SudokuSolver Forum

Working out N4 (mostly)
Outies r123 = +8.
Working out those outies are +8(3)

482631597
136975842
579428316
957284163
613759428
648316975
795842631
361597284
874163759

The usual uncaged killer rules, the number is in the top left corner with top dominating left as usual. There are no single cages and no diagonally-connected cages.
Non-consecutive (NC) horizontally and vertically.

Prelims, based just on positions of totals.
a) 15(?) cage at R1C6 must contain at least R1C678
b) 14(?) cage at R1C9 must contain at least R12C9
c) 7(?) cage at R5C1 must contain at least R5C12, either 7(2) or 7(3), no 7,8,9 in R5C12 also no 3 because R5C12 cannot be {34} NC
d) R4C12 must be 14(2) cage = {59/68}
e) 18(?) cage at R4C3 must contain at least R456C3
f) R67C9 must be 6(2) cage = {15/24} (cannot be 6(3) cage, NC)
g) R67C8 must be 10(2) cage = {19/28/37/46}, no 5
h) R89C4 must be 6(2) cage = {15/24} (cannot be 6(3) cage, NC)
i) 22(?) cage at R3C2 must contain at least R3C234
j) 15(?) cage at R2C3 must contain at least R2C34
k) 10(?) cage at R4C4 must contain at least R4C45
l) 21(?) cage at R5C4 must contain R5C5 and at least one more cell
m) 15(?) cage at R7C7 must contain at least R78C7
n) 25(?) cage at R8C5 must contain R89C5 + R9C6 and at least one of R8C6 or R9C7
o) 26(?) cage at R8C8 must contain at least R89C89
p) 16(?) cage at R1C4 must contain at least R1C45
q) 10(?) cage at R1C2 must contain at least R1C23
r) 13(?) cage at R1C1 must contain at least R123C1
s) 9(?) cage at R3C9 must contain at least R34C9, no 9 in R34C9
t) 20(?) cage at R4C8 must contain R45C8 and at least one more cell
u) 13(?) cage at R2C6 must contain at least R2C67
v) 22(?) cage at R6C1 must contain R67C1 and at least one of R67C2

1. 45 rule on N4 one or two innies total 6 -> either R6C1 = 6
or R6C12 = {24} (cannot be {15} which clashes with R5C12 = 7(2) = {16/25}) => R5C12 = 7(2) = {16}
-> 6 in R5C12 + R6C1, locked for N4
1a. R6C1 = {246}
1b. R4C12 = {59}, locked for R4 and N4
1c. 3,7,8 in N4 only in 18(?) cage at R4C3, 3,7,8 total 18 -> 18(3) cage in R456C3 = {378}, locked for C3 and N4
1d. R456C3 = [738/837], NC
1e. R5C3 = 3 -> no 2,4 in R5C2
1f. R5C12 must be 7(2) cage = {16}, locked for R5 and N4 (cannot be 7(3) cage = {124} with 1 in R5C2, NC)
1g. Naked pair {24} in R6C12, locked for R6 and 22(?) cage at R6C1
1h. Rest of 22(?) cage must total 16 so must contain at least R7C12, no 2,4 in R7C12
1i. 1 and 2 in R5C12 and R6C12 must be diagonally opposite each other (NC) -> no 1 in R7C12 (NC)
1j. 5 and 6 in R4C12 and R5C12 must be diagonally opposite each other (NC) -> no 6 in R3C12 (NC)
1k. 6(2) cage R67C9 = {15}, locked for C9
1l. 10(2) cage at R67C8 = {19/37}/[64/82], no 6,8 in R7C8
1m. 9(?) cage at R3C9 must be 9(2) cage R34C9 = {27/36} (cannot be 9(3) cage = {234}, NC), no 4,8 in R34C9
NC eliminations:
R4C3 = {78} -> no 7,8 in R4C4
R5C3 = 3 -> no 2,4 in R5C4
R6C1 = {24} -> no 3 in R7C1
R6C2 = {24} -> no 3 in R7C2
R6C3 = {78} -> no 7,8 in R6C4

