Prelims
a) R2C34 = {29/38/47/56}, no 1
b) R34C1 = {49/58/67}, no 1,2,3
c) R3C34 = {14/23}
d) R3C67 = {18/27/36/45}, no 9
e) R4C34 = {19/28/37/46}, no 5
f) R4C67 = {29/38/47/56}, no 1
g) R45C8 = {15/24}
h) R56C4 = {16/25/34}, no 7,8,9
i) R56C6 = {59/68}
j) R67C5 = {49/58/67}, no 1,2,3
k) R8C78 = {59/68}
m) R89C9 = {14/23}
n) R9C23 = {49/58/67}, no 1,2,3
o) 24(3) cage at R2C9 = {789}
p) 10(3) cage at R3C5 = {127/136/145/235}, no 8,9
q) 11(3) cage at R7C6 = {128/137/146/236/245}, no 9
r) 32(5) cage at R5C1 = {26789/35789/45689}, no 1
1. 45 rule on R89 1 outie R7C6 = 1, clean-up: no 8 in R3C7
1a. R7C6 = 1 -> R8C56 = 10 = {28/37/46}, no 5
2. Naked triple {789} in 24(3) cage at R2C9, locked for C9
3. 20(4) cage at R5C9 must contain one of 7,8,9 -> R7C8 = {789}
3a. 45 rule on C9 1 outie R7C8 = 1 innie R1C9 + 4 -> R1C9 = {345}
4. 32(5) cage at R5C1 = {26789/35789/45689}, 8,9 locked for N4, clean-up: no 4,5 in R3C1, no 1,2 in R4C4
5. 45 rule on N3 3(2+1) outies R23C6 + R4C9 = 21
5a. Min R23C6 = 12, no 2,3 in R2C6, no 2 in R3C6, clean-up: no 7 in R3C7
6. 45 rule on N78 1 outie R6C5 = 1 innie R9C6 + 4 -> R6C5 = {6789}, R9C4 = {2345}, clean-up: no 8,9 in R7C5
7. 45 rule on C1234 1 outie R9C5 = 1 innie R1C4 + 4, R1C4 = {12345}, R9C5 = {56789}
8. 45 rule on C6789 1 innie R1C6 = 1 outie R8C5 + 1, no 2,6 in R1C6
9. 45 rule on R1 2 outies R2C15 = 7 = {16/25/34}, no 7,8,9
10. 45 rule on N69 2 innies R4C79 = 1 outie R9C6 + 13
10a. Max R4C79 = 17 -> max R9C6 = 4, clean-up: no 9 in R6C5 (step 6), no 4 in R7C5
10b. Min R9C6 = 2 -> min R4C79 = 15 -> min R4C7 = 6, max R4C6 = 5
11. 10(3) cage at R3C5 = {127/136/145/235}, R67C5 = {67}/[85] -> combined cage 10(3) + R67C5 = {127/136}[85]/{145/235}{67}, 5 locked for C5, clean-up: no 1 in R1C4 (step 7), no 2 in R2C1 (step 9)
12. 45 rule on N4 4 innies R4C123 + R5C2 = 13 = {1237/1246/1345}
12a. 7 of {1237} must be in R4C1 -> no 7 in R4C23 + R5C3, clean-up: no 3 in R4C4
13. 45 rule on N14 3 innies R234C3 = 9 = {126/135/234}, no 7,8,9, clean-up: no 2,3,4 in R2C4
14. 45 rule on R1234 3 outies R5C258 = 7 = {124}, locked for R5, clean-up: no 1 in R4C8, no 3,5,6 in R6C4
15. 6 in C9 only in 20(4) cage at R5C9 = {1469/1568/2369/2567} (cannot be {2468/3467} which clash with R89C9)
15a. 1 of {1469} must be in R6C9 -> no 4 in R6C9
16. 45 rule on N7 1 outie R7C4 = 1 innie R8C3 + 5, R7C4 = {6789}, R8C3 = {1234}
16a. R234C3 = 9 (step 13) -> max R2348C3 = 13(4), must contain 1, locked for C3
17. 45 rule on C6789 2 remaining innies R18C6 = 11 = {38/47}/[92] (cannot be [56] which clashes with R56C6), no 5,6, clean-up: no 4 in R8C5 (step 1a)
18. R4C123 + R5C2 (step 12) = {1237/1246/1345}
18a. R4C13 cannot total 10, which would clash with R4C34 (CCC) -> R45C2 cannot total 3 = {12}
18b. R4C123 + R5C2 = {1246} must have one of 1,2 in R4C3 -> no 6 in R4C3, clean-up: no 4 in R4C4
18c. R4C123 + R5C2 = {1345} must have 3,5 in R4C123, locked for R4 => naked pair {24} in R4C68, locked for R4
18d. Taking steps 18b and 18c together, no 4 in R4C3, clean-up: no 6 in R4C4
Step 19, a short forcing chain based on placements for 5 in N8, and step 20, analysis of 10(3) cage at R3C5, have been deleted. They aren’t required after steps 22 and 24.
