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PostPosted: Mon Mar 03, 2014 2:55 am 
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Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1893
Location: Lethbridge, Alberta, Canada
How to Make a TJK Code String

This was posted in the TJK 31 thread.

Quote:
Para: I don't know how to get a PS-string for jigsaw killers from SumoCue. Maybe someone else can provide these.

Mike (mhparker): PS format doesn't support jigsaws - you have to use SumoCue format (which, fortunately, JSudoku supports as well :) ).

Para: Thanks :) How do i get that?

Mike: Using SumoCue, simply select the "Copy (SumoCue)" option to copy the puzzle definition to the Clipboard in SumoCue text format.

With JSudoku, I've just found out it's even easier: the SumoCue format appears to be the default format for jigsaws here, so all you need to do is a normal Copy (Ctrl-C).


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PostPosted: Mon Mar 03, 2014 3:23 am 
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Grand Master
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1893
Location: Lethbridge, Alberta, Canada
Texas Jigsaw Killer 31 V2 by Para (July 2007)
Puzzle Diagrams:
ImageImage

Jigsaw nonet design: Orig 8
Code: Select, Copy & Paste into solver:
SumoCueV1=24J0+0J0=16J0+2J1=18J1+4J1=24J1+6J1=17J2=14J0+0J0+0J0+2J1+4J1=10J1+6J1+8J2+8J2+9J0+9J0=12J0+20J3+20J3+14J3+6J2+6J2+8J2=25J4+27J4=20J4+29J4=15J3=15J2+32J2+32J2=26J5+27J4+29J4+29J4+29J4+31J3=19J5+41J5+41J5+35J5+27J4=15J6+46J6+46J6+31J3+41J5+41J5+35J5+35J5=15J6=30J6+55J6=9J3=20J3+58J3+58J7=6J7+61J7+54J6+54J6+55J8+57J8=12J8=19J8=24J7+69J7+61J7+54J6+55J8+55J8+67J8+67J8+68J8+68J7+69J7+69J7
Solution:
+-------+-------+-------+
| 7 9 4 | 3 6 5 | 1 2 8 |
| 5 6 2 | 9 7 4 | 8 1 3 |
| 1 8 3 | 7 2 6 | 4 9 5 |
+-------+-------+-------+
| 8 5 1 | 4 3 2 | 7 6 9 |
| 9 2 7 | 6 8 1 | 3 5 4 |
| 3 1 9 | 5 4 8 | 2 7 6 |
+-------+-------+-------+
| 4 7 8 | 1 5 9 | 6 3 2 |
| 2 3 6 | 8 9 7 | 5 4 1 |
| 6 4 5 | 2 1 3 | 9 8 7 |
+-------+-------+-------+

Quote:
SSscore: 2.20

Para: I tried to make a V2 of TJK 31 (Ed askes for one :wink:). Because the cage pattern combined wih the jigsaw shapes gave away quick singles i tried to change a few cages to make the opening less obvious.
This one seems to be a bit harder. But there was a different opening i missed in the first run. Which in the end doesn't make it much harder than the original.
TJK 31V2 This one is a lot harder.

TJK 31 V2 was solved as a "tag" by Ed and Richard (rcbroughton).

Para: I am sorry i said TJK 31 was easy. I completely forgot Ruud's has a vindictive side.

(Archive Note) Some posts have been omitted. J-C deleted all his posts when he stopped using Ruud's site. The omitted posts therefore seemed to be meaningless.

Andrew (in 2014): Belated thanks for Para for a nice, challenging V2! I worked in the same key areas as Ed and Richard did in the "tag".

"Tag" walkthrough:
Ed: Para wrote "TJK 31V2 This one is a lot harder."
Oh silly me. Nice one Para.

Here is the start. Haven't looked at any combination crunching yet. Will have to work on this one over the week. Anyone is welcome to join in :D .


TJK 031V2

0. 10(2)r2c6 - no 5
0a. 26(4)r4c9 - no 1
0b. 9(2)r7c4 - no 9
0c. 20(3)r7c5 - no 12
0d. 6(3)r7c8 ={123}
0e. 19(3)r8c6 - no 1

1. 6(3)r7c8 = {123}: all locked for n8

2. "45" n8(r7c7): r79c7 = 15 = h15(2)n8
2a. = {69/78}

3. 24(4)r8c7 must have 4 and 5 for n8 = 45{69/78}

4. "45" n1: r13c3 = 7 = h7(2)n1
4a. ={16/25/34}(no 789)

5. "45" c123: (remembering the h7(2)n1):r456c4 = 15

6. LoL c123: 3 outies r456c4 = 3 innies r8c3 + r9c23
6a. 3 outies = 15 (step 5) -> 3 innies = 15
6b. -> r7c23 = 15 (same cage as 3 innies)
6b r7c23 = h15(2)r7 = {69/78} = [7/9,8/9..]

7. 20(3)r7c5 = {569/578} ({389/479} blocked by h15(2)r7 step 6b)
7a. = 5{69/78}(no 1..4)
7b. 5 locked for r7 and n4(r3c4)
7c. no 4 r8c4

8. Killer quad {6789} in 20(3)r7c5 and h15(2)r7
8a. {6789} locked for r7
8b. no 123 r8c4

9. LoL r789: 3 innies r7c456 = 3 outies r6c234
9a. 3 innies must have 5 -> 3 outies must have 5
9b. -> 15(3)r6c2 must have 5 = 5{19/28/37/46}
9c. 5 locked for r6 & n7(r6c2)

10. Common Peer Elimination (CPE): no 4 in r6c4 since it can 'see' both 4's in r7

11. LoL r123: 3 innies r3c456 = 3 outies r4c678
11a. no 5 in innies (from step 7b) -> no 5 in outies
11b. 15(3)r4c6 = {168/249/267/348}

12. "45" n3(r1c9): 2 innies r3c78 = 13 = h13(2)n3
12a. = {49/58/67}

13. rest of 24(5)r1c7 = 24 - 13 = 11 = h11(3)n2
13a. no 9 r1c78 or r2c7

14. LoL c789: 3 innies r1c78 + r2c7 = 3 outies r456c6
14a. no 9 in innies -> no 9 in outies r456c6

15. "45" c6789: r1c6 = r7c5
15a. -> r1c6 = {56789}

16. 5 in c6 only in r17c6. Here's how.
16a. 2 5's in r7 in c56.
16b. 5 in r7c5 -> 5 in r1c6 (step 15)
16c. or 5 in r7c6
16d. 5 locked for c7

17. LoL c789: 3 innies r1c78 + r2c7 = 3 outies r456c6
17a. no 5 in outies -> no 5 in innies

18. "45" n2: (remembering h11(3)n2): r1c3 = r2c6 = {12346}
18a. no 123 r3c6
18b. no 2 r3c3 (h7(2)n1)

19. LoLr89: 3 outies r7c789 = 3 innies r8c12 + r9c1
19a. no 4 in outies -> no 4 in innies
19b. 15(4)r7c1 = {1239/1248/1347/2346}

20. CPE: no 4 in r6c1 since it sees all 4's in n7(r6c2)

Code:
.-------------------------------.-------------------------------.-------------------------------.
| 123456789 123456789 12346     | 123456789 123456789 56789     | 1234678   1234678   123456789 |
| 123456789 123456789 123456789 | 123456789 123456789 12346     | 1234678   123456789 123456789 |
| 123456789 123456789 13456     | 12346789  12346789  46789     | 456789    456789    123456789 |
:-------------------------------+-------------------------------+-------------------------------:
| 123456789 123456789 123456789 | 123456789 12346789  1234678   | 12346789  12346789  23456789  |
| 123456789 123456789 123456789 | 123456789 12346789  1234678   | 123456789 123456789 23456789  |
| 1236789   123456789 123456789 | 12356789  12346789  1234678   | 12346789  2346789   2346789   |
:-------------------------------+-------------------------------+-------------------------------:
| 1234      6789      6789      | 1234      56789     56789     | 6789      123       123       |
| 1236789   1236789   123456789 | 5678      123456789 2346789   | 456789    456789    123       |
| 1236789   123456789 123456789 | 123456789 123456789 2346789   | 6789      456789    456789    |
'-------------------------------.-------------------------------.-------------------------------'



Richard: I'll jump in on this one - I think there is going to be some significant crunching on this one. I'm struggling to find a few moves to build on your start.

21. LOL on c6-9 - r3789c6 contains no 1 - so no 1 in r12c45 (just as well I got my Jigsaw highlighting working again!)
21a. 18(3) r1c5 now has no 1 - {189} no longer valid

22. 45 rule on c 1-4 r3c5=r9c4 (Let's make a note of this one - could be useful later)
22a. no 5 at r9c4


Couple of extra moves on the train this morning.

23. 45 Rule on n4 - innies r3c456 r7c456 total 30
Cage 20(3) at r7c5 limits r7c56 = 11,12(no 8),13(no 7),14(no 6) = {56}/{57}/{58}/{59}
Cage 12(3) at r3c3 limits r3c45 = 11,9(no 3),8,7,6={29}/{38}/{47}/{18}/{27}/{17}/{26}/{16}/{34}/{24}
23a. combinations = {234579}/{135678}/{125679}/{134589} (must use a 5)
23b. {234579} - r7c56={57} - > r3c345={29}4/{34}9/{24}9
23c. {234579} - r7c56={59} - > r3c345={27}4/{34}7/{24}7
23d. {134589} - r7c56={58} -> r3c345={34}9
23e. {134589} - r7c56={59} -> r3c345={34}8
23f. No other combo with a 4 - > no 4 at r7c4
23g cleanup - no 5 at r8c4

24. Hidden single 4 at r7c1 for r7
24a. 15(4)r7c1 = {1248}/{1347}/{2346} - no 9

25. 15(3)r6c2 - no {168}/{267} - blocked by 15(4)r7c1
25a. no 6 in 15(3)r6c2

26. LOL on r89 - no 9 in r8c12, r9c1 - > no 9 at r7c7
26a. innies on n9 r7c7+r9c7 = 15 - > no 6 at r9c7

27. LOL on r789 - no 6 in r6c234 -> no 6 in r7c456
27a. (from step 22) - no 6 at r1c6


Ed: Nice one Richard. First placement already. This is going to be a breeze. :wink:

First up, alternate step 23, then a bit of combo crunching then finally a couple of eliminations to make the post worthwhile.

Alternative step 23.
23. no [45] in 9(2)r7c4. Here's how.
23a. from step 6b: r7c23 = h15(2)r7 = {69/78} = [6/8,6/7..]
23b. -> r8c3 + r9c23 = 15 = h15(3)r8c3
23c. = {159/249/258/348/357/456} ({168/267} clash with h15(2))
23d. = [4/5..]
23e. from LoL c123: 3 outies r456c4 = 3 innies r8c3 + r9c23
23f. -> 3 outies r456c4 = [4/5..]
23g. -> [45] blocked from 9(2)r7c4

Now some combo. work.
28. r3c78 = h13(2) = {49/58/67}
28a.-> r1c78 + r2c7 = h11(3) = {128/137/146/236}

29. 15(3)r4c5 = {168/249/267/348}

30. "45" c6789: r127c5 = 18 = h18(3)c5
30a. = {279/378/459/567} ({369/468} blocked by 15(3)r4c5 step 29)
30b. since r7c5 = r1c6 (i/oc6789) -> {369/468} also blocked from 18(3)r1c5
30c. 18(3)r1c5 = {279/378/459/567}

A few eliminations! Hope these are correct.
31. no 8 in r8c78, Here's how. (This might be easier to see if they were written as xy chains: might have to edit)
31a. r7c456 = same combinations as 15(3)r6c2 (from LoL r789)
31b. = [1]{59}/[2]{58}/[3]{57}
31c. -> r7c4 + r7c56 + r7c7 = [1]{59}[6]/[2]{58}[7]/[3]{57}[8]
31d. -> r7c7 + r8c4 = [68/77/86]
i. 8 is in r8c4 when 6 is in r7c7 -> no 8 in r8c78
ii. or 8 is in r9c7 when r7c7 = 7 -> no 8 in r8c78
iii. or 8 is in r7c7 -> no 8 in r8c78

32. no 7 in r8c12. Here's how.
32a. from LoL r89: 3 outies r7c789 = 3 innies r8c12 + r9c1
32b. 7 in outies in r7c7 -> 7 in r8c4 (step 31d) -> no 7 in r8c12
-> 7 in innies only fits in r9c1.
32c. if 7 is not in r7c7 -> from LoL r89, no 7 is possible in 8c12


Para: sudokuEd wrote "Nice one Richard. First placement already. This is going to be a breeze. :wink: "

Look who's getting cocky. I never said it was going to be like TJK 18. But this was the easy bit. It get's a bit more challenging from now on. Took me a few days(well mostly nights) to solve it.

ps.
sudokuEd wrote "5. "45" c123: (remembering the h7(2)n1):r456c4 = 15"

Read my walkthrough for V1?


Richard: 33. 45 on n9 r8c3. outies = 25, but r7c23 = 15 so r7c4+r9c7 = 10.
33a. but, since r79c7=15, when r7c4=1 -> r7c7=6, 2->7, 3->8
33b. r7c456=h15(3)
33c. r3c456=h15(3)

34. placement for h15(3)r3c456 and h15(3)r7c456 -
34a. no 2 at r7c4, no 8 at r7c56 because:
{234579} -> needs r7c456=3{57},
{135678} -> needs r7c456=3{57},
{125679} -> needs r7c456=1{59},
{134589} -> needs r7c456=1{59},
34b. (from 21) no 8 at r1c6

35. from 33 - > no 8 at r9c7, no 7 at r7c7
35a. no 7 at r8c4

36. 2 locked in r7c89 for r7 - no 2 at r8c9

37. LOL on r89
37a. No 7 in outies r7c789 - so no 7 at r9c1
37b. 15(4) at r7c1 now = 4{128}/4{236} - must use 2, no 2 in 15(3)r6c2
37c. {258} no longer valid in 15(3) - no 8

[edit - a couple more for Ed to work on overnight]

38. 45 rule on n8 r7c7 - outies total 24.
38a. r7c56=12 -> r89c6 = 12, no 7 = {39}/{84}
38b. r7c56=14 -> r89c9=10, no 8,9 = {37}/{64}
38c - > no 2 in r89c6
Archive note. As alternative way to eliminate 2 from r89c6 is 45 rule on n79 3 innies r8c4+r89c6 1=18 cannot be {279}

39. LOL on c6789 - no 2 in r3789c6 - so no 2 in r12c45

40. outies of c1-4 = r389c6 = h12(3) ={129}/{138}/{147}/{156}/{237}/{246}/{345}
40a. so r127c6=h18(3) = {459}/{567}/{369}/{378}
40b. 15(3) in c5 = {168}/{249}/{267}/{348}
combining 40a/40b. = 33(6)={459}{168}/{459}{267}/{567}{249}/{567}{348}/{378}{249}
={145689}/{245679}/{345678}/{234789} = must use 4, blocked by 1/2/9, 1/4/7, 2/4/6, 3/4/5
40c so h12(3)={138}/{156}/{237} - no 4

41. 45 on c1-4 r9c4 = r3c5 - no 4 at r9c4


Ed: Good going Richard. Step 34 has made a big difference. Here's the next batch of steps.

Think there is a typo in step 40c, so will give an alternate step 40 first.

Alternative step 40. Trying to work out what happened to {129} combo in h12(3)
40. Updating step 30. r127c5 = 18 = h18(3)c5 = {378/459/567} & 15(3)r4c5 = {168/249/267/348}
40a. combining the two cages = {378-249/459-168/459-267/567-249/567-348}
40b. -> 4 locked for c5
40c. "45" c5: r389c5 = 12 = h12(3)c5= {129/138/156/237}

More
42. 3 innies n9(r8c3): r8c4 + r89c6 = 18 = h18(3)n9
42a. = [6]{39/48}(from step 38a)
42b. = [8]{37/46}(from step 38b)
42c. = {369/378/468} = [3/8..]

43. 12(3)r8c5 = {129/156/237}(no 8)({138} blocked by h18(3)n9 step 42c)
43. ->no 8 r3c5 (I/Oc1234)

44. h12(3)c5 (alt.step 40c) = {129/156/237} = [2/6/7..]

45. 15(3)r4c5 = {168/249/348}(no 7) ({267} blocked by h12(3) step 44)

46.from alt. step 23b,c: r8c3 + r9c23 = h15(3)r8c3
46a. = {159/249/357/456}(no 8) ({258} blocked by 12(3)r8c5;{348} blocked by h18(3)n9 step 42)

47. LoL c123: 3 outies r456c4 = 3 innies r8c3 + r9c23
47a. no 8 in innies -> no 8 in outies

48. {159} blocked from h15(3)(r8c3 + r9c23 step 46a). Here's how.
48a. {159} in h15(3) -> r7c23 = {78} -> r7c7 = 6 -> r7c4 = 1 (step 33a)
48b. {159} in h15(3) -> from LoLc123: outies r456c4 = {159}
48c. but this means 2 1's in c4
48d. -> {159} blocked from h15(3)n9

49. updating step 46a. h15(3)n9 = {249/357/456} (no 1)
49a. -> r456c4 (LoLc123) = {249/357/456}(no 1)
49b. -> r456c4:no 9 r45c4 ({249} combo, 9 only can go in r6c4)

50. 1 in n9 now only in 12(3)r8c5
50a. = {129/156}(no 3,7)
50b. -> no 3 or 7 in r3c5 (i/oc1234)
50c. 12(3) = [6/9..]

51. from step 42. 3 innies n9 = h18(3)n9
51a. = [6]{48} ([6]{39} blocked by 12(3)n9 step 50c.)
52b. = [8]{37/46}
52c. = {378/468}(no 9)

53. no 1 in 15(3)r4c5. Here's how.
53a. 1 in 12(3)r8c5 in r9c4 -> 1 in r3c5 (i/oc1234) -> no 1 elsewhere in c5
53b. 1 in 12(3) in r89c5 -> no 1 elsewhere in c5
53c. 1 in c5 only in r389c5.

54. 15(3)r4c5 = {249/348} = 4{29/38}
54a. 4 locked for c5 & n4(r2c4)
54b. no 6 r2c6

Over to you Richard. Should be here.

Code:
.-------------------------------.-------------------------------.-------------------------------.
| 12356789  123456789 12346     | 3456789   356789    579       | 1234678   1234678   123456789 |
| 12356789  123456789 123456789 | 3456789   356789    1234      | 1234678   123456789 123456789 |
| 12356789  123456789 13456     | 1236789   1269      6789      | 456789    456789    123456789 |
:-------------------------------+-------------------------------+-------------------------------:
| 12356789  123456789 123456789 | 234567    23489     1234678   | 12346789  12346789  23456789  |
| 12356789  123456789 123456789 | 234567    23489     1234678   | 123456789 123456789 23456789  |
| 1236789   13579     13579     | 3579      23489     1234678   | 12346789  2346789   2346789   |
:-------------------------------+-------------------------------+-------------------------------:
| 4         6789      6789      | 13        579       579       | 68        123       123       |
| 12368     12368     2345679   | 68        12569     34678     | 45679     45679     13        |
| 12368     2345679   2345679   | 1269      12569     34678     | 79        456789    456789    |
'-------------------------------.-------------------------------.-------------------------------'



Richard: Sorry - but the train journey this evening provided just enough time to finish it.

weren't really any other tricky moves from where you'd left it.

55. 6 Locked in r3c456 in nonet at r3c4. Locked for r3

56. 18(3)r1c5 = {378}/{567} - must use 7. - nowhere else in nonet


57. LOL on r123 - no 4 in r3c456 - so no 4 in r4c678
57a. no 4 in 15(3) r4c6 = 6{18}/{27} - no 3, 9
57b. 6 locked for n3 and r4

58. LOL on c789 - no 7 in r1c78 r2c7 - so no 7 in r456c6

59. LOL on r123 - no 3,9 in r4c678 - so no 3,9 in r3c456

60. 12(3)r3c3 = 4{26}/[381]/[471]/5{16}/[372] - no 1 at r3c3

61. 45 Rule on n2r1c4 - outies r1c3 r3c6 total 10
61a. r1c3 - no 1,6

62. 9 locked in r12c4, 16(3) for c4 = 9{25}/9{34} - no 8

63. Hidden single 9 at r7c6 for c6
63a. 20(3)r7c5 = [596]

64. Hidden single 5 at r1c6 for c6
64a. 18(3)r1c5=5{67}
64b. {67} locked for c5

65. 16(3)r1c3 = {349}
65a. Cleanup - no 3,4 at r1c78

66. 10(2)r23c6 = no 1 at r2c6
66a. 1 locked in r1c78+r2c7 for n2 -> h11(3) = {128}

67. 10(2)r23c6 - no 8 at r3c6

68. 9 locked in 12(3) r8c5 for c5 - 12(3)={129}

69. h15(2) r7c23 = naked pair {78}

70. 15(4) r7c1 = 4{236}
70a. Naked single 5 at r6c4 -> 15(3)r6c2={19}5
70b. {19} locked at c23 for r6

71. 30(5)r7c2 = {78}{456} - no 2,3
71a {456} locked for nonet
71b. naked single 8 at r8c4 - > 9(2) = [18]

72. naked single 2 at r9c4 and r3c5

73. 19(3) r8c6 = {73}9
73a naked single 1 at r9c5 -> r8c5=9

74. Naked single 1 at c9 for r8

75 Naked pair {76} at r3c46
75a. 12(3)r3c3=[462][372] - no 5

76. 24(5)r1c7={128}[49]
76a. Naked single 3 at r3c3 - > r3c4=7, r1c3=4 -> r3c6=6 -> r2c6=4

77. hidden single 4 at r9c2 for nonet

78. hidden single 4 at r8c8 for r8 and nonet

79. hidden single 6 at r4c8 for nonet

80. naked pair {56} at r89c3

81. naked pair {39} at r12c4
81a. r4c4=4
81b. r5c4=6

82. hidden single 6 at r6c9 for row, col and nonet

83. 1 locked in r5c678 for nonet - locked for row5
83a. 1 locked in r4c123 for nonet - locked for row 4
83b. hidden single 1 at r2c8 for nonet

84. 17(4)r1c9 = 1{358} - {358} locked for nonet and col 9
84a. naked single 2 at r4c6 - 15(3) = [276]
84b. r8c7 = 5
84c. r8c3=6, r9c3=5
84d. r7c9=2, r7c8=3
84e r9c9 = 7, r9c8 = 8

84f. more singles:
3 at r9c6,7 at r8c6,6 at r9c1,9 at r4c9,4 at r5c9,2 at r1c8,8 at r2c7,7 at r6c8,5 at r5c8,1 at r1c7,8 at r6c6,1 at r5c6

85. hidden single 4 at r6c5 for row, col, nonet

86. 25(4)r4c1 = {268}9/{358}9/{367}9 -
86a. 9 locked at r5c1 - only leaves {358}9 - 3 locked at r6c1
86b. more singles
2 at r6c7,3 at r5c7,2 at r8c1,3 at r8c2,8 at r5c5,3 at r4c5

87. Hidden single 1 at r4c3 for ror4
87a. more singles: 9 at r6c3, 1 at r6c2

88. hidden single 1 at r3c1
88a. 14(3)r2c1=[518]
88b naked singles to the end

Archive Note. Typos corrected and an alternative way added for step 38.
Para's comments after step 67:
(Archive Note) I hope that the following continues from the "tag" walkthrough and not from some alternative steps posted by J-C. I haven't (yet) worked through the "tag" walkthrough. My impression, from the comments after step 82, is that they may be based on some alternative steps from J-C (which he had since deleted from Ruud's site).