2. 45 rule on N1 3 innies R2C3 + R3C23 = 22 = {589/679}
2a. 7,8 only in R3C2 -> R3C2 = {78}, R23C3 = [69/95] (cannot be [59] because of [589], cannot be [96] because of [976], NC), 9 locked for C3 and N1
NC eliminations:
R3C2 = {78} -> no 7,8 in R2C2 + R3C1
R2C3 = 6 or R3C3 = 5 -> no 5 in R1C3 + R2C2

3. R4C12 = {59}, R5C12 = {16}, R6C12 = {24} -> R456C1 = R456C2 = [514/962] (NC)
3a. 10(?) cage at R1C2 must be 10(2) in R1C23 = [46/64/82]
or 10(3) cage in R1C23 + R2C2 = {127/136/235} (cannot be {145} NC because 5 only in R1C2)
3b. 13(?) cage at R1C1 must be 13(3) cage R123C1
or 13(4) cage R123C1 + R2C2
3c. 13(3) cage R123C1 = {148} (cannot be {157/238/256} which clashes with 10(3) cage R1C23 + R2C3, cannot be {247/346} which clash with R456C1)
or 13(4) cage = {1237/1345} (cannot be {1246} which clashes with 10(2) cage R1C23)
3d. Summarising
13(?) cage = {148/1237/1345}, no 6, 1 locked for N1
10(?) cage = [46/64/82/523], no 2,3,7 in R1C2
3e. 1 in C3 only in R789C3, locked for N7
NC eliminations:
7 of 13(?) cage = {1237} must be in R2C1 -> no 7 in R1C1
1 of 13(?) cage = {1345} must be in R2C1 -> no 2,3,5 in R2C1
1 in R789C3 -> no 2 in R8C3
3f. Consider permutations for R2C3 + R3C23 (step 2a) = [679/985]
R2C3 + R3C23 = [679]
or R2C3 + R3C23 = [985], no 9 in R4C2 (NC) => R456C2 = [514]
-> 10(?) cage at R1C23 = [64/82/523], no 4 in R1C2, no 6 in R1C3
NC elimination:
R1C3 = {24} -> no 3 in R1C4

4. 45 rule on N47, total = 90, no innies -> no outies
4a. 10(?) cage at R7C3 must be completely within N7
4b. 10(?) cage at R1C23 (step 3f) = [64/82/523], consider permutations for R2C3 + R3C23 (step 2a) = [679/985]
R2C3 + R3C23 = [679]
or R2C3 + R3C23 = [985] => R1C23 = [64]
-> 10(?) cage at R7C3 cannot be 10(2) cage in R78C3 = {46}
4c. 10(?) cage at R7C3 must be 10(3) cage R789C3 or R7C3 + R8C23 (cannot be 10(4) cage = {1234} which clashes with R1C3)
4d. 10(3) cage R789C3 or R7C3 + R8C23 = [136/631]/{145} (cannot be [271], NC, cannot be [235] which clashes with R1C23 + R23C3 = [6495]), no 2 in R7C3
4e. Consider permutations for R34C2 = [75/79/85] (cannot be [89], NC)
R34C2 = [75/79] => R2C3 = 6, R1C3 = 2 => R789C3 = {145}
or R34C2 = [85] => 7,9 in C2 must be in R789C2, locked for N7 => 22(?) cage at R6C1 cannot be {24}{79} so must also contain R8C2 => 10(3) cage at R7C3 must be R789C3
-> R789C3 = 10(3) = {145}, locked for C3 and N7
4f. R789C3 = [415/514], NC
4g. R23C3 = [69], R3C2 = 7, R1C3 = 2, R4C3 = 7 (NC) -> R6C3 = 8, clean-up: no 2 in R34C9, no 2 in R7C8
4h. Naked pair {36} in R34C9, locked for C9
[From this point some NCs haven’t been stated where they lead directly to placements, combinations or permutations.]
4i. 22(?) cage at R6C1 must be 22(4) cage R67C12 = {24}[79] (cannot be {24}{68}2, cannot be {24}[76]3, NC) -> R7C12 = [79], R4C12 = [95], R5C12 = [61], R6C12 = [24], R89C1 = [38], R89C2 = [62], R12C2 = [83], clean-up: no 1,3 in R6C8
NC elimination:
R9C3 = {45} -> no 4,5 in R9C4
4j. R9C4 = 1 -> R8C4 = 5
4k. R123C1 = [415/514], NC
NC eliminations:
R2C3 = 6 -> no 7 in R2C4
R3C3 = 9 -> no 8 in R3C4
R4C3 = 7 -> no 6 in R4C4
R6C3 = 8 -> no 9 in R6C4
R8C4 = 5 -> no 4,6 in R7C4, no 4 in R8C5

