I took part in the "tag" solution although I only played a minor part with one significant contribution. That was only about 3 weeks after I joined this site and I was still a Newbie to solving Assassins level killers.
Since I've been working on puzzles in my Unfinished solver I decided to have another go at Uluru, this time as a solo effort.
This time I found a completely different way to solve it. I spotted the first two lines of step 13 fairly early but it took me a few steps before I was in a position to use it.
I'll rate my walkthrough for Uluru at 1.5 because of my permutation analysis in step 13c.
Here is my walkthrough for Uluru. I've included a comment for SudokuEd, who has already seen my walkthrough, plus a couple of comments after I looked at the SudokuSolver log which Ed kindly provided. Please see my next message for an interesting step from the SS log.
Prelims
a) R45C9 = {29/38/47/56}, no 1
b) R56C8 = {29/38/47/56}, no 1
c) R67C9 = {18/27/36/45}, no 9
d) R8C23 = {59/68}
e) 22(3) cage in N1 = {589/679}
f) 22(3) cage in N3 = {589/679}
g) 9(3) cage in N3 = {126/135/234}, no 7,8,9
h) 11(3) cage at R4C3 = {128/137/146/236/245}, no 9
i) 11(3) cage at R5C1 = {128/137/146/236/245}, no 9
j) 21(3) cage in N8 = {489/579/678}, no 1,2,3
k) 14(4) cage at R3C3 = {1238/1247/1256/1346/2345}, no 9
l) 14(4) cage at R7C6 = {1238/1247/1256/1346/2345}, no 9
Steps resulting from Prelims
1a. 22(3) cage in N1 = {589/679}, 9 locked for N1
1b. 22(3) cage in N3 = {589/679}, 9 locked for N3
2. 45 rule on C1 2 outies R19C2 = 4 = {13}, locked for C2
2a. 17(3) cage in N7 = {179/359/368} (only combinations which contain 1 or 3), no 2,4
2b. R9C2 = {13} -> no 1,3 in R89C1
2c. Killer pair 8,9 in 17(3) cage and R8C23, locked for N7
[There was also Killer pair 6,9 in 17(3) cage and R8C23, locked for N7 but I didn’t spot that.]
3. 21(5) cage at R1C3 = {12459/12468/12567/23457} (other combinations clash with R1C2), 2 locked for R1
3a. Killer pair 1,3 in R1C2 and 21(5) cage at R1C3, locked for R1
3b. 21(5) cage at R9C3 = {12459/12468/12567/23457} (other combinations clash with R9C2), 2 locked for R9
3c. Killer pair 1,3 in R9C2 and 21(5) cage at R9C3, locked for R9
4. 45 rule on R9 2 outies R8C19 = 9 = [54/63/72/81], no 9, no 5,6,7,8 in R8C9
4a. 17(3) cage in N7 (step 2a) = {179/359/368}
4b. 9 of {179/359} must be in R9C1 => no 5,7 in R9C1
4c. 16(3) cage in N9 = {169/178/259/268/349/358/367/457}
5. 45 rule on R89 2 outies R7C56 = 13 = {58/67}/[94], no 4 in R7C5, no 1,2,3 in R7C6
6. 45 rule on C9 2 outies R19C8 = 1 innie R3C9 + 13
6a. Min R19C8 = 14, no 4 in R9C8
6b. Max R19C8 = 17 -> max R3C9 = 4
7. 45 rule on C12 3 outies R368C3 = 1 innie R7C2 + 18
7a. Min R7C2 = 2 => min R368C3 = 20 -> no 1,2 in R6C3
7b. Max R368C3 = 24 -> max R7C2 = 6
8. 45 rule on N69 3 innies R8C78 + R9C7 = 11 = {128/137/146/236/245}, no 9
8a. 45 rule on N69 2 outies R78C6 = 1 innie R9C7 + 3
8b. Min R78C6 = 5 -> min R9C7 = 2
[At this stage I missed 9 in C7 only in R4567C7, locked for 32(6) cage, no 9 in R47C8. However I don’t think this made much difference to my solving path; these eliminations are made in steps 15 and 17.]
9. 45 rule on N7 4 innies R7C123 + R9C3 = 14 = {1247/2345} (cannot be {1256} which clashes with R8C23, cannot be {1346} which clashes with R9C2), no 6
10. 45 rule on N4 2(1+1) outies R4C4 + R7C1 = 1 innie R4C1
10a. Min R4C4 + R7C1 = 2 -> min R4C1 = 2
[Ed pointed out that R4C4 and R7C1 cannot both be 1, because there would be no place left for 1 in N4, so min R4C1 = 3. I got this result a different way in the next step.]
