R6C3 = 8 -> R7C3 = 2 (step 9), R78C2 = [42], R9C34 = [12], clean-up: no 6 in R3C6, no 6 in R4C5, no 9 in R6C9, no 7 in R7C9

16. R8C3 = 3 -> R78C4 = 8 = [17], clean-up: no 6 in R34C4, no 8 in R89C5
16a. R34C4 = [85], R45C3 = [45], R67C5 = [35], R78C2 = [95], R7C9 = 8 -> R6C9 = 7, R56C2 = [76], R56C4 = [64], R5C5 = 8 -> R4C5 = 7, R5C9 = 4 -> R4C9 = 6
16b. R7C6 = 3, R7C7 = 7 -> R6C7 = 1 (step 11), R56C8 = [35], R9C8 = 9 -> R9C9 = 3 (cage sum), R9C56 = [64], R9C2 = 8, R8C1 = 4

17. R4C2 = 1, R3C1 = 5 -> R3C2 = 2 (cage sum)

and the rest is naked singles.

Prelims

a) R34C4 = {49/58/67}, no 1,2,3
b) R34C6 = {49/58/67}, no 1,2,3
c) R45C1 = {39/48/57}, no 1,2,6
d) R45C5 = {69/78}
e) R45C9 = {15/24}
f) R67C1 = {17/26/35}, no 4,8,9
g) R67C5 = {17/26/35}, no 4,8,9
h) R67C9 = {69/78}
i) R78C2 = {59/68}
j) R78C8 = {19/28/37/46}, no 5
k) R89C5 = {29/38/47/56}, no 1
l) R9C34 = {16/25/34}, no 7,8,9
m) R9C67 = {18/27/36/45}, no 9
n) 11(3) cage at R5C3 = {128/137/146/236/245}, no 9
o) 11(3) cage at R7C4 = {128/137/146/236/245}, no 9
p) 19(3) cage at R8C1 = {289/379/469/478/568}, no 1

1. 45 rule on C1234 1 innie R2C4 = 9, clean-up: no 4 in R34C4, no 4 in R4C6

2. 45 rule on C6789 1 innie R2C6 = 5, clean-up: no 8 in R34C6, no 8 in R4C4, no 4 in R9C7

3. 19(3) cage at R8C1 = {379/478} (cannot be {289/469/568} which clash with R78C2), no 2,5,6, 7 locked for N7, clean-up: no 1 in R6C1
3a. Killer pair 8,9 in R78C2 and 19(3) cage, locked for N7

4. 45 rule on R1234 3 outies R5C159 = 21 = {489/579} (cannot be {678} because no 6,7,8 in R5C9), no 1,2,3,6, 9 locked for R5, clean-up: no 9 in R4C1, no 9 in R4C5, no 4,5 in R4C9
4a. R5C9 = {45} -> no 4,5 in R5C1, clean-up: no 7,8 in R4C1

5. 45 rule on C12 2 outies R67C3 = 14 = [86/95]
5a. Killer pair 5,6 in R78C2 and R7C3, locked for N7, clean-up: no 2,3 in R6C1, no 1,2 in R9C4
[There’s also a CPE which I only spotted at step 13.]

6. 45 rule on C12 2 innies R56C2 = 9 = {18/27/36/45}, no 9

7. 45 rule on C89 2 outies R67C7 = 8 = {17/26/35}, no 4,8,9

8. 45 rule on C89 2 innies R56C8 = 8 = {17/26/35}, no 4,8,9

9. 9 in C5 only in R45C5 = [69] or in R89C5 = {29} -> R67C5 = {17/35} (cannot be {26}, blocking cages), no 2,6 in R67C3

10. 45 rule on C5 3 innies R123C5 = 11 = {128/146/236} (cannot be {137} which clashes with R67C5), no 7 in R123C5
10a. Killer pair 6,8 in R123C5 and R45C5, locked for C5, clean-up: no 3,5 in R89C5
10b. Killer pair 7,9 in R45C5 and R89C5, locked for C5, clean-up: no 1 in R67C5
10c. Naked pair {35} in R67C5, locked for C5
10d. 1 in C5 only in R123C5, locked for N2
10e. 3 in N2 only in R1C46, locked for R1

[An alternative way to do steps 9 and 10 is
45 rule on C5 3 innies R123C5 = {128/137/146/236}
Killer quad 6,7,8,9 in R123C5, R45C5 and R89C5, locked for C5, clean-up: no 1,2 in R67C5
Naked pair {35} in R67C5, locked for C5 …]

11. 45 rule on N1 3(1+2) outies R1C4 + R4C23 = 8
11a. Min R1C4 = 2 -> max R4C23 = 6, no 6,7,8,9 in R4C23
11b. Min R4C23 = 3 -> no 6,7,8 in R1C4

12. 45 rule on N3 3(1+2) outies R1C6 + R4C78 = 19
12a. Max R1C6 = 8 -> min R4C78 = 11, no 1 in R4C78

13. 5,6 in N7 only in R78C2 + R7C3, CPE no 5,6 in R56C2, clean-up: no 3,4 in R56C2 (step 6)
13a. Killer triple 7,8,9 in R5C1, R56C2 and R6C3, locked for N4, clean-up: no 1 in R7C1
13b. 1 in N7 only in R89C3, locked for C3
13c. 1 in N4 only in R456C2, locked for C2

14. 7 in C3 only in R123C3, locked for N1
14a. Hidden killer pair 8,9 in R123C3 and R6C3 for C3, R6C3 = {89} -> R123C3 must contain one of 8,9
14b. 25(5) cage at R1C3 contains 7 and one of 8,9 = {23479/23578}, no 6

15. R67C3 = [86] (cannot be [95] which clashes with 25(5) cage at R1C3), clean-up: no 4 in R4C1, no 1 in R56C2 (step 6), no 9 in R6C9, no 8 in R78C2, no 7 in R7C9
15a. Naked pair {27} in R56C2, locked for C2 and N4, R5C1 = 9 -> R4C1 = 3, R7C1 = 2 -> R6C1 = 6, R6C9 = 7 -> R7C9 = 8, R56C2 = [72], R5C5 = 8 -> R4C5 = 7
15b. R5C159 (step 4) = {489} (only remaining combination), R5C9 = 4 -> R4C9 = 2
15c. Clean-ups: no 6 in R3C4, no 6 in R3C6, no 1,6 in R56C8 (step 8), no 1 in R7C7 (step 7), no 2,4 in R8C8, no 4 in R89C5, no 1 in R9C6
[No further clean-ups.]

16. R45C3 = [45], R56C8 = [35] -> R67C7 = [17], R67C5 = [35], R6C4 = 4 -> R5C4 = 2 (cage sum), R6C6 = 9, R4C6 = 6 -> R3C6 = 7, R34C4 = [85], R1C4 = 3, R789C4 = [176], R89C3 = [31]
16a. R9C67 = {45} (only remaining combination) -> R9C6 = 4, R9C7 = 5

17. R4C2 = 1 -> R3C12 = 11 = [56]

18. 18(3) cage at R3C8 = {189} (only remaining combination) -> R4C8 = 8, R3C89 = {19}, locked for R3 and N3

and the rest is naked singles.