5. 45 rule on R123 two or three outies total 8 = {14}3 (cannot be [26] because no remaining combinations for 10(?) cage at R4C4 as 10(3) cage) -> R4C67 = {14}, locked for R4, R4C9 = 3 -> R3C9 = 6
5a. 10(?) cage at R4C4 must be 10(2) cage R4C45 = [28]
5b. 22(?) cage at R3C2 must be 22(4) cage R3C2345 = [7942], R13C1 = [45]
5c. R4C67 = {14} must both be part of 16(?) cage at R3C6 = 16(4) cage R34C67 = {38}{14}, R3C8 = 1, clean-up: no 9 in R6C8
[Note. With hindsight it should have been [83]{14} because R3C5 = 2]

6. 45 rule on R789 5 innies R7C45689 = 18, min R7C89 = 4 -> max R7C456 = 14, cannot contain both of 6,8 -> R7C7 = {68} (only other place for 6 or 8 in R7)
6a. R7C69 = [21] (hidden pair in R7) -> R6C9 = 5, R6C8 = 7 -> R7C8 = 3, R7C457 = [846]
6b. Naked pair {79} in R8C56, locked for R8 and N8
6c. R89C56 = {79}{36} must form 25(4) cage
6d. R7C7 = 6, no 9 in R8C7, 26(?) cage at R8C8 requires at least four cells -> 15(3) cage R789C7 = [627/645]
6e. R1256C4 = [6973]
6f. R2C34 must be 15(2) cage = [69]
6g. R12C9 cannot total 14 -> 14(?) cage at R1C9 must also contain R23C8 (since 13(?) cage at R2C6 cannot then reach R3C8)
6h. 13(?) cage at R2C6 must be 13(2) cage R2C67 = {58}, locked for R2 -> R2C5 = 7, R2C67 = [58]
6i. R3C8 = 1, R2C89 = [42] -> R1C9 = 7 (cage sum)
6j. 16(?) cage at R1C4 must be 16(3) cage R1C45 + R2C5 = [637]
6k. 15(?) cage at R1C6 must be 15(3) cage R1C678 = [159]
6l. R79C7 = [67] -> R8C7 = 2 (cage sum)

and the rest is naked singles, without using NC.

The remaining cages must be 10(2) cage in R1C23, 13(4) cage in R123C1 + R2C2, 21(3) cage in R5C456, 20(4) cage in R4C8 + R5C789, 16(4) cage in R67C45, 17(3) cage in R6C67 + R7C6, 19(4) cage in R89C12 and 26(4) cage in R89C89.

The usual uncaged killer rules, the number is in the top left corner with top dominating left as usual. There are no single cages and no diagonally-connected cages.
Non-consecutive (NC) horizontally and vertically.
?? represents a two digit cage total, 10 or greater.

