Thanks Mike for another nice variant; as Para said "a typical Mike Killer".
Para's walkthrough has a neat breakthrough in step 24. However I was concerned about step 23. I don't think that [475] is blocked by 17(3) at R4C5; it's easy to make mistakes when analysing overlapping cages. It should start
45 on R1234: 3 innies: R4C456 = 16 = [475/862/925]: ...
When I showed this to Ed, before asking him to add this post to the archive, he came up with the alternative start to step 24
Quote:
14(3)r4c4 and r9c4 cannot both have 8 -> must have 9 in 14(3)r4c4 (-> 6 in 14(3)r4c6): or must have 6 in r9c4: either way, no 6 in r9c6.
Many thanks Ed for "saving" Para's walkthrough with a slightly different way of expressing his step.
There was one thing which Para, Afmob and I had in common. Our solving paths all used a hidden single in either R9C4 or R9C6 as the key breakthrough but we used different ways to achieve the hidden single; therefore that was a narrow point for this puzzle.
Here is my walkthrough for A69 V1.5.
Prelims
a) R1C34 = {39/48/57}, no 1,2,6
b) R1C67 = {29/38/47/56}, no 1
c) R5C12 = {29/38/47/56}, no 1
d) R5C89 = {49/58/67}, no 1,2,3
e) R78C4 = {14/23}
f) R78C6 = {29/38/47/56}, no 1
g) R8C12 = {19/28/37/46}, no 5
h) R8C89 = {19/28/37/46}, no 5
i) R9C12 = {18/27/36/45}, no 9
j) R9C89 = {14/23}
k) 11(3) cage at R6C1 = {128/137/146/236/245}, no 9
l) 21(3) cage at R6C9 = {489/579/678}, no 1,2,3
1. 45 rule on N8 2 innies R9C46 = 14 = {59/68}
1a. R78C6 = {29/38/47} (cannot be {56} which clashes with R9C46), no 5,6
1b. Hidden killer pair 5,6 in 15(3) cage and R9C46 for N8, R9C46 contains one of 5,6 -> 15(3) cage must contain one of 5,6 = {159/168/267/357} (cannot be {249/348} which don’t contain 5 or 6, cannot be {258/456} which clash with R9C46), no 4
2. 45 rule on C123 4 outies R1239C4 = 26 = {2789/3689/4589/4679/5678}, no 1
3. 45 rule on N7 2(1+1) outies R6C1 + R9C4 = 1 innie R7C3 + 1
3a. Min R6C1 + R9C4 = 6 -> min R7C3 = 5
3b. Max R6C1 + R9C4 = 10 -> max R6C1 = 5
4. 45 rule on N9 2(1+1) outies R6C9 + R9C6 = 1 innie R7C7 + 7
4a. Min R6C9 + R9C6 = 9 -> min R7C7 = 2
5. 45 rule on R89 3 outies R7C456 = 7 = {124}, locked for R7 and N8 -> R8C4 = 3, R7C4 = 2, R7C5 = 1, R7C6 = 4, R8C6 = 7, clean-up: no 9 in R1C3, no 4,7 in R1C7
5a. 1 in N2 only in R23C6, locked for C6
6. 11(3) cage at R6C1 = {137/236} (cannot be {128/146/245} because 1,2,4 only in R6C1), no 5,8
6a. 1,2 only in R6C1 -> R6C1 = {12}