11. 45 rule on N1 2 innies R12C3 = 1 outie R4C1 + 2
11a. Min R12C3 = 5 (cannot be {12} because no 1 in R4C1, cannot be {13} which clashes with R1C2) -> min R4C1 = 3
12. 45 rule on R9 4 innies R9C1289 = 24 = {1689/3489/3579/3678} (other combinations don’t contain 1 or 3 for R9C2)
12a. 9 of {1689} must be in R9C1 (17(3) cage in N7 cannot be [881]), 4 of {3489} must be in R9C9, 9 of {3579} must be in R9C1 -> no 9 in R9C9
13. 45 rule on R19+C19 (counting corner cells twice) 5 innies R19C19 + R3C9 = 31
[Note than R19C19 "see" each other because this is a Killer-X so they must all be different.]
13a. R3C9 = {1234} -> R19C19 = 27,28,29,30 must contain 9 which is only in R1C9 + R9C1, locked for D/, clean-up: no 5 in R8C3
13b. R3C9 + R19C19 = 1{6789}/2{5789}/3{4789/5689}/4{4689/5679}
13c. 3{4789}/4{4689} must have the 4 for R19C19 in R1C1 (16(3) cage in N9 (step 4c) cannot be {349} when 3 in R3C9 and 4 of {457} must be in R8C9) -> no 4 in R9C9
14. R9C1289 (step 12) = {1689/3579/3678}
14a. 9 of {1689} must be in R9C1 (17(3) cage in N7 cannot be [881]), 9 of {3579} must be in R9C1 -> no 9 in R9C8
15. 9 in N9 only in R7C78, locked for R7 and 32(6) cage at R4C7, clean-up: no 4 in R7C6 (step 5)
16. 9 in N6 only in the two 11(2) cages -> one of the 11(2) cages must be {29}, locked for N6, clean-up: no 7 in R7C9
17. R7C7 = 9 (hidden single in C7), placed for D\
17a. R3C2 = 9 (hidden single in N1)
18. 45 rule on N6 2 remaining outies R7C89 = 9 = {18/36/45}/[72], no 2 in R7C8
19. 14(4) cage at R7C6 = {1238/1247/1256/1346/2345}
19a. 7,8 of {1238/1247} must be in R7C6 -> no 7,8 in R8C678
[Even though I used 45s on N69 in step 8, I’ve only now spotted ones which use larger groups of nonets.]
20. 45 rule on N689 1 outie R9C3 = 1 innie R7C4, no 6,8 in R7C4
21. 45 rule on N6789 2 outies R56C4 = 1 innie R7C1 + 12
21a. Min R56C4 = 13, no 1,2,3, no 4 in R5C4
21b. Max R56C4 = 17 -> max R7C1 = 5
22. 45 rule on N6789 4 innies R7C1234 = 14 = {1247/2345}, 2,4 locked for R7, clean-up: no 5,7 in R7C8 (step 18), no 5 in R7C9 (step 18), no 4,5,7 in R6C9
22a. 7 in N9 only in R9C789, locked for R9, clean-up: no 7 in R7C4 (step 20)
23. R45C9 = {29/47/56} (cannot be {38} which clashes with R67C9), no 3,8
24. Hidden killer quad 1,2,3,4 in R3C9, R45C9, R67C9 and R8C9 for C9, R3C9 = {1234}, R67C9 contains one of 1,3, R8C9 = {1234} -> R45C9 must contain one of 2,4
24a. R45C9 (step 23) = {29/47} (cannot be {56} which doesn’t contain 2 or 4), no 5,6
25. 16(3) cage in N9 = {178/358/367/457} (cannot be {268} which clashes with R7C89), no 2, clean-up: no 7 in R8C1 (step 4)
26. R7C3 = 7 (hidden single in N7), placed for D/, clean-up: no 6 in R7C56 (step 5)
27. Naked pair {58} in R7C56, locked for R7 and N8, clean-up: no 1 in R7C89 (step 18), no 1,8 in R6C9, no 5 in R9C3 (step 20)
27a. Naked pair {36} in R7C89, locked for R7 and N9, clean-up: no 6 in R8C1 (step 4), no 3 in R9C3 (step 20)
27b. Naked pair {36} in R67C9, locked for C9
28. 16(3) cage in N9 (step 25) = {178/457}, 7 locked for N9
29. 17(3) cage in N7 (step 2a) = {359/368} -> R9C2 = 3, R1C2 = 1
29a. R89C1 = [59/86], no 8 in R9C1
30. R8C6 = 3 (hidden single in N8)
30a. 14(4) cage at R7C6 (step 19) = {1238/2345}, 2 locked for N9
30b. R7C6 = {58} -> no 5 in R8C78
30c. Naked triple {124} in R8C789, locked for R8 and N9
31. 45 rule on N3 3 innies R123C7 = 14 = {167/248/347} (cannot be {158} which clashes with R9C7, cannot be {257/356} which clash with 22(3) cage), no 5
32. 21(5) cage at R1C3 (step 3) = {23457} (only remaining combination), locked for R1