Para: This finishes it from your position. But there must be something nicer past step 79.

68. 2 in R3 locked in R3C45; R3C5 = R9C4 -> 2 in C4 locked in R39C4 -> R5C4 <> 2

69. LOL C123: R456C4 = R8C3 + R9C23 (no 2)

70. LOL C123: R456C4 = R8C3 + R9C23 = 15 = {159/357/456} (no 8)
70a. 5 locked in R456C4 for C4 -> R12C4 <>5
70b. 5 locked in R8C3 + R9C23 for N9 -> R8C56 + R9C6 <> 5

71. R1C6 = 5(Hidden); R7C5 = 5(hidden)

72. 12(3) at R8C5 = {129/237}: no 6
72a. R9C4 = 2; R3C5 = 2(hidden)

73. 9 in N4 locked for C6.
73a. 9 in N9 locked for R8.

74. 17(4) at R1C9 = {1349/1258/1367}: needs one of {789} in R123C9
74a. 26(4) = {4589/4679/5678}(no 2) : no {2789} would need R456C9 = {789}: clash with step 73.

75. 15(4) at R7C1 = 4{128/236}: when {128}, 8 in R9C1 -->> R8C12 <> 8
75a. 8 in R8 locked for N9

76. 2 in N2 locked in hidden 15(3) at R1C78 + R2C7 = {128/236}: no 4
76a. LOL C123: R1C78 + R2C7 = R456C6: no 4

77. R1C78 + R2C7 and R3C46 together see all 6's in C5 so can't both contain 6.
77a. LOL R123 + C789 -> R456C6 + R4C678 can't both contain 6 -> overlapping cell both sets can't be 6. R4C6 <> 6
77b. R4C6 = 2

78. hidden 11(3) at R456C6 = 2{18/35} : R56C6 = {18/36} = {6|8..}
78a. 26(4) at R4C9 = {4589/4679}: no {5678} -->> {49} locked in 26(4) for N6

Must be something nicer here.
79. R13C3 = [16/34/43]
79a. 8's in N2: R12C4 = 8 or R1C78 + R2C7 = 8
79a. R12C4 = 8 -> R1C3 <> 3
79b. R1C78 + R2C7 = 8 -> R3C78 = {49} -> R3C3 <> 4
79c. R13C3 <> [34]
79d. R1C3 = R2C6 = {14}; R3C3 = {36}
79e. Clean up: R3C6: no 7; R3C4: no 6

80. 16(3) at R1C3 = [178/439]: R12C4 = {39/78} = {3|7...}: no 4,6

81. R456C4 = {159/456}: {357 blocked by step 80): no 3,7
81a. LOL C123: R8C3 + R9C23 = {159/456} = {6|9..} : no 3,7
81b. R7C23 <> {69} blocked by R8C3 + R9C23

82. R7C23 = {78} locked fr R7 and N7
82. R7C67 = [96]

The rest is just basics.


J-C: How didn't I see the move 68 !

Para: I was wondering the same thing after some moves you came up with. But sometimes we just overlook the easy when we are doing difficult things. Guess it's just a question of overthinking things.
Andrew's walkthrough:
It’s so long since I did TJK 31 and TJK 31 V1.5 that I’m starting this without looking at how I solved them.

Prelims

a) R23C6 = {19/28/37/46}, no 5
b) R78C4 = {18/27/36/45}, no 9
c) 20(3) cage at R7C5 = {389/479/569/578}, no 1,2
d) 6(3) cage at R7C8 = {123}
e) 19(3) cage at R8C6 = {289/379/469/478/568}, no 1
f) 26(4) cage at R4C9 = {2789/3689/4589/4679/5678}, no 1

1. Naked triple {123} in 6(3) cage at R7C8, locked for NR7C7
1a. 45 rule on NR7C7 2 innies R79C7 = 15 = {69/78}

2. 45 rule on NR1C1 2 innies R13C3 = 7 = {16/25/34}, no 7,8,9

3. 45 rule on NR6C2 2 innies R7C23 = 15 = {69/78}
3a. R7C23 + 20(3) cage at R7C5 = 35(5) = {56789}, locked for R7, clean-up: R8C4 = {5678}
3b. 5 in R7 only in R7C56, locked for NR3C4

4. Law of Leftovers (LoL) for R789 three outies 15(3) cage at R6C2 must exactly equal three innies R7C456, 5 in R7 only in R7C56 -> 15(3) cage must contain 5, locked for R6 and NR6C2
4a. R7C456 only contains one of 1,2,3,4 -> 15(3) cage must contain one of 1,2,3,4 which must be in R6C23, no 1,2,3,4 in R6C4

5. LoL for R123 three outies 15(3) cage at R4C6 must exactly equal three innies R3C456, no 5 in R3C456 -> no 5 in 15(3) cage

6. 45 rule on NR1C9 2 innies R3C78 = 13 = {49/58/67}, no 1,2,3
6a. 45 rule on NR1C9 3 outies R1C78 + R2C7 = 11 = {128/137/146/236/245}, no 9

7. LoL for C789 three outies R456C6 must exactly equal three innies R1C78 + R2C7, no 9 in R1C78 + R2C7 -> no 9 in R456C6

8. 45 rule on NR1C4 + NR1C9 1 outie R1C3 = 1 innie R2C6 = {12346}, clean-up: no 2 in R3C3 (step 2), no 1,2,3 in R3C6

9. 45 rule on C6789 1 outie R7C5 = 1 innie R1C6 -> R1C6 = {56789}
9a. 5 in R7 only in R7C56, R7C5 = R1C6 -> 5 in R17C6, locked for C5

10. 45 rule on C1234 1 outie R3C5 = 1 innie R9C4, no 5 in R3C5 -> no 5 in R9C4

11. LoL for C789 (step 7), no 5 in R4C456 -> no 5 in R1C78 + R2C7
11a. 1 in C6 only in R2456C6, LoL R4C456 = R1C78 + R2C7 -> 1 must be in R1C78 + R2C67, locked for NR1C4

12. 45 rule on NR6C2 3 outies R8C3 + R9C23 = 15 = {159/249/258/348/357/456} (cannot be {168/267} which clash with R7C23)
12a. 45 rule on NR6C2 + NR8C3 3 innies R8C4 + R89C6 = 18 = {279/369/378/468} (cannot be {459} which clashes with R8C3 + R9C23, cannot be {567} = 5{67} because 19(3) cage at R8C6 cannot be {67}6), no 5, clean-up: no 4 in R7C4

13. R7C1 = 4 (hidden single in R7), placed for NR6C2
13a. 15(4) cage at R7C1 contains 4 = {1248/1347/2346}, no 9
[With hindsight I could have used LoL for R789 (step 4) to lock 9 in 15(3) cage at R6C2 + R7C23 for NR6C2.]

14. 5 in R6 only in 15(3) cage at R6C2 = {159/258/357}, no 6
14a. LoL for R789 (step 4), no 6 in 15(3) cage -> no 6 in R7C56, clean-up: no 6 in R1C6 (step 9)

15. 45 rule on NR6C2 + NR8C3 1 outie R9C7 = 1 innie R8C4 + 1, R8C4 = {678} -> R9C7 = {789}, clean-up: no 9 in R7C7 (step 1a)

16. LoL for R123 (step 5), 15(3) cage at R4C6 -> R3C456 = 15
16a. Hidden killer pair 4,6 in R3C456 and 15(3) cage at R4C5 for NR3C4, neither can contain both of 4,6 (because they don’t contain 5) -> R3C456 and 15(3) cage must each contain one of 4,6 = {168/249/267/348}
16b. R7C456, the remaining hidden 15(3) cage in NR3C4 contains 5 = {159/357} (cannot be {258} which clashes with R3C456), no 2,8, clean-up: no 8 in R1C6 (step 9), no 7 in R8C4, no 8 in R9C7 (step 15), no 7 in R7C7 (step 1a)
16c. LoL for R789 (step 4), R7C456 = {159/357} -> 15(3) cage at R6C2 = {159/357}, no 2,8
16d. Killer pair 7,9 in 15(3) cage at R6C2 and R7C23, locked for NR6C2
16e. 2 in R7 only in R7C89, locked for NR7C7

17. R8C4 + R89C6 (step 12a) = {369/378/468} (cannot be {279} because R8C4 only contains 6,8), no 2
17a. 12(3) cage at R8C5 = {129/147/156/237/246} (cannot be {138/345} which clash with R8C4 + R89C6), no 8, clean-up: no 8 in R3C5 (step 10)

18. R3C5 = R9C4 (step 10) -> R389C5 = 12(3) cage at R8C5 (step 17b) = {129/147/156/237/246}
18a 15(3) cage at R4C5 (step 16a) = {168/249/267/348}
18b. R389C5 = {129/156/237} (cannot be {147/246} which clash with 15(3) cage), no 4
18c. R389C5 = {129/156/237} -> 12(3) cage = {129/156/237}, no 4
18d. 15(3) cage = {168/249/348} (cannot be {267} which clashes with R389C5), no 7
18e. From the combinations for 15(3) cage and R389C5, the remaining hidden cage in C5 R127C5 = {279/378/459/567}
18f. R1C6 = R7C5 (step 9) -> R127C5 = 18(3) cage at R1C5 = {279/378/459/567}

19. R8C4 + R89C6 (step 17) = {378/468} (cannot be {369} which clashes with 12(3) cage at R8C5), no 9, 8 locked for NR8C3

20. LoL for C123 three outies R456C4 must exactly equal three innies R8C3 + R9C23, no 8 in R8C3 + R9C23 -> no 8 in R45C3

21. R1C78 + R2C7 (step 6a) = {128/137/146/236}
21a. R1C3 = R2C6 (step 8) -> R12C4 + R2C6 = 16(3) cage at R1C3 = {178/259/268/349/358/367} (cannot be {169} which clashes with R1C78 + R2C7, cannot be {457} which clashes with 18(3) cage at R1C5)

[Looks like it’s time to start using forcing chains …]
22. R12C4 + R2C6 (step 21a) = {178/259/268/349/358/367}
22a. R1C78 + R2C7 (step 6a) = {128/137/146/236}, LoL for C789 (step 7) R1C78 + R2C7 = R456C6 = {128/137/146/236}
22b. R8C4 + R89C6 (step 19) = {378/468}
22c. Consider placements for 8 in NR8C3
R8C4 = 8 => no 8 in R12C4 + R2C6 => R12C4 + R2C6 cannot contain {178/268/358}
or 8 in R89C6 = {48}, locked for C6 => R23C6 = [19/37], 2 in R456C6 = {236}, locked for C6 => R23C6 = [19] => R12C4 + R2C6 = {178}, 7,8 locked for C4
-> R12C4 + R2C6 = {178/259/349/367}
and 8 in R128C4, locked for C4
[The elimination of 8 from R3C4 is very important for the final breakthrough in step 24a.]
22d. 2 of {259} must be in R2C6 -> no 2 in R12C4
22e. 18(3) cage at R1C5 (step 18f) = {378/459/567} (cannot be {279} which clashes with R12C4 + R2C6), no 2

23. R1C78 + R2C7 (step 6a) = {128/137/146/236}
23a. Consider placement for 2 in NR1C4
2 in R12C4 + R2C6 (step 22c) = {259}, locked for NR1C4 => R1C6 = 7 => R1C78 + R2C7 = {146}
or 2 in R1C78 + R2C7 = {128/236}
-> R1C78 + R2C7 = {128/146/236}, no 7
23b. LoL for C789 (step 7), R1C78 + R2C7 = {128/146/236} -> R456C6 = {128/146/236}, no 7
[Alternatively the elimination of {137} can be obtained from combining R12C4 + R2C6 with 18(3) cage at R1C5.]

24. R12C4 + R2C6 (step 22c) = {178/259/349/367},12(3) cage at R8C5 = {129/156/237}
24a. Remembering that 18(3) cage at R1C5 = R127C5 and 12(3) cage at R8C5 = R389C5, as in earlier steps
Consider combinations for 18(3) cage at R1C5 (step 22e) = {378/459/567}
18(3) cage = {378/567}, 7 locked for NR1C4 => R12C4 + R2C6 = {259/349} => R8C4 = 8 (hidden single in C4)
or 18(3) cage = {459} => R127C5 = {459}, locked for C5 => R389C5 = {237} => 12(3) cage at R8C3 = {237} => 9 in NR8C3 only in R8C3 + R9C23, locked for 30(5) cage at R7C2 => R7C56 = {59} (hidden pair in R7), R7C7 = 6 (cage sum), R9C7 = 9 (step 1a) => R8C4 = 8 (step 15)
-> R8C4 = 8

[I’d originally thought that I’d solved this puzzle using hidden cage R1789C6 = 26 containing 5, but had accidentally omitted one combination, which had made things quite a lot easier. Now I’m back to where I’d reached with that step.
The puzzle is now cracked; the rest is fairly straightforward.]

25. R8C4 = 8, placed for NR8C3, R7C4 = 1, placed for NR3C4, R9C7 = 9 (step 15), R7C7 = 6 (step 1a), both placed for NR7C7, clean-up: no 9 in R7C23 (step 3)
25a. Naked pair {78} in R7C23, locked for R7, NR6C2 and 30(5) cage at R7C2, no 7 in R8C3 + R9C23
25b. LoL for R789 (step 4), R7C456 = 1{59} -> 15(3) cage at R6C2 = {159}, locked for R6 and NR6C2
25c. Naked pair {59} in R7C56, locked for NR3C4, clean-up: no 1 in R2C6, no 1 in R1C3 (step 8), no 6 in R3C3 (step 2)
25d. R8C9 = 1 (hidden single in NR7C7)

26. 15(3) cage at R4C5 (step 18d) = {348}, locked for C5 and NR3C4, clean-up: no 2,6 in R2C6, no 2,6 in R1C3 (step 8), no 1,5 in R3C3 (step 2)
26a. Naked pair {34} in R13C3, locked for C3 and NR1C1
26b. Naked triple {267} in R3C456, locked for R3
26c. Naked quad {3467} in R2389C6, locked for C6
26d. LoL for C789 (step 7), R456C6 = {128} -> R1C78 + R2C7 = {128}
[Alternatively hidden triple in NR1C4.]

27. 18(3) cage at R1C5 = {567} (only remaining combination) -> R1C6 = 5, placed for NR1C4, R12C5 = {67}, locked for C5 and NR1C4 -> R4C5 = 2, R7C56 = [59]
27a. R89C5 = [91], R9C4 = 2, all placed for NR8C3
27b. Naked pair {56} in R89C3, locked for C3, R9C2 = 4 (cage sum)
27c. R89C6 = {37} (hidden pair in NR8C3), locked for C6 -> R23C6 = [46], R3C4 = 7, R3C3 = 3 (cage sum), R1C4 = 4
27d. Naked pair {39} in R12C4, locked for C4 -> R6C4 = 5

28. LoL for R123 (step 5), R3C456 = [726] -> 15(3) cage at R4C6 = {267} = [276] all placed for NR1C9, R4C4 = 4, R5C4 = 6, placed for NR4C1

29. R56C6 = [18], 8 placed for NR4C9
29a. R56C6 = [18] = 9 -> R5C78 + R6C7 = 10 = {235}, locked for NR4C9, 5 also locked for R5

30. R45C4 = [46] = 10 -> R4C3 + R5C23 = 10 = {127} (only possible combination) -> R4C3 = 1, R5C23 = {27}, locked for R5 and NR4C1

31. R56C1 = [93], naked pair {58} in R4C12, locked for R4 -> 15(3) cage at R4C5 = [384]

32. R2C8 = 1 (hidden single in NR1C9), R4C9 = 9, R3C8 = 9 (hidden single in NR1C9), R3C7 = 4 (step 6)

33. R3C12 = {158} -> 14(3) cage at R2C1 = {158} (only possible combination), locked for NR1C1

and the rest is naked singles, without using the nonets.


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PostPosted: Mon Mar 03, 2014 7:20 pm 
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Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1893
Location: Lethbridge, Alberta, Canada
Texas Jigsaw Killer 32 by Ruud (July 2007)
Puzzle Diagrams:
ImageImage

Jigsaw nonet design: Worm by Leonid Kreysin.
Børge's TJK 32 images with udosuk Style Killer Cages and Ruud TJK#33 inspiration:
Image



Image



Image
Cages with cells in 3 jigsaw nonets:  lilac
Cages with cells in 2 jigsaw nonets: green and yellow
Cages with cells in 1 jigsaw nonet: red, blue and grey


See the end of Børge's post for the explanation of this window's title.
Code: Select, Copy & Paste into solver:
SumoCueV1=20J0+0J0+0J1=25J1=9J1+4J1=16J2=19J2+7J2=20J0=8J0+0J0+3J1+3J1+4J3+6J2+6J2+7J2+9J0+10J0+10J1=22J1+3J1=14J3+23J4=10J2+25J2=13J0+9J0+21J5+21J5+21J3=17J3+23J4=15J4+25J2+27J5+27J5=12J5+38J5+32J3+34J4+34J4=16J4+43J4=15J6+38J5=20J5+32J3=17J3+49J4+49J4=15J7+43J7+45J6+45J6+47J5+47J3=30J8+49J8=14J8+60J7+52J7=20J6=10J6+64J6=6J3+58J8+58J8=22J7+60J7+52J7+63J6+63J6+64J6+66J8+66J8+58J8+69J8+69J7+69J7
Solution:
+-------+-------+-------+
| 7 1 4 | 5 2 6 | 8 3 9 |
| 5 4 8 | 9 3 1 | 6 2 7 |
| 6 3 1 | 7 8 2 | 9 4 5 |
+-------+-------+-------+
| 2 9 6 | 4 5 7 | 3 8 1 |
| 3 8 9 | 1 4 5 | 2 7 6 |
| 8 2 7 | 6 9 4 | 1 5 3 |
+-------+-------+-------+
| 1 6 5 | 8 7 3 | 4 9 2 |
| 4 5 2 | 3 6 9 | 7 1 8 |
| 9 7 3 | 2 1 8 | 5 6 4 |
+-------+-------+-------+

Quote:
SSscore: 3.15

Solved as a "tag", started by Caida, then mainly by Mike (mhparker) with a contribution by gary w.

Mike(mhparker): From the above grid position, there are several other smaller moves that scrape off a few candidates without making any real impression on the puzzle, but (inspired by Gary) I would prefer to take the following approach instead: ...
I'll leave the follow-up for now in case someone else has an even better idea, but there's something I like about the above breakthrough combination. True, it may be a hypothetical, but even hypotheticals can be elegant or ugly, and (imho) there's something quite attractive about this one. It's quite compact and the chains don't rely on any modifications to the grid like "tryfurcation" does.

gary w: Lovely,Mike.

Andrew (in 2014): My breakthrough was in the same area, using a short forcing chain and a 45 not used by the "tag". I also used smaller LoLs than in the "tag"; then when I went through the "tag" I realised just how powerful Mike's big LoL (step 21) was.

"Tag" walkthrough:
Caida: Here are my first seven steps for TJK #32. These ones are pretty easy - I need to do some digging before I can figure out the next move.
edited to say: I found one more step without any real digging - hasn't led to anything - but I've added it below.

Guessing this is supposed to be in little wee text as it is the latest TJK - let me know if I should make it bigger.
edited to add: made it bigger

As always, any comments/suggestion/corrections to my walkthrough is most appreciated.
edited to add: made changes in red per Mike's post below

Any additional steps would be really really really appreciated - I'm dying to see this one solved.

TJK #32 Walkthrough:

Nonet Layout

1 1 2 2 2 2 3 3 3
1 1 1 2 2 5 3 3 3
1 1 2 2 2 5 6 3 3
1 1 4 4 5 5 6 6 3
4 4 4 4 5 6 6 6 6
7 4 4 5 5 6 6 9 9
7 7 4 5 8 8 8 9 9
7 7 7 5 8 8 9 9 9
7 7 7 8 8 8 8 9 9

Prelims:

a. 9(3)r1c5 = {126/135/234} (no 7..9)
b. 19(3)r1c8 = {289/379/469/478/568} (no 1)
c. 8(3)r2c2 = {125/134} (no 6..9); no 1 in r2c3 or r3c1 (common peer)
d. 10(3)r3c8 and r8c2 = {127/136/145/235} (no 8,9)
e. 20(3)r2c1 and r6c3 and r8c1 = {389/479/569/578} (no 1,2)
f. 30(4)r7c5 = {6789} (no 1..5)
g. 6(3)r8c4 = {123} (no 4..9)


1. 30(4)r7c5 is fully within n8
1a. {6789} locked within n8

2. r6c7 <> 4,5 as this would eliminate all 4s and 5s from n8

3. LOL for 6789
3a. c789 only have 6789 in n369
3b. r56c6 no 6,7,8,9

4. r2c3 and r3c1 no 1, as this would eliminate all 1s from 8(3)r2c2
This step is no longer needed as it is in the prelims

5. Innies and Outtie n9: r79c7 less r6c9 = 6
5a. max r79c7 = 9(2), max r6c9 = 3
5b. min r6c9 = 1, min r79c7 = 7(2)
5c. r6c9 no 4..9
5d. r79c7 = 7(2)/8(2)/9(2) = {25/34/35/45} (no 1)
5e. LOL for 1 on c789 n369
5f. -> r56c6 no 1

6. 16(3)r5c8: min r5c89 = 13(2) (no 1..3)

7. Innies and Outtie r12: r2c12 less r3c5 = 1
7a. min r2c12 = 4; min r3c5 = 3 (no 1,2)

8. Innies r6789: r6c249 = 11(3) = {128/137/146/236/245} (no 9)


gary w: I've solved this one now.I was able to make the first placement after only................... 1 step.OK I used T&E.The reason I persisted with this was two fold.

1. I hoped that it would give me a better understanding of the puzzle in preparation for,hopefully,a proper attempt at solving it.
2. To have a solution at hand for those trying to solve it..I found my reasoning at times to be rather tortuous,wasn't sure whether my logic was sound and didn't want to spend hours going down the wrong road.

Well,all right,three fold..I just wanted to solve the damn thing!!


