Prelims
a) R1C12 = {17/26/35}, no 4,8,9
b) R2C12 = {18/27/36/45}, no 9
c) R34C5 = {49/58/67}, no 1,2,3
d) R3C67 = {49/58/67}, no 1,2,3
e) R34C8 = {69/78}
f) R5C89 = {16/25/34}, no 7,8,9
g) R67C5 = {69/78}
h) R67C8 = {15/24}
i) R7C67 = {19/28/37/46}, no 5
j) R8C12 = {19/28/37/46}, no 5
k) R9C12 = {49/58/67}, no 1,2,3
l) 10(3) cage at R9C7 = {127/136/145/235}, no 8,9
m) 12(4) cage at R1C4 = {1236/1245}, no 7,8,9
n) 36(8) cage at R3C1 = {12345678}, no 9
Steps resulting from Prelims
1a. 12(4) cage at R1C4 = {1236/1245}, 1,2 locked for N2
1b. 9 in N1 only in R123C3, locked for C3 and 24(4) cage at R1C3, no 9 in R3C4
[and one more in step 3]
2. R34C5 = {49/58} (cannot be {67} which clashes with R67C5), no 6,7 in R34C5
2a. Killer pair 8,9 in R34C5 and R67C5, locked for C5
3. 9 in N4 only in R46C2, locked for C2, clean-up: no 1 in R8C1, no 4 in R9C1
3a. 45 rule on C12 2 innies R46C2 = 14 = {59} (only remaining combination), locked for C2 and N4, clean-up: no 3 in R1C1, no 4 in R2C1, no 8 in R9C1
3b. 36(8) cage at R3C1 = {12345678}, 5 locked for C1, clean-up: no 3 in R1C2, no 4 in R2C2, no 8 in R9C2
3c. Min R4C2 = 5 -> max R4C34 = 7, no 7,8,9 in R4C34
4. R2C12 = {18/36} (cannot be {27} which clashes with R1C12), no 2,7
4a. Killer pair 1,6 in R1C12 and R2C12, locked for N1
5. R3C9 = 1 (hidden single in R3), clean-up: no 6 in R5C8
5a. 40(7) cage at R1C6 = {1456789} (only remaining combination), no 2,3
6. 2,3 in N3 only in 13(3) cage at R1C7 = {238} (only remaining combination), locked for R1 and N3, clean-up: no 6 in R1C12, no 5 in R3C6, no 7 in R4C8
7. Naked pair {17} in R1C12, locked for R1 and N1, clean-up: no 8 in R2C12
7a. Naked pair {36} in R2C12, locked for R2 and N2
8. 12(4) cage at R1C4 = {1245} (only remaining combination) -> R1C45 = {45}, locked for R1 and N2 -> R1C3 = 9, R1C6 = 6, clean-up: no 7,9 in R3C7, no 8,9 in R4C5, no 4 in R7C7
8a. Naked pair {45} in R14C5, locked for C5
9. R3C4 = 3 (hidden single in N2)
9a. R1C3 + R3C4 = [93] = 12 -> R23C3 = 12 = {48}, locked for C3 and N1 -> R3C12 = [52], clean-up: no 8 in R3C6, no 8 in R8C1
9b. 7 in N2 only in R23C6, locked for C6, clean-up: no 3 in R7C7
10. R89C1 = [29] (hidden pair in C1), R8C2 = 8, R9C2 = 4
11. 18(3) cage at R5C3 = {279/369/378/567} (cannot be {189/459/468} because 4,5,8,9 only in R5C4), no 1,4
11a. 5,8,9 only in R5C4 -> R5C4 = {589}
12. 45 rule on R12 1 remaining innie R2C3 = 1 remaining outie R4C9 -> R4C9 = {48}
13. 8 in R9 only in R9C46, locked for N8, clean-up: no 7 in R6C5, no 2 in R7C7
14. Naked quint {13567} in R7C12 + R789C3, CPE no 1,5,6,7 in R7C4
14a. 45 rule on N7 2 innies R7C12 = 1 outie R7C4
14b. Min R7C12 = 4 -> min R7C4 = 4
14c. R7C4 = {49} -> R7C12 = {13/36}, no 7, 3 locked for R7 and N7, clean-up: no 7 in R7C7
14d. 3 in C3 only in R456C3, locked for N4
15. 12(3) cage at R4C2 = {129/156/345} (only combinations contains 5 or 9 for R4C2)
15a. 5 of {156/345} -> no 5 in R4C4
16. 7 in N7 only in R789C3, locked for C3
16a. 18(3) cage at R5C3 (step 11) = {279/369/378/567}
16b. 7 of {279} must be in R5C5 -> no 2 in R5C5
17. 16(3) cage at R6C2 = {169/259/349/358} (cannot be {178/268/367} because R6C2 only contains 5,9, cannot be {457} because 4,7 only in R6C4), no 7
17a. 2 of {259} must be in R6C3 -> no 2 in R6C4
18. R5C5 = 7 (hidden single in N5), clean-up: no 8 in R6C5
19. Naked pair {69} in R67C5, locked for C5 -> R3C5 = 8, R4C5 = 5, R1C45 = [54], R23C3 = [84], R3C7 = 6, R3C6 = 7, R2C6 = 9, R3C8 = 9, R4C8 = 6, R46C2 = [95], R4C9 = 8 (step 12), clean-up: no 1 in R5C8, no 4 in R7C6, no 1 in R7C7, no 1 in R7C8
20. R7C7 = 8 (hidden single in N9), R7C6 = 2, clean-up: no 4 in R6C8
21. Naked pair {13} in R89C5, locked for C5 and N8 -> R2C45 = [12]
22. R89C5 = {13} = 4, 7 in N8 only in 17(4) cage at R7C4 = {1367} (only remaining combination) -> R89C4 = {67}, locked for C4 and N8 -> R7C5 = 9, R7C4 = 4, R4C4 = 2, R4C3 = 1 (cage sum), R6C5 = 6, R89C6 = [58], R7C8 = 5, R6C8 = 1
23. 3 in R4 only in R4C67, locked for 29(6) cage at R4C6 -> R6C6 = 4, R5C6 = 1, R4C6 = 3, R4C7 = 7, R4C1 = 4
24. 4 in N6 only in R5C89 = {34}, locked for R5 and N6 -> R5C12 = [86], R5C34 = [29], R5C7 = 5, R2C7 = 4
25. 37(7) cage contains 5 so must also contain 3, locked for R8 and N9
26. 10(3) cage at R9C7 = {127} (only remaining combination), locked for R9 and N9 -> R7C9 = 6
27. Naked triple {349} in R8C789, locked for 37(7) cage at R6C9 -> R6C9 = 2
and the rest is naked singles.
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