Prelims
a) 9(3) cage at R1C8 = {126/135/234}, no 7,8,9
b) 10(3) cage at R2C3 = {127/136/145/235}, no 8,9
c) 9(3) cage at R4C3 = {126/135/234}, no 7,8,9
d) 10(3) cage at R4C5 = {127/136/145/235}, no 8,9
e) 20(3) cage at R5C8 = {389/479/569/578}, no 1,2
f) 19(3) cage at R6C2 = {289/379/469/478/568}, no 1
g) 11(3) cage at R7C2 = {128/137/146/236/245}, no 9
h) 19(3) cage at R7C3 = {289/379/469/478/568}, no 1
i) 20(3) cage at R8C3 = {389/479/569/578}, no 1,2
1. 45 rule on R1234 3 innies R4C357 = 10 = {127/136/145/235}, no 8,9
2. 45 rule on N3 3 innies R12C7 + R3C9 = 20 = {389/479/569/578}, no 1,2
2a. 45 rule on N3 1 innie R3C9 = 1 outie R2C6 + 2, no 8,9 in R2C6
3. 45 rule on R1 1 innie R1C7 = 1 outie R2C9 + 4, no 3,4 in R1C7, no 6 in R2C9
4. 45 rule on R12 3 innies R2C128 = 21 = {489/579/678}, no 1,2,3
4a. 45 rule on R12 1 innie R2C8 = 1 outie R3C1 + 5, R2C8 = {6789}, R3C1 = {1234}
4b. 45 rule on R12 3 outies R3C178 = 11 = {128/137/146/236/245}, no 9
4c. 16(3) cage at R2C8 = {169/178/268/457} (cannot be {349/358} because R3C178 cannot be 4{34}/3{35}, cannot be {259/367} which clash with 9(3) cage at R1C8), no 3
5. 45 rule on R789 1 outie R6C9 = 1 innie R7C6 + 1, no 1 in R6C9, no 9 in R7C6
6. 45 rule on C12 1 innie R6C2 = 1 outie R1C3 + 2, no 8,9 in R1C3, no 2 in R6C2
7. 45 rule on N7 3 outies R7C45 + R8C4 = 20 = {389/479/569/578}, no 2
7a. 45 rule on N7 1 outie R8C4 = 1 innie R7C3 + 1, no 9 in R7C3
7b. 45 rule on N7 3 innies R789C3 = 19 = {289/379/469/478/568}
7c. 5 of {568} must be in R89C3 (because 20(3) cage at R8C3 cannot be {68}6), no 5 in R7C3, clean-up: no 6 in R8C4
8. 45 rule on N9 1 outie R6C9 = 1 innie R8C7, no 1 in R8C7
9 45 rule on R1 3 outies R2C679 = 14 = {149/167/239/248/257/356} (cannot be {158/347} which clash with R2C128
9a. 8,9 of {149/248} must be in R2C7 -> no 4 in R2C7
10. R12C7 + R3C9 (step 2) = {389/479/569/578}
10a. 4 of {479} must be in R3C9, 7 of {578} must be in R12C7 (R12C7 cannot be {58} because 18(3) cage at R1C7 cannot be {58}5) -> no 7 in R3C9, clean-up: no 5 in R2C6 (step 2a)
11. 45 rule on N78 3 innies R789C6 = 12 = {129/138/147/156/237/246} (cannot be {345} which clashes with R7C45 + R8C4)
11a. 5 of {156} must be in R89C6 (R89C6 cannot be {16} because 13(3) cage at R8C6 cannot be {16}6) -> no 5 in R7C6, clean-up: no 6 in R6C9 (step 5), no 6 in R8C7 (step 8)
[Now to look at more complicated interactions …]
12. R2C679 (step 9) = {149/167/239/248/257/356}
12a. R2C679 = {149/167/239/257/356} (cannot be [284] because 18(3) cage at R1C7 cannot be [828], cannot be [482] because R1C7 = 6 (step 3) clashes with R3C9 = 6 (step 2a) -> {248} combination eliminated), no 8
[… and even more complicated – remember that 16(3) cage at R2C8 no longer contains 3]
12b. R2C679 = {167/239/257/356} (cannot be [194] because R1C7 = 8 (step 3), R3C9 = 3 (step 2a) clash with 16(3) cage at R2C8, cannot be [491] because R1C7 = 5 (step 3), R3C9 = 6 (step 2a) clash with 16(3) cage at R2C8 -> {149} combination eliminated), no 4, clean-up: no 8 in R1C7 (step 3), no 6 in R3C9 (step 2a)
