Prelims
a) R34C2 = {69/78}
b) 11(3) cage at R1C1 = {128/137/146/236/245}, no 9
c) 21(3) cage at R1C3 = {489/579/678}, no 1,2,3
d) 9(3) cage at R1C4 = {126/135/234}, no 7,8,9
e) 11(3) cage at R3C1 = {128/137/146/236/245}, no 9
f) 11(3) cage at R3C3 = {128/137/146/236/245}, no 9
g) 26(4) cage at R1C5 = {2789/3689/4589/4679/5678}, no 1
h) 28(4) cage at R4C8 = {4789/5689}, no 1,2,3
i) 26(4) cage at R6C5 = {2789/3689/4589/4679/5678}, no 1
1. 28(4) cage at R4C8 = {4789/5689}, 8,9 locked for N6
2. 25(6) cage at R2C5 contains 1,2, CPE no 2 in R6C5
3. 45 rule on R9 4(1+3) outies R7C1 + R8C129 = 12
3a. Max R8C129 = 11, no 9 in R8C129
3b. Min R8C129 = 6 -> max R7C1 = 6
3c. Min R7C1 + R8C12 = 6 -> max R8C9 = 6
4. 45 rule on N1 3 innies R3C123 = 13 = {139/148/157/238/256/346} (cannot be {247} which clashes with 21(3) cage at R1C3)
4a. R3C2 = {6789} -> no 6,7,8 in R3C13
5. 45 rule on C4 3 innies R789C4 = 1 outie R6C3 + 20
5a. Max R789C4 = 24 -> max R6C3 = 4
5b. Min R789C4 = 21, no 1,2,3 in R789C4
6. 1 in N8 only in R89C56, CPE no 1 in R8C9
6a. Min R8C9 = 2 -> max R7C1 + R8C12 = 10 (step 3), no 8 in R8C12
7. 33(7) cage at R7C1 = {1234689/1235679/1245678}, 1,2 locked for N7
8. 45 rule on C123 3 innies R678C3 = 1 outie R9C4 + 12
[I originally saw the next sub-step as …
8a. R789C4 = R6C3 + 20 (step 5), max R78C4 = 17 -> R9C4 at least 3 more than R6C3 -> min R78C3 = 15, no 3,4,5 in R78C3
However on looking a bit more closely one can do a bit of algebra…]
8a. Rewriting steps 5 and 8
R78C4 + R9C4 – R6C3 = 20
R78C3 + R6C3 – R9C4 = 12
Adding these together R78C3 + R78C4 = 32
8b. Max R78C3 = 17 -> min R78C4 = 15, no 4,5 in R78C4
8c. Max R78C4 = 17 -> min R78C3 = 15, no 3,4,5 in R78C3
[After looking at Afmob’s walkthrough, I see that I missed that R78C34 are 4(2+2) innies = 32 for C1234.][Don’t think I can get anything by subtracting step 8 from step 5, at least at this stage.]
9. R678C3 = R9C4 + 12 (step 8)
9a. 45 rule on N7 3 innies R7C2 + R78C3 = 1 outie R9C4 + 12
-> R6C3 = R7C2 = {34}
[Or, more simply, 45 rule on N14 1 innie R6C3 = 1 outie R7C2 = {34}.]