21. 45 rule on R12 2 innies R2C29 = 1 outie R3C8 + 7, IOU no 7 in R2C2
22. R23C6 + R4C9 = 21 (step 5) -> min R23C6 = 12
22a. 45 rule on C789 4 outies R2349C6 = 19 = {2359/2368/2467/3457} (cannot be {2458} which clashes with R56C6) -> R23C6 = {57/67/68}/[95] (R2349C6 = {2359} cannot be [9352] which would place 6 in both of R34C7), no 4 in R2C6, no 3,4 in R3C6, clean-up: no 5,6 in R3C7
22b. 5,7,9 of {2359/3457} must be in R23C6 -> no 5 in R4C6, clean-up: no 6 in R4C7
23. Naked triple {789} in R4C479, locked for R4, clean-up: no 6 in R3C1
24. R4C79 = R9C6 + 13 (step 10), R4C479 = {789} = 24 -> R4C4 + R9C6 = 11
24a. 45 rule on N578 5(4+1) innies R4C456 + R5C5 + R9C6 = 20, R4C4 + R9C6 = 11 -> R4C56 + R5C5 = 9 = {126/234} (cannot be {135} which clashes with R56C4), no 5
24b. R4C56 + R5C5 = 9 = {126/234}, 2 locked for N5, clean-up: no 5 in R5C4
25. 5 in N5 only in R56C6 = {59}, locked for C6 and N5, clean-up: no 4 in R3C7, no 1 in R4C3
26. 9 in R4 only in R4C79, locked for N6
26a. R4C79 = {79/89} = 16,17 -> R9C6 = {34} (step 10)
27. R2349C6 (step 22a) = {2368/2467} -> R4C6 = 2, R4C7 = 9, R4C3 = 3, R4C4 = 7, R4C9 = 8
27a. R5C4 = 3, R6C5 = 8 (hidden singles in N5), R7C5 = 5, R6C4 = 4, R45C5 = [61], R3C5 = 3 (cage sum)
27b. R3C4 = 1 (hidden single in N2), R3C7 = 2, R3C6 = 7, R23C9 = [79], R3C3 = 4, R3C1 = 8, R4C1 = 5, R45C8 = [42], R45C2 = [14]
[Routine clean-ups omitted.]
28. R2C6 = 6 (hidden single in C6), R3C8 = 5, R3C2 = 6, R2C2 = 5 (cage sum), R2C3 = 2, R2C4 = 9, R8C3 = 1, R7C4 = 6 (step 16)
29. R8C56 = 10 (step 1a) = [73] (cannot be [28] which clashes with R8C4), R9C56 = [94]
30. R8C7 = 5 (hidden single in R8), R8C8 = 9
30a. Naked pair {28} in R8C24, locked for R8 -> R8C9 = 4, R9C9 = 1
31. R8C1 = 6 -> R8C2 + R9C1 = 9 = [27]
32. R1C9 = 3, R56C9 = {56}, locked for N6 -> R5C7 = 7
and the rest is naked singles.
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