Prelims
a) R1C56 = {13}
b) R34C9 = {59/68}
c) R4C23 = {69/78}
d) R8C56 = {19/28/37/46}, no 5
e) 22(3) cage at R6C3 = {589/679}
f) 14(4) cage at R6C6 = {1238/1247/1256/1346/2345}, no 9
Steps resulting from Prelims
1a. Naked pair {13} in R1C56, locked for R1 and N2
1b. 22(3) cage at R6C3 = {589/679}, 9 locked for R6
2. 45 rule on R6789 2 innies R6C19 = 4 = {13}, locked for R6
3. 45 rule on N4 2 innies R6C23 = 1 outie R5C4 + 2
3a. Min R6C23 = 7 -> min R5C4 = 5
3b. Max R6C23 = 11 -> max R6C2 = 6
[I was a bit slow in spotting this step; I’ve moved it here to simplify the next step.]
4. 45 rule on N12 2 innies R23C6 = 1 outie R4C4 + 16 -> R4C4 = 1, R23C6 = 17 = {89}, locked for C6 and N2, clean-up: no 1,2 in R8C5
4a. R4C4 = 1 -> R3C34 = 11 = {47/56}/[92], no 2,3,8 in R3C3
4b. 33(7) cage at R3C6 must contain 1 -> R3C7 = 1
4c. 3 in N5 only in R45C56, locked for 33(7) cage, no 3 in R4C7
4d. 3 in C4 only in R789C4, locked for N8, clean-up: no 7 in R8C56
4e. Max R2C45 + R3C5 = 18 -> min R2C3 = 2
5. 45 rule on N78 2 outies R6C26 = 1 innie R9C6
5a. Min R6C26 = 6 -> min R9C6 = 6
5b. R9C6 = {67} -> R6C26 = 6,7 = {24/25}, 2 locked for R6
6. Min R9C6 = 6 -> max R9C78 = 8, no 8,9 in R9C7, no 7,8,9 in R9C8
6a. 14(3) cage at R9C6 = {167/257/347/356}
6b. 1 of {167} must be in R9C8, 6 of {356} must be in R9C6 -> no 6 in R9C8
7. 35(6) cage at R1C7 = {236789/245789/345689}, CPE no 8,9 in R2C8
8. 1,3 in N4 only in R4C1 + R5C123 + R6C1 -> 28(6) cage at R4C1 = {123589/123679/134569/134578}
8a. Hidden killer pair 2,4 in 28(6) cage and R6C2 for N4, 28(6) cage contains one of 2,4 -> R6C2 = {24}
9. 31(7) cage at R6C2 = {1234579/1234678}, 1,3,7 locked for N7
9a. 31(7) cage at R6C2 = {1234579/1234678}, CPE no 2,4 in R89C2
9b. R89C2 = {59/68} (other pairs of 5,6,8,9 clash with 31(7) cage)
9c. “Two-zero” killer pairs 5,9 and 6,8 in 31(7) cage and R89C2, locked for N7
[R6C2 must equal R9C3 because R6C2 “sees” all cells in N7 except for R9C3. One can then use the “clone” step that the value {24} in R6C2 and R9C3 must also be in 16(3) cage at R1C1, eliminating 1 from the 16(3) cage => R2C2 = 1 (hidden single in N1). However, since this doesn’t help much, I’ll leave this for now and wait to see whether this result can be obtained without using a “clone” step. The SS score suggests that this ought to be possible.]
10. 9 in N8 only in R89C45, CPE no 9 in R8C2, clean-up: no 5 in R9C2 (step 9b)
11. 14(4) cage at R6C6 = {1238/1247/1256/1346/2345}
11a. 3 of {1238/2345} must be in R7C4 -> no 2,8 in R7C4
[Eliminating 5 from R6C6 at this stage would be too heavy a step.]
12. 45 rule on R123 2 remaining outies R4C89 = 1 innie R3C6 + 3
12a. Max R3C6 = 9, max R4C89 = 12, min R4C9 = 5 -> max R4C8 = 7
13. Hidden killer pair 8,9 in 35(6) cage at R1C7 and R3C89 for N3, 35(6) cage contains one of 8,9 in N3 -> R3C89 must contain one of 8,9
13a. Killer pair 8,9 in R3C6 and R3C89, locked for R3, clean-up: no 2 in R3C4 (step 4a)
14. 35(6) cage at R1C7 = {236789/245789/345689}
14a. 15(3) cage at R2C8 = {249/258/267/348/357/456}
14b. 7 of {267/357} must be in R23C8 (R23C8 cannot be {26/35} which clash with 35(6) cage), no 7 in R4C8
[This next 45 was hard to spot but proves to be the key breakthrough.]
