Step 22 simplified. Thanks Afmob
Prelims
a) R2C23 = {29/38/47/56}, no 1
b) R45C2 = {59/68}
c) R4C89 = {14/23}
d) R56C1 = {19/28/37/46}, no 5
e) R56C3 = {16/25/34}, no 7,8,9
f) R56C8 = {69/78}
g) R56C9 = {69/78}
h) R6C56 = {18/27/36/45}, no 9
i) R8C78 = {49/58/67}, no 1,2,3
j) R9C89 = {59/68}
k) 9(3) cage at R1C1 = {126/135/234}, no 7,8,9
l) 21(3) cage at R1C3 = {489/579/678}, no 1,2,3
m) 10(3) cage at R7C8 = {127/136/145/235}, no 8,9
n) 9(3) cage at R8C6 = {126/135/234}, no 7,8,9
o) And, of course, both 45(9) cages = {123456789}
1. R8C78 = {49/67} (cannot be {58} which clashes with R9C89), no 5,8
1a. Killer pair 6,9 in R8C78 and R9C89, locked for N9
2. Naked quad {6789} in R56C89, locked for N6
2a. 5 in N6 only in R456C7, locked for C7
3. 45 rule on N9 2 innies R79C7 = 8 = [71], clean-up: no 6 in R8C78
3a. Naked pair {49} in R8C78, locked for R8 and N9, clean-up: no 5 in R9C89
3b. Naked pair {68} in R9C89, locked for R9
3c. R9C7 = 1 -> R89C6 = 8 = [35/53/62]
4. 1 in N6 only in R4C89 = {14}, locked for R4 and N6
4a. Naked triple {235} in R456C7, 2,3 locked for C7
5. 6,8 in C7 only in R123C7, locked for N3
5a. Min R23C7 = {46} = 10 -> max R2C6 + R3C8 = 7, no 6,7,8,9 in R2C6 + R3C8 (17(4) cage cannot be 6{46}1)
5b. 6,8 in 45(9) cage at R1C5 only in R1C567 + R2C5, CPE no 6,8 in R1C4
5c. 7 in N3 only in R12C8 + R123C9, locked for 45(9) cage at R1C5, no 7 in R1C56 + R2C5
6. 4,7,9 in R9 only in R9C12345, locked for 45(9) cage at R7C1, no 4,9 in R7C1, no 7 in R8C125
6a. 7 in R8 only in R8C45, locked for 18(4) cage at R6C4, no 7 in R6C4
6b. 18(4) cage cannot contain both of 7,9 -> no 9 in R67C4
7. 45 rule on R123 1 innie R3C6 = 1 outie R4C1 + 7 -> R3C6 = 9, R4C1 = 2, clean-up: no 8 in R56C1, no 5 in R56C3
7a. Max R12C4 = 15 -> min R1C3 = 6
7b. 45(9) cage at R1C5 = {123456789}, 9 locked for N3
8. 2 in C7 only in R56C7 -> 29(5) cage at R5C7 contains 2,7 = {23789/25679} -> R7C5 = 9, R7C6 = {68}
8a. 1 in R7 only in R7C1234, CPE no 1 in R8C3
9. R56C1 = {19/37} (cannot be {46} which clashes with R56C3), no 4,6
9a. Killer pair 1,3 in R56C1 and R56C3, locked for N4
10. 45 rule on N4 2 remaining innies R4C3 + R6C2 = 12 = [57/75/84]
10a. CPE 4 in N4 only in R56C3 + R6C2, CPE no 4 in R7C3
11. 2 in R3 only in R3C89, locked for N3
11a. 45 rule on R12 3 outies R3C789 = 13 contains 2 = {238/247/256}, no 1 in R3C89
11b. R3C7 = {468} -> no 4 in R3C89
11c. 1 in N3 only in R12C89, locked for 45(9) cage at R1C5, no 1 in R1C56 + R2C5
12. 45 rule on N1 1 innie R1C3 = 2 remaining outies R3C45 + 2
12a. Max R3C45 = 7, no 7,8 in R3C45
13. 7 in N2 only in R12C4, locked for C4 and 21(3) cage at R1C3, no 7 in R1C3
13a. 21(3) cage at R1C3 contains 7 = {579/678}, no 4
14. R8C3 = 7 (hidden single in R8), clean-up: no 4 in R2C2, no 5 in R6C2 (step 10)
14a. R9C5 = 7 (hidden single in R9), R4C6 = 7 (hidden single in R4), clean-up: no 2 in R6C56
14b. 4 in N8 only in R79C4, locked for C4
15. 12(3) cage at R6C2 = {147/246} (cannot be {138/156} because R6C2 only contains 4,7, cannot be {237/345} which clash with 10(3) cage at R7C8, ALS block), no 3,5,8 in R7C23, 4 locked for C2
15a. 1 of {147} must be in R7C3 -> no 1 in R7C2
15b. 3,5,8 in N7 only in 45(9) cage at R7C1, no 3,5,8 in R8C5 + R9C4
16. R8C3 = 7 -> 18(4) cage at R6C4 = {1278/1467/2367} (cannot be {2457} which clashes with R9C4), no 5
17. 5 in N8 only in R89C6 (step 3c) = {35}, locked for C6 and N8, clean-up: no 4,6 in R6C5
18. 17(4) cage at R2C6 cannot contain all of 4,6,8, R23C7 = {468} -> no 4 in R2C6
[I’ve noted that 2 in N23 is only in 45(9) cage at R1C5 and 17(4) cage -> 17(4) cage must contain 2. Maybe I’ll use that later.]