12c. R2C679 = {239/257/356} (cannot be [671] because R1C7 = 5 (step 3), R3C9 = 8 (step 2a) clash with 16(3) cage at R2C8, cannot be [761] because R1C7 = 5 (step 3), R3C9 = 9 (step 2a) clash with 16(3) cage at R2C8 -> {167} combination eliminated), no 1, clean-up: no 5 in R1C7 (step 3), no 3 in R3C9 (step 2a)
12d. {356} can only be [653] (cannot be [365] which clashes with R3C9 = 5 (step 2a), cannot be [635] because R1C7 = 9 (step 3), R3C9 = 8 (step 2a) clash with 16(3) cage at R2C8), 9 of {239} must be in R2C7 -> no 3,6 in R2C7
13. 8 in R2 only in R2C128 (step 4) = {489/678}, no 5
13a. 9 of {489} must be in R2C12 (R2C12 cannot be {48} because 16(3) cage at R2C1 cannot be {48}4) -> no 9 in R2C8, clean-up: no 4 in R3C1 (step 4a)
14. 9 in N3 only in R12C7 + R3C9 (step 2) = {479/569}, no 8, clean-up: no 6 in R2C6 (step 2a)
14a. R2C679 (step 12c) = {239/257}, 2 locked for R2
14b. Killer pair 7,9 in R2C128 and R2C679, locked for R2
15. 8 in N3 only in 16(3) cage at R2C8 (step 4c) = {178/268}, no 4,5
15a. 3 in N3 only in 9(3) cage at R1C8 = {135/234}, no 6
[At this stage I originally used some complicated steps starting with outies for C89, a variable combined cage in C7, then variable combined cages in N6 after using R6C9 = R8C7. However I later found some more effective steps so I’ve re-worked and simplified my solving path from here.]
16. 45 rule on R1 3 innies R1C789 = 13 = {139/157/346} (cannot be {247} which clashes with 16(3) cage at R2C8, cannot be {256} because 9(3) cage at R1C8 cannot be {25}2), no 2
17. 16(3) cage at R2C1 = {178/268/349/367} (cannot be {169} because R2C178 = 21 cannot be {69}6)
17a. 15(3) cage at R1C1 = {159/249/258/267/456} (cannot be {168/357} which clash with R1C789, cannot be {348} which clashes with 16(3) cage), no 3, clean-up: no 5 in R6C2 (step 6)
18. 45 rule on N6789 3(1+2) outies R3C9 + R56C6 = 21
18a. R3C9 = {459} -> R56C6 = 12,16,17 = {39/48/57/79/89}, no 1,2,6
18b. R3C9 + R56C6 = 4{89}/5{79}/9{39/48/57}, CPE no 9 in R3C6
19. 45 rule on N36789 3 outies R256C6 = 19 = 2{89}/3{79}/7{39/48}, no 5
19a. R789C6 (step 11) = {129/147/156/246} (cannot be {237} which clashes with R2C6, cannot be {138} which clashes with R256C6), no 3,8, clean-up: no 4,9 in R6C9 (step 5), no 4,9 in R8C7 (step 8)
20. 45 rule on C6789 1 outie R3C5 = 1 innie R1C6 + 2, no 8,9 in R1C6, no 1,2 in R3C5
20a. 45 rule on C6789 3 outies R1C45 + R3C5 = 19 = {289/379/469/478/568}, no 1
20b. 17(3) cage at R1C4 = {179/278/368/458/467} (cannot be {269} which clashes with 15(3) cage at R1C1, cannot be {359} which clashes with R1C789)
20c. 3 of {368} must be in R1C6 (R1C45 cannot be {38} because R1C45 + R3C5 cannot be {38}8), no 3 in R1C45
21. R1C45 + R3C5 (step 20a) = {289/379/469/478/568}
21a. Consider combinations for 18(3) cage at R1C7 = {279/369/567}
18(3) cage = {279} => R1C7 = {79} => R1C45 + R3C5 = {469/478/568} (cannot be {379} which clashes with R1C7)
or 18(3) cage = {369} => R2C6 = 3 => R1C45 + R3C5 = {289/469/478/568}
or 18(3) cage = {567} => R2C6 = 7 => R1C45 + R3C5 = {289/469/568}
[Note. Can also eliminate {469} when R1C7 = 6, but that’s not necessary for this step.]