15. 45 rule on N4578 2 remaining outies R3C6 + R4C7 = 1 innie R9C6 + 8
15a. Min R9C6 = 6, min R3C6 + R4C7 = 14 -> min R4C7 = 5
16. 33(7) cage at R3C6 must contain 2, locked for N5
17. R6C2 = 2 (hidden single in R6), R9C3 = 2 (hidden single in N7)
18. 2 in N1 only in 16(3) cage at R1C1 = {259/268}, no 1,3,4,7
18a. R2C2 = 1 (hidden single in N1)
19. R9C3 = 2 -> 35(6) cage at R8C2 = {236789/245789}, no 1, 7 locked for N8 -> R9C6 = 6, clean-up: no 8 in R8C2 (step 9b), no 4 in R8C56
19a. R89C2 = [59/68] -> R8C4 + R9C45 = {379/478}
19b. Killer pair 8,9 in R8C4 + R9C45 and R8C5, locked for N8
19c. 14(3) cage at R9C6 (step 6a) = {167/356}, no 4
20. R6C26 = R9C6 (step 5), R6C2 = 2, R9C6 = 6 -> R6C6 = 4
21. 4 in N8 only in 35(6) cage at R8C2 (step 19) = {245789} (only remaining combination) -> R89C2 = [59], R8C4 + R9C45 = {478}, locked for N8 -> R8C5 = 9, R8C6 = 1, R1C56 = [13], clean-up: no 6 in R4C3
22. Naked pair {25} in R7C56, locked for R7 -> R7C4 = 3
23. 33(7) cage at R3C6 = {1235679} (only remaining combination), no 8 -> R3C6 = 9, R2C6 = 8, clean-up: no 5 in R4C9
24. 45 rule on N3 2 remaining outies R4C89 = 12 = [39/48], clean-up: no 8 in R3C9
24a. Killer pair 8,9 in R4C23 and R4C89, locked for R4
24b. 2 in R4 only in R4C56, locked for N5
25. R3C8 = 8 (hidden single in R3), R24C8 = 7 = {34} (cannot be {25} because 2,5 only in R2C8), locked for C8
26. 1,2 in N6 only in 15(4) cage at R5C7 = {1239/1248/1257}, no 6
27. 45 rule on N45 1 remaining outie R4C7 = 5, R45C6 = [27], R45C5 = {36}, locked for C5 and N5, R7C56 = [25]
28. 22(3) cage at R6C3 = {589} (only remaining combination, cannot be {679} because 6,7 only in R6C3), locked for R6
[Alternatively R6C78 = {67} (hidden pair in N6), locked for R6.]
29. 17(3) cage at R6C8 = {179/467}, no 8
29a. 7 of {179} must be in R6C8, 4 of {467} must be in R7C9 -> no 6,7 in R7C9
30. 17(3) cage at R8C8 = {278/368/467} (cannot be {458} because R8C8 only contains 2,6,7), no 1,5
31. 5 in N9 only in 14(3) cage at R9C6 (step 19c) = {356} (only remaining combination) -> R9C78 = [35]
31a. 17(3) cage at R8C8 (step 30) = {278/467}, 7 locked for N9
32. R9C1 = 1 (hidden single in R9), R6C19 = [31]
32a. R5C3 = 1 (hidden single in N4)
32b. R8C3 = 3 (hidden single in N7)
33. 16(3) cage at R6C7 = {268} (only remaining combination) -> R6C7 = 6, R78C7 = [82], R6C8 =
33a. Naked triple {467} in R7C123, locked for R7 and N7 -> R7C89 = [19], R8C1 = 8, R4C9 = 8, R3C9 = 6, clean-up: no 5 in R3C34 (step 4a), no 7 in R4C23
33b. R4C23 = [69], R45C5 = [36], R4C8 = 4, R2C8 = 3, R4C1 = 7
34. Naked pair {47} in R3C34, locked for R3 -> R3C2 = 3, R3C5 = 5, R3C1 = 2, R6C5 = 8, R6C3 = 5, R5C12 = [48], R7C1 = 6
35. R12C3 = [86] (hidden pair in C3), R1C4 = 6 (hidden single in R1), R1C2 = 4 (cage sum)
and the rest is naked singles.
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