Prelims
a) R23C1 = {19/28/37/46}, no 5
b) R23C8 = {49/58/67}, no 1,2,3
c) R4C56 = {19/28/37/46}, no 5
d) R5C78 = {39/48/57}, no 1,2,6
e) R78C9 = {12}
f) 19(3) cage at R3C2 = {289/379/469/478/568}, no 1
g) 8(3) cage at R6C5 = {125/134}
h) 14(4) cage at R1C5 = {1238/1247/1256/1346/2345}, no 9
h) 26(4) cage at R8C7 = {2789/3689/4589/4679/5678}, no 1
i) 31(5) cage at R7C4 = {16789/25789/34789/35689/45679}
j) 35(7) cage at R5C2 = {1235789/1245689/1345679/2345678}
Steps resulting from Prelims
1a. Naked pair {12} in R78C9, locked for C9 and N9
1b. 8(3) cage at R6C5 = {125/134}, 1 locked for R6
1c. 31(5) cage at R7C4 contains 9, locked for N8
2. 45 rule on N8 1 outie R9C7 = 7, clean-up: no 5 in R5C8
2a. 45 rule on N9 2 innies R7C78 = 9 = {36/45}, no 8,9
2b. 45 rule on N9 2 outies R6C89 = 13 = {49/58/67}, no 2,3
2c. R14C8 = {12} (hidden pair in C8)
2d. 45 rule on N7 2 outies R56C2 = 10 = [19]/{28/37/46}, no 5, no 9 in R5C2
2e. 35(7) cage at R5C2 contains 5, locked for N7
[This will be the “human solvable” step which SudokuSolver isn’t programmed to find.]
3. 45 rule on the whole grid 3(2+1) innies R13C4 + R5C6 = 25
3a. Max R13C4 = 17 -> min R5C6 = 8
3b. R5C6 = {89} -> R13C4 = 16,17 = {79/89}, 9 locked for C4 and N2
3c. 9 in N8 only in R89C5, locked for C5, clean-up: no 1 in R4C6
3d. 17(3) cage at R4C4 = {278/368/458/467}, no 1
4. 45 rule on N47 2(1+1) outies R3C2 + R6C4 = 12 = {48/57}/[66/93], no 2,3 in R3C2, no 2 in R6C4
4a. 45 rule on N1 1 innie R3C2 = 1 outie R3C5 + 3, R3C2 = {456789} -> R3C5 = {123456}
4b. 45 rule on N36 using R6C89 = 13 (or, if preferred, 45 rule on N3689) 3 innies R346C7 = 11 = {128/146/236/245}, no 9
4c. {128} must have 8 in R4C7 (cannot be [821] which clashes with R4C8), no 8 in R3C7
5. 45 rule on N5 4 innies R5C6 + R6C456 = 18 = {1269/1278/1359/1458/2358} (cannot be {1467/2367/2457/3456} because R5C6 only contains 8,9, cannot be {1368/2349} which clash with R4C56)
5a. R5C6 = {89} -> no 8 in R6C4
5b. 9 in N5 only in R4C56 or R5C6 + R6C456 -> R5C6 + R6C456 = {1269/1359/2358} (cannot be {1278/1458} which clash with R4C56 = [19] (blocking-cages), no 4,7 in R6C456
[Ed told me this is locking-out cages; I’ve never been sure of the difference.]