Very useful in even the t&e solution were the 4 outies of the nonet whose top cell is r3c7.These 4 cells=17 and r7c6 and r6c9 are fairly restricted ..[12345} and {123}respectively.I hoped this would point to a rigorous solving path but I haven't found one yet.


I hope someone will pick up Caida's initiative and "run with it"..my impression is that this will be a very pleasing puzzle to "get to know"


Mike: Caida wrote "edited to say: I found one more step without any real digging - hasn't led to anything - but I've added it below."

Many thanks for making a start on this one. BTW, it's standard practice for team (so-called "tag") solutions to append steps via making a new post, rather than via editing an existing message. Amongst other things, this stresses the team aspect, and generally adds to the excitement and/or interest.

Caida wrote "Guessing this is supposed to be in little wee text as it is the latest TJK - let me know if I should make it bigger."

It may be the latest TJK (although the "About That Number 4" (ATN4?)should have been labelled a TJK really), but it's been around for a long, long time. Not only that, but it's an extreme puzzle, which Ruud clearly intended as a team exercise. [Aside: Ruud posted it shortly after Para's variants for TJK31 had received an unusally high degree of interest and team activity, which Ruud no doubt hoped would continue with the TJK32 if he could make it sufficiently difficult.] For these reasons, normal text size is fine in this case.
(Archive Note): The link will only work for as long as Ruud's website is available; it's a link to Para's TJK 31 V1.5 and TJK 31 V2 puzzles (see archive entries for these puzzles).

Caida wrote "As always, any comments/suggestion/corrections to my walkthrough is most appreciated."

I have two points, both of which relate to understanding your steps:

Quote:
c. 8(3)r2c2 = {125/134} (no 6..9)

Because 1 is locked in this cage, and r2c3 is a common peer of all 3 cells, we can eliminate the 1 from r2c3 here. Maybe you could edit that in to your original post?

Quote:
7a. min r2c12 = 3; min r3c5 = 2 (no 1)

Actually, min. of r2c12 is 4, because the 20(3) cage at r2c1 (which is not listed in the Preliminaries for some reason) cannot contain either of {12}. Thus, by the same logic, r3c5 can't contain a 2 either. If you could update this sub-step accordingly, and add a "Prelim" for 20(3)r2c1, that would be great. For now, I'll carry on assuming that these extra eliminations have been made.

Caida wrote "Any additional steps would be really really really appreciated"

How about one easy one (plus an obvious follow-up) first?:

9. LoL c6789n3689: 4 innies r1234c6 = 4 outies r789c5+r9c4
9a. {45} unavailable in outies
9b. -> no 4,5 in innies

10. 9(3)n25 = {126/135/234}
10a. 4 only available in r1c5
10b. -> no 3 in r1c5

We have thus now (after step 10b) reached the following grid state:

Code:
.-----------------------------------.-----------.-----------------------.-----------.-----------------------.
| 123456789   123456789   123456789 | 123456789 | 12456       1236      | 123456789 | 23456789    23456789  |
:-----------.-----------.           |           '-----------.           |           '-----------.           |
| 3456789   | 12345     | 23456789  | 123456789   123456789 | 1236      | 123456789   123456789 | 23456789  |
|           |           '-----------+-----------.           :-----------'-----------.-----------'-----------:
| 3456789   | 12345       12345     | 123456789 | 3456789   | 1236789     123456789 | 1234567     1234567   |
:-----------+-----------.-----------'           '-----------+-----------.           :-----------.           |
| 123456789 | 3456789   | 123456789   123456789   123456789 | 1236789   | 123456789 | 123456789 | 1234567   |
|           '-----------+-----------------------.-----------+-----------'-----------+-----------'-----------:
| 123456789   123456789 | 123456789   123456789 | 123456789 | 2345        123456789 | 456789      456789    |
:-----------.-----------+-----------.-----------+-----------'-----------------------+-----------.           |
| 123456789 | 12345678  | 3456789   | 12345678  | 123456789   2345        1236789   | 123456789 | 123       |
|           '-----------:           '-----------+-----------.           .-----------'-----------+-----------:
| 123456789   123456789 | 3456789     3456789   | 6789      | 12345     | 2345        123456789 | 123456789 |
:-----------.-----------'-----------.-----------:           '-----------+-----------.           |           |
| 3456789   | 1234567     1234567   | 123       | 6789        6789      | 123456789 | 123456789 | 123456789 |
|           '-----------.           |           '-----------.           |           '-----------'-----------:
| 3456789     3456789   | 1234567   | 123         123       | 6789      | 2345        123456789   123456789 |
'-----------------------'-----------'-----------------------'-----------'-----------------------------------'


From the above grid position, there are several other smaller moves that scrape off a few candidates without making any real impression on the puzzle, but (inspired by Gary) I would prefer to take the following approach instead:

11. LoL r789n789: 3 innies r7c34+r8c4 = 3 outies r6c189
11a. {12} in innies only available in r8c4
11b. -> if r6c9 is either of {12} it must contain the same digit as r8c4

12. r6c9 = 1 -> r79c7 = 7 (step 5) = {25/34} -> r9c45 <> {23} -> r8c4 <> 1 (permutations 6(3))
12a. but this contradicts step 11b
12b. -> no 1 in r6c9

13. r6c9 = 2 -> r79c7 = 8 (step 5) = {35} -> r9c45 <> {3..} -> r8c4 = 3
13a. but this contradicts step 11b
13b. -> no 2 in r6c9

14. Naked single r6c9 = 3


I'll leave the follow-up for now in case someone else has an even better idea, but there's something I like about the above breakthrough combination. True, it may be a hypothetical, but even hypotheticals can be elegant or ugly, and (imho) there's something quite attractive about this one. It's quite compact and the chains don't rely on any modifications to the grid like "tryfurcation" does.

Hope that helps.


gary w: Lovely,Mike.A tiny addition..

15. So r79c7={45}
16. r56c6={45}
17. So r5c89={67}


Mike: Thanks, Gary.

For those attempting to follow this WT in the future, here are Gary's steps in more detailed form:

15. r79c7 = 9 (step 5) = {45}, locked for r7 and n8

16. Hidden pair in c6 at r56c6 = {45}, locked for n6
(Note: also derivable via LoL c789n369)

17. Split 13(3) at r5c89 = {67} (last combo), locked for r5 and n6


Carrying on...

Edit: Fixed minor typo in step 18 (thanks, Caida!)

18. split 8(2) (innies r6789, step 8) at r6c24 = {17/26} (no 4,5,8)

19. 14(3)n56 = {239} (last combo)

20. 17(4)n568 must have one of {45} due to r6c6
20a. -> {1268/1367} blocked
20b. can't have both of {45} (i.e., can't be {1457/2456}), because {456/457} only available in r6c56.
20c. -> no 4,5 in r6c5
20d. additionally, {2357} combo blocked by r6c24 (step 18)
20e. -> possible combos are {1259/1349/1358/2348} (no 6,7)

21. LoL r6789n6789: 7 innies r6c2345+r7c34+r8c4 = 7 outies r3c7+r4c78+r5c6789
21a. outies contain both of {67} (step 17)
21b. -> innies must contain both of {67}
21c. r6c24 (step 18) contains 1 of {67}
21d. only other place for {67} in innies is 20(3)n45,
21e. which must therefore contain 1 of {67}
21f. -> {389} combo blocked
21f. -> 20(3)n45 = {479/569/578} (no 3)

22. LoL r789n789 (step 11): r8c4 = 3
22a. r9c45 = {12}, locked for r9 and n8

23. Naked single (NS) at r7c6 = 3

24. 9(3)n25 (step 10) = {126} (last combo)
24a. -> no 1,2,6 in r2c45 (CPE)

25. 3 in 14(3)n56 (step 19) now locked in r34c7 for c7 and n6

26. Outies n689: r3c6+r6c5 = 11(2) = {29} (last combo), locked for n5
26a. no 9 in r3c5 (CPE)

27. -> split 14(3) at r6c567 = [248/941]
27a. -> r6c6 = 4, r6c7 = {18}

28. NS at r5c6 = 5

29. 17(3)n5 = [647/746] (only possible permutations)
29a. -> r5c5 = 4
29b. r4c6+r6c4 = {67}, locked for n5
29c. no 6,7 in r4c4 (CPE)

30. NS at r2c6 = 1
30a. r1c56 = {26}, locked for r1 and n2

31. Hidden single (HS) in c4 at r6c4 = 6
31a. -> r4c6 = 7 (cage sum), r6c2 = 2 (step 18)
31b. -> split 10(2) at r5c34 = {19}, locked for r5 and n4

32. r6c57 (step 29) = [91]

33. NS at r3c6 = 2
33a. -> r1c56 = [26]

34. r9c45 = [21]

35. HS in r5 at r5c7 = 2
35a. -> r4c8 = 8 (15(3) cage sum)

36. 16(3)n3 = {268/367}
36a. -> r2c7 = 6, r1c7 = {78}, r2c8 = {23}

Now all singles and cage sums to end. :D

And it only took us 4 months to solve it...
Discussion:
gary w: Brilliant Mike.Being new to these type of sudokus I completely missed your move 21.I did learn quite a bit about the structure of this puzzle "solving it" by T&E and following your wt I see that this was the crucial breakthrough.How do you spot these LoLs?
And even the naked singles don't always leap out at you.
I also found the combos for the 17(4) cage to be pivotal and,as I said,the outies of the nonet - top cell r3c7 - 17/4 placed severe restrictions on possible permutations.
So I loved the brevity and clarity of your path through this puzzle.You've put one over on Ruud !!! and it didn't turn out to be another no.18.I'm sure Ruud wouldn't quibble over the placement of your initial 3 at r6c9...or would he?? :wink:

Regards

Gary ;clapclap;
(Archive Note): Second smiley replaced, on Ruud's site it was "salute".


Mike: Thanks for your kind comments, Gary!

Thanks also to both you and Caida, not only for contributing some of the moves, but also for not being intimidated by this puzzle and for providing the initial spark.

gary w wrote "How do you spot these LoLs?"

It's just practice. The best way of learning LoL that I can recommend is to look at the solution, maybe even printing it out on paper like I used to do [Note: don't use SumoCue to do this though, because it doesn't include the nonets in a printout of the solution :(] That way, because you have the digits staring you in the face, you have immediate feedback as to whether you've made a mistake in identifying the matching innies and outies.

gary w wrote "So I loved the brevity and clarity of your path through this puzzle."

Me. too! If someone were to ask me what type of moves give me the most satisfaction, I would reply: "shortcuts". Other moves, like killer triples and so on, are of course satisfying, too. But nothing (for me) matches that "Eureka!" feeling of finding a shortcut and then afterwards watching the latest version of JSudoku struggling through via the tortuous route, or even giving up completely (as it does with this puzzle). BTW, you're pretty damn good at finding these types of moves yourself!

Having said that, I'm not sure that the word shortcut is appropriate in this case, because it's totally uncertain at the moment as to whether a long-winded, traditional approach is even available for this puzzle! Even Sudoku Solver, which (as expected, coming from Richard) has a combination/permutation engine that is second-to-none, gets completely stuck on this one.

The only problem with these more complex moves is that many newcomers tend to use them much too early, when much simpler moves are still available.

gary w wrote "I'm sure Ruud wouldn't quibble over the placement of your initial 3 at r6c9...or would he?? :wink:"

The only thing worth quibbling about would be an error in my logic. Unlike some other Sudoku forums, the main focus of the sudocue.net forums is actually doing the puzzles, not identifying individual moves that can readily be implemented as new solving techniques in software. Of course, that happens here as well, but it's only a by-product, not the main objective. Therefore, anything that works to solve a puzzle is automatically valid. The only reason that some moves may sometimes appear to be "rejected" is that there is a better approach available, although even the interpretation of the word "better" is itself very subjective.

The real question for Ruud is: When can we expect TJK33? To be quite honest, I'm fed up of seeing TJK32 staring at me for the last 4 months (!!) whenever I click on the saw image in the picture bar. This puzzle has been holding us up for far too long now, and I'm tired and just want to finally move on.

All the best.


gary w: Hi Mike,
Agree with everything you say..and I'll take your advice re the LoLs.Without a solution I'm too afraid of mistaking a logical step and going down a blind alley.

Best regards

Gary


Caida: Mike and Gary,

Thank you so much for completing this puzzle!!

I'm so glad it is over :-)

I'm struggling with TJK33 - but relieved to see that Para has posted a walkthrough already - so no need to wait 4 months to be put out of my misery on that one!

Cheers,

Caida


Gary,

To see LOLs I actually have my excel spreadsheet set up so that I can look at only one number at a time.

This means that I have a grid that shows me where all the 1s have been placed, can still be placed, and can not be placed. (with separate grids for the 2s and 3s etc)

I find this helps me to see LOLs - because I can more easily see where a number appears in only 2 nonets for 2 rows (and therefore cannot be in any other row for those particular nonets.

Not sure if I have explained this adequately - but being able to see LOLs really helps me with solving the jigsaws.
Andrew's walkthrough:
Prelims

a) 9(3) cage at R1C5 = {126/135/234}, no 7,8,9
b) 19(3) cage at R1C8 = {289/379/469/478/568}, no 1
c) 20(3) cage at R2C1 = {389/479/569/578}, no 1,2
d) 8(3) cage at R2C2 = {125/134}
e) 10(3) cage at R3C8 = {127/136/145/235}, no 8,9
f) 20(3) cage at R6C3 = {389/479/569/578}, no 1,2
g) 20(3) cage at R8C1 = {389/479/569/578}, no 1,2
h) 10(3) cage at R8C2 = {127/136/145/235}, no 8,9
i) 6(3) cage at R8C3 = {123}
j) 30(4) cage at R7C5 = {6789}

Steps resulting from Prelims
1a. 8(3) cage at R2C2 = {125/134}, CPE no 1 in R2C3
1b. Naked quad {6789} in 30(4) cage at R7C5, locked for NR7C5
1c. 4,5 in NR7C5 only in R7C67 + R9C7, CPE no 4,5 in R6C7

2. 45 rule on R6789 3 innies R6C249 = 11 = {128/137/146/236/245}, no 9

3. 45 rule on NR6C8 2 outies R79C7 = 1 innie R6C9 + 6
3a. Max R79C7 = 9 -> max R6C9 = 3
3b. Min R79C7 = 7, no 1 in R79C7
3c. Max R6C9 = 3 -> min R5C89 = 13, no 1,2,3 in R5C89

4. Law of Leftovers (LoL) for C789 two outies R56C6 must exactly equal two innies R79C7, R79C7 = {2345} -> R56C6 = {2345}
4a. 4,5 in NR7C5 only in R7C67 + R9C7, R79C7 = R56C6 -> 4,5 in R567C6, locked for C6
[Alternatively, and technically simpler, can get the result in step 4a from LoL for C6789.]

5. 45 rule on C123 2 outies R57C4 = 1 innie R4C3 + 3, IOU no 3 in R7C4

6. 45 rule on C789 2 outies R35C6 = 1 innie R6C7 + 6, IOU no 6 in R3C6
6a. Max R35C6 = 14 -> max R6C7 = 8

7. 45 rule on R12 2 innies R2C12 = 1 outie R3C5 + 1
7a. Min R2C12 = 4 -> min R3C5 = 3

8. 9(3) cage at R1C5 = {126/135/234}
8a. 4,5 of {135/234} must be in R1C5 -> no 3 in R1C5

9. 45 rule on NR3C7 + NR6C8 + NR7C5 3 outies R3C6 + R6C5 + R8C4 = 14 (they form a 14(3) cage because they are all in NR2C6)
9a. Max R8C4 = 3 -> min R3C6 + R6C5 = 11, no 1 in R3C6 + R6C5

10. R79C7 = R6C9 + 6 (step 3)
10a. R79C7 + R6C9 = {25}1/{34}1/{35}2/{45}3
10b. LoL for C789 two outies R56C6 must exactly equal two innies R79C7 -> R56C6 + R6C9 = {25}1/{34}1/{35}2/{45}3
10c. 16(3) cage at R5C8 = {69}1/{78}1/{68}2/{67}3 (cannot be {59}2 which is blocked by R56C6 + R6C9 = {35}2, cannot be {49}3/{58}3 which are blocked by R56C6 + R6C9 = {45}3 using LoL), no 4,5 in R5C89
[An alternative way to do step 10c is …]
45 rule on NR6C8 4 outies R5C89 + R79C7 = 22, R56C6 = R79C7 (LoL) -> R56C6 + R5C89 = 22 = {2569/2578/3469/3478/3568/4567} (cannot be {2389/2479} because R56C6 = R79C7 must total at least 7, step 3b), each combination only contains two of 2,3,4,5 -> no 4,5 in R5C89
Then 16(3) cage at R5C8 = {69}1/{78}1/{68}2/{67}3

11. 15(3) cage at R4C8 = {159/249/258/348/357/456} (cannot be {168/267} which clash with 16(3) cage at R5C8)
11a. Consider placements for 1 in NR3C7
1 in 14(3) cage at R3C6 = {149/158/167} (contains one of 4,5,6 in NR3C7) => 15(3) cage cannot be {456}
or 1 in 15(3) cage = {159}
or R6C7 = 1 => R6C9 = {23} => 16(3) cage at R5C8 (step 10c) = {68}2/{67}3, 6 locked for NR3C7
-> 15(3) cage = {159/249/258/348/357}, no 6

[Note. One has to be careful with the LoLs. For example LoL for C1234 four outies R123C5 + R1C6 must exactly equal four innies R6789C4 and LoL for C6789 four outies R789C5 + R9C4 must exactly equal four innies R1234C6. R1C6 and R9C4 are in both of these groups but they don’t necessarily contain the same value, since some cells of R123C5 + R1C6 don’t “see” some cells of R1234C6 and some cells of R6789C4 don’t “see” some cells of R789C5 + R9C4.]

12. 45 rule on C123 5 innies R4567C3 + R6C2 = 29 -> 4 remaining cells in NR3C4 R4C4 + R5C124 = 16, R5C12 are part of 13(3) cage at R4C1 -> R45C4 = R4C1 + 3, IOU no 3 in R5C4
[Alternatively 45 rule on NR1C1 4(2+2) outies R13C3 + R5C12 = 16, LoL for C123 R13C3 exactly equals R45C4 -> R4C4 + R5C124 = 16, …]

13. Min R79C7 = 7 (step 3b) -> min R56C6 = 7 (using LoL for C789)
13a. 14(3) cage at R3C6 = {149/158/167/239/248/257/347} (cannot be {356} because R56C6 = {24} would be less than 7)
13b. 8 of {158/248} must be in R3C6 (R34C7 cannot be {48} which clashes with R56C6 + R5C89 = {3568}, alternative step 10c), no 8 in R34C7

[At this stage I originally did some heavy analysis on 17(4) cage at R6C5. However after finding steps 14 and 15 this proved unnecessary and has been omitted.]

[Then at this stage I incorrectly wrote step 14a as 45 rule on NR6C8 + NR7C5 2 innies R7C6 + R8C4 = 1 outie R6C9 + 3. This happens to give the correct solution to this puzzle, because R6C9, R7C6 and R8C4 are all equal to 3 in the solution. It was only when I started going through the “tag” solution and reached the first placement that I wondered why Mike hadn’t used this 45; then I realised that I’d got the 45 wrong so I re-worked from here with the correct 45.]

14. LoL for R789 3 outies R6C189 must exactly equal 3 innies R7C34 + R8C4
14a. 45 rule on NR6C8 + NR7C5 2 innies R6C9 + R7C6 = 1 outie R8C4 + 3
14b. Consider placements for R6C9
R6C9 = {12} => R8C4 = {12} (LoL because R7C34 don’t contain 1 or 2) => R7C6 = 3
or R6C9 = 3 => R7C6 = R8C4 -> R7C6 = {123}
-> R7C6 = {123}
14c. R79C7 = {45} (hidden pair in NR7C5), locked for C7
14d. R56C6 = {45} (hidden pair in C6), locked for NR3C7
14e. R79C7 = R6C9 + 6 (step 3), R79C7 = {45} = 9 -> R6C9 = 3, placed for NR6C8
14f. R6C9 = 3 -> R5C89 = 13 = {67}, locked for R5 and NR3C7
14g. R6C249 = 11 (step 2), R6C9 = 3 -> R6C24 = 8 = {17/26}
14h. 14(3) cage at R2C6 = {239} (only remaining combination)

15. 45 rule on NR3C7 3(2+1) remaining outies R37C6 + R6C5 = 14 = {23}9/[914/932] -> R6C5 = {249}
15a. R37C6 + R6C5 = {23}9/[914/932], 9 locked for NR2C6
15b. Grouped X-Wing for 9 in R3C6 + R6C5 and 30(4) cage at R7C5 for C56, no other 9 in C5
15c. 17(3) cage at R4C6 = {278/368/467} (cannot be {458} because 4,5 only in R5C5), no 1,5, clean-up: no 7 in R6C2 (step 14g)

16. 17(4) cage at R6C5 = {1259/1349/2348} (cannot be {1358} because R6C5 only contains 2,4,9)
16a. R6C6 = {45} -> no 4 in R6C5
16b. R3C6 + R6C5 + R8C4 = 14 (step 9), max R3C6 + R6C5 = 12 -> min R8C4 = 2
16c. 6(3) cage at R8C4 = {123}, 1 locked for R9 and NR7C5
16d. 17(4) cage = {1259/1349/2348}, 1,8 only in R6C7 -> R6C7 = {18}
16e. Killer pair 1,2 in R6C24 and 17(4) cage, locked for R4

17. Naked triple {239} in R3C6 + R6C5 + R8C4, locked for NR2C6
17a. 17(3) cage at R4C6 (step 15c) = {467} (only remaining combination) -> R5C5 = 4, placed for NR2C6, R4C6 + R6C4 = {67}, locked for NR2C6 -> R2C6 = 1, placed for NR2C6, clean-up: no 6 in R6C2 (step 14g)
17b. Naked pair {58} in R4C5 + R7C4, CPE no 5,8 in R4C4 + R7C5
17c. Naked pair {67} in R4C6 + R6C4, CPE no 6,7 in R4C4
17d. Naked triple {239} in R3C6 + R6C5 + R8C4, CPE no 3 in R3C4
17e. R5C6 = 5 -> R4C8 + R5C7 = 10 = {19/28}, no 3
17f. 3 in NR3C7 only in R34C7, locked for C7 and 14(3) cage at R3C6, no 3 in R3C6
[The puzzle is cracked. The rest is fairly straightforward.]

