Thanks manu for another challenging puzzle!
Afmob wrote:
It's fairly difficult (at least the way I solved it) ...
Congratulations of solving A180 without using "clones".
Having used them for Smooth A180 I was very pleased to have decided to use them again for A180, once I'd spotted the extra step needed for them to be useful. I also used Ronnie's cage blocker (step 1b of her Smooth A180 walkthrough). Thanks Ronnie for posting your walkthrough! I'm not sure that I'd have been able to solve A180 without that very useful step. Maybe I might eventually have found it using permutation analysis as in Afmob's walkthrough.
Here is my walkthrough for A180.
Thanks Afmob for pointing out that step 22 and a couple of earlier details were unnecessary; I've deleted them.Prelims
a) R12C1 = {16/25/34}, no 7,8,9
b) R1C34 = {15/24}
c) R12C5 = {49/58/67}, no 1,2,3
d) R1C89 = {69/78}
e) R23C8 = {16/25/34}, no 7,8,9
f) R23C9 = {18/27/36/45}, no 9
g) R4C23 = {18/27/36/45}, no 9
h) R4C89 = {18/27/36/45}, no 9
i) R5C12 = {19/28/37/46}, no 5
j) R5C34 = {19/28/37/46}, no 5
k) R5C89 = {49/58/67}, no 1,2,3
l) R6C12 = {19/28/37/46}, no 5
m) R67C7 = {18/27/36/45}, no 9
n) R67C8 = {79}
o) R78C1 = {59/68}
p) R78C2 = {14/23}
q) R89C6 = {18/27/36/45}, no 9
r) R89C7 = {49/58/67}, no 1,2,3
s) R9C12 = {17/26/35}, no 4,8,9
t) R9C34 = {49/58/67}, no 1,2,3
u) 8(3) cage in N9 = {125/134}
v) 26(4) cage at R3C1 = {2789/3689/4589/4679/5678}, no 1
Steps resulting from Prelims
1a. Naked pair {79} in R67C8, locked for C8, clean-up: no 6,8 in R1C9, no 2 in R4C9, no 4,6 in R5C9
1b. 8(3) cage in N9 = {125/134}, 1 locked for N9, clean-up: no 8 in R6C7
2. 45 rule on N1 1 outie R4C1 = 1 innie R1C3 + 2
2a. R1C3 = {1245} -> R4C1 = {3467}
3. 45 rule on N69 2 innies R45C7 = 9 = {18/27/36/45}, no 9
3a. 9 in R4 locked in R4C456, locked for N5, clean-up: no 1 in R5C3
4. 45 rule on R1234 1 innie R4C6 = 1 outie R5C5 + 3, no 1,2,3 in R4C6, no 7,8 in R5C5
5. R5C5 “sees” all cells of the 45(9) cage except for R7C6 -> R5C5 = R7C6, no 7,8,9 in R7C6
5a. R4C5 “sees” all cells of the 45(9) cage except for R5C7 and R7C6, but R5C5 = R7C6 -> R4C5 = R5C7, no 9 in R4C5
6. R45C7 = 9 (step 3), R4C5 = R5C7 (step 5a) -> R4C57 = 9
6a. 9 in R4 locked in R4C46
6b. 45 rule on R4 3 remaining innies R4C146 = 18 = {279/369/459} (cannot be {189} because R4C1 only contains 3,4,6,7), no 1,8, clean-up: no 5 in R5C5 (step 4), no 5 in R7C6 (step 5)
6c. 4,7 of {279/459} must be in R4C1, no 4,7 in R4C46, clean-up: no 1,4 in R5C5 (step 4), no 1,4 in R7C6 (step 5)
7. 45 rule on N3 3 innies R123C7 = 14 = {149/158/239/248/257/347} (cannot be {167} which clashes with R1C89, cannot be {356} which clashes with R23C8), no 6
7a. 45 rule on N3 1 outie R1C6 = 1 innie R3C7, no 6 in R1C6
[Now step 1b of Ronnie’s Smooth A180 walkthrough, expressed slightly differently.]
