Continuing working through my backlog, the next variant I tried was A110X V2 which was another one which I found that I hadn't started.
Thanks Ed for another interesting and challenging variant. You have a knack for making good variants by just combining two cages, in this case the 10(3) cage at R5C1 and the 11(2) cage at R6C1.
Ed wrote:
mhparker wrote:
After what was probably about 30 steps
Hope you kept your notes! This "old school" V2 has the same solution as the original, but takes out the "easy" opening (Afmob's step 3a).
It might have helped if I'd looked to see how I'd solved the original but I didn't need to; I solved this as a new puzzle.
udosuk wrote:
Here is my 5-step walkthrough for V2. I haven't read the others' moves but I suspect the critical ones might be similar.
When I went through the walkthroughs posted by Afmob, Mike and udosuk I found that they had all used critical moves based on the innie-outie difference for R8. I therefore found that, even though there were already 3 posted walkthroughs, mine was significantly different because I'd used a different innie-outie difference and related combination analysis.
Edit. Then when I had to re-work my later steps, see below, I found that I needed to use the innie-outie difference for R8 so it appears to be the critical step for this puzzle.Here is my walkthrough for A110X V2.
Thanks Afmob for pointing out that my original step 16 was flawed. I've re-worked from after step 15 and also made a few minor changes to earlier steps. Then thanks again to Afmob for finding that my new step 21 was flawed. I've found an alternative way to do that step.Prelims
a) R1C23 = {49/58/67}, no 1,2,3
b) R1C67 = {19/28/37/46}, no 5
c) R1C89 = {17/26/35}, no 4,8,9
d) R2C23 = {18/27/36/45}, no 9
e) R2C67 = {18/27/36/45}, no 9
f) R2C89 = {29/38/47/56}, no 1
g) R8C12 = {16/25/34}, no 7,8,9
h) R8C78 = {49/58/67}, no 1,2,3
i) R9C12 = {79}
j)R9C34 = {59/68}
k) R9C78 = {17/26/35}, no 4,8,9
l) 19(3) cage in N5 = {289/379/469/478/568}, no 1
Steps resulting from Prelims
1a. Naked pair {79} in R9C12, locked for R9 and N7, clean-up: no 5 in R9C34, no 1 in R9C78
1b. Naked pair {68} in R9C34, locked for R9, clean-up: no 2 in R9C78
1c. Naked pair {35} in R9C78, locked for R9 and N9, clean-up: no 8 in R8C78
2. 45 rule on N14 2 outies R7C12 = 5 = {14/23}
2a. R8C12 = {16/25} (cannot be {34} which clashes with R7C12), no 3,4
2b. Killer pair 1,2 in R7C12 and R8C12, locked for N7
2c. 8 in N7 only in R789C3, locked for C3, clean-up: no 5 in R1C2, no 1 in R2C2
3. 45 rule on N69 2 outies R3C89 = 10 = {19/28/37/46}, no 5
4. 45 rule on N5 3 innies R4C6 + R5C5 + R6C4 = 8 = {125/134}, 1 locked for N5, D/ and 35(7) cage at R3C6, clean-up: no 7 in R1C8, no 6 in R8C1
4a. 45 rule on N2369 2 innies R3C67 = 14 = {59/68}
4b. 45 rule on N1478 2 innies R7C34 = 13 = [49/67] (cannot be {58} which clashes with R3C67), no 1,2,3,5,8, no 4,6 in R7C4
4c. 35(7) cage at R3C6 = {125}{68}[49]/{134}
{59}[67](combining combinations in steps 4, 4a and 4b)
