Prelims
a) 9(3) cage at R1C7 = {126/135/234}, no 7,8,9
b) 10(3) cage at R2C4 = {127/136/145/235}, no 8,9
c) 20(3) cage at R4C5 = {389/479/569/578}, no 1,2
d) 6(3) cage at R6C2 = {123}
e) 10(3) cage at R8C8 = {127/136/145/235}, no 8,9
1. Naked triple {123} in 6(3) cage at R6C2, locked for R6
1a. Min R6C78 = 9 -> max R7C8 = 8
2. Max 10(3) cage at R2C4 + R6C4 = 13 must contain 1, locked for C4
3. 45 rule on N2 1 innie R1C4 = 1 outie R4C4 + 5 -> R1C4 = {6789}, R4C4 = {1234}
4. 45 rule on N1 1 outie R1C4 = 1 innie R2C3 + 2, R1C4 = {6789} -> R2C3 = {4567}
5. 45 rule on whole grid 3(1+2) innies R2C3 + R47C9 = 20
5a. Max R2C3 = 7 -> min R47C9 = 13, no 1,2,3 in R47C9
6. 45 rule on N36 1 outie R7C8 = 1 innie R4C9, no 9 in R4C9, no 1,2,3 in R7C8
6a. 17(3) cage at R6C7 = {458/467}, no 9
6b. 45 rule on N36 3 innies R4C9 + R6C78 = 17 = {458/467}, 4 locked for N6
6c. 45 rule on N3 1 outie R4C8 = 1 innie R3C7, no 4 in R3C7
6d. Min R47C9 = 13 (step 5a), max R4C9 = 8 -> min R7C9 = 5
7. 45 rule on N5 3 innies R456C4 = 13, max R46C4 = 7 -> min R5C4 = 6
8. 45 rule on R89 3 innies R8C7 + R9C78 = 15
8a. 45 rule on N9 (using R8C7 + R9C78 = 15) 3 remaining innies R7C789 = 20 = {389/479/569/578}, no 1,2
8b. R2C3 + R47C9 = 20 (step 5), R7C789 = 20, R7C8 = R4C9 (step 6) -> R2C3 = R7C7 = {4567}
9. R7C789 (step 8a) = {479/569/578}
9a. R8C7 + R9C78 (step 8) = 15 = {168/249/258/348} (cannot be {159/267/357/456} which clash with R7C789), no 7
9b. 10(3) cage at R8C8 = {127/136/235} (cannot be {145} which clashes with R7C789), no 4
10. R7C789 (step 8a) = 20
10a. 45 rule on R789 2 outies R56C1 = 2 innies R7C89
10b. Max R7C7 = 7 -> min R7C89 = 13 -> min R56C1 = 13, no 1,2,3 in R5C1
11. 4,8,9 in N9 only in R7C789 + R8C7 + R9C78, CPE no 4,8,9 in R7C56
11a. R7C789 (step 9) = {479/569/578}
11b. 45 rule on R89 3 outies R7C567 = 14 = {167/257/356} (cannot be {347} = {37}4 which clashes with R7C789 = 4{79}), no 4, clean-up: no 4 in R2C3 (step 8b), no 6 in R1C4 (step 4), no 1 in R4C4 (step 3)
11c. 9 of R7C789 = {569} must be in R7C9 -> no 6 in R7C9
12. R1C4 = R4C4 + 5 (step 3) -> R14C4 = [72/83/94]
12a. 10(3) cage at R2C4 = {136/145/235} (cannot be {127} = {17}2 which clashes with R14C4 = [72], IOD clash), no 7
12b. 4 of {145} must be in R4C4 -> no 4 in R23C4
13. R456C4 = 13 (step 7)
13a. 10(3) cage at R2C4 = {136/145/235}
13b. 3 of {235} must be in R4C4 (because R6C4 = 1, hidden single in C5, max R5C4 = 9 -> min R4C4 = 3) -> no 2 in R4C4, clean-up: no 7 in R1C4 (step 3), no 5 in R2C3 (step 4), no 5 in R7C7 (step 8b)
13c. R7C789 (step 11a) = {479/569/578}
13d. R7C7 = {67} -> no 6,7 in R7C89, clean-up: no 6,7 in R4C9 (step 6)
14. R1C4 = R4C4 + 5 (step 3) = [83/94], 10(3) cage at R2C4 (step 12a) = {136/145/235}
14a. R456C4 = 13 (step 7) = {139/247/346} (cannot be {148} = [481] which clashes with 10(3) cage = {15}4, cannot be {238} = [382] which clashes with R14C4 = [83]), no 8
14b. R456C4 = {139/247/346} = [391/472/463] -> 10(3) cage at R2C4 = {25}3/{15}4 (cannot be {16}3 which clashes with R456C4 = [391]), no 3,6 in R23C4, 5 locked for C4 and N2
14c. R1C4 + R456C4 = 8[391]/9[472]/9[463], 9 locked for C4
14d. 20(3) cage at R4C5 = {389/569/578} (cannot be {479} which clashes with R456C4), no 4
14e. 12(3) cage at R4C6 = {129/138/156/246} (cannot be {147/237/345} which clash with R456C4), no 7
14f. 8,9 of {129/138} must be in R6C6 -> no 8,9 in R45C6
15. R7C789 (step 11a) = {479/569/578}
15a. R7C567 (step 11b) = {167/257} (cannot be {356} = {35}6 which clashes with R7C789 = [659]), no 3, 7 locked for R7