5c. R6C4567 = [6{12}5]/[3{15}2]/[5{13}4][3{25}1], 5 locked for R6, no 3 in R6C7, clean-up: no 8 in R6C89 (step 2b)
5d. R6C4 = {356} -> R3C2 = {679} (step 4), R3C5 = {346} (step 4a)
6. 45 rule on C1 1 innie R1C1 = 1 outie R8C2, no 5 in R8C2 -> no 5 in R1C1
6a. 5 in C1 only in 15(3) cage at R4C1, locked for N4
6b. 19(3) cage at R3C2 = {289/379/469/478}
6c. 13(3) cage at R5C3 = {139/157/238/256/346} (cannot be {148/247} because R6C4 only contains 3,5,6)
6d. 45 rule on N47 4 innies R4C23 + R56C3 = 20
6e. 19(3) cage cannot be [649/694] with R4C23 totalling 13 because R56C3 = 7 doesn’t contain 5, 13(3) cage cannot be [166] and 13(3) cage = {346} clashes with 19(3) cage -> no 6 in R3C2, clean-up: no 3 in R3C5 (step 4a), no 6 in R6C4 (step 4)
6f. R5C6 + R6C456 (step 5b) = {1359/2358}, 3,5 locked for R6 and N5, clean-up: no 7 in R4C56, no 7 in R5C2 (step 2d)
6g. 13(3) cage = {139/157/238/256/346}
6h. 1 of {139/157} must be in R5C3, 3 of {139/238/346} -> no 3,7,9 in R5C3
7. 1 in R5 only in R5C123, locked for N4, 5 in N4 only in R45C1
7a. 45 rule on R67989 4 outies R4C1 + R5C123 = 14 must contain 1 and 5 = {1256}, locked for N4, clean-up: no 7 in R6C2 (step 2d)
7b. 3 in N4 only in R4C23, locked for R4 -> 19(3) cage at R3C2 = {379}, CPE no 9 in R6C2, clean-up: no 1 in R5C2 (step 2d)
7c. 4 in N4 only in R6C123, locked for R6, clean-up: no 9 in R6C89 (step 2d)
7d. Naked pair {67} in R6C89, locked for R6, N6 and 22(4) cage at R6C8, clean-up: no 5 in R5C7, no 3 in R7C78 (step 2a)
7e. Naked pair {45} in R7C78, locked for R7 and N9
7f. Killer pair 8,9 in R5C6 and R5C78, locked for R5
7g. Naked triple {489} in R6C123, locked for N4
7h. Naked pair {37} in R4C23, locked for R4 and 19(3) cage at R3C2 -> R3C2 = 9, R3C5 = 6 (step 4a), placed for disjoint cell at R2C2, R6C4 = 3 (step 4), clean-up: no 1,4 in R2C1, no 4,7 in R2C8, no 1 in R3C1, no 4 in R4C6
7i. R1C4 = 9 (hidden single in C4)
7j. R3C5 = 6 -> R2C23 + R3C3 = 10 = {127/145/235}, no 8
8. R346C7 (step 4b) = {128} (only remaining combination, cannot be {245} which clashes with R7C7) -> R4C7 = 8, R36C7 = {12}, locked for C7, clean-up: no 2 in R4C56, no 4 in R5C78
8a. Naked pair {39} in R5C78, locked for R5 and N6 -> R5C6 = 8
8b. R13C4 + R5C6 = 25 (step 3), R1C3 = 9, R5C6 = 8 -> R3C4 = 8, clean-up: no 2 in R2C1, no 5 in R2C8
8c. R4C6 = 9 (hidden single in N5) -> R4C5 = 1, R14C8 = [12], R36C7 = [21], clean-up: no 8 in R2C1
8d. R34C7 = [28] = 10 -> R23C6 = 8 = {17/35}, no 4
8e. Naked pair {45} in R45C9, locked for C9
8f. R4C8 = 2, R45C9 = {45} = 9 -> R23C9 = 15 = [87], R6C89 = [76], R19C9 = [39]
8g. R1C89 = [13] = 4 -> R12C7 = 11= {56}, locked for C7 and N3 -> R23C9 = [94], R3C1 = 3 -> R2C1 = 7
8h. R23C6 = [35] (only remaining permutation), R6C56 = [52]
8i. Naked triple {146} in R789C6, locked for C6 and N8
8j. R789C6 = {146} = 11, R9C7 = 7 -> R7C5 = 3 (cage sum)
8k. R3C35 = [16] = 7 -> R2C23 = 9 = {45}, locked for R2 and N1
8l. R45C1 = [51] (hidden pair in N4) -> R6C1 = 9 (cage sum)
8m. R789C1 = {2468} -> 20(4) cage at R7C1 = {2468} (only possible combination), locked for N7, 4 also locked for C1
8n. Naked quad {2468} in R1568C2, locked for C2 -> R2C23 = [54]
8o. R6C34 = [83] -> R5C3 = 2 (cage sum)
and the rest is naked singles.