mhparker wrote:Vortex Lite Est. rating 1.25
and
Come on guys, the Vortex Lite is really no more difficult than a typical Assassin!
I'll agree with that although at one time I was struggling a bit. Then I had several restarts when I saw things that I ought to have spotted earlier and it became easier.
As Ruud said, a fun puzzle. Thanks Mike!
Here is my walkthrough for Vortex Lite. No "advanced techniques". Step 34 was combination crunching but not heavy stuff, the rest was normal killer moves.
Thanks Mike for the feedback on step 34. I've also done a bit of minor editing.
1. R1C45 = {17/26/35}, no 4,8,9
2. R45C9 = {29/38/47/56}, no 1
3. R56C1 = {16/25/34}, no 7,8,9
4. R9C56 = {69/78}
5. 20(3) cage in N2 = {389/479/569/578}, no 1,2
6. 11(3) cage at R4C7 = {128/137/146/236/245}, no 9
7. 21(3) cage at R5C3 = {489/579/678}, no 1,2,3
8. 19(3) cage at R6C5 = {289/379/469/478/568}, no 1
9. 22(3) cage at R7C7 = 9{58/67}, 9 locked for N9 and D\
10. 28(4) cage at R2C9 = {4789/5689} = 89{47/56}
10a. CPE no 8,9 in R12C8
11. 13(4) cage at R8C6 = {1237/1246/1345} = 1{237/246/345}, no 8,9
11a. CPE no 1 in R8C9
After the preliminary steps the first placement
12. 45 rule on D/ + D\, R5C5 counts toward both diagonals, total 89 -> R5C5 = 1, locked for D/ and D\, clean-up: no 7 in R1C4, no 6 in R6C1
13. 11(3) cage at R4C7 (step 6) = {128/137/146/236/245}
13a.1 of {128/137} must be in R4C7 -> no 7,8 in R4C7
14. 45 rule on D/ 2 outies R4C4 + R6C6 = 7 = {25/34}
15. 45 rule on D\ 2 outies R4C6 + R6C4 = 15 = {69/78}
16. 12(3) cage in N3 = {237/246/345}, no 8,9
17. 15(3) cage in N1 = {258/267/348} (cannot be {357/456} which clash with R4C4 + R6C6)
18. 45 rule on N1 2 outies R2C4 + R4C1 = 10 = {19/28/37/46/55}
19. 45 rule on N3 2 outies R1C6 + R4C8 = 13 = {49/58/67}, no 1,2,3 in R1C6
20. R8C6 = 1 (hidden single in C6)
20a. 1 in N9 locked in R7C89, locked for R7 and 15(4) cage -> no 1 in R6C9
20b. R7C9 = 1 (hidden single in C9)
[It’s neat the way that fixing R8C6 leads directly to fixing R7C9!]