18. 17(4) cage at R6C5 (step 16) = {1349/2348} -> R6C6 = 4, R7C6 = 3, placed for NR7C5
18a. Naked pair {12} in R9C45, locked for R9 and 6(3) cage at R8C4 -> R8C4 = 3

19. Naked quint {12389} in R5C1234 + R6C2, locked for NR4C3, 8,9 also locked for R5, clean-up: no 1,2 in R4C8 (step 17c)
19a. 20(3) cage at R6C3 = {578} (only remaining combination) -> R7C4 = 8, placed for NR2C6, R67C3 = {57}, locked for C3 and NR4C3
19b. R4C345 = [645], R3C4 = 7 (cage sum), placed for NR1C3, R4C6 = 7, R6C4 = 6
19c. Naked pair {26} in R1C45, locked for R1 and NR1C3

20. 45 rule on NR1C3 2 remaining innies R13C3 = 5 = {14}, locked for C3 and NR1C3 -> R89C3 = [23], R8C2 = 5 (cage total), 2,3,5 placed for NR6C1
20a. 15(3) cage at R6C1 = {168} (only remaining combination) -> R6C1 = 8, placed for NR6C1, R7C12 = {16}, locked for R7 and NR6C1
20b. R6C7 = 1, R5C7 = 2 -> R4C8 = 8 (cage sum)
20c. R6C2 = 2 -> R5C34 = 10 = [91], R5C12 = [38], R4C1 = 2 (cage sum), placed for NR1C1
20d. Naked pair {39} in R34C7, locked for C7 and 14(3) cage at R3C6 -> R3C6 = 2, R1C56 = [26]
20e. Naked pair {89} in R89C6, locked for NR7C5 -> R78C5 = [76], R67C3 = [75]

21. R7C7 = 4 -> R78C8 = 10 = [91]
21a. R6C8 = 5, R7C9 = 2 -> R8C9 = 8 (cage sum)
21b. R9C89 = {46} (hidden pair in NR6C8)

22. R12C7 = [86] (hidden pair in C7), R2C8 = 2 (cage sum), 6 placed for NR1C7

23. 10(3) cage at R3C8 = {145} (only remaining combination) -> R3C89 = [45], R4C9 = 1

24. R3C1 = 6 -> R2C1 + R4C2 = 11 = [59]

and the rest is naked singles, without using the nonets.


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PostPosted: Tue Mar 04, 2014 4:24 am 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1893
Location: Lethbridge, Alberta, Canada
About that number 4 ... by Ruud (November 2007)
Puzzle Diagram:
Image

Note that there is a 45(9) cage at R258C258.

Original diagram no longer available
Jigsaw nonet design: Cabbage by Tim Cieplowski.
Code: Select, Copy & Paste into solver:
SumoCueV1=18J0+0J0+0J0=18J0+3J1=22J1+5J1=14J1+7J1+0J0=45J0+3J0+3J0+10J2+5J1=15J1+10J1+7J3=15J4=16J4+19J0=21J5=12J2+22J2+15J1+15J3=8J3+18J4+18J4=20J6+21J5+21J5+22J2=24J2+33J2+26J3=18J4+10J6+29J6+21J5+10J5=20J5+33J2+10J2+26J3+36J4+29J6+29J6=8J6+41J5+41J5+33J2=18J3+52J3+36J4=9J4+55J7+48J6+48J6+41J5=13J8+60J3+52J3=18J4+10J7+55J7=14J7+10J6=19J8+68J8+10J8=20J8+63J7+63J7+66J7+66J7+68J7+68J8+71J8+71J8+71J8
Solution:
+-------+-------+-------+
| 3 2 8 | 7 6 9 | 5 1 4 |
| 5 6 1 | 4 2 8 | 7 3 9 |
| 4 7 9 | 8 5 3 | 2 6 1 |
+-------+-------+-------+
| 6 5 7 | 1 3 4 | 8 9 2 |
| 1 8 3 | 9 4 2 | 6 7 5 |
| 9 4 6 | 2 7 5 | 1 8 3 |
+-------+-------+-------+
| 8 3 2 | 5 1 6 | 9 4 7 |
| 2 1 4 | 6 9 7 | 3 5 8 |
| 7 9 5 | 3 8 1 | 4 2 6 |
+-------+-------+-------+

Quote:
SSscore: 1.40

Ruud: Here is the fullfillment of the number 4 prophecy.
The blue cells can be seen as a center-dot group or a size-9 cage as shown in the picture.
Number 4 is jumping right at you. The other numbers are not so forthcoming...
Enjoy
PS: It has (4+4) cages of size 4. This would have been TJK32. (4+4)x4=32. :pallid:

Caida: Here is my walkthrough for this jigsaw killer. I found this one quite easy.
As this is my first time doing a walkthrough ...

gary w: This variant turned out to be a jigsaw with a disjoint 9-cage.
It was somewhat easier than any of the assassins I have looked at to date...

Mike (mhparker): Thought I'd join Caida and Gary in doing this puzzle.
Caida found an early move that I missed.
Many thanks to Ruud for once again providing an interesting and enjoyable jigsaw killer. Roll on TJK33!

Andrew (in 2013): As soon as you've done the first step, you will realise what the puzzle name means.
A fun puzzle, particularly when I found step 18a.

Caida's walkthrough:
Here is my walkthrough for this jigsaw killer. I found this one quite easy. I have struggled with all others and have been unable to solve the latest one (Texas Jigsaw Killer 032 - July 18, 2007).

As this is my first time doing a walkthrough I am not sure if I have the notations quite right. Any comments/suggestions would be most appreciated. (also first time posting - so not even sure if I am doing this right)

edited to fix formating/notational errors based on what I've learned so far


Firstly Nonets are as follows
n1 @ r1c1
n2 @ r1c5
n3 @ r2c5
n4 @ r2c9
n5 @ r3c1
n6 @ r3c4
n7 @ r4c3
n8 @ r7c3
n9 @ r7c7
nX special 45 rule at r2c258+r5c258+r8c258

Prelims:
a. 9(3)n58 = {126/135/234} (no 7..9)
b. 8(3)n4 and n7 = {125/134} (no 6..9) (1 locked for n4c9 and for n5)
c. 22(3)n2 = {589/679} (no 1..4) (9 locked for n2)
d. 16(2)n51 = {79} (no 1..6,8) (7,9 locked for r3)
e. 13(2)n94 = {49/58/67} (no 1..3)


1. Innie n6: r5c5 = 4
1a. 4 locked for nX

2. Innies n7: r5c2+r8c5 = 17(2) {89} (no 1..7)
2a. {89} locked for n7 and for NX

3. Innies n3: r2c5+r5c8 = 9(2) = {27/36} (no 1,5)

4. 20(4)n7 = {2567/3467} and 16(2)n51 = {79} (prelim d); both require 7
4a. since both are only in c23 then no 7 anywhere else in c23

5. Innies n5: r37c2+r8c1 = 12(3) = [7]{14}/[7]/{23}/[9]({12}
5a. r7c2 and r8c1 no (5..9)

6. 18(3)n58 = {189/279/369/378/459/468}
6a. if r3c2 = 9 then r5c2 = 8 (step 2) and r8c1 = {1/2} this requires 7 or 8 or 9 in r9c2, doesn’t work
6b. -> r3c23 = [79]
6c. clean up (most using LOL):
6c1. r3c7 no 1
6c2. r2c59, r7c7 no 7
6c3. r2c6 no 9; r2c9 = 9; r567c6 no 9
6c4. r7c8 no 6
6c5. r7c7 no 4
6c6. r5c8 no 2

7. 14(3)n24 = [149]/{23}[9]
7a. r1c89 no (5..8)
7b. r1c8 no 4
7c. cleanups (most using LOL)
7c1. r3c7 no 4
7c2. r3c7 r7c37 no 5
7c3. r7c37 no 6
7c4. r3c7 no 8
7c5. r7c8 no 7,8

8. 12345 locked for r7c23458

9. 9(3)n58 = {12}[6]/{13}[5]/{234}
9a. r8c3 no 1

10. killer pair {45} locked for n4 in 8(3) and r7c8

11. 18(3)n4 = {279/369/378}
11a. killer pair {23} locked for n4 in 18(3) and 8(3)

12. 15(3)n24 = [726]/{186}/[528]
12a. killer pair {12} for n2 in r1c89 and r23c7
12b. no 2 in r1c5
12c. r2c7 no 234
12d. r3c7 no 3
12e. r1c9 = 4; r1c8 = 1
12f. r2c7 no 6,8
12g. r3c7 = 2

13. 8(3)n4 = {125} locked for n4c9
13a. r7c8 = 4
13b. r7c7 = 9
13c. r2c5 = 2 (LOL)
13d. r5c8 = 7 (step 3) (2,7 locked for nX)

14. 18(3)n4 = [873]
14a. r1c6 and r4c8 = 9
14b. r3c8 = 6; r2c7 = 7

15. 22(9) = {589} locked for n2
15a. r2c8 = 3
15b. r1c5 = 6
15c. r8c8 = 5
15d. r9c8 = 2

16. 20(4)n9 = {4268}
16a. r9c7 = 4
16b. cleanup: r8c5 = 9 (LOL)
16c. r5c2 = 8 (step 2)
16d. r5c4, r6c1, r9c2 = 9

17. 19(4)n89 = {1378}
17a. r9c5 = 8
17b. r9c9 = 6
17d. r8c9 = 8

18. 8(3)n7 = {125} (must contain 5) this locks 5 in n7 and 2 in c4
19. {16} locked in nX and c2
19a. r6c2 = 4
19b. {367} locked in r456c3

20. 18(3)n58 = [279]
20a. cleanup: r7c2 = 3
20b. r8c3 = 4
20c. r4c2 = 5
20d. r1c2 = 2
20e. r7c3 = 2

21. 18(3)n5 = [198]

everything else become just clean up of singles

(Archive Note) Typos corrected.
gary w's solving outline:
This was posted in the Assassin 75 thread.

This variant turned out to be a jigsaw with a disjoint 9-cage.
It was somewhat easier than any of the assassins I have looked at to date..it took me about 1.5 hours to solve.

1. r5c5=4
2. r5c2+r8c5=17={89} and r3c23={79}
3. Thus r7c2+r8c1= 3 or 5 if r3c2 is 9/7 respectively.
4. Now 9 at r3c2 -> r5c2=8 r6c2=7 (nowhere else in N at r4c3 for the 7)
and r8c1=1/2...thus cannot complete the 18(3) cage r8c1.Thus
r3c23=79.Thus 9 in 22(3) cage at r1c6 cannot be at r2c6 otherwise cannot place a 9 in r1.therefore in r2 HS r2c9=9.
5. r2c5+r5c8=9.
6. this really helped to crack it...with r3c2=7 r7c2+r8c1=5 so r78c3+r9c12=22 so r8c2(<>4) + r9c5=9 which must be 18

work to do but pretty much a mop up now.

(Archive Note) [89] corrected to {89} in step 2; it's clear from step 4 that r5c2+r8c5 is still {89} after step 2.
Mike's walkthrough:
Thought I'd join Caida and Gary in doing this puzzle (Note: Gary's solving outline can be found here).

Caida found an early move that I missed. I've added a note about that immediately after my step 6.

It's interesting to note that...

Select text in box (e.g., by triple-clicking it) to see what I wrote:
...both Caida and Gary used basically the same hypothetical to get into the puzzle (Gary's step 4 and Caida's step 6a). I missed that, so ended up (instead) having to use the R2 innies to eliminate the {168} combo for the 15(3) cage at R2C7, and hence work out the mappings between R2C59 and R3C37 (LoL R12).


Note that this time I used the "8(3) at R3C9" notation, rather than the "8(3)n4" notation I sometimes use. There's no particular reason for that, and both forms are of course valid.

Many thanks to Ruud for once again providing an interesting and enjoyable jigsaw killer. Roll on TJK33!


"About That Number 4" Walkthrough

Nonet Layout:

111122222
111132224
551633244
557663334
577666334
577766344
558776944
588879999
888889999

Prelims

a) 22(3) at R1C6 = {589/679} (no 1..4); 9 locked for N2
b) 16(2) at R3C2 = {79}, locked for R3; no 7,9 in R12C2 (CPE)
c) 8(3) at R3C9 = {125/134} (no 6..9); 1 locked for C9 and N4
d) 8(3) at R6C4 = {125/134} (no 6..9); 1 locked for N7
e) 9(3) at R7C2 = {126/135/234} (no 7..9)
f) 13(2) at R7C7 = {49/58/67} (no 1..3)

1. Innie N6: R5C5 = 4

2. Innies N7: R5C2+R8C5 = 17(2) = {89}, locked for N7 and 45(9) at R258C258

3. LoL R89: R7C37 = R8C15
3a. {89} unavailable in R7C3 -> R8C5 = R7C7 and hence R8C1 = R7C3
3a. -> R7C7 = {89}, no 7..9 in R8C1
3b. cleanup: R7C8 = {45}; no 1,2 in R9C12

4. R7C8 and 8(3) at R3C9 form killer pair (KP) on {45} -> no 4,5 elsewhere in N4

5. Innies N4: R2C9+R37C8 = 19(3) = {469/478/568} (no 2,3) (other combos unplaceable)

6. Innies N3: R2C5+R5C8 = 9(2) = {27/36} (no 1,5) (other combos unplaceable)

--- Note: missed grouped X-Wing (well-)spotted by Caida here: ---
--- 7 in N7 and 16(2) at R3C2 locked in C23 -> not elsewhere in C23 ---

7. 14(3) at R1C8: min R2C9 = 6 -> max. R1C89 = 8
7a. -> no 8 in R1C89
7b. 1 unavailable in R1C9
7c. -> no 6,7 in R1C8

8. Both of {89} in N2 locked in R1C567+R2C67+R3C7
8a. LoL C89: R1C567+R2C67+R3C7 (6 outies) = R4589C8+R89C9 (6 innies)
8b. -> R4589C8+R89C9 must also contain both of {89}
8c. R89C89 cannot contain both of {89} due to R7C7
8d. -> R4C8 = {89} (only other place available)
8e. other of {89} must go in R8C9+R9C89
8f. -> R7C7 and R89C89 form KP on {89} -> no 8,9 elsewhere in N9 (R89C67)

9. Innies N5: R37C2+R8C1 = 12(3) = {129/147/237} (no 5,6) (other combos unplaceable)
9a. cleanup: no 3,4 in R9C12
9b. no 5,6 in R7C3 (step 3)

10. Hidden quad at R7C1679 = {6789} (no 1..5)

11. 9(3) at R7C2: max. R7C23 = 7 -> min. R8C3 = 2
11a. -> no 1 in R8C3

12. LoL C12345: R123C5 (3 innies) = R567C6 (3 outies)
12a. no 9 in innies
12b. -> no 9 in outies

13. LoL C123: R4C3+R56C23 (5 innies) = R1289C4+R9C5 (5 outies)
13a. no 1 in innies
13b. -> no 1 in outies

14. 18(3) at R5C1 = {189/369/378/459/468/567} (no 2)
(Note: {279} blocked by R3C2)

15. LoL R12: R2C59 (2 innies) = R3C37 (2 outies)
15a. none of {145} in innies
15b. -> no 1,4,5 in R3C7

16. 15(3) at R2C7 = {168/258/267/348} (other combos unplaceable)
16a. must have exactly 1 of {1457}, only available in R2C7
16b. -> R2C7 = {1457} (no 2,3,6,8)

17. Innies R12: R2C2578 = 18(4) = {1467/2367/2457/3456} (other combos unplaceable)
17a. 4 only available in R2C7
17b. -> no 1,5 in R2C7
17c. 15(3) at R2C7 = [438/726]
17d. -> no 6,8 in R3C7

18. LoL R12 (step 15): {23} unavailable in R2C9
18a. -> R3C7 = R2C5, and (hence) R3C3 = R2C9
18b. -> R2C5 = {23} (no 6,7); R2C9 = {79} (no 6,8)
18c. cleanup: no 2,3 in R5C8 (step 6); no 7 in R1C9

--- Note: puzzle cracked now: rest is basically just mop-up ---

19. Innies N4 (step 5) = [784/964] (only possible permutations)
19a. -> R7C8 = 4
19b. -> R7C7 = 9
19c. -> R8C5 = 9 (step 3)
19d. -> R5C2 = 8 (step 2)
19e. cleanup: no 2,3 in R8C3, no 4 in R8C1 (step 3)

20. 8(3) at R3C9 = {125} (no 3) (last combo), locked for C9 and N4

21. 9 in C9 locked in N4 -> not elsewhere in N4 (R6C8)

22. Hidden single (HS) in C8 at R4C8 = 9

23. HS in N6 at R5C4 = 9

24. 5 in R7 locked in 8(3) at R6C4 = {125} (no 3,4), locked for N7
24a. no 5 in R6C4

25. 18(3) at R6C8 = {369/378} (other combos unplaceable)
25a. 6 of {369} must go in R7C9
25b. -> no 6 in R6C89
25c. 3 locked in R6C89 for R6

26. 3 in N7 locked in C3 -> not elsewhere in C3

27. HS in R7 at R7C2 = 3
27a. cleanup: no 6 in R8C3

28. 8 in N5 locked in C1 -> not elsewhere in C1

29. 18(3) at R8C1 = {279} (last combo)
29a. -> R8C1 = 2
29b. R9C12 = {79} (no 5,6), locked for R9 and N8
29c. R7C3 = 2 (step 3)
29d. -> R8C3 = 4 (cage sum)

30. R6C24 = [42] (hidden singles, N7)

31. LoL C123: R1267C4+R78C5 (6 outies) = R7C3+R8C23+R9C123 (6 innies)
31a. outies contain naked pair (NP) = {15}
31b. -> innies must contain both of {15}, only available at R8C2+R9C3
31c. -> R8C2+R9C3 = {15} (no 6,8), locked for N8
31d. innies contain 4 and 7 (from NP on {79} at R9C12)
31e. -> R12C4 = {47} (no 3,5,6,8), locked for C4, N1 and 18(4) cage

32. R3C23 = [79]
32a. -> R2C9 = 9 (step 15)
32b. -> R3C8 = 6 (step 19)

33. Naked single (NS) at R5C8 = 7
33a. -> R2C5 = 2 (step 6)
33b. -> R3C7 = 2 (step 15), R2C7 = 7 (cage sum)
(Archive note) Order of placements reversed in step 33b.

And the rest is singles and cage sums.
Andrew's walkthrough:
Prelims

a) R3C23 = {79}
b) R7C78 = {49/58/67}, no 1,2,3
c) 22(3) cage at R1C6 = {589/679}
d) 8(3) cage at R3C9 = {125/134}
e) 8(3) cage at R6C4 = {125/134}
f) 9(3) cage at R7C2 = {126/135/234}, no 7,8,9
g) There’s a 45(9) centre dot cage R258C258 = {123456789}

Steps resulting from Prelims
1a. 22(3) cage at R1C6 = {589/679}, 9 locked for NR1C5
1b. Naked pair {79} in R3C23, locked for R3
1c. Naked pair {79} in R3C23, CPE no 7,9 in R12C2
1d. 8(3) cage at R3C9 = {125/134}, 1 locked for C9 and NR2C9
1e. 8(3) cage at R6C4 = {125/134}, 1 locked for NR4C3

2. 45 rule on NR3C4 1 innie R5C5 = 4, placed for 45(9) cage R258C258 and NR3C4

3. 45 rule on NR4C3 2 innies R5C2 + R8C5 = 17 = {89}, locked for 45(9) cage R258C258 and NR4C3

4. 45 rule on NR2C5 2 innies R2C5 + R5C8 = 9 = {27/36}, no 1,5
4a. 12(3) cage at R3C5 = {129/138/147/156/345} (cannot be {237/246} which clash with R2C5 + R5C8)
4b. 9 of {129} must be in R4C6 -> no 2 in R4C6

5. 45 rule on NR3C1 3 innies R3C2 + R7C2 + R8C1 = 12
5a. R3C2 = {79} -> R7C2 + R8C1 = 3,5 = {12/14/23}, no 5,6,7,8,9
5b. Max R8C1 = 4 -> min R9C12 = 14, no 1,2,3,4 in R9C12

6. 18(3) cage at R5C1 = {189/369/378/459/468/567} (cannot be {279} which clashes with R3C2), no 2

7. Law of Leftovers (LoL) for R89 two outies R7C37 must exactly equal two innies R8C15, R8C5 = {89} -> R7C7 = {89} (because no 8,9 in R7C3), R8C1 = {1234} -> R7C3 = {1234}, clean-up: R7C8 = {45}
7a. Killer pair 4,5 in 8(3) cage at R3C9 and R7C8, locked for NR2C9

8. LoL for R12 two outies R3C37 must exactly equal two innies R2C59, no 1,4,5 in R2C59 -> no 1,4,5 in R3C7

9. 9(3) cage at R7C2 = {126/135/234}
9a. 5,6 of {126/135} must be in R8C3 -> no 1 in R8C3

10. 15(3) cage at R2C7 = {168/258/267/348} (cannot be {357/456} because 4,5,7 only in R2C7)
10a. 1,4,5,7 only in R2C7 -> R2C7 = {1457}

11. LoL for C6789 three outies R123C5 must exactly equal three innies R567C6, no 9 in R123C5 -> no 9 in R567C6

12. LoL for C123 five outies R1289C4 + R9C5 must exactly equal five innies R45C2 + R345C3, no 1 in R45C2 + R345C3 -> no 1 in R1289C4 + R9C5

13. 45 rule on NR2C9 3 innies R2C9 + R37C8 = 19 = {469/478/568} (cannot be {289/379} because R7C8 only contains 4,5), no 2,3 in R2C9 + R3C8
13a. Min R2C9 = 6 -> max R1C89 = 8, no 7,8 in R1C8, no 8 in R1C9
13b. 14(3) cage cannot be [626] -> no 6 in R1C8

14. 45 rule on NR1C1 2 innies R2C2 + R3C3 = 1 outie R1C5 + 9
14a. Max R2C2 + R3C3 = 15 -> max R1C5 = 6

15. 45 rule on NR7C7 2 innies R7C7 + R8C8 = 1 outie R9C5 + 6
15a. Min R7C7 + R8C8 = 9 -> min R9C5 = 3

16. 45 rule on NR3C1 + NR7C3 1 outie R3C3 = 2 innies R8C2 + R9C5
16a. R7C3 = {79} -> R8C2 + R9C5 = 7,9 = [25/18/27/36/63], no 5,7, in R8C2, no 9 in R9C5
16b. R7C7 + R8C8 = R9C5 + 6 (step 15)
16c. Max R9C5 = 8 -> max R7C7 + R8C8 = 14, no 7 in R8C8
[There’s a shortcut here. R8C2 + R9C5 cannot be [36] because it clashes with R7C7 + R8C8 = 12 = [93]. I’ll leave that for now, since it’s a contradiction move.]

