Prelims
a) R2C12 = {39/48/57}, no 1,2,6
b) R56C7 = {49/58/67}, no 1,2,3
c) R67C8 = {19/28/37/46}, no 5
d) R78C2 = {18/27/36/45}, no 9
e) R78C7 = {12}
f) R9C12 = {19/28/37/46}, no 5
g) 9(3) cage at R3C2 = {126/135/234}, no 7,8,9
1a. 45 rule on N69 1 outie R3C9 = 7
1b. Naked pair {12} in R78C7, locked for C7 and N9, clean-up: no 8,9 in R6C8
1c. 45 rule on N3 2 remaining innies R23C7 = 9 = {36/45}
1d. 14(3) cage at R1C7 = {149/158/239/248} (cannot be {356} which clash with R23C7), no 6
1e. 45 rule on N8 2 remaining innies R7C45 = 5 = {14/23}
1f. Killer pair 1,2 in R7C45 and R7C7, locked for R7, clean-up: no 7,8 in R8C2
1g. 45 rule on R1 3 innies R1C456 = 19 = {379/469/478/568} (cannot be {289} which clashes with 14(3) cage, no 1,2
1h. Hidden killer pair 1,2 in 12(3) cage at R1C1 and 14(3) cage for R1, 14(3) cage contains one of 1,2 -> 12(3) cage must contain one of 1,2 -> no 9 in 12(3) cage
1i. 45 rule on C9 3 remaining innies R124C9 = 11 = {128/146/236/245}, no 9
2a. 45 rule on C123 2 innies R26C3 = 1 outie R7C4 + 15 -> R26C3 = 16,17 = {79/89}, 9 locked for C3
2b. R26C3 = 16,17 -> R7C4 = {12} -> R7C5 = {34} (step 1e)
3. 45 rule on R1234 2 outies R5C48 = 7 = {16/25/34}, no 7,8,9
4a. Consider placements for R7C4 = {12}
R7C4 = 1 => R78C7 = [21] => 1 in N7 only in R9C12 = {19}
or R7C4 = 2 => R78C7 = [12] => 2 in N7 only in R9C12 = {28}
-> R9C12 = {19/28}
[Or, as wellbeback would put it, whichever of 1,2 is in R7C4 must be in R8C7 and also in R9C12.]
4b. R78C2 = {36/45}/[72] (cannot be [81] which clashes with R9C12), no 8 in R7C2, no 1 in R8C2
4c. 45 rule on C12 4 outies R1345C3 = 13 = {1237/1246/1345}, no 8, 1 locked for C3
4d. 45 rule on N78 3 (2+1) innies R78C1 + R7C5 = 14
4e. Consider permutations for R7C45 = [14/23] (step 1e)
R7C45 = [14] => R9C12 = {19} (step 4a)
or R7C45 = [23] => R78C1 = 11, no 1 in R8C1 => 1 in N7 only in R9C12 = {19}
-> R9C12 = {19}, locked for R9, 9 locked for N7
4f. Step 4a must also work in the reverse direction, otherwise there would be no place for 2 in N7 -> R7C4 = 1, R7C5 = 4, R78C7 = [21], clean-up: no 6 in R5C8 (step 3), no 6 in R6C8, no 5 in R8C2
[Alternatively a neat way to get this result is
Consider placement for 2 in N7
2 in R78C12 => grouped X-Wing with R78C7 = {12}, no other 2 in R78
or 2 in R789C3
-> no 2 in R7C4, …]
4g. R7C4 = 1 -> R26C3 (step 2a) = 16 = {79}, 7 locked for C3
4h. 8 in C3 only in R789C3, locked for N7
4i. 17(4) cage at R7C3 contains 1,8 = {268}1/{358}1, no 4
4j. R78C2 = [54/72] (cannot be {36} which clashes with 17(4) cage), no 3,6
4k. R7C5 = 4 -> R78C1 = 10 = {37}/[64], no 2,5, no 6 in R8C1
4l. R78C1 = 10 -> R6C12 = 11 = {29/38/56} (cannot be {47} which clashes with R78C1), no 1,4,7
4m. 4 in N7 only in R8C12, locked for R8
4n. 15(3) cage at R8C4 = {258/267/357}, no 9
4o. 9 in N8 only in R78C6, locked for C6
4p. 38(7) cage at R5C5 contains 4 so must contain 3,8, both locked for N5, clean-up: no 4 in R5C8 (step 3)
5a. 4 in C4 only in 35(6) cage at R1C4 = {245789/345689}, 5,8 locked for C4, 8 locked for N2
5b. 15(3) cage at R8C4 (step 4n) = {267/357} (cannot be {258} because 5,8 only in R8C5), no 8, 7 locked for N8
5c. 5 of {357} must be in R8C5 -> no 3 in R8C5
5d. 45 rule on C1234 2 innies R6C34 = 1 outie R8C5 + 10
5e. R6C34 cannot total 17 -> no 7 in R8C5
5f. Min R8C5 = 2 -> min R6C34 = 12, no 2 in R6C4
5g. 7 in N8 only in R89C4, locked for C4
5h. R8C5 = {256} -> R6C34 = 12,15,16 = [93/96/79], 9 locked for R6 and 38(7) cage at R5C5, clean-up: no 4 in R5C7, no 2 in R6C12 (step 4l)
5i. 21(4) cage at R6C1 (steps 4k and 4l) = {38}[64]/{56}{37}, CPE no 3,6 in R45C1
[At this stage I originally found combined cage R2356C7, with simplification of 17(3) cage at R8C8, but this proved to be unnecessary after my next step, so I’ve omitted it.]