-> R1C45 + R3C5 = {289/469/478/568}, no 3, clean-up: no 1 in R1C6 (step 20)
22. 17(3) cage at R1C4 (step 20b) = {278/368/458/467}, no 9
22a. 9 in N2 only in R3C456, locked for R3, clean-up: no 7 in R2C6 (step 2a)
22b. 18(3) cage at R1C7 (step 21a) = {279/369}, no 5, 9 locked for C7
23. 45 rule on C89 3 outies R379C7 = 18 = {378/468} (cannot be {567} which clashes with R12C7, ALS block), no 1,2,5, 8 locked for C7, clean-up: no 8 in R6C9 (step 8), no 7 in R7C6 (step 5)
23a. Killer pair 6,7 in R12C7 and R379C7, locked for C7, clean-up: no 7 in R6C9 (step 8), no 6 in R7C6 (step 5)
24. 1 in C7 only in R456C7, locked for N6
24a. 2,5 in C7 only in R4568C7, R6C9 = R8C7 (step 8) -> R456C7 + R6C9 contain 2,5, locked for N6
25. Killer pair 3,4 in R456C7 + R6C9 and 20(3) cage at R5C8, locked for N5
26. 18(3) cage at R3C9 = {468/567} (cannot be {459} because 4,5 only in R3C9), no 9, 6 locked for R4 and N6
26a. 16(3) cage at R2C8 (step 15) = {178/268}
26b. 1,2 only in R3C8 -> R3C8 = {12}
27. 17(3) cage at R6C9 = {269/278/359/368/458} (cannot be {179/467} because R6C9 only contains 2,3,5), no 1
27a. 2 of {269/278} must be in R6C9 -> no 2 in R78C9
28. 45 rule on R789 3 outies R6C679 = {158/239/248/257} (cannot be {149} because R6C9 only contains 2,3,5, cannot be {347} which clashes with 20(3) cage at R5C8)
28a. 7,8,9 only in R6C6 -> R6C6 = {789}
28b. 3 of {239} must be in R6C7 (R6C67 cannot be [92] because 13(3) cage at R6C6 cannot be {29}2), no 3 in R6C9, clean-up: no 2 in R7C6 (step 5), no 3 in R8C7 (step 8)
29. 14(3) cage at R4C7 = {149/158/239/248} (cannot be {257} which clashes with R8C7, cannot be {347} which clashes with 20(3) cage at R5C8), no 7
29a. 8,9 only in R5C6 -> R5C6 = {89}
30. 45 rule on C6789 3 innies R134C6 = 14 = {158/248/347/356} (cannot be {149} which clashes with R7C6 cannot be {239} which clashes with R2C6, cannot be {167/257} which clash with R789C6), no 9
30a. 13(3) cage at R8C6 = {157/247/256} (cannot be {139/346} because R8C7 only contains 2,5), no 9
30b. 9 in C6 only in R56C6, locked for N5
30c. 9 in R4 only in R4C12, locked for N4, clean-up: no 7 in R1C3 (step 6)
31. R134C6 (step 30) = {158/248/347/356}
31a. 2 of {248} must be in R34C6 (R34C6 cannot be {48} because 16(3) cage at R3C5 cannot be 4{48}), no 2 in R1C6, clean-up: no 4 in R3C5 (step 20)
32. 19(3) cage at R6C2 = {478/568}, no 2,3, 8 locked for R6, clean-up: no 1 in R1C3 (step 6)
32a. R6C679 (step 28) = {239/257}, no 1,4, 2 locked for R6 and N6
32b. 14(3) cage at R4C7 (step 29) = {149/158}, no 3
33. 45 rule on R6789 3 innies R6C158 = 12 = {147/156} (cannot be {345} which clashes with R6C679), no 3,9