17. 45 rule on R12 4 innies R2C2578 = 18 = {1467/2367/2457/3456}
17a. 4 only in R2C7, 7 of {2367} must be in R2C7 -> R2C7 = {47}
17b. 15(3) cage at R2C7 (step 10) = {267/348}
17c. R3C8 = {68} -> no 6,8 in R3C7

18. R2C2578 (step 17) = {1467/2367/2457/3456}
18a. 5 of {2457/3456} must be in R2C8 (R2C58 cannot be {27/36} which clash with R2C5 + R5C8 (step 4) = {27/36}, CCC), no 5 in R2C2
18b. 5 in 45(9) cage R258C258 only in R28C8, locked for C8 -> R7C8 = 4, placed for NR2C9, R7C7 = 9, placed for NR7C7
18c. 45 rule on R2C9 2 remaining innies R2C9 + R3C8 = 15 = [78/96]

19. 8(3) cage at R3C9 = {125} (only remaining combination), locked for C9 and NR2C9

20. Naked quad {1235} in R7C2345, locked for R7, 5 also locked for NR4C3
20a. 5 in R7 only in R7C45 -> 8(3) cage at R6C4 = {125}, locked for NR4C3

21. 3 in R7 only in R7C23, locked for 9(3) cage at R7C2 = {135/234}
21a. 4,5 only in R8C3 -> R8C3 = {45}
21b. 3 in R7 only in R7C23, CPE no 3 in R8C2

22. 3 in NR2C9 only in R6C89, locked for R6
22a. 3 in NR4C3 only in R45C3, locked for C3
22b. R7C2 = 3 (hidden single in R7), placed for NR3C1

23. 45 rule on NR3C1 2 remaining innies R3C2 + R8C1 = 9 -> R3C2 = 7, R8C1 = 2, both placed for NR3C1, R3C3 = 9, placed for NR1C1
23a. R8C1 = 2 -> R9C12 = 16 = [79], both placed for NR7C3, R5C2 = 8, placed for NR4C3 -> R8C5 = 9
23b. 45 rule on NR1C1 1 outie R1C5 = 1 remaining innie R2C2 -> R1C5 = {126}
23c. 7 in NR4C3 only in R456C3, locked for C3
23d. 7 in NR1C1 only in R12C4, locked for C4

24. R3C3 = R8C2 + R9C5 (step 16)
24a. R3C3 = 9 -> R8C2 + R9C5 = 9 = [18/63]

25. 18(3) cage at R5C1 (step 5) = {189/468} (cannot be {459} because R7C1 only contains 6,8) -> R7C3 = 8, R56C1 = [19/64/91], no 5, no 6 in R6C1
25a. 18(3) cage at R6C8 = {369/378}
25b. R7C9 = {67} -> no 6,7 in R6C89

26. 45 rule on NR1C5 + NR2C9 2 remaining innies R1C5 + R2C8 = 9 = [27/63], clean-up: no 1 in R2C2 (step 23b)
26a. Killer pair 3,7 in R2C5 + R5C8 and R2C8, locked for 45(9) cage R258C258
26b. Killer pair 2,6 in R2C2 and R2C5 + R5C8, locked for 45(9) cage R258C258 -> R8C2 = 1, placed for NR7C3, R8C8 = 5, placed for NR7C7, R78C3 = [24], placed for NR7C3

27. Naked triple {367} in R456C3, locked for C3 and NR4C3 -> R6C2 = 4, clean-up: no 6 in R5C1 (step 25)
27a. Naked pair {19} in R56C1, locked for C1
27b. Naked triple {456} in 15(3) cage at R3C1, 4 locked for C1
27c. Naked quad {2356} in R12C12, locked for NR1C1

28. Naked pair {18} in R12C3, locked for C3 -> R9C3 = 5, R89C4 = 9 = {36}, locked for C4 and NR7C3 -> R9C5 = 8
28a. R8C9 = 8 (hidden single in R8)
28b. 7 in R8 only in R8C67 -> 19(4) cage at R8C6 = {1378}, R9C6 = 1, R8C67 = {37}, locked for R8 and NR7C7 -> R89C4 = [63]

29. R12C4 = {47} = 11 -> R1C5 + R2C3 = 7 = [61], 6 placed for NR1C5, R1C3 = 8, R2C2 = 6 (step 23b), placed for 45(9) cage R258C258, R4C2 = 5, R1C2 = 2

30. 22(3) cage at R1C6 = {589} (only remaining combination) -> R1C67 = [95], R2C6 = 8, R1C1 = 3, R1C8 = 1
30a. R2C8 = 3, R3C7 = 2 (hidden pair in NR1C5)
30b. R6C9 = 3 (hidden single in C9)
30c. R9C8 = 2 (hidden single in R9), R5C8 = 7, placed for NR2C5, R2C5 = 2

31. R2C9 = 9 (hidden single in R2), placed for NR2C9, R1C9 = 4 (cage sum), R12C4 = [74], R2C7 = 7, R3C8 = 6 (cage sum), placed for NR2C9, R34C1 = [46], R7C9 = 7, R6C8 = 8, R4C8 = 9, R7C6 = 6, R8C67 = [73]

32. Naked pair {16} in R56C7, locked for C7 and NR2C5 -> R9C7 = 4, R4C7 = 8

33. Naked pair {12} in R46C4, locked for C4 -> R7C4 = 5

and the rest is naked singles, without using the nonets.


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PostPosted: Thu Mar 06, 2014 4:25 am 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1893
Location: Lethbridge, Alberta, Canada
Toroidal Killer Sudoku #1 by Para (November 2007)
Puzzle Diagrams:
Image   Image
Code: Select, Copy & Paste into solver:
SumoCueV1=19J0+0J0=15J0+2J1=9J1=14J2+5J3=12J4+7J4+0J0=11J1+10J1+2J1+4J2+5J2=19J5+15J4+7J0=18J1+10J1=20J2+20J2=24J2=20J5+23J5+15J6=8J0+18J1+18J2=17J2+20J5+22J5+23J5=14J6+26J6+26J7=7J8+36J2+29J5+29J5+22J6+33J6+33J6=16J7+43J7=10J8+45J3+29J5=16J6+22J6=17J7+33J7=17J7+52J8+45J3=22J3+48J4+48J6+22J7+50J7+50J8=18J8+52J8=14J3+55J4+55J4=11J0=12J7=8J8+61J8+61J3=17J3+63J4+63J4+66J0+66J0+67J1+68J8+68J3+71J3+71J4
Solution:
+-------+-------+-------+
| 9 8 7 | 2 5 3 | 4 1 6 |
| 2 1 3 | 6 4 7 | 8 9 5 |
| 4 7 5 | 8 1 9 | 6 2 3 |
+-------+-------+-------+
| 8 6 9 | 7 2 5 | 3 4 1 |
| 5 2 4 | 3 6 8 | 1 7 9 |
| 6 3 1 | 9 7 4 | 2 5 8 |
+-------+-------+-------+
| 1 9 2 | 5 8 6 | 7 3 4 |
| 7 5 8 | 4 3 1 | 9 6 2 |
| 3 4 6 | 1 9 2 | 5 8 7 |
+-------+-------+-------+

Quote:
SSscore: 1.20

Para: I have been planning to make a toroidal killer sudoku for a while. Well here is my first attempt. So instead of 9 3X3 nonets, we have 9 jigsaw nonets that wrap around the grid. I have coloured the nonets to make it easier to recognize which cells belong together in a nonet. There is also a SumoCue-code available but this makes it a bit harder to spot the seperate nonets.
It was fun to solve. It's not the hardest puzzle, but the first one doesn't really have to be. The toroidal cages make the solving process a bit complex already.
Hope you enjoy it.

Caida: I found this one really interesting.

Mike (mhparker): Well, I finally did it! :-D Got stuck for some time in the middle, though, so hopefully the difficulty of subsequent toroidal killers can be kept at around this level. :wink:
BTW, these toroidal killers are a great invention (thanks, Para!). Some of those CPE moves have to be seen to be believed!
This puzzle appears to have quite an elegant, definite, solving path ("text book" solution). Caida did a better job of finding it than I did. The text in the box below explains why.
Other interesting features about this puzzle were:
* Not many innie/outie ("45") moves available on the nonets, because they were very thin, with (consequently) lots of protruding cages.
* No productive LoL moves, because the nonets were orientated in a diagonal direction."

Andrew (in 2013): A nice puzzle. The hardest part was keeping track of eliminations in the disjoint nonets, even after I’d set up minimised windows so that I could view both my worksheet and Para’s original diagram at the same time.
Dare I saw that Mike's final comments are the reason why I prefer other puzzles to toroidal puzzles.

Caida's walkthrough:
I found this one really interesting. Below is my walkthrough.

I'm putting it in really small text in case anyone else is planning to do a walkthrough and doesn't want to look.

I would love to have any comments/suggestions/corrections on my walkthrough.


edited to add: I've fixed my walkthrough - my original step 5 was faulty logic and wishful thinking - turns out I didn't actually need it! (Thanks Para!)
As always, would appreciate any comments/suggestions/corrections
:)
edited to add: I've made some changes based on suggestions/corrections received (Thanks Mike!!)


Toroidal Killer Sudoku #1

Nonet Numbering
1 1 1 2 2 3 4 5 5
1 2 2 2 3 3 6 5 1
2 2 3 3 3 6 6 7 1
2 3 3 6 6 6 7 7 8
9 3 6 6 7 7 7 8 8
9 4 6 7 7 8 8 8 9
4 4 5 7 8 8 9 9 9
4 5 5 1 8 9 9 4 4
5 5 1 1 2 9 4 4 5

Prelims
a. 19(3)r1c1 and r2c7 = {289/379/469/478/568} (no 1)
b. 9(2) r1c5 = {18/27/36/45} (no 9)
c. 11(3)r2c2 and r8c4 = {128/137/146/236/245} (no 9)
d. 20(3)r3c3 and r3c6 = {389/479/569/578} (no 1,2)
e. 8(3)r3c9 and r8c6 = {125/134} (no 6..9) no 1 in r8c7 (CPE)
f. 14(4)r4c7 = {1238/1247/1256/1346/2345} (no 9)
g. 7(2) r5c1 = {16/25/34} (no 7..9)
h. 16(2)r5c8 = {79} Note {79} now locked for n8 and r5
i. 10(3)r6c1 = {127/136/145/235} (no 8,9)
j. 22(3)r7c2 = {589/679} (no 1..4) no 9 in r7c3 r8c189 r9c2 (CPE)
k. 12(2) r8c5 = {39/48/57} (no 1,2,6)

cleanup
l. r9c5 no 3,5

Walkthrough
1. Outies c12: r28c3 = 11(2) = [29/38/47/56/65] (no 1)
1a. -> r2c3 no 7,8

2. Outies c89: r28c7 = 17(2) = {89} (locked for c7)
2a. -> 20(3)r3c6 no 3 as combo {389} blocked by r2c7
2b. 20(3)r3c6 requires either an 8 or a 9
2c. -> killer pair {89} locked for n6 in r2c7 and 20(3)r3c6 also no 8, 9 in r2c6 (CPE)

3. Outies r12: r3c28 = 9(2) = {18/27/36/45} (no 9)
3a. -> r3c2 no 8

4. Outies r89: r7c28 = 12(2) = {39/48/57} (no 1,2,6)
4a. ->r7c2 no 5 (combo 22(3) = [5]{89} blocked by r8c7)
4b. ->r7c2 no 7 (combo 22(3) = [7]{69} would require r7c8 to equal 5 (step 4) and r8c7 to equal 8, this would given an invalid combo in 18(3)r7c8 of [5]{85})
4c. ->r7c8 no 5,7,8,9
4d. 18(3)r7c8 = [396/486]
4e. -> can’t be [387] as this is blocked by 22(3) = [9]{58/67}
4f. -> can’t be [495] as this is blocked by 22(3) = [8]{59}
4g. ->r8c8 = 6
4h. -> 22(3)r7c2 = [9]{58}/[8]{59}; [9]{67} blocked because 6 is unavailable due to r8c8
4i. cleanup: r8c23 no 7; 22(3)r7c2 = {589}
4j. 5 locked for n5 in r8c23
4k. r8c1, r7c3 and r9c2 no 5,8,9 (can see all of 22(3))
4l. {589} locked for r8 in c237 (naked triple)
4m. cleanup: r9c5 no 4,7: (12(2) = [39/48]

5. 17(3)r8c9: r89c9 can’t be {79} (blocked by r5c9
5a. r9c8 no 1
5b. 17(3)r6c8: r67c9 can’t be {79}
5c. r6c8 no 1

6. 20(3)r3c3: r3c34 no 3 (needs both 8 and 9 and there is no 8/9 in r4c4 b/c step 2c)

7. 18(3)r3c1: r34c1 can’t be {89} (blocked by r9c5 – all in n2)
7a. r4c2 no 1

8. {89} is locked in n4 at r7c2 and r9c8 (no 2..7 in r9c8) (hidden pair)
8a. {89} locked in r9 at r9c58 (naked pair)

9. 19(3)r2c7 = [8]{29}/[9]{28}/[8]{47}
9a. -> [9]{37} blocked as this would lead to r8c8=8, r7c8=4, r7c2 = 8(step4), r9c8 = 9(step 8), r5c8 = 7
9b. -> [9]{46} and [8]{56} blocked by r8c8
9c. -> r23c8 no 3,5
9d. killer triple {789} in r2359c8 (19(3) requires either 7, 8, or 9 in r23) no 7, 8, 9 elsewhere in c8

10. 14(3) r8c1 = {167/347} (no 2)

11. 17(3)r6c8: max of r6c8 = 5
11a. -> min of r67c9 = 12(2) = {93/84}
11b. -> r67c9 no 1,2

12. 8 in r5 locked in r5c56 (I had originally written this as: r5c56 contains only 8s in r5)
12a. -> 8 locked in r5 for n7 (in r5c56)
12b. r2c8 no 2 (as requires either an 8 or 9 in r3c8)
12c. 19(3)r2c7 requires an 8 (step 9)
12d. -> 8 is either in r2c7 or r2c8 (locked for r2)
12e. cleanup: r1c5 no 1 (as no 8 in r2c5)


13. 8 in n1 locked in r1 -> not elsewhere in r1 (I had originally written this as 8 is locked in n1 r1c123 (nowhere else in n1 has option of an 8 so no other n can have an 8 in r1))
13a. cleanup: r2c5 no 1 (as no 8 in r1c5)


14. 8 in n7 is only in c56; 8 in n8 is only in c56
14a. LOL – no other n can have an 8 in c56
Mike suggests writing this as “grouped X-Wing "8 in n78 locked in c56 -> not elsewhere in c56” but I am not yet adept at seeing things as “X-Wings” and still need to see them as LOL
14b. r9c5 = 9, r8c5 = 3
14c. r9c8 = 8; r7c2 = 9
14d. r2c7 = 8; r8c7 = 9; r7c8 = 3

15. r3c4 = 8 as it is the only 8 in c4 hidden single, c4
15a. r4c1 = 8 as it is the only 8 in n2
15b. 18(3)r3c1 = {378/468} (no 1,2,5)
15c. 20(3)r3c3 = {389/578} (no 4,6)
15d. cleanup: r3c3 no 3 pointed out by Mike that the 3 is already gone – so this step is not needed

16a. r6c4 = 9 as it is the only 9 in c4 hidden single, c4
16b. 9 is in only r12 (no other r) for both n1 and n5, no 9 in other ns in r12
Note that this is another X-Wing: grouped X-Wing "9 in n15 locked in r12 -> not elsewhere in r12" and could be written as 9 in n6 locked in c6 -> not elsewhere in c6
16c. 9 in n3 locked for in c3, not elsewhere in c3
16d. n1 is the only n with a 9 available in c1, no other c in n1 can have a 9 Better written as: 9 in c1 locked in n1 -> not elsewhere in n1

17. 16(3)r6c4 = [9]{16/25} (no 4,7)
17a. cleanup: r7c4 no 2
17b. cleanup: r12c5 no 6 missed from step 14b
17c. r9c7 no 4 (would need a {13} in r89c6)

18. 17(3)r8c9 = [8]{27}
18a. -> {27} locked for c9
18b. -> r5c89 = [79]
18c. -> r23c8 = [92]

19. 17(3)r6c8 = {458} no 6
19a. -> 6 in c9 locked in 12(3)r1c8
19b. -> 12(3)r1c8 = [165]

20. 8(3)r3c9 = [341]
20a. cleanup: r7c3 = 2; r6c8 = 5; r9c9 = 7; r8c9 = 2

21. r9c1 = 3, hidden single
21a. 14(3)r8c1 = [734]

22. 8(3) r8c6 = [125]
22a. r8c4 = 4


23. {48} locked for n9 in c9
23a. -> r7c7 no 4; r56c1 no 4

everything from now on is all singles and cage sums


24 (all singles): r6c1 = 6; r123c1 = [924; r5c1 = 5; r7c1 = 1; r6c2 = 3; r1c7 = 4; r78c7 = [27]; r345c7 = [631]; r6c6 = 7
24a. 19(3)r1c1 = [982]
24b. r1c3 = 7; r8c23 = [58]

25. 9(2)r1c5 = [54]
25a. r34c5 = [12]; r1c6 = 3; r1c4 = 2; r3c2 = 7; r4c2 = 6; r2c2 = 1; r5c2 = 2; r5c4 = 3; r5c3 = 4; r6c3 = 1; r9c3 = 6; r9c4 = 1; r2c34 = [36]; r2c6 = 7; r7c4 = 5; r4c4 = 7;

26. 20(3)r3c3 = [587]
26a. 20(3)r3c6 = [965]
26b. 14(4)r4c7 = [3812]
26c. 24(5)r3c5 = [12678]
26d. 17(3)r6c6 = [467]
26e. 17(3)r6c8 = [584
26f. r4c3 = 9

(Archive Note) All the coloured parts were in Caida's post, after feedback from Mike and Para.
Mike's walkthrough:
Well, I finally did it! :-D Got stuck for some time in the middle, though, so hopefully the difficulty of subsequent toroidal killers can be kept at around this level. :wink:

BTW, these toroidal killers are a great invention (thanks, Para!). Some of those CPE moves have to be seen to be believed!

This puzzle appears to have quite an elegant, definite, solving path ("text book" solution). Caida did a better job of finding it than I did. The text in the box below explains why.

"I got stuck immediately after step 14. I finally managed to find step 15 (and the follow-up steps) to break the deadlock. However, the "intended" way forward at this point was probably the killer triple on {789} in C8, which sets up a grouped X-Wing on 8 in N78 and C56, which in turn fixes 12(2) at R89C5. This corresponds to the route Caida took. I eventually saw both of these moves (steps 17b and 24), but too late. Nevertheless, I've decided to post my WT, in order to show an alternative path.

Other interesting features about this puzzle were:
* Not many innie/outie ("45") moves available on the nonets, because they were very thin, with (consequently) lots of protruding cages.
* No productive LoL moves, because the nonets were orientated in a diagonal direction."

Now to the WT:

Edited to include feedback from Para. Many thanks.

Toroidal Killer 1 Walkthrough

Nonet Layout:

1 1 1 2 2 3 4 5 5
1 2 2 2 3 3 6 5 1
2 2 3 3 3 6 6 7 1
2 3 3 6 6 6 7 7 8
9 3 6 6 7 7 7 8 8
9 4 6 7 7 8 8 8 9
4 4 5 7 8 8 9 9 9
4 5 5 1 8 9 9 4 4
5 5 1 1 2 9 4 4 5

Prelims:

a) 19(3) at R1C1 and R2C7 = {289/379/469/478/568} (no 1)
b) 9(2) at R1C5 = {18/27/36/45} (no 9)
c) 11(3) at R2C2 and R8C4 = {128/137/146/236/245} (no 9)
d) 20(3) at R3C3 and R3C6 = {389/479/569/578} (no 1,2)
e) 8(3) at R3C9 and R8C6 = {125/134} (no 6..9) -> no 1 in R78C7 (CPE)
f) 14(4) at R4C7 = {1238/1247/1256/1346/2345} (no 9)
g) 7(2) at R5C1 = {16/25/34} (no 7..9)
h) 16(2) at R5C8 = {79}, locked for R5 and N8
i) 10(3) at R6C1 = {127/136/145/235} (no 8,9)
j) 17(3) at R6C6 = {269/278/359/368/458/467} (no 1); no 2 in R7C7
(Note: {179} blocked, and no 2 in R7C7, because {79} only available in R7C7)
k) 17(3) at R6C8 = {179/269/278/359/368/458/467}; no 1 in R67C9
(Note: no 1 in R67C9 because {79} unavailable in R6C8)
l) 22(3) at R7C2 = {589/679} (no 1..4) -> no 9 in R7C3, R8C189, R9C2 (CPE!)
m) 12(2) at R8C5 = [39/48/57/84] (no 1,2,6; no 3,5 in R9C5)

1. Outies R89: R28C7 = 17(2) = {89}, locked for C7
1a. {89} in R2C7 blocks {389} combo for 20(3) at R3C6 (Prelim d)
1b. -> 20(3) at R3C6 = {479/569/578} (no 3) = {(8/9)..}
1c. R2C7 and 20(3) at R3C6 form killer pair on {89} within N6
1d. -> no 8,9 elsewhere in N6
1e. CPE(N6): no 8,9 in R2C6

2. {89} now unavailable in R4C4 -> {389} combo for 20(3) at R3C3 (prelim d) must have 3 in R4C4
2a. -> no 3 in R3C34

3. 8 in R5 locked in N7 -> not elsewhere in N7

4. Outies R12: R3C28 = 9(2) = {27/36/45} (no 1,8,9)
(Note: {18} blocked, because neither of these 2 digits available in R3C8)
4a. 19(3) at R2C7 = {289/379/469/478/568}
4b. {89} only available in R2C78
4c. -> no 2 in R2C8

5. Outies C12: R28C3 = 11(2) = [29/38/47/56/65] (no 1,7,8 in R2C3)
5a. 11(3) at R2C2 = {137/146/236/245} (no 8)
(Note: {128} blocked, because {18} only available in R2C2)
5b. 1 only available in R2C2
5c. -> no 7 in R2C2

6. Outies R89: R7C28 = 12(2) = [57/75/84/93] (no 1,2,6; no 8,9 in R7C8)

7. Innies R89: R8C2378 = 28(4) = {(47/56)89} (no 1,2,3)
7a. 8 locked for R8
7b. {12} now unavailable to 18(3) at R7C8 = {369/378/459/468}
(Note: {567} blocked, because 18(3) must have 1 of {89} due to R8C7)
7c. possible permutations are: [396/387/495/594/486]
7d. -> no 7 in R7C8, no 8 in R8C8
7e. cleanup: no 5 in R7C2 (step 6), no 4 in R9C5

8. Hidden pair (HP) in N4 at R7C2+R9C8 = {89} (no 1..7)
8a. -> no 8 in R9C2 (CPE)
8b. cleanup: no 5 in R7C8 (step 6)
8c. 18(3) at R7C8 (step 7c) = [396/387/495/486]
8d. -> no 4 in R8C8

9. 4 now unavailable to R89 innies (step 7)
9a. -> R8C2378 = {5689} (no 7)
9b. {56} locked for R8
9c. cleanup: no 7 in R9C5, no 4 in R2C3 (step 5)

10. 22(3) at R7C2 = {589} (no 6) (last combo)
10a. 5 locked in R8C23 for N5 and R8
10b. no 8 in R7C3 (CPE)
10c. cleanup: no 5 in R2C3 (step 5)

11. Naked single (NS) in R8 at R8C8 = 6

12. Naked pair (NP) on {89} at R9C58 -> no 8,9 elsewhere in R9

13. {589} unavailable to 14(3) at R8C1
13a. -> 14(3) at R8C1 = {(16/34)7} (no 2)

14. 17(3) at R8C9 = {179/269/278/368} (no 4)
(Note: {467} blocked because 17(3) requires 1 of {89} due to R9C8)
14a. 6 only available in R9C9
14b. -> no 3 in R9C9

15. 12(2) at R8C5 requires either 3 in R8 or 8 in R9
15a. -> [386] permutation blocked for 17(3) at R8C9
15b. -> no 3 in R8C9
15c. 17(3) at R8C9 = {179/269/278} = {(1/2)..}

At this point, Para commented (via PM):

"Glad you picked up on this step, as it was in my intended walk-through as well, except you missed the follow-through. The 17(3) now either needed a 9 in C8 or a 7 in C9, which blocked the [97] combination for 16(2) in R5C89. Found that combo of moves pretty cool."