16. R1C4 = R4C4 + 5 (step 3), R1C4 = R2C3 + 2 (step 4) -> R2C3 = R4C4 + 3, R2C3 = R7C7 (step 8b) -> R7C7 = R4C4 + 3 -> R4C4 + R7C7 = [36/47]
16a. R7C8 = R4C9 (step 6) which “sees” R4C4 -> R7C78 cannot be [74] -> R7C789 (step 11a) = {569/578} (cannot be {479} = [749]), no 4, 5 locked for R7 and N9, clean-up: no 4 in R4C9 (step 6)
16b. R7C567 (step 15a) = {167} (only remaining combination), locked for R7 and 29(6) cage at R7C5, 1 also locked for N8
16c. 4 in N6 only in R6C78, locked for R6
16d. 4 in C9 only in R123C4, locked for N3
16e. 4 in R7 only in R7C1234, locked for 31(6) cage at R5C1
16f. 9(3) cage at R1C7 = {126/135}, 1 locked for N3, clean-up: no 1 in R4C8 (step 6c)
17. 17(3) cage at R6C7 (step 6a) = {458} (only remaining combination, cannot be {467} because R7C8 only contains 5,8)
17a. Naked triple {458} in R4C9 + R6C78, locked for N6, clean-up: no 5,8 in R3C7 (step 6c)
18. R2C3 + R47C9 = 20 (step 5) = 6[59]/7{58}, 5 locked for C9
19. Hidden killer pair 6,7 in R5C4 and R89C4 for C4, R89C4 cannot contain both of 6,7 (which would clash with R7C56) -> R89C4 contains one of 6,7 and R5C5 = {67}
19a. Killer pair 6,7 in R89C4 and R7C56, locked for N8
[Cracked. The rest is fairly straightforward.]
20. R1C4 = 9 (hidden single in C4) -> R2C3 = 7 (step 4), R4C4 = 4 (step 3), placed for D\, R7C7 = 7 (step 8b), placed for D\, clean-up: no 7 in R4C8 (step 6c)
20a. Naked pair {16} in R7C56, locked for N8
20b. R5C4 = 6 (hidden single in C4) -> R6C4 = 3 (step 14a), placed for D/, clean-up: no 3 in R4C8 (step 6c)
20c. Naked triple {278} in R789C4, locked for C4 and N8
20d. Naked pair {15} in R23C4, locked for N2
20e. Naked pair {12} in R6C12, locked for N4
21. 10(3) cage at R8C8 = {136} (only remaining combination), locked for N9
21a. R8C7 + R9C78 (step 9a) = {249} (only remaining combination), locked for N9
21b. Naked pair {58} in R7C89, locked for R7 -> R7C4 = 2
21c. Naked pair {58} in R47C9, locked for C9
21d. Naked triple {349} in R7C123, locked for N7 and 31(6) cage at R5C1)
22. 8 in N3 only in R13C8, locked for C8 -> R67C8 = [45], R6C7 = 8, R47C9 = [58]
23. R5C4 = 6 -> R45C3 = 12 = {39}/[84]
23a. Killer pair 4,9 in R45C3 and R7C3, locked for C3
24. 20(3) cage at R4C5 (step 14d) = {578} (only remaining combination), locked for C5 and N5 -> R6C6 = 9, placed for D\, R45C6 = {12}, locked for C6 -> R7C56 = [16]
25. 5 in C6 only in R89C6, locked for 33(6) cage at R8C6
25a. Naked pair {78} in R78C4, CPE no 7,8 in R9C1
25b. 33(6) cage at R8C6 must contain 5 for C6 and 7,8 for R9 = {135789/345678}, no 2, 3 locked for N8
26. R7C1234 = {349}2 = 18 -> R56C1 = 13 = [76/85]
26a. 17(3) cage at R4C1 = {359/467} (cannot be {368} which clashes with R56C1, cannot be {458} because 4,5 only in R5C2), no 8
26b. 4,5 only in R5C2 -> R5C2 = {45}
27. 5 in C7 only in 9(3) cage at R1C7 = {135} -> R2C8 = 1, placed for D/, R23C4 = [51], R12C7 = [53]
28. R1C4 = 9 -> R1C23 = 9 = {18/36}, no 2,4
28a. 13(3) cage at R1C1 = {148/256/346} (cannot be {139/238} which clash with R1C23), no 9
28b. R3C2 = 9 (hidden single in N1) -> R2C2 + R3C3 = 7 = [25], 5 placed for D\ -> R5C5 = 8, placed for both diagonals, R46C5 = [75], R56C1 = [76]
28c. R4C6 = 2, placed for D/ -> R3C7 = 6, placed for D/
28c. 6 in N1 only in R1C23 = {36}, locked for R1 and N1 -> R1C1 = 1, placed for D\
29. 1 in R9 only in 33(6) cage at R8C6 = {135789}, no 4,6
29a. R8C5 = 4 (hidden single in N8), R1C5 = 2 -> R12C6 = 15 = [78]
and the rest is naked singles, without using the diagonals.