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3x3::k:1537:1537:6402:6402:5379:6916:6916:6916:1797:6918:6402:6402:3079:5379:5379:2824:6916:1797:6918:6918:3079:3079:5379:5379:2824:2824:5129:6918:6918:7178:7178:3595:3595:5129:5129:5129:6668:6668:6668:7178:7178:7178:5129:2317:2317:6670:6668:6668:7183:1552:1552:4369:4114:4114:6670:2579:7183:7183:4116:4369:4369:4373:4373:6670:2579:7183:7183:4116:5910:5910:3351:4373:6670:6670:6670:4116:4116:5910:5910:3351:4373:

that R6C4 can be quite quickly (relatively speaking) reduced to one possibility. Don't ask me how: it involved looking at the relations between this cell and others all over.

I'll rate my walkthrough for A210 at 1.5 to Hard 1.5; I used a contradiction move and several forcing chains, plus I found it hard to spot several steps which is why I haven't just said 1.5.

Prelims

a) R1C12 = {15/24}
b) R12C9 = {16/25/34}, no 7,8,9
c) R4C56 = {59/68}
d) R5C89 = {18/27/36/45}, no 9
e) R6C56 = {15/24}
f) R6C89 = {79}
g) R78C2 = {19/28/37/46}, no 5
h) R89C8 = {49/58/67}, no 1,2,3
i) 11(3) cage in N3 = {128/137/146/236/245}, no 9
j) 27(4) cage at R1C6 = {3789/4689/5679}, no 1,2

1. Naked pair {79} in R6C89, locked for R6 and N6, clean-up: no 2 in R5C89

2. 45 rule on R6789 2 innies R6C23 = 8 = {26/35}
2a. Killer pair 2,5 in R6C23 and R6C56, locked for R6

3. 45 rule on N3 1 outie R1C6 = 1 innie R3C9, no 1,2 in R3C9
3a. 45 rule on N3 4 innies R1C78 + R2C8 + R3C9 = 27 = {3789/4689/5679}

4. 28(5) cage at R4C3 = {34579/34678} (cannot be {13789/14689/15679/23689/24589/24679/25678} which clash with R4C56), no 1,2

5. 2 in N5 only in R6C56 = {24}, locked for R6 and N5, clean-up: no 6 in R6C23 (step 2)
5a. Naked pair {35} in R6C23, locked for R6 and N4
5b. R6C4 = 1 (hidden single in N5)

6. 28(5) cage at R4C3 (step 4) = {34579/34678} -> R4C3 = 4

7. 45 rule on R6789 3 outies R5C123 = 18 = {189/279}, no 6, 9 locked for R5 and N4
7a. 6 in N4 only in R4C12 + R6C1, CPE no 6 in R23C1

8. R6C4 = 1 -> 28(5) cage at R6C4 = {13789/14689/15679}, no 2

9. 45 rule on N6 1 outie R3C9 = 1 innie R6C7 -> R3C9 = {68}, R1C6 = {68} (step 3)
9a. Naked pair {68} in R6C17, R3C9 = R6C7 -> naked pair {68} in R3C9 + R6C1, CPE no 8 in R3C1

10. 45 rule on N7 2 innies R78C3 = 1 outie R6C1 + 9
10a. R6C1 = {68} -> R78C3 = 15,17 = {69/78/89}, no 3,5

11. R5C123 = 18, R4C4 + R5C456 = 24
11a. 45 rule on R5 1 outie R4C4 = 1 innie R5C7 + 6, R4C4 = {789}, R5C7 = {123}

12. 3 in N5 only in R5C456, locked for R5, clean-up: no 9 in R4C4, no 6 in R5C89

13. 9 in R4 only in R4C56 = {59}, locked for R4 and N5

14. R5C89 = {45} (hidden pair in R5)

15. 11(3) cage in N3 = {128/137/146/245} (cannot be {236} which clashes with R12C9)
15a. 8 of {128} must be in R23C7 (R23C7 cannot be {12} which clashes with R5C7) -> no 8 in R2C8

16. 45 rule on C1234 2 outies R5C56 = 1 innie R9C4 + 8
16a. Hidden killer pair 3,6 in R5C4 and R5C56 for R5, R5C56 must contain one of 3,6 (cannot be {36} = 9 because no 1 in R9C4) -> R5C4 = {36}
16b. R5C56 = {37/38/67/68} = 10,11,13,14 -> R9C4 = {2356}