True. Wish I'd seen that. Ah well, what might have been...

(Archive Note) Typo corrected in Para's comment.

16. 8(3) at R3C9 cannot have both of {12} within C9 due to 17(3) at R8C9 (step 15c)
16a. -> no 5 in R4C8

17. 6 unavailable to 19(3) at R2C7 (step 4a) = {289/379/478} (no 5)
17a. -> 19(3) must have 2 of {789}, 1 of which must be within R23C8
17b. -> R23C8, R5C8 and R9C8 form killer triple on {789} within C8
17c. -> no 7,8,9 elsewhere in C8

18. HS in C8 at R6C8 = 5
18a. split 12(2) at R67C9 = {39/48} (no 2,6,7) = {(3/4)..}
18b. -> R67C9 and R7C8 form killer pair on {34} within N9
18c. -> no 3,4 elsewhere in N9
18d. cleanup: no 3,4 in R5C2

19. 5 in C9 locked in N1 -> not elsewhere in N1

20. 8(3) at R8C6 = {125} (no 3,4) (last combo)
20a. no 5 in R7C7 (CPE)

21. {34} in R9 locked in R9C1234
21a. R9C34 cannot contain both of {34} due to 11(3) cage sum
21b. -> R9C12 must contain at least 1 of {34}
21c. -> 14(3) at R8C1 (step 13a) = {347} (no 1,6)

22. CPE: R7C7 sees all 6's in N8
22a. -> no 6 in R7C7

23. Naked single (NS) at R7C7 = 7
23a. -> split 10(2) at R67C6 = {28/46} (no 3)

24. Grouped X-Wing on 8 as follows:
24a. 8 in N78 locked in C56
24b. -> no 8 elsewhere in C56
24c. cleanup: no 1 in R12C5

25. R89C5 = [39]
25a. cleanup: no 6 in R12C5

26. NS at R9C8 = 8
26a. -> split 9(2) at R89C9 = {27} (last combo), locked for C9
26b. no 2 in R9C7 (CPE)

27. R5C89 = [79]

28. NS at R7C2 = 9
28a. -> split 13(2) at R8C23 = {58}, 8 locked for R8 and N5

29. NS at R8C7 = 9
29a. -> R7C8 = 3 (cage sum)

30. NS at R2C7 = 8
30a. -> split 11(2) at R23C8 = [92] (last permutation)

31. 2 in 8(3) at R8C6 locked in R89C6 -> not elsewhere in C6 and N9
31a. -> R67C6 (step 23a) = {46} (no 8), locked for C6 and N8
31b. cleanup: no 5 in R5C2

32. R4C89 = [41]
32a. -> R3C9 = 3 (cage sum)

The rest is now naked singles and a few cage sums.
Andrew's walkthrough:
A nice puzzle. The hardest part was keeping track of eliminations in the disjoint nonets, even after I’d set up minimised windows so that I could view both my worksheet and Para’s original diagram at the same time. It’s possible that I may have missed some CPEs.

Prelims

a) R12C5 = {18/27/36/45}, no 9
b) R5C12 = {16/25/34}, no 7,8,9
c) R5C89 = {79}
d) R89C5 = {39/48/57}, no 1,2,6
e) 19(3) cage at R1C1 = {289/379/469/478/568}, no 1
f) 11(3) cage at R2C2 = {128/137/146/236/245}, no 9
g) 19(3) cage at R2C7 = {289/379/469/478/568}, no 1
h) 20(3) cage at R3C3 = {389/479/569/578}, no 1,2
i) 20(3) cage at R3C6 = {389/479/569/578}, no 1,2
j) 8(3) cage at R3C9 = {125/134}
k) 10(3) cage at R6C1 = {127/136/145/235}, no 8,9
l) 22(3) cage at R7C2 = {589/679}
m) 11(3) cage at R8C4 = {128/137/146/236/245}, no 9
n) 8(3) cage at R8C6 = {125/134}
o) 14(4) cage at R4C7 = {1238/1247/1256/1346/2345}, no 9

1. Naked pair {79} in R5C89, locked for R5 and NR4C9, clean-up: no 3,5 in R9C5
1a. Max R67C6 = 14 -> min R7C7 = 3
1b. 17(3) cage at R6C6 cannot be {179}, because 7,9 only in R7C7 -> no 1 in R67C6

2. 17(3) cage at R6C8 cannot be {179} which clashes with R5C9 (because no 7,9 in R6C8), no 1

3. 8(3) cage at R8C6 = {125/134}, CPE no 1 in R8C7

4. 22(3) cage at R7C2 = {589/679}, CPE no 9 in R7C3 + R8C189 + R9C2

5. 45 rule on R12 2 outies R3C28 = 9 = [18]/{27/36/45}, no 8 in R3C2, no 9 in R3C8

6. 45 rule on R89 2 outies R7C28 = 12 = [57/75/84/93], no 6 in R7C2, R7C8 = {3457}
6a. 45 rule on R89 4 innies R8C2378 = 28 = {4789/5689}, no 1,2,3, 8,9 locked for R8, clean-up: no 4 in R9C5
6b. R8C78 cannot be {47} because 18(3) cage at R7C8 cannot be 7{47}, R8C78 cannot be {48} because no 6 in R7C8, R8C78 cannot be [94] because 22(3) cage at R7C2 cannot contain both of 7,8 -> no 4 in R8C78
6c. R8C2378 = {5689} (only remaining combination), locked for R8, clean-up: no 7 in R9C5
6d. 18(3) cage at R7C8 = {369/468/567} (cannot be {459} = [495] because 22(3) cage at R7C2 cannot contain both of 6,8), 6 locked for R8
6e. 18(3) cage = {369/468/567}, CPE no 6 in R1C7
6f. 22(3) cage = {589} (only remaining combination), no 7, clean-up: no 5 in R7C8
6g. Naked triple {589} in 22(3) cage, CPE no 5,8 in R7C3 + R8C8 + R9C2 -> R8C8 = 6, placed for disjoint nonet, no 6 in R6C2 + R7C1, clean-up: no 3 in R3C2 (step 5)

7. 45 rule on C12 2 outies R28C3 = 11 = [29/38/65]

8. 45 rule on C89 2 outies R28C7 = 17 = {89}, locked for C7
8a. 5 in R8 only in R8C23, locked for 22(3) cage at R7C2, no 5 in R7C2, clean-up: no 7 in R7C8 (step 5)
8b. 5 in R8C23, locked for disjoint nonet, no 5 in R1C89 + R2C8 + R9C19
8c. 19(3) cage at R2C7 = {289/379/478}, no 5, clean-up: no 4 in R3C2 (step 5)
8d. 1,2 in NR2C7 only in R4C5 + R5C34 + R6C3, CPE no 1,2 in R4C3

9. 20(3) cage at R3C6 = {479/569/578} (cannot be {389} which clashes with R2C7), no 3
9a. Killer pair 8,9 in R2C7 and 20(3) cage, locked for NR2C7
9b. 8,9 in NR2C7 only in R2C7 and R34C6, CPE no 8,9 in R2C6
9c. 20(3) cage at R3C3 = {389/479/569/578}
9d. 3 of {389} must be in R4C4 -> no 3 in R3C34
9e. 3 in NR2C7 only in R4C45 + R5C34 + R6C3, CPE no 3 in R4C3

10. 8 in R5 only in R5C56, locked for NR3C8, no 8 in R3C8 + R6C45 + R7C4, clean-up: no 1 in R3C2 (step 5)
10a. 19(3) cage at R2C7 (step 8c) = {289/379/478}
10b. 2 of {289} must be in R3C8 -> no 2 in R2C8

11. R7C2 + R9C8 = {89} (hidden pair in disjoint nonet)
11a. Naked pair {89} in R9C58, locked for R9
11b. 5 in C8 only in R46C8, CPE no 5 in R4C9 + R6C45

12. 19(3) cage at R2C7 (step 8c) = {289/379/478}
12a. Killer triple 7,8,9 in 19(3) cage, R5C8 and R9C8, locked for C8
12b. Max R6C8 = 5 -> min R67C9 = 12, no 2
12c. Hidden killer pair 8,9 in 17(3) cage at R6C8 and R8C7 for disjoint nonet, R8C7 = {89} -> 17(3) cage must contain one of 8,9 = {269/278/359/368/458} (cannot be {467}

13. 17(3) cage at R8C9 = {179/269/278/368} (cannot be {467} because R9C8 only contains 8,9), no 4
13a. 6 of {368} must be in R9C9 -> no 3 in R9C9
[I didn’t spot Mike’s {368} = [368] clashes with R89C5 = [39/48] which, as Para pointed out, would have led to R5C89 = [79] (cannot be [97] which clashes with 17(3) cage).]

14. 11(3) cage at R2C2 = {137/236/245} (cannot be {128/146} because 1,4,8 only in R2C2), no 8
14a. 5,7 of {137/245} must be in R3C2 -> no 5,7 in R2C2

15. Hidden killer pair 8,9 in R1C9 + R2C8 and R8C23 for disjoint nonet, R8C23 contains one of 8,9 -> R1C9 + R2C8 must contain one of 8,9
15a. 8,9 in Para’s red disjoint nonet only in R1C123 + R2C19
15b. Double killer pair 8,9 in R1C9 + R2C8, R1C123 + R2C19 and R2C7, locked for R12, no 8,9 in R1C456 + R2C45, clean-up: no 1 in R12C5
15c. Max R12C4 = 13 -> min R1C3 = 2
[Cracked. The rest is fairly straightforward.]

16. 9 in C6 only in R34C6, locked for NR2C7 -> R2C7 = 8, placed for NR2C7, R8C7 = 9, placed for disjoint nonet, no 9 in R67C9, R7C8 = 3 (cage sum), placed for disjoint nonet, no 3 in R56C1 + R6C9 + R89C6, R7C2 = 9 (step 6), clean-up: no 2 in R2C3 (step 7), no 6 in R3C2 (step 5), no 4 in R5C2
16a. Naked pair {58} in R8C23, locked for disjoint nonet, no 8 in R1C9
[I missed that R7C2 was placed for a disjoint nonet, but got the result in the next step.]

17. R9C8 = 8 (hidden single in C8), R9C5 = 9, placed for disjoint nonet, no 9 in R34C1, R8C5 = 3, placed for NR4C9, clean-up: no 6 in R12C5

18. 11(3) cage at R2C2 (step 14) = {137/236} (cannot be {245} because R2C3 only contains 3,6), no 4,5, clean-up: no 4 in R3C8 (step 5)
18a. 11(3) cage at R2C2 = {137/236}, 3 locked for R2 and disjoint nonet, no 3 in R1C4 + R34C1
18b. R3C2 = {27} -> no 2 in R2C2
18c. Naked pair {27} in R3C28, locked for R3

19. R34C1 must contain 8 (only remaining positions in disjoint nonet), locked for C1 and 18(3) cage at R3C1, no 8 in R4C2
19a. 18(3) cage = {378/468}, no 1,2,5
19b. 3 of {378} must be in R4C2 -> no 7 in R4C2

20. 9 in C1 only in R12C1, locked for disjoint nonet, no 9 in R1C3 + R2C9
20a. 19(3) cage at R1C1 = {289/379/469}, no 5
20b. 8 of {289} must be in R1C2 -> no 2 in R1C2

21. R9C8 = 8 -> R89C9 = 9 = {27}, locked for C9 -> R5C89 = [79], R3C8 = 2, placed for NR3C8, R2C8 = 9 (cage sum), R3C2 = 7, placed for disjoint nonet, no 7 in R1C45 + R2C4 + R4C1, clean-up: no 2 in R2C5
21a. R3C2 = 7 -> R2C23 = 4 = [13], 1 placed for disjoint nonet, no 1 in R1C4, R8C3 = 8 (step 7), R8C2 = 5, clean-up: no 2,6 in R5C1

22. R1C1 = 9 (hidden single in R1)

23. R4C1 = 8 (hidden single in R4) -> 18(3) cage at R3C1 (step 19a) = {468} (only remaining combination), no 3
23a. Naked pair {46} in R3C1 + R4C2, CPE no 4,6 in R3C345

24. R1C2 = 8 (hidden single in C2), R2C1 = 2 (cage sum), placed for disjoint nonet, no 2 in R1C3 + 11(3) cage at R8C4

25. R9C1 = 3 (hidden single in C1), placed for disjoint nonet, no 3 in R1C9, R8C1 + R9C2 = 11 = {47} -> R8C1 = 7 placed for disjoint nonet, no 7 in R1C7, R9C2 = 4, placed for disjoint nonet, no 4 in R1C89 + R7C3, R8C9 = 2, placed for disjoint nonet, no 2 in R1C7 + R6C2 + R9C7, R9C9 = 7, placed for disjoint nonet, no 7 in R7C3, R4C2 = 6, placed for NR1C6, R3C1 = 4, placed for disjoint nonet, no 4 in R1C45 + R2C4, clean-up: no 5 in R2C5

26. R6C2 = 3, placed for disjoint nonet, no 3 in R1C7, R5C2 = 2, placed for NR1C6, R5C1 = 5, placed for disjoint nonet, no 5 in R6C9 + R7C79 + R9C6, R7C1 = 1, placed for disjoint nonet, no 1 in R19C7, R6C1 = 6, placed for disjoint nonet, no 6 in R7C79

27. R9C7 = 5, R89C6 = 3 = [12]
27a. R3C7 = 6, placed for NR2C7, R1C7 = 4, R7C7 = 7

28. Naked pair {13} in R45C7, locked for C7 and NR3C8, R6C7 = 2, placed for NR4C9, R5C6 = 8 (cage sum)

29. R7C7 = 7 -> R67C6 = 10 = [46], 4 placed for NR4C9 -> R4C9 = 1, R45C7 = [31]

30. R6C3 = 1 (hidden single in R6), R9C3 = 6, R5C345 = [436], R4C3 = 9 (cage sum)

and the rest is naked singles, without using the nonets.


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PostPosted: Thu Mar 06, 2014 9:20 pm 
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Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1893
Location: Lethbridge, Alberta, Canada
Texas Jigsaw Killer 33 by Ruud (November 2007)
Puzzle Diagrams:
Image   Image

Jigsaw nonet design: Orig 8.
Børge's TJK 33 images with udosuk Style Killer Cages and Ruud TJK#33 inspiration:
Image



Image



Image
Cages with cells in 3 jigsaw nonets:  pink and orange
Cages with cells in 2 jigsaw nonets: green, yellow and brown
Cages with cells in 1 jigsaw nonet: red, blue and grey


See the end of Børge's post for the explanation of this window's title.
Code: Select, Copy & Paste into solver:
SumoCueV1=21J0+0J0=20J0+2J1=29J1=13J1+5J1+5J1=10J2=19J3+0J0+0J0+2J1+4J1+5J1=13J2+15J1+8J2+9J3+9J3=14J0+20J0+4J4+4J4+15J2=23J2+8J2=20J3+9J3+20J0+4J0+4J4=22J4+32J2+25J2+25J2+27J3=19J3+37J3=18J4+39J4+32J4=18J4=11J5+25J5+27J6+37J6+39J3+39J6+39J6+42J4+42J5+43J5=14J5=17J6+37J6+37J6=20J6+57J7=14J7+59J7+43J5+53J5+54J8+54J8=21J8+65J6+57J7+57J7+59J7=19J5+70J5=12J8+72J8+72J8+65J8+65J8=18J8+77J7+77J7+70J7
Solution:
+-------+-------+-------+
| 5 3 8 | 7 9 6 | 2 4 1 |
| 2 9 4 | 5 8 1 | 6 3 7 |
| 8 5 7 | 6 1 3 | 4 9 2 |
+-------+-------+-------+
| 7 4 1 | 2 6 9 | 8 5 3 |
| 9 1 3 | 4 2 5 | 7 8 6 |
| 4 7 6 | 1 5 8 | 3 2 9 |
+-------+-------+-------+
| 3 6 2 | 8 7 4 | 9 1 5 |
| 6 8 5 | 9 3 2 | 1 7 4 |
| 1 2 9 | 3 4 7 | 5 6 8 |
+-------+-------+-------+

Quote:
SSscore: 2.15

Para: This was a fun puzzle. About the difficulty level i like for a Texas Jigsaw Killer. Not too hard but needs that extra bit of thinking. My walk-through is the revised version, took out a lot of steps and analysis out that wasn't really necessary. Also edited the order a bit. In the end the clean up kinda disappeared.

Caida: I had started this walkthrough and kept getting stumped so I went through Para's walkthrough looking for clues as to where I could go next.
I found that I didn't see Para's step 12 and 14a & b - so had to get an explanation. (Thanks Para!!!)
Armed with the explanation I needed, I tackled the puzzle again. I did use Para's step 12 and 14a&b (my step 1), but I think I came up with a different enough solution path that I am posting it below.

Andrew (in 2014): This was the last Texas Jigsaw Killer posted by Ruud.
I agree with part of Para's comment; this puzzle definitely needs that extra bit of thinking. It doesn't take too long to find the key area but then it takes time to work out how to break in. The three of us did that in different ways.
While quite a hard puzzle, the SSscore seems too high; it's more like a 1.5 puzzle.

Para's walkthrough:
This was a fun puzzle. About the difficulty level i like for a Texas Jigsaw Killer. Not too hard but needs that extra bit of thinking. My walk-through is the revised version, took out a lot of steps and analysis out that wasn't really necessary. Also edited the order a bit. In the end the clean up kinda disappeared.


Walk-through TJK 33

Nonet Numbering:
111222223
411222323
441155333
441155333
444555566
774775666
777788866
999788866
999999888


1. 20(3) at R1C3, R4C1 = {389/479/569/578}: no 1,2

2. 13(4) at R1C6 = {1237/1246/1345}: no 8,9; 1 locked for N2

3. 10(3) at R1C9 = {127/136/145/235}: no 8,9

4. 22(3) at R4C6 = {589/679}: no 1,2,3,4; 9 locked -> pointing R4C5 + R5C7: no 9

5. 18(5) at R5C4 = {12348/12357/12456}: no 9; 1,2 locked -> pointing R6C6: no 1,2

6. 11(3) at R5C8 = {128/137/146/236/245}: no 9

7. R67C9 = {59/68}: no 1,2,3,4,7

8. 19(3) at R8C8 = {379/478}: {289/469/568} blocked by R67C9: no 1,2,5,6; 7 locked in cage -> pointing: R5C9, R8C567: no 7
8a. Killer Pair {89} in R67C9 + 19(3) cage -> pointing: R5C9: no 8.9

9. 45 on C9: 3 outies: R348C8 = 21 = {489/579/678}: no 1,2,3

10. 45 on C6789: 2 innies: R38C6 = 5 = {14/23}

11. 45 on N1: 2 innies: R1C3 + R4C4 = 10 = {37/46}/[82/91]: no 5; R4C4: no 8,9

12. 19(5) at R5C2 = {12349/12358/12367/12457/13456}: 1 locked.
12a. 1 locked in 19(5) and 18(5) at R5C4: 1 locked within R5C2345 + R6C3 and R6C245 + R7C23 -> 1 locked for N7 within R6C245 + R7C23

13. LOL R89: R8C489 = R7C567: no 1

14. LOL R6789: R5C89 = R6C36: no 9
14a. (from step 12a.) 1 locked within R5C234589 for R5 -> R5C7: no 1
14b. 18(3) at R5C7 = [279]/{36}[9]/{45}[9]/{378/468/567}: no 1; R6C7: no 2

15. 45 on N3: 1 outie and 4 innies: R2C8 + 22 = R3C8 + R4C789: R2C8: no 9

16. 45 on C89: 3 innies: R129C8 = 13
16a. Cageoverlap: R19C8 = R23C7
16b. LOL C89: R2346C7 = R129C8 + R9C9
16c. From step 16a.:LOL C89: R19C8 = R23C7; R2C8 + R9C9 = R46C7: no 2
16d. When R9C9 = 9, R6C7 = 9(hidden single N6) -> R4C7: no 9(When R9C9 no 9, R4C7 no 9 through LOL C89)
16e. 9 in 22(3) at R4C6 locked for C6 and N5

17. 9 in N6 either in R6C7, 14(2) at R6C9 or 19(3) at R8C8
17a. R6C7 -> R9C9 = 9(step 16b), so either 14(2) = {59} or 19(3) = {739}: R6789C9 = {3|5..}
17b. 10(3) at R1C9 = {127/136/145}: {235} blocked by step 17a.: 1 locked for C9 and N3
17c. Clean up: R19C8: no 1(LOL C89)

18. 13(3) at R2C7 = {238/247/256/346}: no 9
18a. Clean up: R9C8: no 9(LOL C89)

19. 9 in N3 locked within 23(4) at R3C8 -> 23(4) = {2489/2579/3569} = {2|3..}(in R45C9),{4|5..}: {3479} blocked by 10(3) at R1C9
19a. 10(3) at R1C9 = {127/136}= {2|3..}: {145} blocked by 23(4): no 4,5
19b. Killer Pair {23} in 10(3) at R1C9 + R45C9 -> locked for C9
19c. 19(3) at R8C8 = {478}: no 9 -> pointing: R567C9 + R8C567: no 4,8
19d. R67C9 = {59}(last combo) -> locked for C9 and N6
19e. Jigsaw Triple {478} in R7C567(LOL R89) + R9C9 -> locked for N8
19f. R7C567 can't use 3 of {478} -> R8C4: no 4,7,8(LOL R89)
19g. Clean up: R23C7: no 8(LOL C89); R6C36: no 5(LOL R6789)

20. 45 on R789: 4 innies: R7C2389 = 14 = {126}[5]/{234}[5]: no 7,8,9; R7C9 = 5; 2 locked for R7
20a. 1 in R7 locked within innies: R7C238 = {126} -> locked for R7
20b. R6C9 = 9
20c. Clean up: R8C4: no 2,5,6(LOL R89)

21. 11(3) at R5C8 = {128/137/146} = {4|7|8..}: {236} blocked by R5C9
21a. Killer Triple {478} in 11(3) at R5C8 + R8C89 -> locked for N6
21b. Clean up: R2C8 = {36}; R4C7 = {78}; R9C9 = {78} (all through LOL C89)
21c. 4 in R9 locked for N9
21d. 4 in R8 locked for N6
21e. 4 in N8 locked for R7
21f. 4 in N7 locked for R6
21g. Clean up: R567C8: no 6; R45C6: no 7,8

22. 45 on N8: 1 innie and 2 outies: R9C9 + 7 = R7C4 + R9C6: R7C4 + R9C6 = [95/86/87/96]: [78] blocked by IOU: R7C4 = {89}; R9C6 = {567}

23. LOL R89: R7C567 needs 2 of {478} and 1 of {39}
23a. 18(3) at R9C6 = [792]/[7]{56}: [693] blocked by R7C567; R9C6 = 7; R9C7 = {569}; R9C8 = {256}
23b. R9C9 = 8; R7C4 = 8(step 22); R6C6 = 8(hidden)
23c. R4C7 = 8(hidden); R5C8 = 8(hidden)
23d. Clean up: R45C6 = {59} -> locked for C6 and N5

24. 9 in C7 locked for N8

25. 14(3) at R7C6 = [491]: {347} blocked by R7C5; [392] blocked by step 23(LOL R89)
25a. R7C5 = 7(hidden); R7C1 = 3; R8C4 = 9

26. 6 in C9 locked within R12345C9 -> CPE: R34C8: no 6
26a. 6 in C8 locked within R129C8 -> locked within R236C7(LOL C89) for C7
26b. R9C78 = [56]; R2C8 = 3; R6C7 = 3(LOL C89); R5C7 = 7; R5C9 = 6(hidden)
26c. 6 in R7 locked for N7
26d. R6C3 = 6(hidden); R7C2 = 6(hidden); R8C1 = 6; R8C2 = 8; R8C3 = 5(hidden)

Now it is really down to basics.

