Prelims
a) R1C34 = {69/78}
b) R1C67 = {49/58/67}, no 1,2,3
c) R34C1 = {49/58/67}, no 1,2,3
d) R34C9 = {69/78}
e) R67C1 = {69/78}
f) R67C9 = {49/58/67}, no 1,2,3
g) R9C34 = {49/58/67}, no 1,2,3
h) R9C67 = {69/78}
i) 10(3) cage at R1C8 = {127/136/145/235}, no 8,9
j) 10(3) cage at R4C2 = {127/136/145/235}, no 8,9
k) 10(3) cage at R4C8 = {127/136/145/235}, no 8,9
1. R1C67 = {49/58} (cannot be {67} which clashes with R1C34), no 6,7
1a. Killer pair 8,9 in R1C34 and R1C67, locked for R1
2. R34C1 = {49/58} (cannot be {67} which clashes with R67C1), no 6,7
2a. Killer pair 8,9 in R34C1 and R67C1, locked for C1
3. R67C9 = {49/58} (cannot be {67} which clashes with R34C9), no 6,7
3a. Killer pair 8,9 in R34C9 and R67C9, locked for C9
4. R9C34 = {49/58} (cannot be {67} which clashes with R9C67), no 6,7
4a. Killer pair 8,9 in R9C34 and R9C67, locked for R9
5. 45 rule on N1 1 outie R4C1 = 1 innie R1C3 -> R1C3 = {89}, R4C1 = {89}, clean-up: no 8,9 in R1C4, no 8,9 in R3C1
6. 45 rule on N3 1 outie R1C6 = 1 innie R3C9 -> R1C6 = {89}, R3C9 = {89}, clean-up: no 8,9 in R1C7, no 8,9 in R4C9
7. 45 rule on N7 1outie R9C4 = 1 innie R7C1 -> R7C1 = {89}, R9C4 = {89}, clean-up: no 8,9 in R6C1, no 8,9 in R9C3
8. 45 rule on N9 1 outie R6C9 = 1 innie R9C7 -> R6C9 = {89}, R9C7 = {89}, clean-up: no 8,9 in R7C9, no 8,9 in R9C6
9. Hidden killer pair 8,9 in R4C1 and R456C3 for N4, R4C1 contains one of 8,9 -> R456C3 must contain one of 8,9
9a. Killer pair 8,9 in R1C3 and R456C3, locked for C3
10. Hidden killer pair 8,9 in R456C7 and R6C9 for N6, R6C9 contains one of 8,9 -> R456C7 must contain one of 8,9
10a. Killer pair 8,9 in R456C7 and R9C7, locked for C7
11. R1C6 = R3C9 (step 6), R36C9 = {89} -> R1C6 + R6C9 form naked pair {89} (locking cages), CPE no 8,9 in R6C6
12. R9C4 = R7C1 (step 7), R47C1 = {89} -> R4C1 + R9C4 form naked pair {89} (locking cages), CPE no 8,9 in R4C4
13. R4C1 = R1C3 (step 5), R47C1 = {89} -> R1C4 + R6C1 form naked pair {67} (locking cages using 15(2) cage sums), CPE no 6,7 in R1C1 + R6C4
14. R6C9 = R9C7 (step 8), R36C9 = {89} -> R4C9 + R9C6 form naked pair {67} (locking cages using 15(2) cage sums), CPE no 6,7 in R4C6 + R9C9
15. R1C6 = R3C9 (step 6), R36C9 = {89} -> R1C7 + R7C9 form naked pair {45} (locking cages using 13(2) cage sums), CPE no 4,5 in R1C9 + R78C7
16. R9C4 = R7C1 (step 7), R47C1 = {89} -> R3C1 + R9C3 form naked pair {45} (locking cages using 13(2) cage sums), CPE no 4,5 in R23C3 + R9C1
17. R1C3 = R4C1 (step 5), R1C36 = {89} -> R1C7 + R3C1 form naked pair {45} (locking cages using 13(2) cage sums), CPE no 4,5 in R1C12 + R3C78
18. R6C9 = R9C7 (step 8), R9C47 = {89} -> R7C9 + R9C3 form naked pair {45} (locking cages using 13(2) cage sums), CPE no 4,5 in R7C23 + R9C89
19. R1C6 = R3C9 (step 6), R1C36 = {89} -> R1C4 + R4C9 form naked pair {67} (locking cages using 15(2) cage sums), CPE no 6,7 in R1C9 + R4C4
20. R7C1 = R9C4 (step 7), R9C47 = {89} -> R6C1 + R9C6 form naked pair {67} (locking cages using 15(2) cage sums), CPE no 6,7 in R6C6 + R9C1
21. 14(3) cage at R1C1 can only contain one of 1,2,3, R1C1 = {123} -> no 1,2,3 in R1C2 + R2C1
21a. Naked pair {67} in R1C24, locked for R1
22. 14(3) cage at R9C9 can only contain one of 1,2,3, R1C1 = {123} -> no 1,2,3 in R8C9 + R9C8
22a. Naked pair {67} in R9C68, locked for R9
23. 10(3) cage at R1C8 = {127/136/235} (cannot be {145} which clashes with R1C7), no 4
24. 13(3) cage at R8C1 can only contain one of 1,2,3 (because it doesn’t contain 8 or 9) -> no 1,2,3 in R8C1 + R9C2
24a. Naked pair {45} in R9C23, locked for R9 and N7
24b. Naked pair {67} in R68C1, locked for C1
24c. Naked pair {45} in R23C1, locked for C1 and N1
25. 10(3) cage at R4C2 = {127/136/235} (cannot be {145} which clashes with R9C2), no 4
[Cracked. The rest is fairly straightforward.]
26. R9C2 = 4 (hidden single in C2), R9C3 = 5, R9C4 = 8, R9C7 = 9, R9C6 = 6, R6C9 = 9 (hidden single in N6), R9C7 = 4, R3C9 = 8, R4C9 = 7, R1C6 = 8 (step 6), R1C7 = 5, R1C3 = 9, R1C4 = 6, R4C1 = 9 (hidden single in N4), R3C1 = 4, R7C1 = 8, R6C1 = 7
27. R8C1 = 6, R9C2 = 4, R9C1 = 3 (cage sum), R8C9 = 5, R9C8 = 7, R9C9 = 2 (cage sum), R9C5 = 1, R1C2 = 7, R2C1 = 5, R1C1 = 2 (cage sum)
28. Naked pair {13} in R1C89, locked for R1 and N3, R2C9 = 6
29. R1C5 = 4, R23C5 = 8 = [35], R2C23 = [81]
30. R5C1 = 1, R5C9 = 3, R1C89 = [31]
30a. 10(3) cage at R4C2 = {235} (only remaining combination), locked for C2 and N4 -> R3C23 = [63]
31. 10(3) cage at R4C8 = {145} (only remaining combination), locked for C8 and N6 -> R78C8 = [68]
31a. Naked pair {29} in R23C8, locked for N3 -> R23C7 = [47]
32. 18(3) cage at R5C5 = {279/567} (cannot be {459} which clashes with R5C8, cannot be {468} which clashes with R5C3), no 4,8
32a. R5C3 = 8 (hidden single in R5)
32b. 6 in R5 only in R5C57 -> 18(3) cage = {567} (only remaining combination) -> R5C7 = 6, R5C56 = [75], R5C2 = 2, R5C8 = 4, R5C4 = 9
32c. Naked pair {29} in R78C5, locked for C5 and N8
33. 13(3) cage at R2C4 = {247} (only remaining combination) -> R2C4 = 7, R4C4 = 4, R3C4 = 2
and the rest is naked singles.
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