4d. 4 only in R4C6 + R5C5 + R6C4 + R7C3, locked for
D/, clean-up: no 7 in R2C9
4e. 9 only in R3C67 + R7C4, CPE no 9 in R3C4
4f. 5 in R7 only in R7C56, locked for N8
4g. 8 on D/ only in R2C8 + R3C7, locked for N3, clean-up: no 2 in R1C6, no 1 in R2C6, no 3 in R2C8, no 2 in R3C89 (step 3)
5. 45 rule on C123 3 innies R789C3 = 17 = {368/458}
5a. 3,5 only in R8C3 -> R8C3 = {35}
5b. R9C3 = 8 (hidden single in C3), R9C4 = 6
5c. 4 in N7 only in R7C123, locked for R7
5d. 6 in N7 only in R7C3 + R8C2, locked for D/, clean-up: no 2 in R1C8, no 5 in R2C9, no 8 in R3C6 (step 4a)
5e. 6 in 35(7) cage at R3C6 only in R3C6 + R7C3, CPE no 6 in R3C3
6. 45 rule on N9 2 innies R7C78 = 1 outie R6C9 + 2
6a. Min R7C78 = {16} = 7 (cannot be {12} which clashes with R7C12) -> min R6C9 = 5
6b. Max R7C78 = 11 -> R7C78 must contain one of 1,2
6c. Killer pair 1,2 in R7C12 and R7C78, locked for R7
7. 45 rule on C1 2 innies R89C1 = 2 outies R5C23 + 7
7a. Max R89C1 = 14 -> max R5C23 = 7, no 7,8,9
8. 45 rule on C789 3 innies R123C7 = 16 = {169/178/259/268/457} (cannot be {349} which clashes with R3C89, cannot be {358} which clashes with R9C7, cannot be {367} because no 3,6,7 in R3C7), no 3, clean-up: no 7 in R1C6, no 6 in R2C6
9. 17(4) cage in N8 = {1259/1349/1457/2348} (cannot be {1358/2357} because R9C56 must contain two of 1,2,4)
9a. 17(4) cage only contains two of 1,2,4 -> no 1,2,4 in R8C6
9b. 5 of {1457} must be in R7C6 -> no 7 in R7C6
10. Hidden killer pair 3,5 in R78C6 and R7C5 + R8C45 for N8, R78C6 contains one of 3,5 -> R7C5 + R8C45 must contain one of 3,5 -> 18(4) cage at R7C5 must contain both of 3,5 = {1359/2358}, no 4,7, CPE no 3 in R8C6
10a. 3 in R8 only in R8C345, locked for 18(4) cage, no 3 in R7C5
10b. Hidden killer pair 3,5 in R7C5 + R8C45 and R7C6 for N8, R7C5 + R8C45 contains one of 3,5 -> R7C6 = {35}
11. 4 in N8 only in R9C56, locked for R9
11a. Killer pair 1,2 in R7C78 and R9C9, locked for N9
12. 45 rule on N8 4 remaining innies R78C45 = 22 = {1579/2389/2578}
12a. 5 of {1579/2578} must be in R7C5, 9 of {2389} must be in R7C4 -> no 9 in R7C5
13. 45 rule on R89 2 outies R7C56 = 2 innies R89C9 + 3
13a. R7C56 = [53/85] = 8,13 -> R89C9 = 5,10 = [41/82/91], no 6,7 in R8C9
14. 5,8 in R7 only in R7C56789
14a. 45 rule on R7 5 remaining innies R7C56789 = 27 = {15678/23589}, R7C56 = [53/85] -> R7C789 = {167/289} = 14,19
14b. R7C78 = {16/17/28/29} = 7,8,10,11 -> R6C9 (step 6) = {5689}, no 7
15. 22(4) cage at R6C9 = {1489/1678/2578} (cannot be {4567} which doesn’t contain one of 1,2, cannot be {1579/2479/2569} which clash with R8C78)
, 8 locked for C915a. 1,4 of {1489} must be in R89C9 -> no 9 in R8C9
15b. 7 of {1678} must be in R7C9 -> no 6 in R7C9
Re-worked from here16. R7C789 = {167/289} (step 14a)
16a. 22(4) cage at R6C9 = {1489/2578} (cannot be {1678} which clashes with R7C789, CCC), no 6
16b. R6C9 = {589} -> R7C78 (step 6) = 7,10,11 = {16/28/29} (step 14b), no 7
[Looks like I need to use the same step as the other walkthroughs used although it’s simpler here after my earlier work.]
