The positions of seven NC cells are given (R1C5, R1C8, R2C8, R5C1, R5C5, R6C7 and R8C2; alternatively find them given that they are 2236789.
Prelims
a) R1C67 = {29/38/47/56}, no 1
b) R2C67 = {14/23}
c) R3C34 = {69/78}
d) R34C8 = {39/48/57}, no 1,2,6
e) R34C9 = {15/24}
f) R4C23 = {17/26/35}, no 4,8,9
g) R67C7 = {19/28/37/46}, no 5
h) R78C6 = {18/27/36/45}, no 9
i) 20(3) cage at R3C1 = {389/479/569/578}, no 1
j) 22(3) cage at R8C7 = {589/679}
k) 8(3) cage at R8C8 = {125/134}
l) 26(4) cage at R1C8 = {2789/3689/4589/4679/5678}, no 1
1a. 45 rule on N9 3 innies R789C7 = 23 = {689}, locked for C7 and N9, R9C6 = {57}, clean-up: no 2,3,5 in R1C6
1b. 22(3) cage at R8C7 = {589/679}, 9 locked for N9
1c. 8(3) cage at R8C8 = {125/134}, 1 locked for N9
1d. R7C7 = {68} -> R6C7 = {24}
2a. 45 rule on N478 2(1+1) innies R4C1 + R9C6 = 16 = [97] -> R89C7 = {69}, R7C7 = 8 -> R6C7 = 2, clean-up: no 9 in R1C6, no 4 in R1C7, no 3 in R2C6, no 3 in R3C8, no 4 in R3C9, no 2 in R7C6, no 1,2 in R8C6
2b. 45 rule on N1 1 outie R4C1 = 1 remaining innie R3C3, R4C1 = 9 -> R3C3 = 9 -> R3C4 = 6, clean-up: no 5 in R1C7, no 3 in R4C8
2c. R4C1 = 9 -> R3C12 = 11 = {38/47}, no 5
3a. 9 in C6 only in R56C6, locked for N5
3b. 16(3) cage at R5C6 = {169/259/349}, no 7,8
3c. 1,5 of {169/259} must be in R5C7 -> no 1,5 in R56C6
4a. 6,9 in N3 only in 26(4) cage at R1C8 = {3689/4679}, no 2,5
4b. Killer pair 3,7 in R1C7 and 26(4) cage, locked for N3, clean-up: no 2 in R2C6, no 5 in R4C8
4c. R3C9 = 2 (hidden single in N3) -> R4C9 = 4, clean-up: no 8 in R3C8
4d. Naked triple 1,4,5 in R2C7 + R3C78, 1 locked for C7, 5 locked for R3, 4 locked for N3
4e. 26(4) cage at R1C8 = {3689}, 3 locked for N3, R1C7 = 7 -> R1C6 = 4, R2C67 = [14], R3C8 = 5 -> R4C8 = 7, R3C7 = 1, clean-up: no 1 in R4C23, no 5 in R7C6, no 5,8 in R8C6
4f. 16(3) cage at R5C6 (step 3b) = {259} (only remaining combination) -> R5C67 = [25], R6C6 = 9, R4C7 = 3, R3C6 = 8, R4C6 = 5 (cage sum)
4g. Naked pair {26} in R4C23, 6 locked for R4 and N4
4h. Naked pair {18} in R4C45, locked for N5, R3C5 = 3 (cage sum)
4i. Naked pair {47} in R3C12, 7 locked for N1
4j. Naked pair {36} in R78C6, locked for N8
4k. 12(3) cage at R1C2 = {138/156}, no 2, 1 locked for R1
4l. 7 in N9 only in 14(3) cage at R8C8 = {257/347}
4m. 2,4 only in R8C8 -> R8C8 = {24}
5a. 45 rule on N8 2 remaining innies R78C4 = 1 outie R9C3 + 8
5b. R78C4 cannot total 16 -> no 8 in R9C3
6a. 21(5) cage at R7C5 = {12459/12468/13458} (cannot be {12369} because 3,6 only in R9C3)
6b. Consider combinations for 21(5) cage
21(5) cage = {12459} => R8C4 = 8 (hidden single in N8) => R4C4 = 1
or 21(5) cage = {12468} => R78C4 = {59} (hidden pair in N8)
or 21(5) cage = {13458} => R78C4 = {29} (hidden pair in N8)
-> no 1 in R7C4, no 1,4 in R8C4
6c. 1 in N8 only in R78C5 + R9C45, locked for 21(5) cage, no 1 in R9C3
[Seems like I need to start using the NC cells.]