6b. 11(3) cage at R6C1 = {137/236}, 3 locked for R7 and N7, clean-up: no 6 in R9C12
7. 14(3) cage at R4C6 = {239/356}, no 8, 3 locked for C6 and N5, clean-up: no 8 in R1C7
8. 3 in N2 only in 13(3) cage = {238/346}, no 5,7,9
8a. 7 in N2 only in R123C4, locked for C4
9. 15(3) cage in N8 (step 1b) = {159} (only remaining combination, cannot be {168} which clashes with 13(3) cage in N2), locked for C5 and N8
10. Naked pair {68} in R9C46, locked for R9, clean-up: no 1 in R9C12
11. Killer pair 2,4 in R9C12 and R9C89, locked for R9
12. 45 rule on R9 3 remaining innies R9C357 = 17 = {179/359}
12a. 3 of {359} must be in R9C7 -> no 5 in R9C7
13. 1 in N5 only in 14(3) cage at R4C4 = {149/158}, no 6
14. 16(3) cage at R8C3 = {169/178} (cannot be {259/457} because R9C4 only contains 6,8, cannot be {268} because no 2,6,8 in R9C3), no 2,4,5, 1 locked for C3 and N7, clean-up: no 9 in R8C12
14a. R9C4 = {68} -> no 6,8 in R8C3
14b. 9 in N7 only in R789C3, locked for C3
15. 16(3) cage at R8C7 = {169/178/358} (cannot be {259/349/457} because R9C6 only contains 6,8, cannot be {268} because no 2,6,8 in R9C7, cannot be {367} because 3,7 only in R9C7), no 2,4
15a. R9C6 = {68} -> no 6,8 in R8C7
16. Naked triple {159} in R8C357, locked for R8
17. R6C1 + R9C4 = R7C3 + 1 (step 3)
17a. Min R6C1 + R9C4 = 7 -> min R7C3 = 6
18. 5 in N7 only in R9C12 = {45}, locked for R9 and N7 -> R89C5 = [59], clean-up: no 6 in R8C12, no 1 in R9C89
18a. Naked pair {28} in R8C12, locked for R8 and N7
18b. Naked pair {46} in R8C89, locked for N9
18c. Naked pair {23} in R9C89, locked for N9
19. 1 in N9 only in R89C7, locked for C7
19a. R5C4 = 1 (hidden single in R5)
20. 21(3) cage at R6C9 = {489/579/678}
20a. 4,6 of {489/678} must be in R6C9 -> no 8 in R6C9
20b. 5 of {579} must be in R7C89 (R7C89 cannot be {79} which clashes with R89C7, ALS block) -> no 5 in R6C9
20c. Hidden killer pair 5,8 in R7C7 and 21(3) cage at R6C9 for R7, 21(3) cage must contain one of 5,8 in R7C89 -> R7C7 = {58}
21. 45 rule on R1234 2 innies R4C56 = 1 remaining outie R6C4 + 3, IOU no 3 in R4C6
21a. 45 rule on R1234 3 innies R4C456 = 16 = {259/268/457}
21b. 8 of {268} must be in R4C4 -> no 8 in R4C5
21c. 7 of {457} must be in R4C5 -> no 4 in R4C5
[Can probably make further candidate eliminations within N5, looking at the interactions between R4C456 and the three vertical cages but I’ll leave that until later when I hope that I’ll have been able to reduce the number of combinations in the cages in N5.]