6a. R1C456 (step 1g) = {379/469/478/568}, R23C7 (step 1c) = {36/45}
6b. R1C456 = {379/469/568} (cannot be {478} = [874] because no combination for 21(4) cage at R1C5)
6c. Consider placement of 6 in N1
6 in 12(3) cage at R1C1 => R1C456 = {379}
or 6 in R3C123 => no 3 in R2C7 => R1C456 = {379/469} (cannot be {568} = 8{56} because no combination for 21(4) cage at R1C5)
-> R1C456 = {379/469}, no 5,8, 9 locked for R1 and N2
6d. 9 in N3 only in 15(3) cage at R2C8 = {159/249}, no 3,6,8, 9 locked for C8, clean-up: no 1 in R6C8
6e. 9 in C7 only in R45C7, locked for N6
6f. 8 in N3 only in 14(3) cage at R1C7 = {158/248}, no 3,6, 8 locked for R1
6g. R23C7 = {36} (hidden pair in N3), locked for C7, clean-up: no 7 in R56C7
6h. R9C7 = 7 (hidden single in C7) -> R89C8 = 10 = [64], clean-up: no 3 in R6C8
6i. R8C4 = 7 (hidden single in N8) -> R8C5 + R9C4 = 8 = [26/53], no 2 in R9C4, clean-up: no 3 in R7C1 (step 4k)
6j. R8C5 = {25} -> R6C34 (step 5h) = 12,15 = [93/96] -> R6C3 = 9, R2C3 = 7, clean-up: no 5 in R2C12
6k. Naked pair {36} in R69C4, locked for C4
7a. Hidden killer pair 8,9 in R2C12 and R3C1, R2C12 contain one of 8,9 -> R3C1 = {89}
7b. 45 rule on N1 1 remaining innie R3C1 = 1 outie R4C3 + 5, R3C1 = {89} -> R4C3 = {34}
7c. 9(3) cage at R3C2 = {135/234} (cannot be {126} which doesn’t contain one of 3,4), no 6
7d. 6 in N1 only in 12(3) cage at R1C1 = {156/246}, no 3
7e. R1C56 = {37} (hidden pair in R1) -> R1C4 = 9 (step 6c)
7f. R1C56 = {37} = 10 -> R2C67 = 11 = [56], R3C7 = 3
7g. 9(3) cage = {135/234} -> R4C3 = 3 -> R3C1 = 8
7h. R8C1 = 3 (hidden single in N7) -> R7C1 (step 4k) = 7, R7C2 = 5 -> R8C2 = 4
7i. R78C1 = [73] = 10 -> R6C12 = 11 = [56] -> R69C4 = [36], R8C5 = 2 (cage sum)
7j. R3C1 = 8 -> R4C12 = 9 = [27]
7k. R2C12 = [93] (hidden pair in N1) -> R9C12 = [19], R5C123 = [481]
7l. R23C5 = [16] -> R2C8 = 2
7m. R6C8 = 7 -> R7C8 = 3, R5C8 = 5, R5C7 = 9 -> R6C7 = 4
and the rest is naked singles.