10. R789C4 = R6C3 + 20 (step 5)
10a. R6C3 = {34} -> R789C4 = 23,24 = {689/789}, no 4,5, 8,9 locked for C4 and N8
11. 16(4) cage at R4C4 = {1357/1456/2347} (cannot be {1267} because R6C3 only contains 3,4, cannot be {2356} which clashes with 9(3) cage at R1C4, ALS block)
11a. Killer pair 6,7 in 16(4) cage and R789C4, locked for C4
12. 9(3) cage at R1C4 = {135/234}, 3 locked for C4 and N2
13. 45 rule on C12 2 outies R9C34 = 1 innie R2C2 + 1
13a. Min R9C34 = 7 -> min R2C2 = 6
13b. Max R9C34 = 10 -> max R9C3 = 4
14. Killer quad 1,2,3,4 in 11(3) cage at R3C3, R6C3 and R9C3, locked for C3
15. 21(3) cage at R1C3 = {579/678}, 7 locked for N1, clean-up: no 8 in R4C2
15a. R3C123 (step 4) = {139/148/238/346} (cannot be {256} which clashes with 21(3) cage), no 5
16. 25(6) cage at R2C5 = {123469/123478/123568/124567}
16a. Double killer pair 1,2 in R123456C4 and 25(6) cage, locked for N25
17. 45 rule on N3 2 outies R3C6 + R4C7 = 1 innie R1C7 + 7, IOU no 7 in R3C6
18. 24(6) cage at R8C9 = {123459/123468/123567}, 1 locked for R9
18a. Hidden killer triple 7,8,9 in 33(7) cage at R7C1 and 24(6) cage for R9, 24(6) cage contains one of 7,8,9 -> 33(7) cage must contain two of 7,8,9 in R9 -> no 7 in R8C12 (because 33(7) cage only contains two of 7,8,9)
19. 5 in N7 only in 33(7) cage at R7C1 = {1235679/1245678}, 7 locked for R9
20. 24(6) cage at R8C9 = {123459/123468}
20a. Killer triple 2,3,4 in R9C3 and 24(6) cage, locked for R9
20b. Hidden killer triple 2,3,4 in R9C3 and 24(6) cage for R9, R9C3 = {234} -> 24(6) cage contains two of 2,3,4 in R9 -> R8C9 = {234} (only other place in cage for 2,3,4)
21. R8C9 = “sees” all 2,3,4 in N8 except for R7C56, R8C9 = {234} -> R7C56 must contain one of 2,3,4 -> 26(4) cage at R6C5 = {2789/3689/4589/4679} (cannot be {5678} which doesn’t contain any of 2,3,4)
21a. One of 2,3,4 in R7C56 -> no 3,4 in R6C5
21b. 8,9 of {4589} must be in R6C5 + R7C4 -> no 5 in R6C5
22. R9C34 = R2C2 + 1 (step 13)
22a. Min R9C34 = 8 -> min R2C2 = 7
22b. Max R9C34 = 10 -> max R9C4 = 8
22c. 9 in C4 only in R78C4, CPE no 9 in R7C3
23. 1 in C3 only in 11(3) cage at R3C3 = {128/137/146}, no 5
24. 5 in C3 only in 21(3) cage at R1C3 (step 15) = {579} (only remaining combination), locked for N1, clean-up: no 6 in R4C2
24a. Naked pair {79} in R24C2, locked for C2
25. 30(5) cage at R7C3 = {15789/24789/25689/34689/35679/45678}
25a. R7C3 + R8C34 must contain three of 6,7,8,9 -> R8C56 = {15/24/25/34/35/45}, no 6,7
25b. 45 rule on C1234 1 innie R7C4 = 2 outies R8C56 + 2
25c. Min R8C56 = 6 -> min R7C4 = 8
25d. R7C4 = {89} -> R8C56 = 6,7 = {15/24/25/34}
25e. 30(5) cage = {15789/24789/25689/34689} (cannot be {45678} because max R8C56 = 7), 9 locked for R8
25f. 30(5) cage = {15789/24789/25689/34689} -> R8C3 = {89} because one of 8,9 must be in the cell which doesn’t “see” R7C4
25g. 26(4) cage at R6C5 (step 21) = {2789/3689/4679} (cannot be {4589} which clashes with R8C56), no 5
26. 21(4) cage at R5C2 = {2469/2478/3459/3468/3567} (cannot be {1569/1578/2568} because R7C2 only contains 3,4, cannot be {1479/2379} which clash with R4C2, cannot be {1389} which clashes with R34C2, combo blocker), no 1
26a. 21(4) cage = {2469/2478/3459/3468} (cannot be {3567} which clashes with R34C2, combo blocker because 9 in N4 only in R4C2 + R6C1)
