Prelims
a) R12C1 = {29/38/47/56}, no 1
b) R1C23 = {59/68}
c) R12C9 = {19/28/37/46}, no 5
d) R2C34 = {17/26/35}
e) R23C5 = {49/58/67}, no 1,2,3
f) R2C67 = {15/24}
g) R34C6 = {18/27/36/45}, no 9
h) R56C7 = {16/25/34}, no 7,8,9
i) 19(3) cage at R6C4 = {289/379/469/478/568}, no 1
j) 19(3) cage at R8C2 = {289/379/469/478/568}, no 1
1a. 45 rule on C789 1 outie R2C6 = 4 -> R2C7 = 2, clean-up: no 7,9 in R1C1, no 6,8 in R1C9, no 6 in R2C34, no 9 in R23C5, no 8 in R2C9, no 5 in R34C6, no 5 in R56C7
1b. 45 rule on N9 1 outie R6C9 = 5
1c. 45 rule on N3 or N6, 1 outie R4C9 = 1 innie R3C7, no 5 in R3C7, no 2 in R4C9
1d. R12C1 = [29/38/47/83] (cannot be {56} which clashes with R1C23), no 5,6
1e. 45 rule on N8 2(1+1) outies R6C4 + R8C3 = 10 = {28/37/46}/[91], no 5,9 in R8C3
2. 45 rule on N2 3 remaining innies R2C4 + R3C46 = 15 = {159/258/267} (cannot be {168/357} which clash with R23C5), no 3, clean-up: no 5 in R2C3, no 6 in R4C6
2a. 5,7 of {159/258/267} must be in R2C4 -> R2C4 = {57}, clean-up: no 7 in R2C3
2b. 9 of {159} must be in R3C4 -> no 1 in R3C3
2c. Killer pair 5,7 in R2C4 and R23C5, locked for N2, clean-up: no 2 in R4C6
2d. 3 in N2 only in 13(3) cage at R1C4, locked for R1, clean-up: no 8 in R2C1, no 7 in R2C9
2e. 13(3) cage = {139/238}, no 6
2f. Killer pair 8,9 in R1C23 and 13(3) cage, locked for R1, clean-up: no 3 in R2C1, no 1 in R2C9
2g. 7 in R1 only in R1C789, locked for N3, clean-up: no 7 in R4C9 (step 1c)
3. 45 rule on C6 3 innies R156C6 = 18 = {189/279/369} (cannot be {378} which clashes with R34C6, cannot be {567} because no 5,6,7 in R1C6), no 5, 9 locked for C6
3a. 5 in C6 only in 14(3) cage at R7C6, locked for N8
3b. 14(3) cage = {158/257/356}
[A couple of IOUs.]
4a. 45 rule on N1 2 outies R4C13 = 1 innie R2C3 + 8, no 8 in R4C1
4b. 45 rule on N7 2 outies R6C23 = 1 outie R8C3 + 6, no 6 in R6C2
[I’ve analysed 28(6) cage at R2C2 by looking at the values missing from combinations.]
5. 28(6) cage = {123589/124579/124678/134578/234568} (cannot be {123679} because cannot place both of 5,8 in R1C23, cannot be {134569} because cannot place both of 2,7 in R12C1)
5a. 2 in R1 only in R1C1 and 13(3) cage at R1C4 (step 2e) = {238}
5b. 28(6) cage = {124579/124678/134578/234568} (cannot be {123589} because R1C1 = 4 + R1C23 = {68} clash with 13(3) cage = {238})
5c. 28(6) cage = {124579/124678/134578} (cannot be {234568} because R2C1 = 7 clashes with R2C34 = [17])
[And then interactions between the innies and outies for N1.]
5d. 28(6) cage = {124678/134578} (cannot be {124579} because R12C1 = [29] clashes with R4C13 = {29} and R12C1 = [47] clashes with R4C13 = {47}), no 9
5e. 28(6) cage = {134578} (cannot be {124678} because R1C23 = {59} -> R12C1 = [47] clashes with R4C13 = {47}), no 2,6
5f. R1C1 = 2 (hidden single in N1), placed for D\, R2C1 = 9, clean-up: no 5 in R1C23, no 8 in 13(3) cage at R1C4 (step 2e), no 1 in R1C9
5g. 13(3) cage = {139}, 1 locked for R1 and N2, clean-up: no 8 in R4C6
5h. Naked pair {68} in R1C23, locked for R1 and N1
5i. 28(6) cage = {134578} -> R4C3 = 8, R1C23 = [86], clean-up: no 8 in R3C7 (step 1c), no 2,4 in R6C4 (step 1e)
5j. Naked triple {457} in R1C789, locked for N3, clean-up: no 4 in R4C9 (step 1c)
5k. 45 rule on N1 1 innie R2C3 = 1 remaining outie R4C1 -> R4C1 = {13}
5l. 19(3) cage at R8C2 = {379/469}, no 2,5, 9 locked for N7
5m. 8 in C1 only in 14(3) cage at R7C1 = {158} (only remaining combination), locked for C1 and N7 -> R4C1 = 3, R2C3 = 3 -> R2C4 = 5, R2C9 = 6 -> R1C9 = 4, placed for D/, clean-up: no 8 in R2C5, no 7,8 in R3C5, no 6 in R3C6, no 3 in R3C7 (step 1c), no 7,9 in R6C4 (step 1e)
5n. R23C5 = [76], R2C2 = 1, placed for D\, R2C8 = 8, placed for D/, clean-up: no 2 in R8C3 (step 1e)
6. 45 rule on N2 3 remaining outies R4C456 = 16 = {457} (only possible combination, cannot be {169} which clashes with R4C9, cannot be {259/268} because R4C6 only contains 1,7) = [457], 4 placed for D\, 7 placed for D/, R3C6 = 2, R3C4 = 8
[Cracked. The rest is straightforward.]
6a. 1 in R4 only in R4C789, locked for N6, clean-up: no 6 in R56C7
6b. Naked pair {34} in R56C7, locked for C7 and N6
7. R6C9 = 5 -> 20(4) cage at R6C9 = {1568/2567}, no 3,9
7a. 6 in 20(4) cage only in R8C8 -> R8C8 = 6, placed for D\
7b. R4C7 + R5C9 = [68] (hidden pair in N6), clean-up: no 1 in 20(4) cage
7c. Naked pair {27} in R78C9, locked for N9
7d. R3C3 + R7C7 = [75] (hidden pair on D\), R7C3 = 2, R78C9 = [72], R8C3 = 4 -> R6C4 = 6 (step 1e), R9C3 = 9, R8C2 = 3, placed for D/, R9C2 = 7 (cage sum), R3C12 = [45]
7e. R5C5 = 9, placed for D/, R3C7 = 1
7f. R56C1 = [67] = 13 -> R45C2 = 6 = [24]
7g. R6C4 = 6 -> R7C45 = 13 = [94]
and the rest is naked singles, without using the diagonals.