Seems like I've been able to make some progress with some interesting techniques. Can someone check through to make sure its all valid? [EDIT: Rewritten from step 22 + new marks pic. Thanks Mike for finding the mistake. EDIT2: problem with new step 27 and other typos fixed: thanks Para and Mike] Add some more too if you like . I probably won't be able to look at it again till the weekend.
Seems like we need to add Killer LoL to our jargon to account for the effect of cage combinations on LoL. I also use 'overlap' to describe the congruence between combinations. Might need to find a different name so it is not confused with the normal 'overlap' technique. Names and the need for them is not my forte.....
Cheers
Ed
Texas Jigsaw Killer 18
1. 23(3)r3c6 = {689}: all locked for n3
2. "45" r123: r4c6 - 7 = r3c9
2a. r3c9 = {12}
2b. r4c6 = {89}
2c. 6 in n3 only in r3: 6 locked for r3
3. Already resorting to nibbles.
3a. no 1 r2c1. Here's how.
3b. 1 in r2c1 -> r3c12 = {89}
3c. But this clashes with r3c67.
4. Killer LoL (KLoL) chain r123: no 2 r2c1. Here's how.
4a. LoLr123: 4 innies r2c1 + r3c129 = 4 outies r4c456 + r5c5
4b. only one 2 is possible in outies -> only one 2 is possible in innies.
4c. 2 in r2c1 -> r3c12 = {79} -> r4c6 = 9 -> r3c9 = 2 (step 2)
4d. but this means two 2's in r123 innies. Not possible.
5.KLoL r123: no {567} combo in 18(3)r4c6. Here's how.
5a.LoLr123 same as step 4a. Outies must have 8/9 (r4c6) -> innies must have 8/9
5b. ->18(3)r2c1 {567} combo. blocked
6. KLoL r123: no 2 r3c12. Here's how.
6a. 2 in r3c12 -> rest of 18(3) = {79}(no 8)
6b. 2 in r3c12 -> r3c9 = 1 -> r4c6 = 8 (step 2)
6c. But we know from LoLr123 that there can be no 8 in outies when 2 in r3c12.
7. 22(3)r5c3 = 9{58/67}
7a. 9 locked for n6(r5c3)
8. "45" c12: r4c3 + 4 = r6c2
8a. r4c3 = 1..5
9. "45" r89: 3 innies r8c136 = 8 = h8(3)r8
9a. = 1{25/34}
9b. 1 locked for r8
10. 1 in n9(r8c2) only in r9: 1 locked for r9
11. 21(3)r8c8 = {489/579/678} = [4/7..]
12. KLoL r89: no {457} combo in 16(3)r8c7. Here's how.
12a. LoL r89: 6 innies r8c1367 + r9c78 = 6 outies r56c67 + r7c78
12b. and both = 24 ie h24(6)r89innies = 24(6)r89outies
12c. 6 outies must have 1,2 & 3 for n8(r5c6)
12d. -> 6 innies must have 1,2 & 3
12e. note: 6 innies cannot have any repeats since the 6 outies are all in the same nonet.
12f. -> when h8(3)r8 = {125}, 16(3)r8c7 must be 3....
12g. -> when h8(3)r8 = {134}, 16(3) must be 2...
12h. -> {457} blocked from 16(3)
13. KLoL r89: no {358} combo. in 16(3)r8c7
13a. since the 6 outies in LoL r89(step 12) are all in the same nonet, there can be no repeats in the 6 innies
13b. since h8(3)r8 = {125/134} = [3/5..]
13c. -> {358} combo blocked from 16(3)
13d. -> the valid combinations in the h24(6)r89innies are
i. {125/349} = {123459}
ii. {125/367} = {123567}
iii. {134/259} = {123459}
iv. {134/268} = {123468}
13e. = 123{459/468/567} = [4/7..] not both.
14. KLoL r89: {147} combo blocked from both 12(3) cages in n8(r5c6): clash with 21(3)r8c8 = 4/7..] step 11 (thanks Mike for this much easier way).
15. 16(3)r8c7 = {259/268/349/367}
15a. ->{456} blocked from 15(3)r7c5
15b. ->{239/356} blocked from 14(3)r5c4
16. "45" r789: r6c7 + 3 = r7c9
16a. min r7c9 = 4, max r6c7 = 6
17. "45" c1234: r6c5 - 2 = r4c4
17a. max r4c4 = 7, min. r6c5 = 3
18. "45" c789: r135c7 = 19 = h19(3)c7
18a. no 1
19. LoLc789: 4 innies r189c7 + r9c8 = 4 outies r3456c6
19a. no 1 in innies -> no 1 in r56c6 [edit typo]
20. 1 in n8 only in 12(3)r6c7 = {129/138/156}(no 4,7)
21. 12(3)r5c6 = {237/246/345}(no 8,9)
22. KLoL Over-lap c789: no {259} combo. in 16(3)r8c7. Here's how.
22a. Lol c789: 4 innies r189c7 + r9c8 = 4 outies r3456c6
22b. the 12(3)r5c6 has 2/4 cells in the outies.