17. 5 in N7 only in 26(6) cage at R6C1 = {123569/123578/124568/134567}, 1 locked for N7, clean-up: no 9 in R78C2

18. 45 rule on C789 4 outies R1789C6 = 25 = {1789/2689/3679/4678} (cannot be {3589/4579} which clash with R4C6), no 5
18a. R1789C6 = {1789/2689/4678} contain 8, R1789C6 = {3679} => R5C6 = 8 -> R15789C6 must contain 8, locked for C6

19. 45 rule on N14 2(1+1) remaining innies R3C3 + R6C1 = 1 outie R1C4 + 2
19a. Min R3C3 + R6C1 = 7 -> min R1C4 = 5
19b. Max R1C4 = 9 -> max R3C3 + R6C1 = 11, min R6C1 = 6 -> max R3C3 = 5

20. 17(3) cage at R6C7 = {269/278/368/458/467} (cannot be {179/359} because R6C7 only contains 6,8), no 1

21. Hidden killer pair 7,9 in R6C9 and R789C9 for C9, R6C9 = {79} -> R789C9 must contain one of 7,9
21a. 17(4) cage in N9 = {1259/1349/1367/2357} (other combinations don’t contain 7 or 9, cannot be {1457} which clashes with R89C8), no 8
21b. 17(4) cage only contains one of 7,9 -> no 7,9 in R7C8

22. 8 in C9 only in R34C9, locked for 20(5) cage at R2C9, no 8 in R4C78

23. 45 rule on N2 4 innies R123C4 + R1C6 = 24 = {2589/2679/3489/4569/4578} (cannot be {3579} because R1C6 only contains 6,8, cannot be {3678} because {367}8 clashes with R4C5 and {378}6 clashes with R4C4)
23a. R1C6 = {68} -> no 6,8 in R123C4

24. 12(3) cage at R2C4 = {129/147/237/345}
24a. 1,3 of {129/237} must be in R3C3 (R23C4 cannot be {37} because no combination of R123C4 + R1C6 (step 23) contains both of 3,7) -> no 2 in R3C3

25. 45 rule on N9 3 innies R789C7 = 15 = {249/258/267/348} (cannot be {168} which clashes with R6C7, cannot be {456} which clashes with R89C8, cannot be {159/357} which clash with 17(4) cage), no 1

26. 25(4) cage at R1C3 = {1789/2689/3589/3679} (cannot be {4579} which clashes with R1C12, cannot be {4678} because R12C3 = {68} clashes with R78C3), no 4
[The {4579} clash was easy but I’ve only just spotted the {4678} clash.]
26a. 2 of {2689} must be in R12C3 (R12C3 cannot be {68} which clashes with R78C3) -> no 2 in R2C2

[The next step was even harder to spot. I’d realised for a while that 7 in N1 must be in either 25(4) cage at R1C3 or in 27(5) cage at R2C1 but it still took time to spot ...]
27. Consider the placement for 7 in N1
7 in 25(4) cage at R1C3 in N1 => no 7 in R1C4
or 7 in 27(5) cage at R2C1 in N1 => no 7 in R4C12 => R4C4 = 7 (hidden single in R4) => no 7 in R1C4
27b. -> no 7 in R1C4

28. 25(4) cage at R1C3 (step 26) = {1789/2689/3589/3679}, R1C12 = {15/24}
28a. Consider combinations for R1C12
28b. R1C12 = {15} => R3C3 = 3 => killer triple 1,2,3 in R1C12 + R3C3 + 25(4) cage at R1C3, locked for N1
or R1C12 = {24}, locked for N1
28c. -> no 2 in R2C1 + R3C12

[Getting a bit heavier, since I can’t see anything easier.]
29. R789C7 (step 25) = {249/258/267/348}
29a. Consider combinations for R789C6
R789C6 = {249}, locked for N9, 8 in N9 only in R89C8 = {58}, locked for C8 => R5C89 = [45], R12C9 = {16/34}, R4C9 = 2 (hidden single in C9) => R6C7 = 8 (hidden single in N6)
or R789C7 {258/267/348}, killer pair 6,8 in R6C7 and R789C7, locked for C7
29b. -> 8 must be in R6789C7, locked for C7