26b. 7,9 of {2469/2478/3459} must be in R6C1, 6 of {3468} must be in R6C1 (R567C2 cannot be {368/468} which clash with R3C2, R567C2 cannot be {346} which clashes with R34C2, combo blocker because 9 in N4 only in R4C2 + R6C1) -> R6C1 = {679}
26c. 21(4) cage = {2469/2478/3459/3468}, 4 locked for C2
27. 21(4) cage at R5C2 (step 26a) = {2469/2478/3459/3468}, R6C3 = R7C2 (step 9a) -> R5C2 + R6C123 = {2469/2478/3459/3468}, 4 locked for N4
28. 5 in N4 only in 11(3) cage at R3C1
or in 21(4) cage at R5C2
5 in 11(3) cage at R3C1 = 4{25}, 2 locked for N4 => 21(4) cage at R5C2 (step 26a) = {3459/3468}
or 5 in 21(4) cage at R5C2 (step 26a) = {3459}
-> 21(4) cage at R5C2 = {3459/3468}, no 2,7, 3 locked for C2
28a. 21(4) cage at R5C2 = {3459/3468} -> R5C2 + R6C123 = {3459/3468}, 3 locked for N4
28b. R6C1 = {69} -> no 6 in R56C2
29. 33(7) cage at R7C1 contains 1 in C12 -> the other 1 in C12 must be either in 11(3) cage at R1C1
or 11(3) cage at R3C1
29a. Consider combinations for 11(3) cage at R3C1
11(3) cage at R3C1 contains 1 -> no 1 in 11(3) cage at R1C1
or 11(3) cage at R3C1 = {236} = 3{26} => R6C1 = 9, R4C2 = 7, R3C2 = 8 => R1C2 = 6 (hidden single in N1)
or 11(3) cage at R3C1 = {245} = 4{25} => R3C123 = 4[63/81] (step 4) => R1C2 = 2 (hidden single in N1)
-> no 1 in R1C2
30. R8C2 = 1 (hidden single in C2)
30a. R8C56 (step 25d) = 6,7, R8C9 = {234} -> max R8C569 = 11 must contain 2, locked for R8
30b. 1 in N8 only in R9C56, locked for R9
30c. 1 in N9 only in R7C789, CPE no 1 in R5C7
[Just spotted R78C3 = 15,17 (cannot be 16 = [79] which clashes with 21(3) cage at R1C3, ALS block) -> R78C4 = 15,17, step 8a.
Alternatively R78C3 and R78C4 cannot be 16 because the 7s in R7C3 and R8C4 “see” each other.
]
31. R78C3 = 15 = [69/78] => R78C4 = 17 = [98/89]
or R78C3 = 17 = [89] => R78C4 = 15 = [96]
-> no 7 in R8C4
32. 7 in R8 only in R8C78, locked for N9 and 23(5) cage at R6C6, no 7 in R6C67
33. Killer pair 7,9 in 21(3) cage at R1C3 and R78C3, locked for C3)
34. 11(3) cage at R3C3 (step 23) = {128/146}, no 3
34a. 21(4) cage at R5C2 (step 28a) = {3459} (only remaining combination, cannot be {3468} which clashes with 11(3) cage at R3C3) -> R6C1 = 9, R567C2 = {345}, 5 locked for C2 and N4, R4C2 = 7, R3C2 = 8, R2C2 = 9, R9C2 = 6, R1C2 = 2
35. Naked pair {57} in R12C3, locked for C3 -> R78C3 = [89], R7C4 = 9, R8C4 = 6
35a. R9C4 = 8 (hidden single in C4)
36. R1C2 = 2 -> R12C1 = 9 = {36}, locked for C1
37. 11(3) cage at R3C1 = {128} (only remaining combination) -> R3C1 = 1, R45C1 = {28}, locked for C1 and N4, R3C3 = 4, R6C3 = 3
38. Naked pair {45} in R78C1, locked for N7 -> R7C2 = 3, R9C13 = [72]
39. R8C78 = {78} (hidden pair in R8)
39a. R8C78 = {78} = 15 -> R6C67 + R7C7 = 8 = {125} -> R6C6 = 5, R67C7 = {12}, locked for C7, R56C2 = [54]
40. 16(4) cage at R4C4 (step 11) = {2347} (only remaining combination), 2,4,7 locked for C4 and N5
41. Naked triple {135} in 9(3) cage at R4C1, locked for N2
42. 45 rule on N5 1 remaining innie R6C5 = 1 outies R23C5 + 2
42a. Min R23C5 = [42] = 6 -> R6C5 = 8, R23C5 = [42], R7C5 = 7, R7C6 = 2 (cage sum), R67C7 = [21]
43. R12C6 = {78} (hidden pair in N2)
43a. R12C6 = {78} = 15 -> R1C57 = 11 = [65], R3C6 = 9
44. Naked triple {367} in R3C789, locked for R3 and N3 -> R2C7 = 8
45. R3C6 + R4C7 = R1C7 + 7 (step 17), R1C7 = 5, R3C6 = 9 -> R4C7 = 3
45a. R8C78 = [78], R3C7 = 6, R2C8 = 2 (cage sum)
46. 30(5) cage at R7C3 (step 25f) = {34689} (only remaining combination) -> R8C56 = [34]
and the rest is naked singles.