Normal cages are sums. Minus sign indicates difference between the numbers in those two-cell cages. Non-consecutive including diagonally (just given as NC). Repeats are allowed in the central cage.
Prelims
a) No 5 in 5- cages -> no 5 in R4C12, no 5 in R6C34, no 5 in R4C6789
b) 7- cages must be {18/29} -> R67C6 = {18/29}, R89C4 = {18/29}
1. R6 contains three 5- cages from {16/27/38/49}
1a. R6C12 is 4- = {15/59} (cannot be {26/37/48} which only allow two 5- cages in R6), 5 locked for R6
1b. R6C12 contains one of 1,9 -> R6C5 = {46} (to allow three 5- cages in R6)
NC clean-ups:
R6C12 contain 5 -> no 4,6 in R57C12
R6C5 = {46} -> no 5 in R5C456 + R7C45
2. R4 contains three 4- cages from {15/26/37/48/59}
2a. R4C12 is 5- = {16/49} (cannot be {27/38} which would leave two of the three 4- cages as {15} and {59})
2b. R4C5 = {16/49} with three 4- cages -> R4C5 = {28}
3. C4 contains three 2- cages from {13/24/35/46/57/68/79}
3a. R89C4 = {18/29}
R89C4 = {18} gives three 2- cages from {24/35/46/57/79} => two must be {35} and {79} with the third one of {24/46} => R5C4 = {26}
R89C4 = {29} gives three 2- cages from {13/35/46/57/68} => two must be {13} and {57} with the third one of {46/68} => R5C4 = {48}
-> R5C4 = {2468}
3b. Similarly from R67C6 -> R5C6 = {2468}
3c. 30(5) cage at R4C5 contains even numbers in R46C5 and R5C46 -> R5C5 must also be even = {2468}
3d. 30(5) cage contains repeats but since it’s in R5 and C5 no number can be repeated more than twice = {26688/44688} (can only make total of 30 with 8 repeated) -> R4C5 = 8, also 8 in R5C46, locked for R5
3e. The non-repeated number must be in R5C5 -> R5C5 = {26} -> no 2 in R5C46
3f. 6 of {26688} must be in one of R5C46 and in R6C5, 6 of {44688} must be in R5C5) -> 6 in R5C456, locked for R5, 6 in R56C5, locked for C5
3g. R4C5 = 8 -> no 4 in 4- cages in R4 -> 4 in R4 only in R4C12 = {49}, 9 locked for R4
NC clean-ups:
R4C5 = 8 -> no 7,9 in R3C456, no 7 in R4C46
Naked pair {49} in R4C12 -> no 3,5,8 in R3C12, no 3,5 in R5C12
Overlapping cages
4a. R4C4 = {12356} -> no 2,6 in R3C4, no 3 in R4C3
4b. R4C6 = {12356} -> no 2,6 in R3C6, no 3 in R4C7
4c. R6C6 = {1289} -> R6C7 = {3467}
5. Consider combinations for R67C6 = {18/29}
R67C6 = {18}, 8 locked for C6
or R67C6 = [29] => R6C67 = [27] (5- cage) => no 8 in R5C6 (NC)
or R67C6 = [92] => no 8 in R5C6 (NC)
-> R5C6 = {46}
5a. 30(5) cage at R4C5 must contain 8 twice -> R5C4 = 8, no 1 in R89C4 (7- cage)
5b. Naked pair {29} in R89C4, locked for C4
5c. 2- cage at R67C4 = {13/46}, no 7
5d. 7 in C4 only in 2- cage at R12C4 = {57}, 5 locked for C4
NC clean-ups:
R5C4 = 8 -> no 7 in R4C3, no 7,9 in R56C3
R1C4 = {57} -> no 6 in R12C3
R2C4 = {57} -> no 6 in R3C3
R5C6 = {46} -> no 5 in R4C67 + R5C7
Naked pair {57} in R12C4 -> no 8 in R12C3
Naked pair {29} in R89C4 -> no 1,3,8 in R89C3, no 1,3 in R89C5
Overlapping cages
6a. No 7 in R4C3 -> no 3 in R4C4, no 1 in R3C4
6b. No 2,5 in R4C4 -> no 1,6 in R4C3
6c. No 5 in R4C6 -> no 3 in R3C6
6d. No 5 in R4C67 -> no 1 in R4C67
6e. No 7,8 in R6C4 -> no 2,3 in R6C3
6f. No 9 in R6C34 -> no 4 in R6C34 -> no 6 in R7C4 (2- cage)
Also
6g. 7 in R4 only in R4C789 -> no 6 in R4C8 (NC) -> no 2 in R4C9 (2- cage)
6h. 6 in C4 only in R46C4 -> no 5 in R5C3 (NC)
7. 30(5) cage at R4C5 (step 3d) = {26688/44688}, must have the same numbers in R5C6 and R6C5
7a. Consider permutations for R34C4 = [13/46]
R34C4 = [13] => R6C4 = 6 => R6C5 = 4
or R34C4 = [46] => R3C34 = [26] (2- cage) => R4C67 = [37] (2- cage) => no 6 in R5C6 (NC) => R5C6 = 4
-> 30(5) cage = {44688} -> R5C5 = 6, R5C6 = 4, R6C5 = 4, no 3 in R67C4, no 3 in R6C7 (NC)
[Cracked. A lot easier now.]
