Disjoint 30(5) cage R2C5689 + R3C9
Prelims
a) R23C2 = {18/27/36/45}, no 9
b) R2C34 = {15/24}
c) R5C12 = {19/28/37/46}, no 5
d) R67C9 = {29/38/47/56}, no 1
e) R7C23 = {39/48/57}, no 1,2,6
f) R89C1 = {16/25/34}, no 7,8,9
g) R8C78 = {19/28/37/46}, no 5
h) R89C9 = {17/26/35}, no 4,8,9
j) 21(3) cage at R1C1 = {489/579/678}, no 1,2,3
j) 19(3) cage at R7C5 = {289/379/469/478/568}, no 1
k) 14(4) cage at R1C3 = {1238/1247/1256/1346/2345}, no 9
l) 27(4) cage at R5C4 = {3789/4689/5679}, no 1,2
1a. 45 rule on R89 1 outie R7C5 = 6, clean-up: no 5 in R6C9
1b. R7C5 = 6 -> R8C56 = 13 = {49/58}
1c. 27(4) cage at R5C4 = {3789/4689/5679}, CPE no 9 in R4C4
2a. 45 rule on R1 2 outies R2C17 = 11 = {47/56}/[83/92], no 1,8,9 in R2C7
2b. 45 rule on R12 2 outies R3C29 = 11 = [29]/{38/47/56}, no 1 in R3C2, no 1,2 in R3C9, clean-up: no 8 in R2C2
2c. 45 rule on N6 2 innies R6C79 = 11 = {29/38/47}/[56], no 1,6 in R6C7
2d. 45 rule on N69 1 outie R9C6 =2, clean-up: no 5 in R8C1, no 6 in R8C9
2e. R9C6 = 2 -> R9C78 = 13 = {49/58/67}, no 1,3
3a. 45 rule on N3 4 outies R23C56 = 26 = {2789/3689/4589/4679/5678}, no 1
3b. 45 rule on N23 2 outies R12C3 = 1 innie R3C4 + 1
3c. Min R12C3 = 3 -> min R3C4 = 2
4. 45 rule on N1234 2(1+1) outies R4C4 + R7C1 = 7 = {25/34}/[61], no 7,8,9, no 1 in R4C4
[I missed R4C4 + R7C1 cannot be [52] when R4C4 ‘sees’ all remaining 5s in N4. This forces 37(7) cage at R8C2 to contain both of 2,6 in N7 and leads to a much simpler solution.]
5a. 37(7) cage at R8C2 = {1246789/1345789/2345689} must contain 4,8,9
5b. R8C4 + R9C45 cannot contain both of 4,8 (which clashes with R8C56) -> R89C23 must contain at least one of 4,8 -> R7C23 = {39/57} (cannot be {48} which clashes with R89C23), no 4,8
5c. 8 in N7 only in R89C23, locked for 37(7) cage
5d. 45 rule on N9 3 remaining innies R7C789 = 14 = {149/158/239/248} (cannot be {257} which clashes with R89C9), cannot be {347} which clashes with R7C23), no 7, clean-up: no 4 in R6C9, no 7 in R6C7 (step 2c)
[First time through I then made a careless elimination which wasn’t valid, so have re-worked with a heavier step to reach the same result.]
5e. Consider combinations for R89C9 = {17/35}/[26]
R89C9 = {17/35} => R7C789 = 14 = {149/239/248} (cannot be {158} which clashes with R89C9)
or R89C9 = [26] => 37(7) cage at R8C2 = {1345789} (only remaining combination), R78C1 = [26] (hidden pair in N7) => R9C1 = 1, R8C4 = 1 (for the 37(7) cage) then
either R7C23 = {39}, locked for N7 => 9 in 37(7) cage in R9C45, 9 locked for R9 => R9C78 = {58}, locked for N9 => R7C789 = {149}
or R7C23 = {57}, locked for R7 => R7C789 = {149}
-> R7C789 = {149/239/248}, no 5, clean-up: no 6 in R6C9, no 5 in R6C7 (step 2c)
5f. 45 rule on N78 3 remaining innies R7C146 = 13 = {139/148/157/238} (cannot be {247} which clashes with R7C789)
5g. R9C78 (step 2e) = {58/67} (cannot be {49} which clashes with R7C89), no 4,9
5h. 9 in R9 only in R9C2345, locked for 37(7) cage at R8C2
6a. R7C789 (step 5e) = {149/239/248}
6b. Consider combinations for R8C56 = {49/58}
R8C56 = {49} => 4 in N9 only in R7C789 = {149/248}
or R8C56 = {58} => 8 in R7 only in R7C789 = {248}
-> R7C789 = {149/248}, no 3, 4 locked for R7 and N9, clean-up: no 3 in R4C4 (step 4), no 8 in R6C9, no 3 in R6C7 (step 2c), no 6 in R8C78
6c. 6 in N9 only in R9C789, locked for R9, clean-up: no 1 in R8C1
6d. Hidden killer pair 5,6 in R89C9 and R9C78 for N9, R9C78 contains one of 5,6 -> R89C9 must contain one of 5,6 = [26]/{35}, no 1,7
7a. R7C146 (step 5f) = {139/157/238}
7b. Consider combinations for R7C23 = {39/57}
R7C23 = {39}, 9 locked for N7 => 9 in 37(7) cage at R8C2 in R9C45, locked for N8 => R8C56 = {58}, 5 locked for N8