30. 11(3) cage in N3 (step 15) = {137/146/245}
Hidden killer pair 1,2 in R12C9 and 11(3) for N3, 11(3) cage contains one of 1,2 -> R12C9 must contain one of 1,2 -> R12C9 = {16/25}, no 3,4

31. R1C78 + R2C8 + R3C9 (step 3a) = {3789/4689} (cannot be {5679} which clashes with R12C9), no 5
[Alternatively 8 in N3 only in R1C78 + R2C8 + R3C9 ...]
31a. 3 of {3789} must be in R12C8 (R12C8 cannot be {79} which clashes with R6C8 -> no 3 in R1C7

32. 1 in N9 only in 17(4) cage (step 21a) = {1259/1349/1367}
32a. 1 of {1259} must be in R7C8 (R789C9 cannot be {129/159} which clash with R12C9) -> no 2,5 in R7C8

[I’m probably missing something better but the only way forward I can see is a contradiction move; while contradiction moves can often be replaced by forcing chains, I couldn’t see how to do that this time.]
33. 45 rule on N2 2 innies R1C46 = 1 outie R3C3 + 12
33a. R1C12 cannot be {15}, here’s how
R1C12 = {15}, locked for R1 and N1 => R3C3 = 3, R1C46 = 15 = [96] => 27(4) cage at R1C6 = {4689} = [6489], 25(4) cage at R1C3 (step 26) = {2689} => R1C3 = 2 => no remaining candidate for R1C9
33b. -> R1C12 = {24}, locked for R1 and N1, clean-up: no 5 in R2C9

[Followed by something similar. This was intended to be a forcing chain but happens to contain a contradiction; still it’s better than my original contradiction move which gave the same result.]
34. R1C46 = R3C3 + 12 (step 33), R3C3 = {135} -> R1C46 = 13,15,17 = [58/96/98]
34a. Consider the permutations for R1C46
R1C46 = [58/98] = 27(4) cage at R1C6 = {3789}
R1C46 = [96] => 9 in 27(4) cage at R1C6 must be in R2C8 => 27(4) cage cannot be {4689}
34b. -> 27(4) cage at R1C6 = {3789} (only remaining combination), no 4,6 -> R1C6 = 8, R1C78 + R2C8 = {379}, locked for N3, 3 also locked for C8, R3C9 = 8 (step 3), R6C7 = 8 (step 9), R6C1 = 6

35. 8 in C8 only in R89C8 = {58}, locked for C8 and N9 -> R5C89 = [45], clean-up: no 2 in R2C9
35a. Naked pair {16} in R12C9, locked for C9 and N3 -> R3C8 = 2, R23C7 = {45}, locked for C7

36. 1,4 in N9 only in 17(4) cage (step 21a) = {1349} (only remaining combination) -> R7C8 = 1, R789C9 = {349}, locked for C9 and N9 -> R4C9 = 2, R5C7 = 1, R4C78 = [36], R6C89 = [97], R4C4 = 7 (step 11a)
36a. R5C7 = 8 (hidden single in N5)

37. 45 rule on C1234 2 remaining innies R59C4 = 9 = {36} (locked for C4)

38. 45 rule on N7 2 remaining outies R78C4 = 12 = {48}, locked for C4, N8 and 28(4) cage at R6C4, no 8 in R78C3

39. R2C4 = 2 (hidden single in C4) -> 12(3) cage at R2C4 (step 24) = {129} (only remaining combination) -> R3C34 = [19], R1C4 = 5

40. R1C7 = 9 (hidden single in N3)
40a. R1C4 = 5 -> 25(4) cage at R1C3 (step 26) = {3589} (only remaining combination) -> R1C3 = 3, R2C23 = {89}, locked for N1, R6C23 = [35], clean-up: no 7 in R78C2

41. Naked pair {57} in R23C1, locked for C1 and N1 -> R3C2 = 6, clean-up: no 4 in R78C2
41a. Naked pair {28} in R78C2, locked for C2 and N7 -> R1C12 = [24], R2C23 = [98], R4C12 = [81], R5C123 = [972], R9C2 = 5, R89C8 = [58]