8. R1C89 = [69/87]
8a. R5C89 = [49/58/85] (cannot be [67] which clashes with the permutations for R1C89 in either C8 or C9), no 6,7
8b. R4C89 = {18/36}/[27] (cannot be {45} which clashes with R5C89), no 4,5
8c. R45C7 (step 3) = {18/27/36} (cannot be {45} which clashes with R5C89), no 4,5, clean-up: no 4,5 in R4C5 (step 5a)
8d. 4 in R4 locked in R4C123, locked for N4, clean-up: no 6 in R5C12, no 6 in R5C4, no 6 in R6C12
8e. 4 in C1 locked in R1234C1, CPE no 4 in R3C23
9. 45 rule on R5 3 innies R5C567 = 12 = {156/237/246} (cannot be {138} which clashes with R5C12, cannot be {147} because R5C5 only contains 2,3,6, cannot be {345} which clashes with R5C89), no 8, clean-up: no 8 in R4C5 (step 5a), no 1 in R4C7 (step 3)
9a. 4,5 of {156/246} must be in R5C6 -> no 1,6 in R5C6
10. Hidden killer pair 4,5 in R5C89 and R6C79 for N6, R5C89 contains one of 4,5 -> R6C79 must contain one of 4,5
10a. 45 rule of N9 3 outies R6C789 = 14 = {149/257/347} (cannot be {158/248/356} because R6C8 only contains 7,9, cannot be {167/239} which don’t contain 4 or 5), R6C79 = {12345}, clean-up: no 2,3 in R7C7
10b. 6 in N6 locked in R45C7 or R4C89 -> 3 locked in R45C7 or R4C89 (9(2) locking cages), locked for N6, clean-up: no 6 in R7C7
11. 6 in C7 locked in R4589C7
11a. R45C7 = {18/27/36}, R89C7 = {49/58/67} -> combined cage R4589C7 = {1867/3649/3658} (other combinations don’t contain 6) -> no 2,7 in R45C7, clean-up: no 2,7 in R4C5 (step 5a)
12. R5C567 (step 9) = {156/237/246}
12a. 3 of {237} must be R5C7 -> no 3 in R5C56, clean-up: no 3 in R7C6 (step 5)
12b. 4,7 of {237/246} must be in R5C6 -> no 2 in R5C6
13. 45 rule on N7 3 innies R789C3 = 18 = {279/378/459/468} (cannot be {189/369/567} which clash with R78C1), no 1
14. 10(2) cages in N4 = {19/28/37} -> combined cage R56C12 = {1928/1937/2837}
14a. 45 rule on N4 3 innies R4C1 + R56C3 = 16 = {169/268/367/457} (cannot be {178/259/349/358} which clash with R56C12)
14b. 1 of {169} must be in R6C3 -> no 9 in R6C3
14c. R4C23 = {36/45} (cannot be {18/27} which clash with R56C12)
15. 8 in R4 locked in R4C789, locked for N6, clean-up: no 5 in R5C89
15a. R5C89 = [49], R1C9 = 7, R1C8 = 8, R67C8 = [79], clean-up: no 5,6 in R2C5, no 3 in R23C8, no 1,2 in R23C9, no 2 in R4C8, no 1 in R4C9, no 1 in R5C12, no 6 in R5C3, no 1 in R5C4, no 3 in R6C12, no 5 in R7C7, no 5 in R8C1, no 4 in R89C7
15b. 9 in C7 locked in R123C7 -> R123C7 (step 7) = {149/239}, no 5, clean-up: no 5 in R1C6 (step 7a)
16. Naked quad {2378} in R5C1234, locked for R5 -> R5C567 = [651], R4C6 = 9, R4C5 = 1 (step 5a), R7C6 = 6 (step 5), clean-up: no 4 in R123C7 (step 15b), no 1,4 in R1C6 (step 7a), no 7 in R2C5, no 9 in R3C7 (step 7a), no 8 in R4C9, no 8 in R7C7, no 8 in R8C1, no 3,4 in R89C6, no 7 in R9C3
17. Naked pair {36} in R4C89, locked for R4 -> R4C4 = 2, R4C1 = 7 (step 6b), R1C3 = 5 (step 2), R1C4 = 1, R4C23 = [54], R4C7 = 8, clean-up: no 2 in R1C1, no 2,6 in R2C1, no 8 in R2C5, no 3 in R5C12, no 5 in R89C7, no 3 in R9C1, no 1 in R9C2, no 8,9 in R9C4
18. Naked triple {239} in R123C7, locked for C7 and N3 -> R6C7 = 5, R7C7 = 4, R6C9 = 2, clean-up: no 5 in R23C8, no 6 in R23C9, no 1 in R8C2
18a. Naked pair {28} in R5C12, locked for R5 and N4 -> R5C34 = [37]
18b. R6C3 = 6 (hidden single in R6)
[I’ll detour slightly to look at larger cages and it proves to be very helpful.]
19. R4C1 = 7 -> 26(4) cage at R3C1 = {2789/4679}, no 3, 9 locked for R3 and N1
20. 32(7) cage at R2C6 = {1235678} (only remaining combination) -> R3C5 = 5, 7 locked for C6, clean-up: no 2 in R89C6
20a. R23C9 = [54], clean-up: no 6 in R3C12 (step 19)
20b. Naked triple {289} in R3C123, locked for R3 and N1 -> R3C67 = [73], R2C6 = 2, R1C6 = 3, R12C7 = [29], R12C5 = [94], R23C4 = [86], R23C8 = [61], R4C89 = [36]
21. R9C9 = 1 (hidden single in C9), R89C6 = [18], R6C6 = 4, R6C4 = 3, R7C3 = 7 (cage sum)
and the rest is naked singles.