27. 23(4) at R3C8 = 6{59}[3] -> R4C9 = 3; R34C8 = {59} -> locked for C8
27a. R8C9 = 4(hidden); R8C8 = 7; R1C8 = 4(hidden); R1C7 = 2
27b. R12C6 = {16} -> locked for N2 and C6

28. 20(3) at R1C3 = [8]{57}: R1C3 = 8; R12C4 = {57} -> locked for C4 and N2
28a. R12C5 = [98]; R6C5 = 5(hidden); R1C6 = 6(hidden); R2C7 = 6(hidden)
28b. R2C6 = 1; R3C7 = 4; R1C2 = 3(hidden); R5C3 = 3(hidden); R9C4 = 3(hidden)
28c. R9C5 = 4; R5C4 = 4(hidden)

29. 6 in C5 locked for 29(6) cage at R1C5
29a. R4C5 = 6(hidden); R3C4 = 6(hidden)
29a. R34C3 = {17}(last combo within 14(3)) -> locked for C3 and N1

And the rest is all naked singles.

(Archive Note) Typos corrected; the addition to step 14a was in Para's posted walkthrough.
Caida's walkthrough:
I had started this walkthrough and kept getting stumped so I went through Para's walkthrough looking for clues as to where I could go next.

I found that I didn't see Para's step 12 and 14a & b - so had to get an explanation. (Thanks Para!!!)
Armed with the explanation I needed, I tackled the puzzle again. I did use Para's step 12 and 14a&b (my step 1), but I think I came up with a different enough solution path that I am posting it below.


TJK #33 Walkthrough

Nonets:

111222223
411222323
441155333
441155333
444555566
774775666
777788866
999788866
999999888


Preliminaries:

a. 20(3)n12 and n47 = {389/479/569/578} (no 1,2)
b. 13(4)n2 = {1237/1246/1345} (no 8,9) 1 locked for n2 in 13(4) -> no 1 elsewhere in n2
c. 10(3)n3 = {127/136/145/235} (no 8,9)
d. 22(3)n53 = {589/679} (no 1..4) -> no 9 in r4c5 or r5c7 (common peers)
e. 11(3)n6 = {128/137/146/236/245} (no 9)
f. 14(2)n6 = {59/68} (no 1..4,7)
g. 19(3)n68 = {289/379/469/478/568} (no 1)
h. 18(5)n547 = {12348/12357/12456} (no 9) -> no 1,2 in r6c6 (common peers)

My step 1 is steps 12, 13, and 14a&b in Para’s walkthrough (Thanks Para for helping me to see this!!)
1. 19(5)n47 = {12349/12358/12367/12457/13456}
1a. -> 18(5)n547 and 19(5)n47 both require 1 (prelim h)
1b. r7c14 and r8c4 no 1 (common peers)
1c. 1 is locked in n7 within either 18(5) or 19(5); therefore one of the 1s in 18(5) and 19(5) must be in n7 (group 1)
1d. therefore the other 1 must be in r5c2345 + r6c3 (group 2); meaning either there is a 1 in r5c2345 OR in r6c3
1e. LOL move on r6789: r5c89 = r6c36; this means that if there is a 1 in r6c3 then there MUST be a 1 in r5c89
1f. -> and if there is NOT a 1 in r6c3 then there must be a 1 in r5c2345 (step 1f)
1h. -> 1 locked for r5 in r5c234589; no 1 in r5c7
1i. LOL r89: r7c567 = r8c489; no 1

2. 19(3)n68 = {379/478} (other combinations {289/469/568}blocked by 14(2)n6)
2a. -> 19(3)n68 no 2,5,6
2b. -> r5c9 and r8c567 no 7 (common peers)
2c. killer pair {89} in 19(3)n68 and 14(2)n6
2d. -> r5c9 no 8,9 (common peer)

3. Innies r789: r7c2389 = 14(4) = {1238/1256/1346/2345} (no 7,9)
Note: combo 1247 not possible because of r7c9
3a. r7c238 no 8 (any 8 must be in r7c9)
3a. cleanup: r6c9 no 5

4. Outties c9: r348c8 = 21(3) = {489/579/678} (no 1..3)

5. Innies c6789: r38c6 = 5(2) = {14/23} (no 5..9)

6. 14(2)n6 and 19(3)n68 = {59}/{478} or {68}/{379}
6a. -> either r7c9 contains a 5 OR r89c9 contains a 3
6b. -> this blocks combo {235} from 10(3)n3
6c. -> 10(3)n3 = {127/136/145} -> 1 locked for n3 and c9

7. Innies and Outties n6: r5c9 + r6c7 – r9c9 = 1
7a. -> r6c7 no 1, 9
7b. 1 now locked in 11(3)n6 for c8 = {128/137/146} no 5

8. LOL r6789: r6c36 = r5c89 (step 1e) no 9

9. 18(3)n56 = {378/468/567} no 2

10. Innies and Outies n3: r3c8+r4c789 – r2c8 = 22
10a. ->max r3c8+r4c789 = {6789} = 30; max r2c8 = 8 (no 9)
10b. LOL c789: r1c78+r2c8+r5c7 = r78c56 no 9
10c. r9c9 no 9 (CPE with all 9s in n6)

11. Innies c89: r129c8 = 13(3) = {238/247/256/346} (no 9)
11a. LOL: c89: r2346c7 = r129c8+r9c9 (no 9)
11b. -> 9 locked in 22(3)n53 within r45c6 - > no 9 elsewhere in n5 or c6

12. 9 locked in n3 in 23(4)n36 = {2489/2579/3569}
Note: {3479} blocked by 10(3)n3
12a. 10(3)n3 = {127/136} no 4,5
Note: combo {145} blocked by 23(4)n36
12b. killer pair {23} locked in 10(3)n3 and 23(4)n6 for c9 -> no 2,3 elsewhere in c9
12c. 19(3)n68 = {478} no 9 -> no 4,8 in r567c9 or r8c567 (CPE)
12d. 14(2)n6 = [95]
12e. 23(4)n36: r34c8 no 6,7 (combos with 6 or 7 {3569/2579} – must have {59} in r34c8 b/c of 14(2)n6)

13. Innies n1: r1c3+r4c4 = 10(2) = {19/28/37/46} (no 5)
13a. r4c4 no 8,9

14. revisit of step 3: Innies r789: r7c238 = 9(3) = {126/234}
14a. 2 locked for r7 in c238
14b. LOL r89: r7c567 = r8c489; no 2,5

15. looking again at the 19(3)n68
15a. -> if r9c9 = 4 then 4 in n9 is in r8c123; if r9c9 <> 4 then r8c89 = 4 then 4 in n9 is in r9c123456
15b. -> no possibility of 4 in r9c78 or r8c4
15c. same can be said of 7 and 8
15d. summarizing: r9c78 and r8c4 no 4,7,8

16. revisit of step 7: Innies and Outties n6: r5c9 + r6c7 – r9c9 = 1
16a. -> max of r9c9 = 8; max of r5c9+r6c7 = 9
16b. -> r6c7 no 4, 8

17. 5 locked in r6 for n7 -> no 5 elsewhere in r6

18. revisit of step 9: 18(3)n56 = {378/468/567}
18a. -> no 6 in r5c7 (either 6 is in r6c7 for {468} or 5 is in r5c7 for {567})

19. if 10(3)n3 = {136} then r5c9 = 2 otherwise 10(3)n3 = {127}
19a. r4c9 no 2

20. 18(3)n98 = [792/396/693/495/756/765]
20a. r9c6 = {3467} no 1,2,5,8; r9c7 = {569} no 1,2,3
20b. 1 locked in r9 for n9

21. Innies and Outties n8: r7c4 + r9c6 – r9c9 = 7
21a. -> r7c4 no 3,6,7; r9c6 no 4

22. re-revisit of step 3(and 14): Innies r789: r7c238 = 9(3) = {126/234}
22a. 1 locked for r7 in c238
22b. -> r7c278 = {126} -> locked for r7
22c. LOL r89: r7c567 = r8c489; no 6

23. 11(3)n6 = {128/137/146} and r8c89 must contain 2 of {478}
23a. -> killer triple {478} locked in n6 in 11(3)n6 and r8c89
23b. -> r6c7 no 7

24. re-revisit of step 9(and step 18): 18(3)n56 = {378/468/567}
24a. r5c7 and r6c6 = {78/48}/[57] no 3,6
24b. looking at 22(3)n35 and 18(3)n56
24c. -> r4c7 no 5,6 -> blocked by 18(3)n56
24d. ->cleanup: r45c6 no 7,8

I’m not sure if this step wound up be necessary – but I was pleased with myself for spotting it
25. within n3 7 and 8 are locked in 10(3) 23(4) and r4c7 (if 10(3)={127} then r4c7 = {8}; if 10(3)= {136} then 23(4)in n3 = {489/579} and r4c7 = {7/8}
25a. -> r23c7 no 7,8

26. Outies r9: r8c3489 = 25(4) = [59]{47}

Now everything falls into place

26a. r9c9 = 8
26b. 11(3)n6 = {128}
26c. 23(4)n36 = {3569}
26d. 10(3)n3 = {127}
26f. r8c89 = [74]
26g. 22(3)n35 = {589}
26h. HS: r9c7 = 5 -> 18(3)n98 = [756]
26i. 12(3)n9 = {129}
Archive note Insert 13(3)n32={346}
26j. 18(3)n56 = [783]
26k. r45c9 = [36]
26l. 17(3)n79 = [3]{68}
26m. 14(3)n8 = [491]
26n. 20(4)n78 = [87]{23}
26o. HS: r5c8 = 8
26p. 13(4)n2 = {1246}
26q. r12c8 = [43]; r1c7 = 2
26r. 20(3)n12 = [8]{57}
26s. r12c5 = [98]
26t. HS: r1c2 = 3; r5c3 = 3; r9c4 = 3, r9c5=4
26u. HS: r5c4 = 4; r6c5 = 5
26v. 18(5)n457 = {12456} no 7
26w. 19(5)n47 = {12367} no 4
26x. 20(3)n47 = [794]
26y. 19(4)n4 = {2458}
26z. Everything left is now singles

(Archive Note) Typos corrected; also a couple of omissions from step 26.
Andrew's walkthrough:
Prelims

a) R67C9 = {59/68}
b) 20(3) cage at R1C3 = [389/479/569/578}, no 1,2
c) 10(3) cage at R1C9 = {127/136/145/235}, no 8,9
d) 20(3) cage at R4C1 = [389/479/569/578}, no 1,2
e) 22(3) cage at R4C6 = {589/679}
f) 11(3) cage at R5C8 = {128/137/146/236/245}, no 9
g) 19(3) cage at R8C8 = {289/379/469/478/568}, no 1
h) 13(4) cage at R1C6 = {1237/1246/1345}, no 8,9
i) 18(5) cage at R5C4 = {12348/12357/12456}, no 9

Steps resulting from Prelims
1a. 13(4) cage at R1C6 = {1237/1246/1345}, 1 locked for NR1C4
1b. 22(3) cage at R4C6 = {589/679}, CPE no 9 in R4C5+R5C7
1c. 18(5) cage at R5C4 = {12348/12357/12456}, CPE no 1,2 in R6C6

2. 19(3) cage at R8C8 = {379/478} (cannot be {289/469/568} which clash with R67C9), no 2,5,6
2a. 19(3) cage = {379/478}, CPE no 7 in R5C9+R8C567
2b. Killer pair 8,9 in R67C9 and 19(3) cage, no 8,9 in R5C9

3. 45 rule on C9 3 outies R348C8 = 21 = {489/579/678}, no 1,2,3

4. 45 rule on C6789 2 innies R38C6 = 5 = {14/23}

5. 45 rule on R789 4 innies R7C2389 = 14 = {1238/1256/1346/2345} (cannot be {1247} because no 1,2,4,7 in R7C9) , no 7,9, clean-up: no 5 in R6C9
5a. 8 of {1238} must be in R7C9 -> no 8 in R7C238

6. 45 rule on NR1C1 2 innies R1C3 + R4C4 = 10 = {37/46}/[82/91], no 5, no 8,9 in R4C4

7. 45 rule on NR2C1 + NR6C1 + NR7C5 + NR8C1 1 innie R9C9 = 2 outies R5C45 + 2
7a. Max R5C45 = 7, no 7,8 in R5C45

8. 45 rule on NR7C5 2(1+1) outies R7C4 + R9C6 = 1 innie R9C9 + 7, IOU no 7 in R7C4

9. Law of Leftovers (LoL) for R6789 two outies R5C89 must exactly equal two innies R6C36, no 9 in R5C89 -> no 9 in R6C6

10. 45 rule on R123 4 outies R4C2345 = 1 innie R3C8 + 4
10a. Max R4C2345 = 13, no 8,9, 1 locked for R4
10b. Min R4C2345 = 10 -> min R3C8 = 6

11. 45 rule on NR5C8 2 innies R5C9 + R6C7 = 1 outie R9C9 + 1, IOU no 1 in R6C7

12. 19(5) cage at R5C2 and 18(5) cage at R5C4 both contain 1, 1 in NR5C8 only in 11(3) cage at R5C8 + R5C9
12a. Grouped Swordfish for 1 in 19(5) cage at R5C2, 18(5) cage at R5C4 and 11(3) cage at R5C8 + R5C9, no other 1 in R567
12b. LoL for R89, no 1 in R7C567 -> no 1 in R8C4

13. 1 in R7 only in R7C238 -> R7C2389 (step 5) = {1238/1256/1346}
13a. R7C2389 is locked for R7, for any of these combinations -> min two other cells in R7 = 7 -> min R7C67 = 7 -> no 8,9 in R8C7

14. Deleted, replaced by the simpler step 12b.

15. 10(3) cage at R1C9 = {127/136/145/235}, 11(3) cage at R5C8 = {128/137/146/236/245}
15a. Consider combinations for 19(3) cage at R8C8 (step 2) = {379/478}
15b. 19(3) cage = {379}, 3 locked for C9 => 10(3) cage = {127/145}, 1 locked for C9 => 1 in NR5C8 only in 11(3) cage = {128/137/146}
or 19(3) cage = {478} => R67C9 = [95]
-> 11(3) cage = {128/137/146/236}, no 5
[With hindsight I ought to have eliminated {235} from the 10(3) cage, but the next step does this and more.]

[At this stage I spent some time looking at LoL for C9, five outies R234C7 + R34C8 must exactly equal five innies R56789C9, R348C8 = 21 (step 3) -> R34C8 cannot equal R89C9 -> at least one of R34C8 must be in R67C9, but I couldn’t find a way to use this.]

16. 19(3) cage at R8C8 (step 2) = {379/478}
16a. Consider combinations for 11(3) cage at R5C8 (step 15b) = {128/137/146/236}
11(3) cage = {128/146/236} => R67C9 = [95]
or 11(3) cage = {137}, locked for NR5C8 => 19(3) cage = {48}7 => R67C9 = [95]
-> R67C9 = [95], both placed for NR5C8
16b. 19(3) cage at R8C8 = {478} (only remaining combination), CPE no 4,8 in R5C9+R8C567
16c. 5 in NR1C9 only in R234C7 + R4C8, CPE no 5 in R2C8

17. 10(3) cage at R1C9 = {127/136}, no 4, 1 locked for C9 and NR1C9
17a. 1 in NR5C8 only in 11(3) cage at R5C8 (step 15b) = {128/137/146}, 1 locked for C8

18. R7C9 = 5 -> R7C2389 (step 13) = {1256} (only remaining combination), locked for R7

19. 13(3) cage at R2C7 = {238/247/256/346}, no 9

20. 6 in C9 only in R12345C9, CPE no 6 in R34C8

21. R348C8 (step 3) = {489/579}, 9 locked for C8 and NR1C9
21a. 7 of {579} must be in R8C8 -> no 7 in R34C8

22. 22(3) cage at R4C6 = {589/679}, 9 locked for C6 and NR3C5
22a. 9 in C7 only in R79C7, locked for NR7C5

23. R7C4 + R9C6 = R9C9 + 7 (step 8)
23a. Min R9C9 = 4 -> min R7C4 + R9C6 = 11 -> min R9C6 = 2
23b. R9C9 = {478} -> R7C4 + R9C6 = 11,14,15 -> no 4 in R9C6 (because no 7 in R7C4)

24. 18(3) cage at R9C6 = {279/369/378/459/468/567} (cannot be {189} because 1,9 only in R9C7)
24a. 9 of {279} must be in R9C7 -> no 2 in R9C7
24b. 1 in NR7C5 only in R8C567, locked for R8

25. 45 rule on C9 2 innies R45C9 = 1 outie R8C8 + 2, IOU no 2 in R4C9

26. R5C9 + R6C7 = R9C9 + 1 (step 11)
26a. R9C9 = {478} -> R5C9 + R6C7 = 5,8,9 = {23/26/36} (cannot be [27] which clashes with 19(3) cage at R8C8 = {47}8), no 4,7,8 in R6C7
[Or, with hindsight, hidden killer triple 2,3,6 in R5C9 + R6C7 and 11(3) cage at R5C8 for NR5C8, 11(3) cage (step 17a) contains one of 2,3,6 -> R5C9 + R6C7 = {236}.]

27. LoL for C9 five outies R234C7 + R34C8 must exactly equal five innies R56789C9
27a. 13(3) cage at R2C7 (step 19) = {238/247/256/346}
27b. 8 of {238} must be in R23C7 (R23C7 cannot be {23} because only one of 2,3 in R56789C9), no 8 in R2C8

28. LoL for R789 four outies R6C1245 must exactly equal four innies R78C89
28a. R7C9 = 5 -> R6C1245 must contain 5, locked for R6 and NR6C1
28b. No 3 in R78C89 -> no 3 in R6C1245

29. 45 rule on R9 3 innies R9C459 = 15 = {168/249/258/267/348/357/456} (cannot be {159} because R9C9 only contains 4,7,8)
29a. 7 of {267/357} must be in R9C9 -> no 7 in R9C45

30. 18(3) cage at R5C7 = {378/468/567}, no 2
30a. R6C7 = {36} -> no 3,6 in R5C7 + R6C8

31. LoL for R6789 two outies R5C89 must exactly equal two innies R6C36
31a. R5C9 = {236}, R6C6 = {478} cannot be equal -> R5C8 = {478}, R6C3 = {236}
31b. Naked triple {478} in R5C8 + R8C89, locked for NR5C8
[Or 11(3) cage at R5C8 can only contain one of 4,7,8, R5C8 = {478} -> no 4,7,8 in R6C8.]

32. 11(3) cage at R5C8 (step 17a) = {128/137/146}, R348C8 (step 21) = {489/579} -> combined cage R345678C8 = {128}{579}/{137}{489}/{146}{579}, 7 locked for C8
32a. 7 in C8 only in R58C8, locked for NR5C8

33. LoL for R89 three outies R7C567 must exactly equal three innies R8C489, no 2,6 in R7C567 -> no 2,6 in R8C4

34. 25 rule on R9 4 outies R8C3489 = 25 = {4579/4678} (cannot be {2689} because 2,6 only in R8C3, cannot be {3589/3679} because R8C89 only contain 4,7,8), no 2,3, 4,7 locked for R8
34a. 5,6 only in R8C3 -> R8C3 = {56}
[I originally missed that step 34 locked both of 4,7 (I’d only spotted 7) for R8, so I’ve re-worked my steps from here.]