7b. R3C4 = 3 (hidden single in C4) -> R4C4 = 1 (2- cage), R67C4 = [64], R4C3 = 5 (4- cage), no 4 in R4C2 (NC)
7c. R6C7 = 7 -> R6C6 = 2 (5- cage) -> R7C6 = 9 (7- cage), no 1 in R5C7, no 8 in R6C8 (NC)
7d. R6C4 = 6 -> R6C3 = 1 (5- cage), no 2 in R5C3 (NC)
7e. R4C12 = [49], R6C12 = [95], R5C3 = 3
7f. R4C67 = [62] (4- cage) -> R3C6 = 8 (2- cage)
7g. R6C89 = [38], R4C89 = [73]
7h. R5C7 = 9
NC clean-ups:
R3C4 = 3 -> no 2,4 in R23C3, no 2 in R23C5
R3C6 = 8 -> no 7,9 in R2C5, no 7 in R2C6
R4C3 = 5 -> no 4,6 in R3C2
R4C6 = 6 -> no 5 in R3C57
R4C7 = 2 -> no 1 in R3C78 + R5C8
R4C8 = 7 -> no 6 in R3C789
R4C9 = 3 -> no 2,4 in R3C89, no 2 in R5C89
R5C3 = 3 -> no 2 in R5C2
R6C1 = 9 -> no 8 in R7C12
R6C2 = 5 -> no 6 in R7C3
R6C3 = 1 -> no 2 in R7C23
R6C4 = 6 -> no 7 in R7C35
R6C6 = 2 -> no 1,3 in R7C57
R6C7 = 7 -> no 6,8 in R7C78
R6C8 = 3 -> no 2 in R7C89
R6C9 = 8 -> no 7 in R7C9
8a. R3C7 = 4
8b. R357C8 = [951]
8c. R23C3 = [97]
8d. R3C1259 = [6215]
8e. R7C3579 = [8256]
8f. R5C1 = 2 (hidden single in R5), no 1 in R5C2 (NC)
8g. R5C29 = [71] -> R7C12 = [73], no 2,4 in R8C3 (NC)
8h. R8C3 = 6
NC clean-ups:
R2C3 = 9 -> no 8 in R12C2
R3C2 = 2 -> no 1,3 in R2C1, no 1 in R2C2
R3C3 = 7 -> no 6 in R2C2
R2C2 = 4 -> no 3,5 in R1C1, no 5 in R2C1
R3C7 = 4 -> no 3,5 in R2C6, no 3 in R2C7
R3C8 = 9 -> no 8 in R2C78
R7C1 = 7 -> no 8 in R8C2
R7C5 = 2 -> no 1,3 in R8C6
R7C7 = 5 -> no 4 in R8C8
R7C8 = 1 -> no 2 in R8C89
R7C9 = 6 -> no 7 in R8C9
[Now to use the regular cages. I had realised earlier, before I found the power of step 7, that they cannot contain more than two consecutive numbers since each cells is adjacent to at least three others; that isn’t needed now.]
9a. R2C123 = [849], R3C2 = 2 -> R1C2 = 6 (cage sum)
9b. R789C2 = [318] -> R8C1 = 5
9c. R2C6 = 1 -> R2C789 = [627], R3C8 = 9 -> R1C8 = 4 (cage sum)
9d. R789C8 = [186], R8C7 = 3 -> R8C9 = 4 (cage sum)
9e. R2C6 = 1 -> R1C6 = 3 (2- cage)
and the rest is naked singles.
