Prelims
a) R12C1 = {16/25/34}, no 7,8,9
b) R12C2 = {16/25/34}, no 7,8,9
c) R12C8 = {18/27/36/45}, no 9
d) R12C9 = {19/28/37/46}, no 5
e) R34C1 = {29/38/47/56}, no 1
f) R34C9 = {29/38/47/56}, no 1
g) R45C5 = {49/58/67}, no 1,2,3
h) R5C12 = {59/68}
i) R5C89 = {16/25/34}, no 7,8,9
j) R67C5 = {89}
k) 11(3) cage at R1C5 = {128/137/146/236/245}, no 9
l) 19(3) cage at R1C6 = {289/379/469/478/568}, no 1
m) 19(3) cage at R2C6 = {289/379/469/478/568}, no 1
n) 19(3) cage at R6C1 = {289/379/469/478/568}, no 1
o) 22(3) cage at R8C3 = {589/679}
p) 14(4) cage at R7C1 = {1238/1247/1256/1346/2345}, no 9
q) 23(6) cage at R3C8 = {123458/123467}, no 9
Steps resulting from Prelims
1a. Naked pair {89} in R67C5, locked for C5, clean-up: no 4,5 in R45C5
1b. Naked pair {67} in R45C5, locked for C5 and N5
1c. 22(3) cage at R8C3 = {589/679}, 9 locked for N7
2. 45 rule on C5 2 innies R89C5 = 4 = {13}, locked for C5 and N8, CPE no 1,3 in R9C78
2a. Naked triple {245} in 11(3) cage at R1C5, locked for N2
2b. 1 in N2 only in R123C4, locked for C4
3. 45 rule on R12 2 innies R2C46 = 1 outie R3C5 + 11
3a. R3C5 = {245} -> R2C46 = 13,15,16 = {67/69/78/79}, no 1,3 in R2C46
4. 45 rule on R89 2 outies R7C19 = 9 = {18/27/36/45}, no 9 in R7C9
5. 45 rule on N7 2 innies R7C23 = 9 = {27/36/45}/[81], no 8 in R7C3
6. 45 rule on C89 3 innies R349C8 = 19 = {289/379/469/478/568}, no 1
6a. 9 of {289} must be in R9C8 -> no 2 in R9C8
6b. 23(6) cage at R3C8 = {123458/123467}, 1 locked for C7
7. 45 rule on C1234 2 outies R9C56 = 8 = [17/35]
7a. 45 rule on C1234 2 innies R89C4 = 10 = {28/46}, no 5,7,9
8. 45 rule on N5 2 outies R4C37 = 1 innie R6C5 + 3
8a. Min R6C5 = 8 -> min R4C37 = 11, no 1 in R4C3
8b. 1 in N4 only in R4C2 + R56C3, locked for 25(6) cage at R3C2, no 1 in R3C2 + R7C3, clean-up: no 8 in R7C2 (step 5)
9. 1 in R3 only in 16(3) cage at R2C4 = {169/178}, no 2,3,4,5
10. 19(3) cage at R6C1 = {379/469/478/568} (cannot be {289} which clashes with R6C5), no 2, clean-up: no 7 in R7C3 (step 5)
10a. Killer pair 8,9 in 19(3) cage and R6C5, locked for R6
10b. Killer pair 8,9 in R5C12 and 19(3) cage, locked for N4, clean-up: no 2,3 in R3C1
10c. Min R4C37 = 11 (step 8a), max R4C3 = 7 -> min R4C7 = 4
10d. 9 in N6 only in R4C79, locked for R4
10e. Max R6C89 = 13 -> min R7C8 = 4
11. 1 in N7 only in 14(4) cage at R7C1 = {1238/1247/1346} (cannot be {1256} which clashes with 22(3) cage at R8C3), no 5, clean-up: no 4 in R7C9 (step 4)
11a. 7,8 of {1238/1247} must be in R789C1 (R789C1 cannot be {123/124} which clashes with R12C1), no 7,8 in R8C2