42. R78C3 = R6C1 + 9 (step 10)
42a. R6C1 = 6 -> R78C3 = 15 = {69}, locked for C3 -> R9C3 = 7

43. R7C5 = 5 (hidden single in N8), R4C56 = [95]
43a. R7C5 = 5 -> 16(4) cage in N8 = {1357/2356}, 3 locked for N8
43b. 7 of {1357} must be in R8C5 -> no 1 in R8C5

44. R6C7 = 8 -> 17(3) cage at R6C7 = {278} (only remaining combination) -> R7C67 = {27}, locked for R7 -> R78C2 = [82], R78C4 = [48], R7C1 = 3, R7C9 = 9, R78C3 = [69]

45. 6 in C7 only in R89C7 -> 23(4) cage at R8C6 = {1679} (only remaining combination) -> R9C6 = 9, R8C6 = 1, R9C7 = 6, R8C7 = 7

and the rest is naked singles.

NOTE: this is an optimised solution so only the important steps are included. However, I try and do clean-up as I go. Let me know please of any corrections or clarifications. Thanks Andrew!

Prelims
i. 6(2)n1 = {15/24}(no 3,6..9)
ii. 27(4)r1c6: no 1,2
iii. 7(2)n3: no 7,8,9
iv. 11(3)n3: no 9
v. 14(2)n5 = {59/68}(no 1..4,7)
vi. 9(2)n6: no 9
vii. 6(2)n5 = {15/24}(no 3,6..9)
viii. 16(2)n6 = {79}
ix. 10(2)n7: no 5
x. 13(2)n9: no 1,2,3

1. 16(2)n6 = {79}: both locked for r6 and n6
1a. no 2 in 9(2)n6

2. "45" on r6789: 2 innies r6c23 = 8 = {26/35}(no 1,4,8) = [2/5, 3/6..]

3. 6(2)n5 = {15/24} = [2/5..]
3a. Killer pair 2,5 with h8(2)r6c23 (step 2)
3b. both locked for r6

4. 7 in n5 only in 28(5)r4c3 -> no 7 in r4c3

5. 14(2)n5 = {59/68}= [6/9..]
5a. "45" on n5: 1 outie r4c3 - 3 = 1 innie r6c4
5b. but [63] blocked by h8(2)r6c23 (step 2)
5c. [96] blocked by 14(2)n5 (step 5)
5d. = [41] only. r4c3 = 4, r6c4 = 1

6. 6(2)n5 = {24}: locked for n5 & r6
6a. h8(2)r6c23 = {35}: locked for n4 & r6

7. "45" on r1234: 1 outie r5c7 + 6 = 1 remaining innie r4c4
7a. r5c7 = (123), r4c4 = (789)

8. 4 in r5 only in 9(2)n6 = {45} only: locked for n6 and r5

9. "45" on n6: 1 outie r3c9 = 1 innie r6c7 = (68)

10. Hidden killer pair 7,9 in c9 -> r789c9 must have 7/9 for c9
10a. -> min. any 3 cells in 17(4)n9 = {127} = 10
10b. -> no 8 in 17(4)

11. 8 in c9 only in r34c9: locked for 20(5)r3c9
11a. no 8 in r4c78

12. "45" on n3: r1c6 = r3c9 (important for cracking step) = (68)

Now for the cracker (Clone IOD)
13. "45" on c89: 2 outies r1c67 + 1 = 4(1+3) innies r34c89
13a. since r1c6 = r3c9 (step 12)(ie 1 of the outies = 1 of the innies) -> 1 remaining outie r1c7 + 1 = 3(1+2) remaining innies r3c8 + r4c89.
13b. -> max. remaining 3 innies = 10.
13c. Min. r34c8 = 3 -> max r4c9 = 7 (no 8)

14. r3c9 = 8 (hidden single c9)

Much easier now.

I usually work with only the lower diagram visible (with notes about 45s, etc. below the diagram). Ed's neat 45 on C89 is much more visible on his originally posted diagram and my upper diagram.

I had spotted 45 rule on C9 3 outies R457C8 = 2 innies R34C9 +4 with 8 in C9 only in R34C9 but wasn't able to make any use of that.

After a lot of hard work my third placement was in R1C6, the clone of R3C9; it's interesting how, with such different ways to make the breakthrough, those were still the two key cells.