or R7C23 = {57}, 5 locked for R7
-> no 5 in R7C46
7c. R7C146 = {157} must be [571], no 7 in R7C6
7d. 5 in R7 only in R7C123, locked for N9, clean-up: no 2 in R8C1
7e. 37(7) cage at R8C2 (step 5a) must contain both or neither of 2,6
7f. Consider placement of 2,6 in N7
R789C1 = [261] => R7C46 = {38}, 3 locked for R7 => R7C23 = {57}
or R8C23 = {26}, 6 locked for N7 => R89C1 = {34}, 3 locked for N7 => R7C23 = {57}, 5 locked for N7 => R7C1 = 1
-> R7C1 = {12}, R7C23 = {57}, 7 locked for R7 and N7, 1 in R79C1, locked for C1 and N7, clean-up: no 2,4 in R4C4 (step 4), no 9 in R5C2
7g. R7C146 = {139/238} -> R7C46 = {38/49}, 3 locked for N8
7h. 9 in N7 only in R9C23, locked for R9
7i. Consider combinations for R8C56 = {49/58}
R8C56 = {49} => R7C46 = {38} => R9C45 = {15/17} (cannot be {57} which clashes with R9C78), 1 locked for R9 => R89C1 = {34}
or R8C56 = {58}, R7C46 = {39} => R7C1 = 1, R89C1 {34}
-> R7C1 = 1, R89C1 = {34}, locked for C1 and N7 clean-up: no 7 in R2C7 (step 2a)
7j. R7C1 = 1 -> R7C46 = {39}, 9 locked for R7 and N8, R4C4 = 6 (step 4), clean-up: no 6,7 in R5C2, no 2 in R6C9, no 9 in R6C7 (step 2c), no 4 in R8C56
7k. Naked pair {58} in R8C56, locked for R8, 5 locked for N8
7l. Naked pair {26} in R8C23, 2 locked for R8, R8C9 = 3 -> R9C9 = 5, R89C1 = [43], clean-up: no 6,8 in R3C2 (step 2b), no 1,3 in R2C2, no 8 in R6C9 (step 2c), no 8 in R7C9, no 7 in R8C78, no 8 in R9C78
7m. R8C4 = 7 (hidden single in R8)
7n. 27(4) cage at R5C4 = {3789} (only remaining combination) -> R5C5 = 7, R567C4 = {389}, locked for C4, 8 locked for N5, clean-up: no 3 in R5C2
7o. 1 in N1 only in R123C3, locked for C3
7p. 2,5 in C4 only in R123C4, locked for N2
7q. 12(3) cage at R6C5 = {129/345}
7r. R7C6 = {39} -> R6C56 = [21]/{45}
7s. Killer pair 2,4 in R6C56 and R6C7, locked for R6
7t. R7C1 = 1 -> R6C12 = 11 = {56}/[83]
7u. 12(3) cage at R4C5 = 2{19}/{345}, no 1,9 in R4C5
7v. 2 in N5 only in R46C5 -> 12(3) cage at R4C5 = 2{19} or 12(3) cage at R6C5 = [219] -> 1,9 in R4567C6, locked for C6 (locking-cages)
8a. R23C56 (step 3a) = {3689/4679}
8b. 6,7 of {4679} must be in R23C6 -> no 4 in R23C6
8c. Hidden killer pair 7,8 in R1C56 and R23C56 for N2, R23C56 contains one of 7,8 -> R1C56 must contain one of 7,8
8d. 14(4) cage at R1C3 = {1238/1247}, no 5,6, 1,2 locked for R1
8e. One of 7,8 in R1C56 -> no 7,8 in R1C3
8f. 7 of {1247} must be in R1C6 -> no 4 in R1C6
8g. 4 in C6 only in R456C6, locked for N5
9a. 45 rule on C12 3 innies R789C2 = 18 = [729] (only possible permutation) -> R789C3 = [568], clean-up: no 1 in R2C4, no 4,9 in R3C9 (step 2b), no 8 in R5C1
9b. 1 in N2 only in R1C45, locked for R1
9c. 31(6) cage at R3C3 contains 6 and 7,9 for C3 = {135679/234679}, 3 locked for C3
9d. 45 rule on N1 4 innies R12C3 + R3C13 = 15 = {1239/1248/1257} (cannot be {1347/1356/2346} which clash with R23C2), no 6
9e. 3 of {1239} must be in R3C3 -> no 9 in R3C3
9f. 5 of {1257} must be in R3C1 -> no 7 in R3C1
10a. 13(3) cage at R3C1 = {157/238/247/256} (cannot be {139/148/346} because 1,3,4 only in R4C2), no 9
10b. 3 of {238} must be in R4C2 -> no 8 in R4C2
10c. 7 of {157} must be in R4C1, 5 of {256} must be in R4C2 -> no 5 in R4C1
10d. 9 in N1 only in R12C1, locked for C1, clean-up: no 1 in R5C2
10e. R4C2 = 1 (hidden single in N4) -> R34C1 = [57], clean-up: no 4 in R23C2
10f. R23C2 = [63] -> R3C9 = 8 (step 2b)
10g. R2C1 = 8 (hidden single in R2) -> R2C7 = 3 (step 2a), R1C123 = [942], R2C3 = 1 -> R2C4 = 5, R6C12 = [65]
10h. R6C5 = 2 -> R67C6 = 10 = [19], R6C7 = 4 -> R6C9 = 7 (step 2c), R7C9 = 4
10i. R5C9 = 1 (hidden single in C9) -> R56C8 = 15 = [69]
and the rest is naked singles.