Another variant which I've just tried for the first time; in spring and summer 2007 I didn't try variants because we were moving from Calgary to Lethbridge.
Prelims
a) 9(3) cage at R1C4 = {126/135/234}, no 7,8,9
b) 21(3) cage at R1C9 = {489/579/678}, no 1,2,3
c) 20(3) cage at R2C4 = {389/479/569/578}, no 1,2
d) 23(3) cage at R7C1 = {689}
e) 10(3) cage at R7C8 = {127/136/145/235}, no 8,9
f) 10(3) cage at R8C2 = {127/136/145/235}, no 8,9
1. Naked triple {689} in 23(3) cage at R7C1, locked for N7
2. 45 rule on R89 2 innies R8C19 = 11 = [65/83/92]
3. 45 rule on N4 1 innie R4C2 = 1 outie R4C4 + 1, no 1 in R4C2, no 9 in R4C4
4. 45 rule on N6 1 outie R4C6 = 1 innie R4C8 + 4, no 1,2,3,4 in R4C6, no 6,7,8,9 in R4C8
5. 45 rule on N7 1 outie R9C4 = 1 innie R7C3 + 1, no 1,7,9 in R9C4
6. 45 rule on N9 1 innie R7C7 = 1 outie R9C6 + 1, no 1 in R7C7, no 9 in R9C6
7. 45 rule on N7 3 innies R789C3 = 12 = {147/237/345}
7a. 2 of {237} must be in R89C3 (cannot be 2{37} because 13(3) cage at R8C3 cannot be {37}3), no 2 in R7C3, clean-up: no 3 in R9C4 (step 5)
8. 45 rule on N1 1 innie R3C3 = 1 outie R4C2 + 1, no 9 in R4C2, clean-up: no 8 in R4C4 (step 3)
9. 45 rule on N3 1 outie R4C8 = 1 innie R3C7 + 2, R3C7 = {123}, R4C8 = {345}, clean-up: no 5,6 in R4C6 (step 4)
10. 45 rule on R123 3 outies R4C258 = 12 = {138/147/156/237/246/345} (cannot be {129} because R4C8 only contains 3,4,5), no 9
10a. R4C258 = {147/156/237/246/345} (cannot be {138} = [813] because R4C24 = [87], step 3, clashes with R4C68 = [73], step 4), no 8, clean-up: no 9 in R3C3 (step 8), no 7 in R4C4 (step 3)
10b. R4C258 = {147/237/246/345} (cannot be {156} = [615] because R4C24 = [65], step 3, clashes with R4C8)
10c. R4C258 = {147/246/345} (cannot be {237} = {27}3 which clashes with R4C68 = [73]), 4 locked for R4, clean-up: no 5 in R4C2 (step 3), no 6 in R3C3 (step 8)
10d. 1 of {147} must be in R4C5 -> no 7 in R4C5
10e. 3 of {345} must be in R4C2 (cannot be 4{35} which clashes with R4C24 = [43], step 3), 7 of {147} must be in R4C2, 4 of {246} must be in R4C8 -> no 4 in R4C2, no 3 in R4C58, clean-up: no 5 in R3C3 (step 8), no 1 in R3C7 (step 9), no 3 in R4C4 (step 3), no 7 in R4C6 (step 4)
11. 45 rule on N3 3 innies R2C8 + R3C78 = 10 = {127/136/235} (cannot be {145} because R3C7 only contains 2,3), no 4,8,9
11a. 2 of {235} must be in R3C7 (cannot be [235/532] which clash with 12(3) cage at R2C8 = [255/525]), no 2 in R23C8
12. 45 rule on C1234 2 innies R18C4 = 7 = {16/25/34}, no 7,8,9
13. 45 rule on C6789 2 innies R18C6 = 6 = {15/24}
14. 45 rule on C5 3 innies R189C5 = 18 = {189/279/369/378/459/468/567}
14a. 1,2 of {189/279} must be in R1C5 -> no 1,2 in R89C5
15. 45 rule on C12 1 innie R1C2 = 1 outie R6C3 + 1, no 1 in R1C2, no 9 in R6C3
16. 45 rule on C89 1 outie R6C7 = 1 innie R1C8 + 2, no 1,2 in R6C7, no 8,9 in R1C8
17. 45 rule on N36 3 outies R234C6 = 20 = {389/479/569/578}, no 1,2
17a. 8 of {389/578} must be in R4C6 (cannot be {38}9 because 14(3) cage at R2C6 cannot be {38}3), no 8 in R23C6
18. 45 rule on N14 3 outies R234C4 = 18 = {189/279/369/459/567} (cannot be {378} because no 3,7,8 in R4C4, cannot be {468} because 20(3) cage at R2C4 cannot be {48}8)