[And at last some placements …]
34. R6C7 = [93] (hidden pair in R6), R6C9 = 2 (hidden single in R6), R7C6 = 1 (step 5), R8C7 = 2 (step 8), R5C6 = 8
34a. Naked triple {479} in 20(3) cage at R5C8, locked for N6
34b. Naked pair {68} in R4C89, locked for R4, R3C9 = 4 (cage total), R2C6 = 2 (step 2a)
34c. 4 in N6 only in R56C8, locked for C8
35. 18(3) cage at R1C7 (step 22b) = {279} (only remaining combination) -> R12C7 = {79}, locked for C7 and N3
35a. Naked pair {68} in R24C8, locked for C8
35b. R3C8 = 2 (hidden single in N3)
36. 2 in N1 only in 15(3) cage at R1C1 = {249/258/267}, no 1
37. R6C9 = 2 -> R78C9 = 15 = {69/78}, no 3,5
37a. Naked quad {6789} in R45C9 + R78C9, locked for C9
38. 12(3) cage at R7C7 = {138/147/156/345}, no 9
38a. 1 of {147} must be in R8C8 -> no 7 in R8C8
39. 19(3) cage at R6C2 (step 32) = {568} (only remaining combination, cannot be {478} which clashes with R6C8), locked for R6
40. 2 in N8 only in R9C45, locked for R9
40a. 13(3) cage at R8C5 contains 2 = {238/247/256}, no 9
41. 18(3) cage at R3C3 = {189/279/369/378/459/468/567}
41a. 1 of {189} must be in R4C4 -> no 1 in R3C34
41b. 1 in R3 only in R3C12, locked for N1
42. 14(3) cage at R5C1 = {167/257/347} (cannot be {356} because R6C1 only contains 1,4,7), 7 locked for N4
43. 9 in N4 only in 17(3) cage at R3C2 = {179/269/359}, no 4,8
43a. 7 of {179} must be in R3C2 -> no 1 in R3C2
44. R3C1 = 1 (hidden single in N1)
44a. R3C178 (step 4b) = {128} (only remaining combination) -> R3C7 = 8, R2C6 = 6, R4C89 = [86], clean-up: no 9 in R78C9 (step 37)
44b. Naked pair {78} in R78C9, locked for C9 and N9 -> R5C9 = 9
45. R9C8 = 9 (hidden single in N9), R9C79 = 7 = [43/61], no 5 in R9C9
45a. 5 in N9 only in R78C8, locked for C8
46. R3C9 = 1 -> R2C12 = 15 = {78}, locked for R2 and N1 -> R12C7 = [79], R2C9 = 3 (step 3), R1C89 = [15], R9C9 = 1, R9C7 = 6 (step 45), R7C7 = 4
47. 2,9 in N1 only in 15(3) cage at R1C1 = {249}, locked for R1 and N1 -> R2C3 = 5
47a. Naked pair {68} in R1C45, locked for N2 -> R1C6 = 3, R3C5 = 5 (step 20), R34C6 = [74], R89C6 = [65]
48. R3C4 = 9 -> R3C3 + R4C4 = [63]
49. R3C2 = 3 -> R4C12 = 14 = {59}, locked for R4 and N4 -> R45C7 = [15], R4C3 = 2, R46C5 = [71], R5C45 = [62], 19(3) cage at R6C2 = [685]
50. 19(3) cage at R7C3 = {379} (only remaining combination) -> R7C5 = 9, R7C34 = [37]
51. R15C3 = [41], R89C3 = [97], R8C4 = 4 (cage sum)
52. R7C89 = [58], R7C2 = 2, R8C2 = 1 (hidden single in N7), R9C2 = 8 (cage sum)
and the rest is naked singles.
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