12. 19(3) cage at R2C6 = {289/379/469/478/568}
12a. 2,4,5 of {289/469/478/568} only in R3C7 -> no 6,8 in R3C7
13. 45 rule on C1 2 innies R56C1 = 1 outie R8C2 + 13
13a. Max R56C1 = 17 -> max R8C2 = 4
13b. Min R56C1 = 15 (cannot be 14 which clashes with R5C12, CCC), no 3,4,5 in R56C1, clean-up: no 9 in R5C2
13c. Min R56C1 = 15 -> min R8C2 = 2
13d. 1 in N7 only in R789C1, locked for C1, clean-up: no 6 in R12C1
14. 5 in C1 only in R12C1 = {25} or in R34C1 = {56} -> R34C1 = {47/56}/[83] (cannot be [92], locking-out cages), no 2,9 in R34C1
14a. 9 in C1 only in R56C1, locked for N4
15. 2 in N4 only in R4C2 + R456C3, CPE no 2 in R7C3, clean-up: no 7 in R7C2 (step 5)
15a. R7C23 = {36/45}, killer pair 5,6 in R7C23 and 22(3) cage at R8C3, locked for N7, clean-up: no 3 in R7C9 (step 4)
16. 19(3) cage at R6C1 (step 10) = {379/469/478} (cannot be {568} which clashes with R5C2), no 5, clean-up: no 4 in R7C3 (step 5)
16a. 3 of {379} must be in R7C2 -> no 3 in R6C2
16b. 6 of {469} must be in R7C2 (R6C12 cannot be [96] which clashes with R5C12), no 6 in R6C12
17. R12C2 = {16/25} (cannot be {34} which clashes with 19(3) cage at R6C1), no 3,4 in R12C2
18. 45 rule on C9 2 outies R58C8 = 1 innie R6C9
18a. Min R58C8 = 3 -> min R6C9 = 3
18b. Max R6C9 = 7 -> max R58C8 = 7 -> max R8C8 = 6
19. 17(3) cage at R6C8 = {179/269/278/359/368/458/467}
19a. 8,9 of {359/458} must be in R7C8 -> no 5 in R7C8
20. 9 in C2 only in R39C2
20a. 45 rule on C12 3 innies R349C2 = 18 = {189/279/369/459}
20b. 1 in C2 only in R12C2 = {16} or in R349C2 = {189} -> R349C2 = {189/279/459} (cannot be {369}, locking-out cages), no 3,6 in R349C2
21. 3 in C2 only in R78C2, locked for N7, clean-up: no 6 in R7C2 (step 5), no 6 in R7C9 (step 4)
22. 19(3) cage at R6C1 (step 16) = {379/478}, 7 locked for R6 and N4, clean-up: no 4 in R3C1
22a. R7C2 = {34} -> no 4 in R6C2
22b. Max R6C9 = 6 -> max R58C8 (step 18) = 6, no 6 in R58C8, clean-up: no 1 in R5C9
23. 17(3) cage at R6C8 = {269/359/368/458/467} (cannot be {179/278} because 7,8,9 only in R7C8), no 1
23a. 7,8,9 only in R7C8 -> R7C8 = {789}
24. 45 rule on C9 2 innies R56C9 = 1 outie R8C8 + 7
24a. Max R56C9 = 11 -> max R8C8 = 4
[I’d used this 45 earlier but then replaced it by the one in step 18, which was more effective at that stage.]
25. 19(4) cage at R4C3 = {2359/2368/2458}, CPE no 2 in R4C6
[There's a "chainy" way to eliminate {2368}; step 27 provides an easier way to eliminate this combination.]