21. 45 rule on N7 2 outies R6C2 + R9C4 = 4 = [13/22]
22. 45 rule on N9 1 remaining outie R6C9 = 4, clean-up: no 9 in R1C6 (step 19), no 3 in R4C4 (step 14), no 7 in R45C9, no 3 in R5C1
22a. 15(4) cage at R6C9 = 14{28/37}, no 5,6
22b. 4 in N9 locked in 13(4) cage at R8C6 = 14{26/35}, no 7
[Alternatively killer pair 7,8 in R7C8 + R8C9 and 22(3) cage, locked for N9]
23. 28(4) cage at R2C9 (step 10) = {4789/5689}
23a. 4 of {4789} must be in R3C8 -> no 7 in R3C8
24. 11(3) cage at R4C7 (step 6) = {128/137/146/236/245}
24a. 4 of {245} must be in R5C6 -> no 5 in R5C6
25. 17(3) cage in N7 = {269/278/359/458} (cannot be {368/467} which clash with R4C6 + R6C4)
26. 45 rule on R123 3 outies R4C158 = 20 = {389/479/569/578}, no 1,2, clean-up: no 8,9 in R2C4 (step 18)
27. 45 rule on R789 2 remaining outies R6C25 = 9, R6C2 = {12} -> R6C5 = {78}
28. 45 rule on C123 3 outies R259C4 = 11 = {128/137/236/245} (cannot be {146} because R9C4 only contains 2,3), no 9
28a. 7 of {137} must be in R5C4 -> no 7 in R2C4
28b. 6 of {236} must be in R5C4 -> no 6 in R2C4
28c. Clean-up: no 3,4 in R4C1 (step 18)
29. 45 rule on C789 2 remaining outies R15C6 = 9 = [54/63/72], clean-up: no 5,9 in R4C8 (step 19)
30. 28(4) cage at R2C9 (step 10) = 89{47/56}, 9 locked in R2C9 + R3C89 for N3
30a. 4/5 now in N3 -> 12(3) cage in N3 (step 16) = {237/246} (cannot be {345} which clashes with 28(4) cage) = 2{37/46}, no 5, 2 locked for N3 and D/
31. Killer pair 6,7 in 12(3) cage in N3 and R4C6 + R6C4, locked for D/
32. 5 on D/ locked in 17(3) cage, locked for N7
32a. 17(3) cage (step 25) = {359/458}
[Step 31 wasn’t actually needed but I saw it before I checked for an earlier step with the 17(3) cage.]
33. 45 rule on C1234 6 remaining outies R1348C5 + R46C6 = 25, min R46C6 = 8 -> max R1348C5 = 17, no 9
33a. R1348C5 = 14, 15, 16 or 17 = {2345/2346/2347/2348/2356/2357/2456}, 2 locked for C5
34. 18(4) cage at R1C6 must have 1 in R1C78 + R2C7 = {1368/1458/1467}
34a. If {1368} -> R1C78 + R2C7 = {138} => 12(3) cage = {246} -> R2C9 + R3C89 => {579} clashes with all combinations for 28(4) cage at R2C9
34b. If {1467} with R1C6 = 6 => R1C78 + R2C7 = {147} clashes with 12(3) cage
34c. If {1467} with R1C6 = 7 => R1C78 + R2C7 = {146} => R2C9 + R3C89 => {589} => R4C8 = 6 => no 6 in R1C8
34d. -> 18(4) cage at R1C6 = {1458/1467} = 14{58/67}, no 3, 4 locked for N3, no 6 in R1C6 (steps 34a and 34b) no 6 in R1C8 (step 34c), clean-up: no 6 in 12(3) cage (step 30a), no 7 in R4C8 (step 19) , no 3 in R5C6 (step 29)
See discussion of this after the walkthrough
35. Naked triple {237} in 12(3) cage in N3, locked for D/ and N3, clean-up: no 8 in R4C6 + R6C4 (step 15), no 9 in 17(3) cage in N7 (step 32a)
36. 11(3) cage at R4C7 (step 6) = {128/146/236/245} (cannot be {137} because R5C6 only contains 2,4), no 7
36a. 7 in N6 locked in 15(3) cage = 7{26/35}, no 1,8,9
37. R4C7 = 1 (hidden single in N6)
37a. R5C67 = 10 = [28/46]
38. R1C8 = 1 (hidden single in C8), clean-up: no 7 in R1C5
38a. R9C8 = 4 (hidden single in C8)
39. Naked pair {68} in R4C8 + R5C7, locked for N6, clean-up: no 3,5 in R45C9