17. 45 rule on R8 2 innies R8C69 = 1 outie R7C5 + 7
17a. R7C5 = {58} -> R8C69 = 12,15 = [84/78], no 9, 8 locked for R8
18. 4 in N8 only in 17(4) cage (step 9) = {1457/2348} -> R78C6 = [38/57]
18a. 45 rule on N2 3 innies R123C6 = 17 = {179/269/368/467} (cannot be {278} because no 2,7,8 in R3C6, cannot be {359} which clashes with R7C6, cannot be {458} which clashes with R78C6), no 5, clean-up: no 4 in R2C7, no 9 in R3C7
19. 35(7) cage at R3C6 (step 4c) = {125}[68][49]/{134}[95][67], 5 locked for D/, clean-up: no 3 in R1C8, no 6 in R2C9, no 2 in R8C1
19a. Killer pair 2,6 in 35(7) cage at R3C6 and R8C2, locked for D/, clean-up: no 6 in R1C8, no 9 in R2C9
19b. R3C89 (step 3) = {19/46} (cannot be {37} which clashes with R1C9), no 3,7
19c. Killer pair 6,9 in R3C6 and R3C89, locked for R3
19d. 3 in N3 only in R12C9, locked for C9
19e. 5 in C9 only in R456C9, locked for N6
20. R1C67 = {19/46}/[82] (cannot be [37] which clashes with R1C9), no 3,7
20a. R123C7 (step 8) = {178/259/268} (cannot be {169} because R3C7 only contains 5,8, cannot be {457} which clashes with R1C89), no 4, clean-up: no 6 in R1C6
20b. R3C7 = {58} -> no 5 in R2C7, clean-up: no 4 in R2C6
20c. 7 of {178} must be in R2C7 -> no 1 in R2C7, clean-up: no 8 in R2C6
[Then I had to do further re-work, in this case only step 21, because my original step 21 was flawed.]
21. R2C67 = [27/36/72], R2C89 = [74/83/92] -> combined cage R2C6789 = [2783/3674/3692/7283]
21a. R123C7 (step 20a) = {259/268} (cannot be {178} = [178] which clashes with R2C6789 = [2783], CCC), no 1,7, 2 locked for C7 and N3, clean-up: no 9 in R1C6, no 2 in R2C6, no 9 in R2C8
[Having found that, I saw that Afmob used the simpler
9 on D/ only in R2C8 + R9C1, CPE no 9 in R2C1
9 in N1 only in R1C123, locked for R1 ...]22. R9C1 = 9 (hidden single in D/), R9C2 = 7, clean-up: no 6 in R1C3, no 2 in R2C3
23. 9 in N1 only in R1C23 = {49}, locked for R1 and N1, clean-up: no 1 in R1C6, no 6 in R1C7, no 5 in R2C23
23a. R1C67 = [82], R2C7 = 6, R2C6 = 3, R2C9 = 4, R2C8 = 7, R1C9 = 3, placed for D/, R1C8 = 5, R3C7 = 8, R3C6 = 6 (step 4a), R78C6 = [57], R7C34 = [49], R1C23 = [49], R7C5 = 8, R7C9 = 7, R8C9 = 8, R7C7 = 1, R9C9 = 2, both placed for D\, R6C9 = 5 (step 16a), R7C8 = 6, R9C78 = [53], R2C23 = [81]
24. Naked pair {23} in R7C12, locked for N7 -> R8C3 = 5, R8C12 = [16]
25. R5C5 = 5, naked pair {12} in R4C6 + R6C4, locked for N5
26. Naked pair {17} in R1C45, locked for R1 and N2, R2C4 = 5 (cage sum)
27. Naked triple {169} in R345C9, locked for 23(5) cage at R3C9
27a. R345C9 = {169} = 16 -> R5C78 = 7 = [34], R56C6 = [94], R6C5 = 6 (cage sum)
and the rest is naked singles.