22. 13(3) cage in N2 (step 8) = {238/346}
22a. 7 in C4 only in R123C4 -> R1239C4 (step 2) = {4679/5678}
22b. Consider placements for 4 in N2
4 in R123C4 => R1239C4 = {4679} = {479}6, no 6 in R23C4
or 4 in 13(3) cage = {346}, 6 locked for N2
-> no 6 in R23C4
23. R9C4 = 6 (hidden single in C4) -> R89C3 (step 14) = [91], R9C6 = 8, R89C7 = [17], clean-up: no 3 in R1C7
24. 21(3) cage at R6C9 = {489/579} (cannot be {678} because 6,7 only in R6C9), no 6
24a. 4,7 only in R6C9 -> R6C9 = {47}
25. 25(4) cage at R5C3 = {2689/3679/4579/4678} (cannot be {1789} because 1,9 only in R6C2, cannot be {3589} because R7C3 only contains 6,7), no 1
25a. 9 of {2689/3579} must be in R6C2 -> no 2,3 in R6C2
26. 45 rule on N47 3 remaining innies R4C123 = 14 = {149/167/239/347} (cannot be {158/248/257/356} which clash with R4C456), no 5
27. 45 rule on N69 3 remaining innies R4C789 = 15 = {159/168/348/357} (cannot be {249/258/267/456} which clash with R4C456), no 2
28. 2 in N6 only in 18(4) cage at R5C7 = {2358/2457} (cannot be {1269/2349/2367} because R7C7 only contains one of 5,8, cannot be {1278} because 1,7 only in R6C8, cannot be {2457} which clashes with R6C9), no 1,6,9, CPE no 5 in R4C7
29. R6C1 = 1 (hidden single in R6), 11(3) cage at R6C1 (step 6) = {137} (only remaining combination) -> R7C12 = {37}, locked for R7, R7C3 = 6
30. R7C3 = 6 -> 25(4) cage at R5C3 (step 25) = {2689/3679/4678}, no 5
31. 5 in N4 only in R5C12 = {56}, locked for R5 and N4, clean-up: no 7,8 in R5C89
31a. Naked pair {49} in R5C89, locked for R5 and N6 -> R6C9 = 7
31b. 21(3) cage at R6C9 (step 24) = {579} (only remaining combination), 5,9 locked for R7 -> R7C7 = 8
32. Naked triple {235} in R5C7 + R6C78, locked for N6, 5 also locked for R6 -> R4C7 = 6, clean-up: no 5 in R1C6
32a. Naked pair {18} in R4C89, locked for R4, CPE no 1,8 in R3C8
32b. Naked pair {23} in R5C67, locked for R5
33. 17(3) cage at R4C5 = {278} (only remaining combination, cannot be {467} because 4,6 only in R6C5), locked for C5 and N5 -> R5C6 = 3, R5C7 = 2, clean-up: no 9 in R1C6, no 5 in R4C4 (step 13)
33a. Naked pair {49} in R46C4, locked for C4 and N5 -> R4C6 = 5, R6C6 = 6, R1C6 = 2, R1C7 = 9, clean-up: no 3,8 in R1C3
33b. Naked pair {35} in R6C78, locked for R6
34. 4 in C7 only in R23C7, locked for N3
35. 19(4) cage at R3C6 contains 6 = {1369/1468}, no 5
36. 5 in C3 only in R123C3, locked for N1
37. 15(4) cage at R2C9 = {1257/1356}, no 8 -> R4C9 = 1, R4C8 = 8
37a. 7 of {1257} must be in R3C8 -> no 2 in R3C8
38. 19(4) cage at R3C6 (step 35) contains 6,8 = {1468} (only remaining combination) -> R3C67 = [14], R2C6 = 9
39. 15(4) cage at R2C9 (step 37) = {1257} (only remaining combination, cannot be {1356} which clashes with R2C7) -> R3C8 = 7, R23C9 = {25}, locked for C9 and N3 -> R2C7 = 3, R6C78 = [53], R7C89 = [59], R5C89 = [94], R8C89 = [46], R9C89 = [23], R1C9 = 8, clean-up: no 4 in R1C3
39a. Naked pair {57} in R1C34, locked for R1
40. 9 in R3 only in R3C12, locked for 21(4) cage at R2C1, no 9 in R4C1
40a. 9 in N4 only in R46C2, locked for C2
41. R3C1 = 9 (hidden single in R3) -> 21(4) cage at R2C1 = {2379/2469}, no 8
41a. 3 of {2379} must be in R3C2 (R24C1 cannot be [73] which clashes with R7C1) -> no 3 in R4C1
42. 3 in N4 only in R4C23, locked for 23(4) cage at R3C3, no 3 in R3C3
42a. 23(4) cage = {3578} (only remaining combination, cannot be {3479} because R3C4 only contains 5,8), no 2,4,9
42b. Naked pair {37} in R4C23, locked for R4 and N4 -> R5C3 = 8
and the rest is naked singles.
Rating Comment. I'll rate my walkthrough for A69 V1.5 at Easy 1.5. The forcing chain in step 22 is short and localised, almost combination analysis.