18a. 6 of {369/567} must be in R4C4 (cannot be {67}5 because 20(3) cage at R2C4 cannot be {67}7), no 6 in R23C4
19. 45 rule on N14 3 innies R345C3 = 19 = {289/469} (cannot be {379/478} which clash with R789C3, cannot be {568} = 8{56} because 17(3) cage at R4C3 cannot be {56}6), no 1,3,5,7, 9 locked for C3 and N4, clean-up: no 2,6 in R4C2 (step 8), no 1,5 in R4C4 (step 3)
19a. R3C3 = {48} -> no 4,8 in R45C3
19b. Killer pair 2,4 in R345C3 and R789C3, locked for C3, clean-up: no 3,5 in R1C2 (step 15)
19c. 17(3) cage at R4C3 = {269} (only remaining combination), CPE no 2,6 in R4C1
20. R4C258 (step 10c) = {147/345} (cannot be {246} because 2,6 only in R4C5), no 2,6
21. 20(3) cage at R2C4 = {389/479/578}
21a. R3C3 = {48} -> no 4,8 in R23C4
22. 8 in N2 only in R23C5 -> 13(3) cage at R2C5 = {148} (only remaining combination, cannot be {238} because no 2,3,8 in R4C5), locked for C5
22a. Killer pair 1,4 in 9(3) cage at R1C4 and 13(3) cage, locked for N2
23. 2 in N2 only in 9(3) cage at R1C4, locked for R1, clean-up: no 1 in R6C3 (step 15), no 4 in R6C7 (step 16)
23a. 9(3) cage = {126/234}, no 5, clean-up: no 2 in R8C4 (step 12), no 1 in R8C6 (step 13)
24. 45 rule on N6 3 innies R4C78 + R5C7 = 14
24a. 45 rule on N36 3 innies R345C7 = 12 = {129/138/237} (cannot be {147/156} because R3C7 only contains 2,3, cannot be {246} = 2{46} because R4C78 + R5C7 cannot be {46}4, cannot be {345} = 3{45} because R4C78 + R5C7 cannot be {45}5), no 4,5,6
25. 45 rule on N5789 3 innies R4C456 = 15 = {168/249}
25a. 45 rule on R6789 3 outies R5C456 = 15 = {168/249/357} (cannot be {159/258/267/348/456} which clash with R4C456)
25b. 45 rule on R6 3 innies R6C456 = 15 = {168/249/357} (cannot be {159/258/267/348/456} which clash with R4C456)
[Step 25 reminded me of the top row of A74 Brick Wall, which also had three 15(3) cages. This step was actually more powerful until I realised that I’d overlooked a killer pair which reduced the number of combinations in step 25. In its original form, this step was longer but reduced the three 15(3) hidden cages to the same combinations. That’s why I’ve kept it in even though, in the simplified form, there aren’t any immediate candidate eliminations.]26. 45 rule on N1 3 innies R2C2 + R3C23 = 16
26a. 15(3) cage at R2C2 = {267/357} (cannot be {159/168/249/258/456} because R4C2 only contains 3,7, cannot be {348} = {48}3 because R2C2 + R3C23 cannot be {48}4), no 1,4,8,9, 7 locked for C2, clean-up: no 6 in R6C3 (step 15)
27. 45 rule on C12 3 outies R126C3 = 14 = {158/167/356}
27a. 3 of {356} must be in R6C3 (cannot be {36}5 because 15(3) cage at R1C2 cannot be 6{36}), no 3 in R12C3
27b. R126C3 = {158/167} (cannot be {356} = {56}3) which clashes with 15(3) cage at R2C2), no 3, 1 locked for C3 and N1, clean-up: no 4 in R1C2 (step 15), no 2 in R9C4 (step 5)
28. R789C3 (step 7) = {237/345}, 3 locked for N7
28a. 4 of {345} must be in R89C3 (cannot be 4{35} because 13(3) cage at R8C3 cannot be {35}5), no 4 in R7C3, clean-up: no 5 in R9C4 (step 5)
29. 10(3) cage at R8C2 = {127/145}
29a. 7 of {127} must be in R9C1 -> no 2 in R9C1
30. 15(3) cage at R1C2 = {159/168}, no 7, 1 locked for N1
30a. Killer pair 5,6 in 15(3) cage at R1C2 and 15(3) cage at R2C2, locked for N1