40. Naked pair {29} in R45C9, locked for C9 and N6, clean-up: no 8 in R7C8 (step 22a)
41. R3C8 = 9 (hidden single in N3)
42. R7C7 = 9 (hidden single in N7)
43. 28(4) cage at R2C9 = {5689}
43a. 5 locked in R23C9, locked for C9 and N3, clean-up: no 8 in R8C8 (step 9)
44. R4C8 = 8 (hidden single in C8), R5C7 = 6, R5C6 = 4, R1C6 = 5 (step 29), clean-up: no 3 in R1C45, no 2 in R89C7 (step 22b), no 2 in R4C4, no 3 in R6C6 (both step 14) -> R4C4 = 5, R6C6 = 2, 2,5 locked for D\, R6C2 = 1, no 5 in R5C1, no 3 in R6C1 -> R56C1 = [25], R45C9 = [29] , clean-up: no 8 in R9C9 (step 9)
[Clean-up edited]
44a. R5C8 = 5 (hidden single in R5)
44b. R9C3 = 1 (hidden single in R9)
45. R3C7 = 2, R6C7 = 7 (hidden singles in C7), R6C5 = 8, R6C8 = 3, clean-up: no 7 in R9C6
46. R2C8 = 7, R7C8 = 2, R8C8 = 6, R1C9 = 3, R89C9 = [87], 6,7 locked for D\, clean-up: no 8 in R9C6
47. R5C4 = 7, R5C23 = [38], R4C5 = 3 (hidden single in N5)
48. R4C5 = 3 -> R3C45 = 11 = [47]
48a. R7C6 = 7 (hidden single in C6), R7C5 = 4, R7C3 = 5, R8C2 = 4, R9C1 = 8
48b. R8C5 = 5 (hidden single in C5), R89C7 = [35]
49. R7C4 = 8 (hidden single in C4), R8C4 = 2
and the rest is naked singles and a cage sum
4 9 7 6 2 5 8 1 3
6 8 2 1 9 3 4 7 5
1 5 3 4 7 8 2 9 6
9 7 4 5 3 6 1 8 2
2 3 8 7 1 4 6 5 9
5 1 6 9 8 2 7 3 4
3 6 5 8 4 7 9 2 1
7 4 9 2 5 1 3 6 8
8 2 1 3 6 9 5 4 7
Step 34. Mike commented that the logic was too complicated and should be broken down into smaller steps, for example
34. 18(4) cage at R1C6 must have 1 in R1C78 + R2C7 = {1368/1458/1467}
34a. If {1458}, 5 must be in R1C6 -> no 5 in R1C78+R2C7
34b. -> 5 in N3 locked in 28(4) at R2C9 = {5689} (no 4,7) -> no 6 in R12C8.
34c. Hidden killer pair on {47} in N3, as follows. 12(3) at R1C9 contains 1 of {47}. Only other place for {47} in N3 = 18(4) at R1C6 = {(4/7)..} = {1458/1467} (no 3) -> 4 locked in R1C78+R2C7 for N3
34d. Clean-up: no 6 in R1C6 (step 19), no 6 in 12(3) cage (step 30a), no 3 in R5C6 (step 29)
There is a reason why I missed his step 34a, which I ought to have seen. After typing step 34 with "must have 1" I used Ruud's combination calculator and forgot to enter that so I got {1368/1458/1467/3456}. When I was part way through my sub-steps I realised why {3456} wasn't allowed but forgot to go back and look at {1458} again, having already accepted that it was valid.
Mike's step 34c is a difficult one to find. You either spot it immediately or have to work hard to find it.
One can combine thoughts from both sets of sub-steps and come up with
34. 18(4) cage at R1C6 must have 1 in R1C78 + R2C7 = {1368/1458/1467}
34a. If {1368} -> R1C78 + R2C7 = {138} => 12(3) cage = {246} -> R2C9 + R3C89 => {579} clashes with all combinations for 28(4) cage at R2C9
34b. 18(4) cage at R1C6 = {1458/1467}, no 3, 4 locked for N3, clean-up: no 6 in 12(3) cage (step 30a)
34c. 28(4) at R2C9 (step 10) = {5689} (only remaining combination), no 7 -> CPE no 6 in R1C8
34d. Clean-up: no 6 in R1C6 (step 19), no 3 in R5C6 (step 29)
I'll leave it to you to decide whether Mike's step 34c or my step 34a is the easier one to find.