32a. 22(3) cage in N3 = {589/679}
32b. 5,7 only in R2C9 -> R2C9 = {57}
33. Killer pair 6,8 in R1C1 and 22(3) cage, locked for N1
34. R89C1 (step 29a) = [59] (cannot be [86] which clashes with R1C1) -> R8C1 = 5, R9C1 = 9, placed for D/, R1C9 = 8, placed for D/, R1C1 = 6, placed for D\, R8C2 = 6, placed for D/, R8C3 = 8, R1C8 = 9, R2C9 = 5 (step 32b), R9C9 = 7, placed for D\, R2C2 = 8, R3C3 = 5, both placed for D\, clean-up: no 4 in R45C9
34a. Naked pair {29} in R45C9, locked for C9 and N6
35. R19C19 = [6897] = 30 -> R3C9 = 1 (step 13b), R8C9 = 4, R9C8 = 5 (step 28), R9C7 = 8, clean-up: no 6 in R56C8
35a. R3C9 = 1 -> R23C8 = 8 = {26} -> R2C8 = 2, placed for D/, R3C8 = 6, R7C8 = 3, R67C9 = [36], R8C8 = 1, placed for D\, R8C7 = 2, R7C6 = 8 (step 30a), R7C5 = 5, clean-up: no 8 in R56C8
36. Naked pair {47} in R56C8, locked for N6 -> R4C8 = 8
37. 1,8 in C1 only in 11(3) cage at R5C1 = {128} (only remaining combination), locked for C1
38. R6C3 = 9 (hidden single in C3), R456C2 = 14 = {257} (only remaining combination), locked for C2 and N4 -> R7C2 = 4, R6C4 = 5
39. R7C1 = 2 (hidden single in C1), R7C4 = 1, R5C4 = 9 (cage sum), R45C9 = [92], R8C45 = [79], R9C3 = 1
40. 11(3) cage at R4C3 = {236} (only remaining combination) -> R4C4 = 2, R45C3 = {36}, locked for C3 and N4, R2C3 = 4, R1C3 = 2, R4C1 = 4, R6C6 = 4, R5C5 = 3, placed for D/
and the rest is naked singles.
Ruudiculous tag Killer - Uluru
Thanks Ed for the SSv3.3 log for Uluru. It used an interesting breakthrough step. Here is the position after step 100.
First SS found
101. Conjugate pair r2c9=5=r9c9 in c9
101a. Candidate 5 removed from r2c2
101b. Cage sum in cage 22(3) n1 - removed 8 from r3c3
which in my words is 5 in c9 only in r29c9 -> no 5 in r2c2 (using the diagonal), then no 8 in r3c3
Next SS found the interesting
102. X-Cycle on candidate 5 at r2c9=r3c8 - r3c3=r9c9
102a. Removed candidate 5 from r3c1456
A complicated step, possibly some sort of "Fish". In my words it seems to be
5 in c9 only in r29c9, 5 in n3 only in r2c9+r3c8, 5 in d\ only in r3c3+r9c9 -> 5 in r3 only in r3c38, locked for r3
Code: Select all
.-------------------------------.-------------------------------.-------------------------------.
| 68 1 23457 | 23457 23457 2457 | 2347 689 89 |
| 234578 5678 234578 | 3456789 123456789 12456789 | 1234678 1234 57 |
| 234578 9 5678 | 12345678 12345678 1245678 | 12347 23456 124 |
:-------------------------------+-------------------------------+-------------------------------:
| 23456789 245678 12345678 | 1234 56789 1247 | 1345678 1345678 2479 |
| 1234678 245678 1245678 | 5689 1234 1245678 | 1345678 23456789 2479 |
| 1234678 245678 123456789 | 5689 56789 124 | 1345678 23456789 36 |
:-------------------------------+-------------------------------+-------------------------------:
| 12 247 1247 | 124 58 58 | 9 36 36 |
| 58 568 689 | 679 679 3 | 124 124 14 |
| 69 3 124 | 12469 12469 12469 | 58 578 578 |
'-------------------------------.-------------------------------.-------------------------------'
101. Conjugate pair r2c9=5=r9c9 in c9
101a. Candidate 5 removed from r2c2
101b. Cage sum in cage 22(3) n1 - removed 8 from r3c3
which in my words is 5 in c9 only in r29c9 -> no 5 in r2c2 (using the diagonal), then no 8 in r3c3
Next SS found the interesting
102. X-Cycle on candidate 5 at r2c9=r3c8 - r3c3=r9c9
102a. Removed candidate 5 from r3c1456
A complicated step, possibly some sort of "Fish". In my words it seems to be
5 in c9 only in r29c9, 5 in n3 only in r2c9+r3c8, 5 in d\ only in r3c3+r9c9 -> 5 in r3 only in r3c38, locked for r3