19. 29(5) cage at R3C6 contains 7,9 = {23789/25679/34679} (cannot be {14789} because 1,4 only in R5C6), no 1
19a. 2,4 only in R5C6 -> R5C6 = {24}
19b. 5 of {25679} must be in R4C7 -> no 5 in R4C5
20. 9(3) cage at R1C1 = {126/135/234}
20a. 2 of {126} must be in R1C2 -> no 6 in R1C2
20b. 5 of {135} must be in R12C1 (R12C1 cannot be {13} which clashes with R56C1), no 5 in R1C2
21. 12(3) cage at R6C2 (step 15) = {147/246} -> R7C23 = {14/26}
21a. Law of Leftovers (LoL) for N7, because of 45(9) cage at R7C1, 2 remaining outies R8C5 + R9C4 must exactly equal 2 remaining innies R7C23, R7C23 = {14/26} -> R8C5 + R9C4 = {14/26} = [14/62]
21b. 2 in N8 only in R789C4, locked for C4
21c. R6C7 = 2 (hidden single in R6)
22. 9 in N5 only in 23(4) cage at R4C3 = {1589/2489/3569}
22a. 45 rule on R4 3 remaining innies R4C234 = 1 outie R5C6 + 18
22b. R5C6 = {24} -> R4C234 = 20,22 = {569/589} (cannot be {389} because 23(4) cannot contain both of 3,8), 5 locked for R4 -> R4C7 = 3
23. R5C7 = 5, R9C6 = 6 (cage sum), R8C5 = 1, R9C4 = 4 (step 21a), clean-up: no 3 in R6C5, no 8 in R6C6
23a. R78C4 = {28}, locked for C4, R8C3 = 7 -> R6C4 = 1 (cage sum), R6C6 = 4 -> R6C5 = 5
23b. R5C6 = 2 -> R4C5 = 8 (cage sum)
[Routine clean-ups omitted from here.]
24. R4C3 = 5, R45C2 = {68} (only remaining combination) = [68], R6C3 = 3 -> R5C3 = 4
25. R6C2 = 7 -> R7C23 = 5 = [41]
26. R1C6 = 8
26a. 21(3) cage at R1C3 (step 13a) = {579} (only remaining combination) -> R1C3 = 9, R12C4 = {57}, 5 locked for C3
26b. 2 in N2 only in R12C5, locked for 45(9) cage at R1C5
27. R2C6 = 1, R3C8 = 2 (hidden single in R3) -> R23C7 = 14 = {68}, 6 locked for C3
28. R3C789 (step 11a) = {256} (only remaining combination, cannot be {238} which clashes with R3C34, ALS block) -> R3C79 = [65], R3C12345 = [71834]
29. 9(3) cage at R1C1 = {234} (only remaining combination) -> R1C2 = 2, R12C1 = {34}, 3 locked for C1 and N1
30. R4C4 = 9, R5C45 = [63], R56C1 = [19]
31. R1C7 = 4, R8C78 = [94], R4C89 = [14], R2C2 = 5, R2C4 = 7
32. Naked pair {23} in R78C9, locked for C9 and N9 -> R2C9 = 9, R5C9 = 7 -> R6C9 = 8
and the rest is naked singles.
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