Prelims, based just on positions of totals.
a) ??(?) cage at R1C6 must contain at least R1C678
b) 14(?) cage at R1C9 must contain at least R12C9
e) 7(?) cage at R5C1 must contain at least R5C12, either 7(2) or 7(3), no 7,8,9 in R5C12 also no 3 because R5C12 cannot be {34} NC
d R4C12 must be 14(2) cage = {59/68}
e) 18(?) cage at R4C3 must contain at least R456C3
f) R67C9 must be 6(2) cage = {15/24} (cannot be 6(3) cage, NC)
g) R67C8 must be 10(2) cage = {19/28/37/46}, no 5
h) R89C4 must be 6(2) cage = {15/24} (cannot be 6(3) cage, NC)
i) 22(?) cage at R3C2 must contain at least R3C234
j) ??(?) cage at R2C3 must contain at least R2C34
k) 10(?) cage at R4C4 must contain at least R4C45
l) 21(?) cage at R5C4 must contain R5C5 and at least one more cell
m) 15(?) cage at R7C7 must contain at least R78C7
n) 25(?) cage at R8C5 must contain R89C5 + R9C6 and at least one of R8C6 or R9C7
o) 26(?) cage at R8C8 must contain at least R89C89
o) ??(?) cage at R1C4 must contain at least R1C45
p) ??(?) cage at R1C2 must contain at least R1C23
q) 13(?) cage at R1C1 must contain at least R123C1
r) 9(?) cage at R3C9 must contain at least R34C9, no 9 in R34C9
s) 20(?) cage at R4C8 must contain R45C8 and at least one more cell
t) ??(?) cage at R2C6 must contain at least R2C67
u) 22(?) cage at R6C1 must contain R67C1 and at least one of R67C2

[Same step 1 as for the original version.]
1. 45 rule on N4 one or two innies total 6 -> either R6C1 = 6
or R6C12 = {24} (cannot be {15} which clashes with R5C12 = 7(2) = {16/25}) => R5C12 = 7(2) = {16}
-> 6 in R5C12 + R6C1, locked for N4
1a. R6C1 = {246}
1b. R4C12 = {59}, locked for R4 and N4
1c. 3,7,8 in N4 only in 18(?) cage at R4C3, 3,7,8 total 18 -> 18(3) cage in R456C3 = {378}, locked for C3 and N4
1d. R456C3 = [738/837], NC
1e. R5C3 = 3 -> no 2,4 in R5C2
1f. R5C12 must be 7(2) cage = {16}, locked for R5 and N4 (cannot be 7(3) cage = {124} with 1 in R5C2, NC)
1g. Naked pair {24} in R6C12, locked for R6 and 22(?) cage at R6C1
1h. Rest of 22(?) cage must total 16 so must contain at least R7C12, no 2,4 in R7C12
1i. 1 and 2 in R5C12 and R6C12 must be diagonally opposite each other (NC) -> no 1 in R7C12 (NC)
1j. 5 and 6 in R4C12 and R5C12 must be diagonally opposite each other (NC) -> no 6 in R3C12 (NC)
1k. 6(2) cage R67C9 = {15}, locked for C9
1l. 10(2) cage at R67C8 = {19/37}/[64/82], no 6,8 in R7C8
1m. 9(?) cage at R3C9 must be 9(2) cage R34C9 = {27/36} (cannot be 9(3) cage = {234}, NC), no 4,8 in R34C9
NC eliminations:
R4C3 = {78} -> no 7,8 in R4C4
R5C3 = 3 -> no 2,4 in R5C4
R6C1 = {24} -> no 3 in R7C1
R6C2 = {24} -> no 3 in R7C2
R6C3 = {78} -> no 7,8 in R6C4

[Moving on to what HATMAN has indicated is a key step.]
2. 45 rule on R123 two or three outies total 8 = {14}3 (cannot be [26] because no remaining combinations for 10(?) cage at R4C4 as 10(3) cage) -> R4C67 = {14}, locked for R4, R4C9 = 3 -> R3C9 = 6
2a. 10(?) cage at R4C4 must be 10(2) cage R4C45 = [28], R46C3 = [78], R4C8 = 6
2b. R89C4 = {15}, locked for C4 and N8
2c. 10(2) cage at R67C8 = [19/73/91], no 2,4,7 in R7C8
NC eliminations:
R3C9 = 6 -> no 7 in R2C9, no 5,7 in R3C8
R4C4 = 2 -> no 3 in R3C4
R4C5 = 8 -> no 7,9 in R35C5
R4C8 = 6 -> no 5,7 in R5C8
R4C9 = 3 -> no 2,4 in R5C9
R6C3 = 8 -> no 9 in R6C4 + R7C3
R5C5 = {45} -> no 4,5 in R5C6, no 5 in R6C5
2d. Naked pair {79} in R5C46, locked for R5 and N5 -> R5C9 = 8
2e. R4C8 = 6, R5C9 = 8 -> 20(?) cage at R4C8 must be R4C8 + R5C789 = 6{24}8, 4 locked for R5 and N6 -> R4C67 = [41], R5C5 = 5, R67C9 = [51],
2f. 10(2) cage at R67C8 = [73] -> R6C7 = 9
2g. 21(?) cage at R5C4 must be R5C456 = {579}
NC eliminations:
R4C6 = 4 -> no 3,5 in R3C6
R4C7 = 1 -> no 2 in R35C7
R5C5 = 5 -> no 6 in R6C5
R6C7 = 9 -> no 8 in R7C7
R7C8 = 3 -> no 2,4 in R7C7 + R8C8
R7C9 = 1 -> no 2 in R8C9
R6C5 = {13} -> no 2 in R7C5
R7C7 = {567} -> no 6 in R7C6 + R8C7
2h. R5C78 = [42]