7a. R2C7 = 4, R3C8 = 5 -> R2C8 (a specified NC cell) = {89}, no 8,9 in R1C8 + R2C9
7b. R1C7 = 7 -> R1C8 (a specified NC cell) = 3, R2C9 = 6
7c. Naked triple {124} in R789C8, 1 locked for C8 and N9
7d. 12(3) cage at R1C2 (step 4k) = {138/156} = {18}3/{16}5, no 5 in R1C23, no 8 in R2C3
7e. 12(3) cage = {18}3/{16}5 -> 13(3) cage at R1C1 = 6{25}/{28}{238}, no 5 in R1C1
7f. R1C6 = 4 -> R1C5 (a specified NC cell) = {29}
7g. R1C4 = 5 (hidden single in R1)
7h. 5 in N8 only in R789C5, locked for 21(5) cage at R7C5, no 5 in R9C3
7i. 21(5) cage at R7C5 (step 6a) contains 1,5 in N8 = {12459/13458}, no 6
8a. R5C5 (a specified NC cell) = {467} -> R56C6 = {46/47} (cannot be {67}), 4 locked for C5 and N5
8b. Naked pair {37} in R56C4, 7 locked for C4 and N5
8c. R2C5 = 7 (hidden single in N2)
8d. 21(5) cage at R7C5 (step 7i) = {12459} (only remaining combination, cannot be {13458} because then R78C4 = {29} (hidden pair in N8) clashes with R2C4), no 3,8, 4 locked for R9, 9 locked for N8
8e. R8C4 = 8 (hidden single in N8), R4C45 = [18]
8f. Naked pair {24} in R7C48, locked for R7
8g. 45 rule on N7 R7C13 + R9C3 = 12 = {37}2/{17}4/{35}4, no 6
8h. 25(5) cage at R6C2 = {14578/23578}
8i. R7C4 = {24} -> no 4 in R6C23
8j. 45 rule on N4 2 innies R6C23 = 1 outie R7C1 + 5, R6C23 cannot total 10 = {37} which clashes with R6C4 -> no 5 in R7C1
8k. 23(5) cage at R5C1 = {13478} (only possible combination), no 5
8l. 5 in N4 only in R6C12, locked for 25(5) cage
8m. R7C13 + R9C3 = 12 = {37}2/{17}4, 7 locked for R7 and N7
8n. R8C9 = 7 (hidden single in N9)
8o. 25(5) cage = {14578/23578}, CPE no 7 in R5C3
8p. 7 in C4 only in R67C4, locked for 25(5) cage, no 7 in R6C2
9a. R5C5 (a specified NC cell) = {46} -> R5C45 = [36/74] (cannot be [34/76])
9b. 23(5) cage at R5C1 = {13478}
9c. R67C1 cannot be [47] which clashes with R3C1 -> R5C123 must contain at least of 4,7
9d. R5C45 = [36] (cannot be [74] which clashes with R5C123), R6C45 = [74]
9e. R6C8 = 6 (hidden single in N6)
9f. R7C3 = 7 (hidden single in C3)
9g. 23(5) cage at R5C1 = {13478}, 3 locked for C1
10a. R7C13 + R9C3 (step 8m) = [174/372]
10b. 45 rule on N8 1 remaining innie R7C4 = 1 remaining outie R9C3, R7C1 + R9C3 = [14/32] -> R7C14 = [14/32]
10c. R8C9 = 7 -> R7C89 = 7 = [25] (cannot be [43] which clashes with R7C14) -> R7C14 = [14], R7C5 = 9, R9C34 = [42], R1C5 = 2, R9C58 = [51], R5C3 = 8, R25C8 = [89]
10d. R6C1 = 3 -> R5C1 (a specified NC cell) = 7
11. Consider placements for R7C2 = {36}
R7C2 = 3 => R2C12 = {25}, R1C1 = 6 (cage sum)
or R7C2 = 6, R9C1 = 8 => R1C1 = 6
-> R19C1 = [68], R1C23 = [81], R6C23 = [15], R2C3 = 3
12. R8C2 (a specified NC cell) = 9 (cannot be 2,3,5,6 because of NC with R7C2 + R8C13)
and the rest is naked singles, without using NC.