22c. the 16(3)r8c7 has 3/4 cells in the innies
22d. -> the valid combination in the 12(3) must overlap with 2 of the candidatates from the 16(3) combination
(IF that combo has at most 1 of 6/8/9 corresponding to r34c6 in outies: this is what I missed first WT)
22e. since 12(3) combinations are {237/246/345} and since {259} combo in 16(3)r8c8 has ONLY 1 of 6/8/9 AND does not have 2 candidates overlapping with 12(3)
22f. ->{259} combo blocked
23. 16(3)r8c8 = {268/349/367}(no 5)
24. KLoL + (double)Overlap c789: r56c6 no 5 [edit out invalid part]. Here's how.
24a...d (same reasoning as steps 22a..d)
24e. the 12(3) combinations are {237/246/345}
24f. the {349/367} combo's from 16(3) must have a double-overlap candidates with the 12(3) -> r56c6 = {34/37}
24g. for the {268} combo from 16(3), r3456c3 = {68}2{3/4/7} -> r56c6 = {23/24/27}(no 5)
24h. finally, the {268} combo from 16(3) could also double-overlap with 12(3) -> r56c6 = {26}
24h. In summary: r56c6 = {23/24/26/27/34/37}(no 5)
25. LoLc789: no 5 in outies -> no 5 in r1c7
26. "45" c6789: 3 innies r789c6 = 13 = h13(3)c6 [edit:typo]
26a. = {139/148/157/256} others blocked by r56c6 (step 24h.)
27. {148} combo blocked from h13(3)c7. Here's how. [edit: true but flawed reason: see Mikes preamble to step 37]
27a. 15(3)r7c5 = {159/168/249/258/267/357} ({348} blocked by 16(3)r8c7)
[edit: following steps are still valid]
28. h13(3)c6 = {139/157/256}(no 4,8)
28a. r78c6 + r9c6 = [913]/{15}[7]/[751]/{25}[6]/[625]
28b. ({13}[9]/[931]/{17}[5]/{56}[2] blocked by combinations unavailable in 15(3))
28c. no 3 r78c6, no 2 or 9 in r9c6 [edit:typos]
29. 15(3)r7c5 = [591]/9{15}/[375]/8{25}/[762] = {159/258/267/357} = [5/7..]
29a. r7c5 = {35789}
30. 14(3)r5c4: {257} blocked by 15(3)
Code: Select all
.-----------------------.-----------------------.-----------.-----------------------.-----------------------.
| 123456789 123456789 | 12345678 12345678 | 123456789 | 123456789 2346789 | 123457 123457 |
:-----------. :-----------. | | .-----------'-----------. |
| 3456789 | 123456789 | 123456789 | 12345678 | 123456789 | 123456789 | 123457 123457 | 123457 |
| '-----------: '-----------: :-----------'-----------. :-----------:
| 1345789 1345789 | 12345789 12345789 | 12345789 | 689 689 | 123457 | 12 |
:-----------------------'-----------.-----------'-----------: .-----------'-----------: |
| 123456789 123456789 12345 | 1234567 12345678 | 89 | 123456789 123456789 | 123456789 |
:-----------------------.-----------+-----------. :-----------'-----------. | |
| 123456789 123456789 | 56789 | 123456789 | 12345678 | 23467 234567 | 123456789 | 123456789 |
| .-----------' | '-----------: .-----------+-----------'-----------:
| 123456789 | 56789 56789 | 123456789 3456789 | 23467 | 12356 | 123456789 123456789 |
:-----------'-----------.-----------'-----------.-----------'-----------: '-----------. |
| 12345678 12345678 | 12345678 12345678 | 35789 125679 | 1235689 1235689 | 456789 |
| .-----------: .-----------+-----------. :-----------.-----------'-----------:
| 12345 | 23456789 | 12345 | 456789 | 23456789 | 125 | 2346789 | 456789 456789 |
:-----------' :-----------' | '-----------: '-----------. |
| 123456789 123456789 | 456789 456789 | 123456789 13567 | 2346789 2346789 | 456789 |
'-----------------------'-----------------------'-----------------------'-----------------------'-----------'