3. 45 rule on N9, total = 45, no innies -> no outies
3a. 15(?) cage at R7C7 must be 15(3) cage R789C7 -> 26(?) cage at R8C8 must be 26(4) cage R89C89
3b. 26(4) cage cannot be {2789}, NC
3c. 2,6 in N7 only in 15(3) cage = {267} -> R8C7 = 2, R79C7 = {67}, 7 locked for C7 and N9
3d. Naked triple {358} in R123C7, 5,8 locked for N3
3e. R1C9 = 7 (hidden single in C9)
3f. 14(?) cage at R1C9 can only be 14(4) cage R1C9 + R2C89 + R3C8 -> R2C9 = 2, R23C8 = [41], R1C8 = 9
3g. R2C8 = 4 -> R2C7 = 8 (NC)
NC eliminations:
R2C7 = 8 -> no 7,9 in R2C6
R7C7 = {67} -> no 7 in R7C6
R9C7 = {67} -> no 6,7 in R9C6

4. 17(?) cage at R6C6 must contain 9 in R6C7 and at least one more cells R7C7 -> R67C6 = [62]
4a. R6C45 = [31]
4b. 16(?) cage at R6C4 must be 16(4) cage R67C45 -> R7C45 = 12 = [84]
4c. 6 in C4 only in R12C4, locked for N2
4d. Naked pair {23} in R13C5, 3 locked for C5 and N2
4e. R9C6 = 3 (hidden single in C6)
[From this point some NCs haven’t been stated where they lead directly to placements, combinations or permutations.]
4f. 25(?) cage at R8C5 must be 25(4) cage = {3679} -> R9C5 = 6, R8C56 = [97]
4g. R89C4 = [51], R89C7 = [67], R89C8 = [85], R89C9 = [49]
4h. R7C3 = 5 -> R8C3 = 1, R9C3 = 4 -> 10(?) cage at R7C3 must be 10(3) cage in R789C3
4i. R2C5 = 7 -> R2C4 = 9, R123C3 = [269], R13C4 = [64], R13C5 = [32], R13C7 = [53], R123C6 = [158]

5. 22(?) cage at R3C2 must be 22(4) cage R3C2345, R3C345 = [942] -> R3C2 = 7 (cage total)
5a. R3C1 = 5 -> 13(?) cage at R1C1 must be 13(4) cage -> R1C1 = 4, R2C12 = [13]
5b. R1C2 = 8, R4C12 = [95], R5C12 = [61], R5C46 = [79], R6C12 = [24], R7C12 = [79], R8C12 = [36], R9C12 = [82]
5c. ??(?) cage at R1C4 must total at least 10 so must be R1C45 + R2C5

The remaining cages must be ?? cage in R1C23, ?? cage in R1C678, ?? cage in R2C34, 22(4) cage in R67C12, 19(4) cage in R89C12 and 26(4) cage in R89C89.

I cannot see how to determine whether ?? cage at R2C6 is R2C67 with ?? cage at R3C6 R34C67 or ?? cage at R2C6 is R2C67 + R3C7 and ?? cage at R3C6 is R3C6 + R4C67. Both versions give cage totals containing two digits.
I've solved all the numbers and got the same result as for the original version. However for this version I haven't solved all the cages. I need a total in either R2C6 or R3C6 to do that.