Prelims courtesy of SudokuSolver

Preliminaries

Cage 4(2) n7 - cells ={13}

Cage 6(2) n1 - cells only uses 1245

Cage 14(2) n2 - cells only uses 5689

Cage 13(2) n9 - cells do not use 123

Cage 13(2) n6 - cells do not use 123

Cage 9(2) n56 - cells do not use 9

Cage 9(2) n8 - cells do not use 9

Cage 9(2) n3 - cells do not use 9

Cage 11(2) n45 - cells do not use 1

Cage 21(3) n2 - cells do not use 123

Cage 20(3) n8 - cells do not use 12

Cage 13(4) n9 - cells do not use 89

Cage 14(4) n12 - cells do not use 9

Cage 26(4) n7 - cells do not use 1

No routine clean-up done unless stated

1. "45" on n7: 1 outie r6c1 = 1

1a. "45" on n3: 1 outie r4c9 = 1

2. 4(2)n7 = {13} only: both locked for n7 and r7

3. "45" on n8: 1 innie r7c6 + 6 = 1 outie r9c7 = [28] only permutation

3a. r7c6 = 2 -> r6c56 = 11 (no 9)

3b. no 7 in 9(2)n8, no 6,8 in r8c5

3c. no 5 in 13(2)n9

4. 21(3)n2: {489} blocked by 14(2)n2 needing an 8 or 9

4a. = {579/678}(no 4)

4b. must have 7: 7 locked for n2 and c6

4c. no 4 in r6c5 (sp11(2))

5. 14(2)r23c5 = {59/68}; 9(2)r78c5 = [81]/[63]/{45}

5a. if 14(2) = {68} -> 9(2) = {45}; or 14(2) = {59}

5b. -> 5 locked in one of those two cages: locked for c5 (combined cages)

5c. no 6 in r6c6 (sp11(2))

6. "45" on n2: 1 innie r3c4 + 4 = 1 outie r1c3

6a. r1c3 = (5678), r3c4 = (1234)

6b. hidden quad 1,2,3,4 in n2 -> r123c4+r1c5 = {1234}

6c. 17(3)r3c4 can only have one of 1,2,3,4 -> no 2,3,4 in r4c45

7. 20(3)r7c4 = {389/479/569/578} = two of 6..9

7a. -> r456c4 must have two of 6..9 for c4

7b. r4c5 = (6789)

7c. sp11(2)r6c56 must have one of 6,7,8

7d. -> killer quad 6,7,8,9 in those four areas: all locked for n5 (hidden killer quad)

7e. r4c7 = (456)

Ready for the first advanced step - which totally cracked one of the previous versions of this puzzle

8. 21(3)n2 = {579}/678}

8a.if {579} -> 14(2)n2 = {68} (combined cage)

8b. r9c7 = 8 -> r89c6 + r9c5 = 14 = {149/167/347/356}

8c. if {149} -> 9(2)n8 = {36} only (combined cage)

8d. 9 in c6 in one of 21(3)n2 or r89c6 -> 14(2)n2 = {68} or 9(2)n8 = {36}

8e. must have 6: 6 locked for c5

8f. no 5 in r6c6 (sp11(2))

9. 6 in c5 in 14(2)n2 = {68} or 6 in 9(2)n8 -> no {18} in 9(2)n8 since it would block all 6 in c5 (locking-out cages)

9a. 9(2) = [63]/{45}

10. 8 in n8 only in 20(3) = {389/578}(no 4,6) = 3 or 5

10a. 8 locked for c4

11. 9(2)n8 = [63]/{45} = 3 or 5

11a. killer pair 3,5 with 20(3) (step 10): both locked for n8

12. 3 in c6 only in n5: 3 locked for n5

12a. no 8 in r6c6 (sp11(2))

13. 8 in c6 only in 21(3)n2 = {678} only: 6,8 locked for n2, 6 for c6

14. 14(2)n2 = {59}: both locked for c5

14a. r78c5 = [63]

14b. 13(2)n9 = {49} only: both locked for n9 and r7

15. "45" on n9: 2 remaining innies r78c9 = 11 = [56] only permutation

16. 16(4)r6c1: must have 7 or 8 for r7c1 = [1]{258} only

16a. R7C1 = 8

16b. r8c12 = {25} both locked for r8

16c. r7c4 = 7

17. r8c4 = 8 (hidden single n8)

17a. r9c4 = 5 (cage sum)

18. 17(3)r3c4 must have 6 or 9 for r4c4, and 7 or 8 for r4c5 = {179/368/467}(no 2)

18a. no 6 in r1c3 (IODn2=-4)

Next key step

19. "45" on n2: 3 outies r1c3 + r4c45 = 21 = [5][97]/[7][68]/[8][67]:

19a. note: must have 5 in r1c3 or 6 in r4c4

19b. note2: must have 7 in r1c3 or r4c5

19c. note: no eliminations yet (can take 7 from r4c3 but not important to this optimised solution)

20. 11(2)r6c3: [56] blocked by step 19a

20a. = {29}/[74] = 4 or 9

21. 13(2)n6: {49} blocked by 11(2)n4

21a. = [58]/{67}(no 4,9; no 5 in r6c8)

22. hidden pair 5,6 in r6. 13(2)n6 can only have one of 5,6 -> r6c2 = (56)

23. "45" on r6: 2 remaining innies r6c29 = 9 = [54/63]

24. hidden pair 2,9 in r6 -> 11(2)r6c3 = {29} only

25. "45" on n1: 3 innies r1c3 + r23c1 = 17

25a. but {458} blocked by 6(2)n1 needing 4 or 5

25b. must have 5,7,8 for r1c3 = {278/359/368/467}

25c. note: if has 7 in r1c3 must have {46} in r23c1

25d. note: no eliminations yet

The final crack by removing 6 from r6c2

26. from step 19b. must have 7 in r1c3 or r4c5

26a. if in r1c3 -> r23c1 = {46} (step 25c) -> no 6 in r6c2 (same cage)

26b. if in r4c5 -> 7 in r6 only in 13(2)r6c78 = {67}

26c. -> both places have 6 -> no 6 in r6c2

27. r6c2 = 5

27a. r6c9 = 4 (h9(2)r6c29), r6c56 = [83], r4c5 = 7 -> r34c4 = 10 = [19/46] (no 3)

27b. no 7 in 1c3 (IODn2=-4)

27b. r8c12 = [52]

28. 31(6)r2c1 = {23489}[5]/{23678}[5]

28a. must have 8 -> r5c2 = 8

28b. must have 2 & 3 which are only in c1: locked for c1

29. naked pair {14} in r3c24: both locked for r3

30. naked pair {67} in r6c78: locked for n6

30a. r4c67 = [45], naked pair {19} in r89c6: 1 locked for n8 and c6, r9c5 = 4, r5c6 = 5

31. 22(5)r5c3 must have 1 for n5 = {13459/13567}(no 2)

31a. must have 3: 3 locked for r5

31b. r5c5 = 1

32. naked pair {29} in r5c89: both locked for r5 and n6

Straightforward now.