Prelims
a) R12C1 = {15/24}
b) R1C89 = {19/28/37/46}, no 5
c) R23C9 = {39/48/57}, no 1,2,6
d) R34C8 = {19/28/37/46}, no 5
d) R4C45 = {14/23}
e) R45C6 = {89}
f) R45C7 = {19/28/37/46}, no 5
g) R45C9 = {17/26/35}, no 4,8,9
h) R56C1 = {69/78}
i) R56C3 = {19/28/37/46}, no 5
j) R5C45 = {15/24}
k) R67C2 = {39/48/57}, no 1,2,6
l) R78C1 = {14/23}
m) R89C9 = {17/26/35}, no 4,8,9
n) R9C12 = {29/38/47/56}, no 1
1. Naked pair {89} in R45C6, locked for C6 and NR3C5, clean-up: no 1,2 in R4C7
2. R12C1 = {15} (cannot be {24} which clashes with R78C1), locked for C1, clean-up: no 4 in R78C1, no 6 in R9C2
2a. Naked pair {23} in R78C1, locked for C1, clean-up: no 8,9 in R9C2
2b. Naked pair {15} in R12C1, CPE no 1,5 in R2C23
3. 45 rule on C1 2 outies R39C2 = 7 = {25/34}, no 1,6,7,8,9, clean-up: no 4 in R9C1
4. 4 in C1 only in R34C1, locked for NR2C1, clean-up: no 6 in R56C3, no 3 in R9C2 (step 3), no 8 in R9C1
4a. 15(3) cage at R3C1 contains 4 = {249/348/456}, no 7
5. R23C9 = {39/48} (cannot be {57} which clashes with the pair of 8(2) cages in C9), no 5,7
6. 45 rule on C9 2 outies R17C8 = 9 = {18/27/36}/[45], no 9, no 4 in R7C8, clean-up: no 1 in R1C9
7. 45 rule on R123 2 outies R4C18 = 11 = [47/83/92], clean-up: R3C8 = {378}
7a. R17C8 (step 6) = {18/36}/[45] (cannot be {27} which clashes with R34C8), no 2,7, clean-up: no 3,8 in R1C9
7b. Killer pair 3,8 in R23C9 and R34C8, locked for NR2C7, clean-up: no 2,7 in R5C7, no 5 in R5C9
8. 45 rule on R789 2 outies R6C29 = 10 = [37/46/73/82/91], clean-up: no 7 in R7C2
9. 45 rule on C123 1 outie R3C4 = 1 innie R1C3 + 2, no 8,9 in R1C3, no 1,2 in R3C4
10. 45 rule on C789 1 outie R7C6 = 1 innie R9C7 + 4, R7C6 = {567}, R9C7 = {123}
11. R39C2 (step 3) = {25}/[34], R67C2 = {39/48}/[75] -> combined cage R39C2 + R67C2 = {25}{39}/{25}{48}/[34][75], 5 locked for C2
12. Hidden killer pair 4,8 in R17C9 and R23C9 for C9, R23C9 contains both or neither of 4,8 -> R17C9 must contain both or neither of 4,8, no 8 in R1C9 -> no 4 in R7C9
12a. 4 in C9 only in R123C9, locked for NR1C9, clean-up: no 6 in R5C7
[If I’d spotted step 14 earlier, step 12 wouldn’t have been needed – which would have been a pity.]
13. 28(5) cage at R1C5 must contain at least one of 8,9 -> R1C5 = {89}
13a. 28(5) cage = {15679/24679/25678/34579/34678}, CPE no 7 in R6C6
13b. 1 of {15679} must be in R123C6 (R123C6 cannot be {567} which clashes with R7C6), no 1 in R3C5
13c. 7 in NR3C5 only in R3C56, locked for R3 and 28(5) cage, no 7 in R12C6, clean-up: no 5 in R1C3 (step 9), no 3 in R4C8
13d. 6 in NR3C5 only in R3C56 + R6C6, CPE no 6 in R12C6
14. The two 8(2) cages in C9 + R6C9 must contain 3, locked for C9, clean-up: no 9 in R23C9
14a. Naked pair {48} in R23C9, locked for C9 and NR1C9 -> R3C8 = 3, R4C8 = 7, placed for NR1C9, clean-up: no 1 in R1C3 (step 9), no 6 in R1C8, no 3 in R5C7, no 1 in R5C9, no 6 in R7C8 (step 6), no 4 in R9C2 (step 3), no 7 in R9C1
14b. Naked pair {25} in R39C2, locked for C2, clean-up: no 7 in R6C2
[Cracked. The rest is fairly straightforward.]
15. R56C1 = {78} (only remaining combination, cannot be {69} which clashes with R9C1), locked for C1
15a. Naked pair {78} in R56C1, CPE no 7,8 in R6C3, clean-up: no 2,3 in R5C3
16. Killer pair 1,4 in R5C45 and R5C7, locked for R5, clean-up: no 9 in R6C3
17. Naked triple {789} in R5C136, locked for R5, clean-up: no 1 in R4C9
18. 16(3) cage at R4C2 = {169/268/349/358} (cannot be {259} because 2,5 only in R4C3)
18a. R5C2 = {36} -> no 3,6 in R4C23
19. 3 in R4 only in R4C45 = {23}, locked for R4, clean-up: no 6 in R5C9
20. 1 in R4 only in 16(3) cage at R4C2 (step 18) = {169} (only remaining combination) -> R5C2 = 6, placed for NR2C1, R34C1 = {49}, locked for NR2C1 -> R4C2 = 1, placed for NR2C1, R2C1 = 5, placed for NR2C1, R1C1 = 1, placed for NR1C1, R3C2 = 2, placed for NR2C1, R4C3 = 9, placed for NR1C1, R34C1 = [94], R45C6 = [89], R4C7 = 6, placed for NR1C9, R5C7 = 4, placed for NR3C5, R4C9 = 5, placed for NR1C9, R5C9 = 3, placed for NR5C8, R3C7 = 1, R6C3 = 3, R5C3 = 7, R5C1 = 8, R6C1 = 7 placed for NR6C1, clean-up: no 4 in R1C8, no 9 in R1C9, no 2 in R5C45, no 9 in R7C2, no 5 in R7C6 (step 10)
20a. Naked pair {15} in R5C45, locked for R5 and NR3C5 -> R5C8 = 2
[Some other clean-ups missed; they aren’t important.]
21. R1C8 = 8, placed for NR1C4, R1C9 = 2, placed for NR1C9, R2C7 = 9, R1C5 = 9, R7C8 = 1 (step 6) -> R67C9 = 15 = [69], all placed for NR5C8, R6C6 = 2, placed for NR3C5, R4C5 = 3, R4C4 = 2, placed for NR1C1
22. Naked pair {45} in R68C8, locked for C8 and NR5C8 -> R2C8 = 6, R1C7 = 3 (cage sum), both placed for NR1C4, R6C7 = 8, R6C8 = 5 (cage sum), R8C8 = 4, R9C7 = 2, placed for NR7C5
22a. Naked pair {57} in R78C7, locked for NR7C5 -> R7C6 = 6, placed for NR7C5, R3C56 = [67]
23. R6C6 = 2 -> R6C45 = 10 = [91], 1 placed for NR6C1, R6C2 = 4, R7C2 = 8, both placed for NR6C1
24. Naked pair {48} in R2C39, locked for R2 -> R2C6 = 1, R1C6 = 5 (cage sum)
and the rest is naked singles, without using the nonets.
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