25a. Killer pair 8,9 in 19(4) cage and R6C5, locked for N5
26. 1 in N5 only in 16(4) cage at R4C6 = {1249/1258/1348/1357/1456} (cannot be {1267} because 6,7 only in R4C7)
26a. 6,7,8,9 only in R4C7 -> R4C7 = {6789}
27. R4C37 = R6C5 + 3 (step 8)
27a. R6C5 = {89} -> R4C37 = 11,12 must contain one of {2345} -> R4C3 = {2345}
27b. 19(4) cage at R4C3 (step 25) = {2359/2458}, CPE no 5 in R4C6
27c. 4 of {2458} must be in R4C3 (R456C4 cannot be {248/458} which clash with R89C4), no 4 in R456C4
27d. 4 in N5 only in R456C6, locked for C6
27e. 16(4) cage at R4C6 (step 26) = {1249/1348/1456}, no 7
27f. 4 in C4 only in R7C4 or in R89C4 = {46} -> no 6 in R7C4 (locking-out cages)
28. 25(6) cage at R3C2 = {123469/123568/124567} (cannot be {123478} because R7C3 only contains 5,6), 6 locked for C3
29. 22(3) cage at R8C3 = {589} (only remaining combination), locked for N7 -> R7C3 = 6, R7C2 = 3 (step 5), clean-up: no 1 in R7C9 (step 4)
29a. 19(3) cage at R6C1 (step 22) = {379} (only remaining combination) -> R6C12 = [97], R67C5 = [89], clean-up: no 5 in R5C2
29b. Naked pair {68} in R5C12, locked for R5 and N4 -> R45C5 = [67], clean-up: no 5 in R3C1, no 5 in R3C9
30. 19(4) cage at R4C3 (step 27b) = {2359} (only remaining combination) -> R5C4 = 9, R4C3 + R5C46 = {235}, CPE no 3 in R4C6
31. 16(4) cage at R4C6 (step 27e) = {1249/1348}, no 5
31a. 5 in N5 only in R46C4, locked for C4 and 19(4) cage at R4C3, no 5 in R4C3
32. R35C1 = {68} (hidden pair in C1), clean-up: no 4 in R4C1
33. 4 in N4 only in R4C2 + R45C3, locked for 25(6) cage at R3C2, no 4 in R3C2 + R7C4
33a. 25(6) cage at R3C3 (step 28) = {123469/124567}, no 8
34. 4 in C4 only in R89C4 = {46}, locked for C4 and N8
34a. 8 in C4 only in R123C4, locked for N2
35. 16(3) cage at R2C4 (step 9) = {178} (only remaining combination), no 9, CPE no 7 in R2C6
36. 19(3) cage at R2C6 = {379/469}, no 2,5
37. R349C2 (step 20b) = {189/459}, no 2
37a. 2 in N4 only in R456C3, locked for C3
38. 2 in N1 only in R12C1 = {25} or in R12C2 = {25}, 5 locked for N1 (locking cages) -> R3C2 = 9, clean-up: no 2 in R4C9
39. R3C2 = 9 -> 25(6) cage at R3C3 (step 33a) = {123469} (only remaining combination) -> R7C4 = 2, R4C2 + R56C3 = {134}, locked for N4, 3 also locked for C3 -> R4C3 = 2, R4C1 = 5, R3C1 = 6, R46C4 = [35], clean-up: no 2 in R12C1, no 1 in R12C2, no 8 in R3C9, no 7 in R7C19 (step 4)
[Routine clean-ups omitted from here]
40. Naked pair {25} in R12C2, locked for C2 -> R8C2 = 4, R9C2 = 8, R7C1 = 1, R7C9 = 8 (step 4), R7C8 = 7, R7C67 = [54], R9C6 = 7, R9C5 = 1 (step 7)
40a. R4C9 = 7 (hidden single in R4), R3C9 = 4
41. R12C9 = {19} (only remaining combination), locked for C9 and N3
41a. R12C8 = {36} (only remaining combination), locked for C8 and N3
42. R34C8 = [28], R6C8 = 4, R6C9 = 6 (cage sum)
43. R7C9 = 8 -> R8C89 + R9C9 = 9 = {135} (only remaining combination) -> R8C8 = 1, R89C9 = {35}, locked for C9 and N9
44. R3C67 = [37] -> R2C6 = 9 (cage sum)
45. Naked pair {18} in R3C34, locked for 16(3) cage at R2C4 -> R2C4 = 7
and the rest is naked singles.
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