30b. 14(3) cage at R1C1 = {239/347} (cannot be {248} which clashes with R3C3), no 8, 3 locked for C1 and N1
31. 13(3) cage at R4C1 = {148/157/238/247} (cannot be {256} which clashes with R45C3, ALS block, cannot be {346} because no 3,4,6 in R4C1), no 6
31a. 14(3) cage at R6C1 = {158/167/248/257/356} (cannot be {347} which clashes with R4C2)
31b. Variable hidden killer quad 1,4,5,8 in 13(3) cage and 14(3) cage for N4 -> each cage must contain two of 1,4,5,8 or one cage must contain one and the other three
31c. 14(3) cage = {158/248/257/356} (cannot be {167} because 13(3) cage cannot contain all of 4,5,8)
31d. 3 of {356} must be in R6C2 -> no 6 in R6C2
31e. 13(3) cage = {148/157/247} (cannot be {238} which clashes with 14(3) cage), no 3
[At this stage I originally used a short forcing chain
14(3) cage at R6C1 (step 31c) = {158/248/257/356}
Consider placement for 6 in 23(3) cage at R7C1
6 in R78C1 -> 14(3) cage = {158/248/257}
or 6 in R7C2 -> no 6 in R1C2, no 5 in R6C3 (step 15) -> 14(3) cage = {158/248/257}
-> 14(3) cage = {158/248/257}, no 3,6
Then, when I went through Ed’s walkthrough, I realised that he’d done more work in N6 than I had; I hadn’t looked at the 15(3) and 16(3) cages, so …]32. 14(3) cage at R6C1 (step 31c) = {158/248/257/356}
32a. 18(3) cage at R4C6 = 8{19/37}/9{18/27}
32b. 16(3) cage at R6C7 = {169/349/367} (cannot be {178} which clashes with 18(3) cage at R4C6, cannot be {259/268/358/457} which clash with 14(3) cage at R6C1), no 2,5,8, clean-up: no 3,6 in R1C8 (step 16)
32c. 14(3) cage at R6C1 = {158/248/257} (cannot be {356} which clashes with 16(3) cage at R6C7), no 3,6
33. R4C2 = 3 (hidden single in N4), R3C3 = 4 (step 8), R4C4 = 2 (step 3)
33a. Naked pair {69} in R45C3, locked for C3
[Cracked. The rest is straightforward, routine clean-ups have been omitted.]
34. R3C3 = 4 -> R23C4 = 16 = {79}, locked for C4 and N2
35. R4C258 (step 20) = {345} (only remaining combination) -> R4C5 = 4, R4C8 = 5, R4C6 = 9 (step 4), R45C3 = [69], R3C6 = 3 (step 9)
35a. R4C8 = 5 -> R23C8 = 7 = {16}, locked for C8 and N3, clean-up: no 7,8 in R6C7 (step 16)
36. Naked pair {56} in R23C6, locked for C6 and N2, clean-up: no 1 in R1C6 (step 13)
36a. Naked pair {24} in R18C6, locked for C6
37. Naked pair {18} in R23C5, locked for N2
37a. Naked triple {234} in 9(3) cage at R1C4, locked for R1 -> R1C8 = 7, R6C7 = 9 (step 16), R1C1 = 9, clean-up: no 2 in R8C9 (step 2)
37b. Naked pair {68} in R78C1, locked for C1 and N7 -> R7C2 = 9
38. R1C23 = [61] (hidden pair in R1), R2C3 = 8 (cage sum)
39. R2C1 = 3 (hidden single in N1), R3C1 = 2 (cage sum)
40. Naked pair {58} in R1C79, locked for N3 -> R3C9 = 9, R2C9 = 4, R1C9 = 8 (cage sum), R12C7 = [52]
41. R4C7 = 8 (hidden single in R4), R5C7 = 1 (cage sum), R4C19 = [17]
41a. R4C9 = 7 -> R5C89 = 8 = [26], R6C89 = [43], R7C8 = 3, R8C9 = 5, R7C9 = 2 (cage sum), R8C1 = 6 (step 2), R9C9 = 1
[And now to use those hidden 15(3) cages before it’s too late
]
42. R6C456 (step 25b) = {168} (only remaining combination) -> R6C5 = 6, R6C46 = {18}, locked for R6 and N5
43. R6C5 = 6 -> R57C5 = 8 = [35]
and the rest is naked singles.
I'll rate my walkthrough for A56 V2 at 1.5. There was a lot of analysis of interactions between hidden cages.