35. LoL for R9 three outies R8C123 must exactly equal three innies R9C789
35a. R8C3 = {56} -> R9C78 must contain at least one of 5,6 -> 18(3) cage at R9C6 (step 24) = {369/459/468/567} (cannot be {279/378} which don’t contain 5 or 6), no 2
35b. 9 of {369} must be R9C7 -> no 3 in R9C7

[And more important, but I saw the {56} first …]
36. LoL for R9 three outies R8C123 must exactly equal three innies R9C789, no 4,7 in R8C123 -> no 4,7 in R9C789
36a. R9C9 = 8, placed for NR7C5 -> R8C89 = [74], R5C8 = 8, R67C8 = 3 = {12}, locked for C8 and NR5C8
36b. R348C8 = 21 (step 3), R3C8 = 9, R8C8 = 7 -> R4C8 = 5 (cage sum), placed for NR1C9
36c. R34C8 = [95] = 14 -> R45C9 = 9 = {36}, locked for C9
36d. Naked triple {127} in 10(3) cage at R1C9, locked for NR1C9

37. LoL for R89, three outies R7C567 must exactly equal three innies R8C489, no 8 in R7C567 -> R8C4 = 9, R7C7 = 9
37a. R8C89 = [74] -> R7C56 = {47}, locked for R7
37b. Naked pair {38} in R7C14, locked for NR6C1
37c. R8C3489 (step 34) = {4579} (only remaining combination) -> R8C3 = 5, placed for NR8C1

38. R9C7 = 5 (hidden single in R9) -> R9C68 = 13 = [76], 6 placed for NR7C5, R7C56 = [74], R6C6 = 8

39. R8C12 = {68} (hidden pair in R8), R7C1 = 3 (cage sum), R7C4 = 8

40. 9 in R9 only in 12(3) cage at R9C1 = {129}, locked for R9

41. 22(3) cage at R4C6 = {589} (only remaining combination) -> R4C7 = 8, R45C6 = [95], 5 placed for NR3C5

42. Naked triple {346} in 13(3) cage at R2C7, 6 locked for C7 and NR1C9 -> R45C9 = [36], R6C7 = 3, R5C7 = 7 (cage sum)
42a. Naked pair {46} in R23C7, locked for C7 and 13(3) cage -> R12C8 = [43], both placed for NR1C4
42b. Naked triple {126} in R12C6 + R1C7, locked for NR1C4

43. Naked pair {57} in R12C4, locked for C4 and NR1C4, R1C3 = 8 (cage sum), R12C5 = [98]

44. R7C6 = 4, R7C7 = 9, R8C7 = 1 (cage sum), R1C7 = 2
44a. Naked pair {16} in R12C6, locked for C6

45. 29(6) cage at R1C5 = {123689} (only remaining combination), no 4

46. R6C5 = 5 (hidden single in C5)
46a. 18(5) cage at R5C4 = {12456} (only remaining combination), no 3, 6 locked for R6

47. R1C3 = 8 -> R4C4 = 2 (step 6), placed for NR1C1, R3C6 = 3, R8C56 = [32], R9C45 = [34]
47a. R5C5 = 2 (hidden single in C5), R6C3 = 6
47b. R3C4 = 6 (hidden single in C4), placed for NR1C1, R34C5 = [16], R23C7 = [64], R3C3 = 7, R4C3 = 1 (cage sum), both placed for NR1C1

48. R7C3 = 2, R9C3 = 9, R2C3 = 4, R5C3 = 3, R7C8 = 1, R7C2 = 6

49. R5C3 = 3, R7C23 = [62] -> R67C2 = 8 = [17]

and the rest is naked singles, without using the nonets.


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PostPosted: Fri Mar 07, 2014 3:37 am 
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Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1893
Location: Lethbridge, Alberta, Canada
Toroidal Killer Sudoku #2 by Para (December 2007)
Puzzle Diagrams:
Image   Image

Jigsaw nonet design: "H" by Bob & Debbie Scott
Code: Select, Copy & Paste into solver:
SumoCueV1=14J0=26J1+1J0+1J1+1J2=10J3+5J3=9J3+7J4+0J0+0J0=27J0=14J1+12J2=17J3=28J2+15J3+7J5=15J0=12J5+11J0+11J6+12J2+14J2+15J2+15J3=17J5+18J5+19J5+19J0+11J6=21J2+31J6=11J2+15J7+26J5=8J7=22J5+37J8+11J6+31J6=25J6+33J2+33J7+26J5+36J7=18J5+37J8+31J6+31J8+41J6=15J4+51J7=10J7+36J7+46J1+46J8=10J8=18J8+41J6+41J4+51J7+53J4=18J7+46J1+46J8+57J1+58J8+58J3+41J4=12J4+70J4+63J0+63J1=9J1+74J1=19J8+76J3+76J4+76J3+70J4
Solution:
+-------+-------+-------+
| 5 7 6 | 4 9 2 | 8 1 3 |
| 8 1 4 | 2 7 9 | 6 3 5 |
| 9 2 3 | 6 5 8 | 4 7 1 |
+-------+-------+-------+
| 6 3 7 | 5 1 4 | 2 8 9 |
| 4 8 5 | 9 2 1 | 3 6 7 |
| 3 4 9 | 8 6 7 | 1 5 2 |
+-------+-------+-------+
| 1 6 2 | 7 4 3 | 5 9 8 |
| 7 5 1 | 3 8 6 | 9 2 4 |
| 2 9 8 | 1 3 5 | 7 4 6 |
+-------+-------+-------+

Quote:
SSscore: 1.70

Para: Sinterklaas didn't forget about all off you. Here's his present for you.
It's a Toroidal Jigsaw Sudoku with a delicious chocolate nonet in the center.
Enjoy this treat.

Caida: I found it really difficult - mostly just in keeping the nonet patterns in mind. I had to restart about a dozen times after coming up with conflicts on the home stretch - still not sure what I was doing wrong those times - but each restart helped me in simplifying my solution.
Really enjoyable puzzle Para - Thanks!!
(hope I got it right :-))

Andrew (in 2013): A nice puzzle. The hardest part was keeping track of eliminations in the disjoint nonets, even after I’d set up minimised windows so that I could view both my worksheet and Para’s original diagram at the same time. I probably wouldn’t have spotted step 5a without doing that.
It felt hard at first but, once I started using the nonets, it wasn’t too difficult.

Caida's walkthrough:
Here's my walkthrough for Toroidal TJK #2.

I found it really difficult - mostly just in keeping the nonet patterns in mind. I had to restart about a dozen times after coming up with conflicts on the home stretch - still not sure what I was doing wrong those times - but each restart helped me in simplifying my solution.

Really enjoyable puzzle Para - Thanks!!
(hope I got it right :-))

Toroidal TJK2

Nonets:

121234445
111234346
161733346
661737386
869777386
869797588
829997585
829294555
122294545

Preliminaries:

a. 26(4)r1c2 = {2789/4589/4679/5678} (no 1,3)
Note: {3689} blocked by 10(2)r1c6
b. 10(2)r1c6 and r6c9 and r7c4= {19/28/37/46} (no 5)
c. 9(3)r1c8 = {126/135/234} (no 7..9)
d. 17(3)r2c6 = {89} -> locked for c6; -> r2c57 and r3c8 no 8,9 (CPE)
e. 15(2)r3c1 = {69/78} (no 1..5)
f. 11(3)r4c7 = {128/137/146/236/245} (no 9)
g. 8(3)r5c1 = {125/134} (no 6..9) -> 1 locked for c1 and n8
h. 22(3)r5c2 = {589/679} (no 1..4)
i. 18(5)r6c2 = {12348/12357/12456} (no 9)
j. 9(2)r9c3 = {18/27/36/45} (no 9)

k. cleanup: 10(2)r1c6: r1c7 no 1,2
l. cleanup: 10(2)r6c9: r7c9 no 9
m. cleanup: 18(3)r7c5: r78c5 no 1 (needs both {89})

1. Outies c1: r29c2 = 10(2) = {19/28/37/46} (no 5)

2. Outies c9: r18c8 = 3(2) = {12} -> locked for c8
2a. -> r1c9 and r8c6 no 1,2 (CPE)
2b. -> 1 in r1 locked in n4 -> no 1 elsewhere in n4 (r9c6 no 1)

3. 11(3)r4c7: min r5c8 = 3 -> max r45c7 = 8(2) (no 8)

4. Innie and Outie r1: r1c1 = r2c9
4a. -> r1c1 no 7,8,9
4b. -> r2c9 no 1

6. 9(3)r1c8 = [162/135/153/234/243]
6a. -> r2c9 no 6
6b. -> r1c1 no 6 (step 4)

7. Innie and Outie r9: r8c1 – r9c9 = 1
7a. -> r9c9 no 9

8. 12(3)r8c8 = [192]/[1]{56}/[291]/[2]{37/46}
Note: [1]{38/47} blocked by 9(3)r1c8
8a. -> r89c9 no 8
8b. -> r8c9 no 1,2
8c. -> r8c1 no 9 (step 7)

9. Innies and Outies r6789: r6c345 – r5c16 = 18
9a. -> min r5c16 = 3 -> min r6c345 = 21 (no 1,2,3)
9b. -> max r6c345 = 24 -> max r5c16 = 6 (no 6,7)

10. Innies and Outies r1234: r5c49 - r4c567 = 9
10a. -> min r4c567 = 6 -> min r5c49 = 15 (no 1,2,3,4,5)
10b. -> max r5c49 = 17 -> max r4c567 = 8 (no 6,7,8,9)
10c. -> r4c567 = 6(3)/7(3)/8(3) = {123/124/125/134} -> 1 locked for r4 -> no 1 elsewhere in r4

11. 21(5)r4c5
11a. -> min r6c45 = 12 (from step 9a 21 less 9) -> max r4c56+r5c5 = 9 (no 7,8,9)
11b. -> 8 and 9 in n7 is locked in c4 -> no 8,9 elsewhere in c4
11c. -> cleanup: 10(2)r7c4: no 1,2
11d. -> cleanup: 9(2)r9c3: r9c3 no 1
11e. -> 9 in n2 is locked in r19c2 -> no 9 elsewhere in c2
11f. -> 9 in 22(3)r5c2 is locked in r56c3 -> no 9 elsewhere in c3 and n9
11g. -> 9 in c5 is locked in n3 -> no 9 elsewhere in n3
11h. -> single: r3c6 = 8 –> r2c6 = 9
11i. -> 9 in n1 is locked in c1 -> no 9 elsewhere in c1
11j. -> 9 in c7 is locked in n5 -> no 9 elsewhere in n5
11k. -> 9 in c8 is locked in n8 -> no 9 elsewhere in n8
11l. cleanup: 10(2)r1c6: no 1
11m. cleanup: 10(2)r7c9: no 1
11n. hidden single: r1c8 = 1; r8c8 = 2

12. 9(3)r1c8: no 4
12a. -> r1c1 no 4 (step 4)
12b. 12(3)r8c8: no 1, 5
12c. -> r8c1 no 3,6 (step 7)
12d. hidden single: r3c9 = 1

13. 17(3)r3c9 = [1]{79}
13a. -> {79} locked for c9 and n6 -> no 7,9 elsewhere in c9 and n6
13b. -> 12(3)r8c8 = [2]{46} -> {46} locked for c9 and n5 -> no 4,6 elsewhere in c9 and n6
13c. -> 9(3)r1c8 = [1]{35} -> {35} locked for c9 -> no 3,9 elsewhere in c9
13d. -> r1c1 no 2 (step 4)
13e. 10(2)r6c9 = [28]

14. 8(3)r5c1 = {134} -> locked for c1 and n8
14a. -> r1c1 and r2c9 = 5 (step 4)
14b. 15(2)r3c1 = [78/96] -> r3c1 no 6
14c. single: r1c9 = 3
14d. cleanup: 10(2)r1c6: no 7

15. 22(3)r5c2 = [6]{79}/[8]{59}
15a. -> r56c3 no 6,8

16. r8c1 = 7; r9c9 = 6 (step 7)
16a. single: r8c9 = 4
16b. r34c1 = [96]
16c. single: r5c2 = 8
16d. triple {234} locked in r346c2 -> no 2,3,4 elsewhere in c2

17. 18(3)r8c1 = [729]
17a. 14(3)r1c1 = [581]

18. 12(3)r3c2 = {23}[7] -> 2,3 locked for c2
18a. single: r6c2 = 4
18b. r45c9 = [97]

19. 5 locked in r78c2 for c2 and n2 -> no 5 elsewhere in n2
19a. 5 locked in c4 for n7 -> no 5 elsewhere in n7
19b. 5 locked in c5 for n3 -> no 5 elsewhere in n3
19c. 5 locked in c6 for n4 -> no 5 elsewhere in n4

20. hidden single: r1c7 = 8; r1c6 = 2
20a. hidden single: r1c5 = 9

21. 9(2)r9c3 = [81]

22. 1 locked in n5 in c7 - > no 1 elsewhere in c7
22a. hidden single: r4c5 = 1
22b. hidden singles: r7c3 = 2; r2c4 = 2; r8c3 = 1; r5c5 = 2; r8c4 = 3; r3c5 = 5; r8c5 = 8; r8c7 = 9; r7c8 = 9


Singles and cage sums to the end

(Archive Note) Typos corrected.
Andrew's walkthrough:
A nice puzzle. The hardest part was keeping track of eliminations in the disjoint nonets, even after I’d set up minimised windows so that I could view both my worksheet and Para’s original diagram at the same time. I probably wouldn’t have spotted step 5a without doing that.

It felt hard at first but, once I started using the nonets, it wasn’t too difficult.

I must admit I’m not particularly keen on toroidal puzzles. Toroidal killers take away many 45s. Toroidal jigsaws take away the use of Law of Leftovers.

Nonets are numbered according to the highest cell, for example NR1C5. For disjoint nonets, they are described by the highest cell on the grid, for example disjoint nonet NR1C1.

Prelims

a) R1C67 = {19/28/37/46}, no 5
b) R23C6 = {89}
c) R34C1 = {69/78}
d) R67C9 = {19/28/37/46}, no 5
e) R78C4 = {19/28/37/46}, no 5
f) R9C34 = {18/27/36/45}, no 9
g) 9(3) cage at R1C8 = {126/135/234}, no 7,8,9
h) 11(3) cage at R4C7 = {128/137/146/236/245}, n o 9
i) 8(3) cage at R5C1 = {125/134}
j) 22(3) cage at R5C2 = {589/679}
k) 26(4) cage at R1C2 = {2789/3689/4589/4679/5678}, no 1
l) 18(5) cage at R6C2 = {12348/12357/12456}, no 9

1. Naked pair {89} in R23C6, locked for C6, clean-up: no 1,2 in R1C7
1a. Naked pair {89} in R23C6, CPE no 8,9 in R2C57 using NR1C5
[I missed CPE no 8,9 in R3C8 using NR1C6]

2. 8(3) cage at R5C1 = {125/134}, 1 locked for C1 and disjoint nonet NR5C1, no 1 in R4567C8 + R6C9, clean-up: no 9 in R7C9

3. 18(3) cage at R7C5 = {189/279/…}
3a. 1 of {189} must be in R8C6 -> no 1 in R78C5

4. 45 rule on C1 2 outies R29C2 = 10 = {19/28/37/46}, no 5

5. 45 rule on C9 2 outies R18C8 = 3 = {12}, locked for C8
5a. Naked pair {12} in R18C8, CPE no 1,2 in R1C9 using disjoint nonet NR1C9
[I missed CPE no 1,2 in R8C6 using disjoint nonet NR1C6]
5b. 1 in R1 only in R1C68, locked for disjoint nonet NR1C6, no 1 in R89C6
5c. Min R67C8 = 7 -> max R6C7 = 8

6. 9(3) cage at R1C8 = {126/135/234}
6a. 6 of {126} must be in R1C9 -> no 6 in R2C9

7. 11(3) cage at R4C7 = {128/137/146/236/245}
7a. 8 of {128} musty be in R5C8 -> no 8 in R45C7

8. 18(3) cage at R8C1 cannot be {89}1 which clashes with R23C1-> no 1 in R9C2, clean-up: no 9 in R2C2 (step 4)

9. 45 rule on R1 1 outie R2C9 = 1 innie R1C1, no 6,7,8,9 in R1C1, no 1 in R2C9

10. 45 rule on R9 1 outie R8C1 = 1 innie R9C9 + 1, no 9 in R9C9

11. 45 rule on R6789 3 innies R6C345 = 2 outies R5C16 + 18
11a. Min R5C16 = 3 -> min R6C345 = 21, no 1,2,3
11b. Max R6C345 = 24 -> max R5C16 = 6, no 6,7 in R5C6
11c. Min R6C345 = 21 -> min R6C45 = 12 -> max R4C56 + R5C5 = 9, no 7,8,9 in R4C56 + R5C5
11d. 8,9 in NR3C4 only in R3456C4, locked for C4, clean-up: no 1,2 in R78C4, no 1 in R9C3

12. 45 rule on R1234 2 outies R5C49 = 3 innies R4C567 + 9
12a. Max R5C49 = 17 -> max R4C567 = 8 = {123/124/125/134}, no 6,7, 1 locked for R4
12b. Min R4C567 = 6 -> min R5C49 = 15 = {69/78/79/89}, no 1,2,3,4,5
12c. Min R4C23 = 5 -> max R3C2 = 7

13. 9(3) cage at R1C8 = {126/135/234} -> R12C9 = {26/34/35}
13a. 45 rule on C9 4 innies R1289C9 = 18 = {1269/1359/2367/3456} (cannot be {1278/1458/1467/2457} which don’t contain {26/34/35} for R12C9, cannot be {1368/2349} which clash with R67C9, cannot be {2358} because 12(2) cage at R8C8 cannot be 2{28}), no 8, clean-up: no 9 in R8C1 (step 10)
13b. 9 of {1269/1359} must be in R8C9 -> no 1 in R8C9
13c. 17(3) cage at R3C9 = {179/278/458/467} (cannot be {269/359/368} which clash with R1289C9), no 3
13d. 1 of {179} must be in R3C9 -> no 9 in R3C9

14. 45 rule on R1 3 innies R1C189 = 9 = {126/135/234}
14a. R1C67 = {19/28/37/46}
14b. R1C189 + R1C67 must contain 3, locked for R1

15. 45 rule on C6789 2 innies R48C6 = 1 outie R9C5 + 7
15a. Max R48C6 = 12 -> max R9C5 = 5
15b. Min R48C6 = 8, max R4C6 = 5 -> min R8C6 = 3

16. 9 in disjoint nonet NR1C2 only in R19C2, locked for C2
16a. 22(3) cage at R5C2 = {589/679}, 9 locked for C3 and NR5C3, no 9 in R678C5

17. 18(3) cage at R7C5 = {378/468/567}, no 2

18. 9 in C5 only in R13C5, locked for NR1C5, no 9 in R3C67 -> R3C6 = 8, placed for NR1C5, R2C6 = 9, placed for disjoint nonet NR1C6, clean-up: no 1 in R1C6, no 7 in R4C1

19. R1C8 = 1 (hidden single in R1), R12C9 = 8 = {35}/[62], no 4, clean-up: no 4 in R1C1 (step 9)
19a. R8C8 = 2, placed for disjoint nonet NR1C9 (in the main part at NR6C7), clean-up: no 8 in R6C9, no 3 in R8C1, no 1 in R9C9 (both step 10)
19b. R8C8 = 2 -> R89C9 = 10 = {37/46}, no 5,9, clean-up: no 6 in R8C1 (step 10)
19c. Killer pair 3,6 in R12C9 and R89C9, locked for C9, clean-up: no 4,7 in R67C9

20. 2 in C9 only in R2346C9, CPE no 2 in R6C2 using disjoint nonet NR2C9

21. 9 in C8 only in R467C8, locked for disjoint nonet NR4C8 -> R6C9 = 2, placed for disjoint cage NR4C8, R7C9 = 8, placed for disjoint nonet NR6C7, clean-up: no 2 in R1C1 (step 9), no 6 in R1C1 (step 19), no 5 in 8(3) cage at R5C1
21a. Naked pair {35} in R12C9, locked for C9, clean-up: no 7 in R89C9 (step 19b), no 4,8 in R8C1 (step 10)
21b. Naked pair {46} in R89C9, locked for C9 and disjoint nonet NR1C9, no 4,6 in R6789C7

22. Naked pair {79} in R45C9, locked for C9 and disjoint nonet NR2C9, no 9 in R4C1, no 7 in R3456C2, clean-up: no 6 in R3C1
22a. R3C9 = 1 (hidden single in C9), placed for disjoint nonet NR2C9, no 1 in R6C2

23. Naked triple {134} in 8(3) cage at R1C1, locked for C1 and disjoint nonet NR4C8, no 3,4 in R4567C8 -> R1C1 = 5, placed for disjoint nonet R1C1, R1C9 = 3, placed for disjoint nonet R1C9, no 3 in R6789C7, R8C1 = 7, placed for disjoint nonet R4C8, no 7 in R4567C8, R9C9 = 6 (step 10), R8C9 = 4, R3C1 = 9, R4C1 = 6, placed for disjoint nonet NR3C2, no 6 in R356C2, clean-up: no 7 in R1C67, no 3,6 in R7C4, no 3 in R9C34
23a. R1C1 = 5 -> R2C12 = 9 = [27/81]
23b. R8C1 = 7 -> R9C12 = 11 = [29/83]

24. R2C9 = 5, placed for disjoint nonet R3C2, no 5 in R3456C2 -> R5C2 = 8, R56C3 = 14 = {59}, locked for C3 and NR5C3, clean-up: no 4 in R9C4
24a. Naked triple {234} in R346C2, locked for C2 -> R9C2 = 9, R9C1 = 2 (cage sum), placed for disjoint nonet R1C1, no 2 in R1234C3, R2C1 = 8, R2C2 = 1 (cage sum), 8 placed for disjoint nonet R1C1, no 8 in R14C3, clean-up: no 7 in R9C34

25. Naked quad {3467} in R1234C3, locked for C3 -> R9C3 = 8, R78C3 = [21]

26. R1C7 = 8 (hidden single in R1), R1C6 = 2, R1C5 = 9 (hidden single in R1)

27. R2C4 = 2 (hidden single in disjoint nonet NR1C2), R23C5 = 12 = [75], both placed for NR1C5, no 5,7 in R2345C7

28. Naked quad {1579} in R6789C7, 1 locked for C7
28a. 11(3) cage at R4C7 = {236/245}, 2 locked for C7
28b. R5C8 = {56} -> no 6 in R4C8

29. Naked quad {5689} in R4567C8, locked for C8

30. Naked triple {346} in R2C78 + R3C7, locked for 28(5) cage at R2C7 -> R3C8 = 7, R4C8 = 8 (cage sum)

31. R3C2 = 2 (hidden single in R3), R4C23 = 10 = [37], R45C9 = [97], R6C2 = 4

32. Naked triple {567} in R178C2, locked for disjoint nonet NR1C2 -> R1C4 = 4, R78C4 = [73], R1C23 = [76]

33. R345C4 = [659]

34. R4C5 = 1 (hidden single in NR1C5), R4C6 = 4, R6C45 = [86], R5C5 = 2 (cage sum)

35. R56C3 = [59], R5C8 = 6, R67C8 = [59], R6C7 = 1 (cage sum)

36. R5C8 = 6, R4C7 = 2, R5C7 = 3 (cage sum)

and the rest is naked singles, without using the nonets.


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PostPosted: Wed Mar 19, 2014 2:33 am 
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Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1893
Location: Lethbridge, Alberta, Canada
Jigsaw Killer by Ruud (May 2006)
Puzzle Diagrams:
Image   Image
Code: Select, Copy & Paste into solver:
SumoCueV1=14J0=13J0+1J0=18J3=5J3+4J3=16J1=19J1+7J1+0J0+3J0+3J0+3J3=16J3+13J3+6J1+7J1+7J1+0J5=16J0+19J0=15J3+21J3=11J2+6J6=12J1+25J1=19J5=19J5+19J0+21J3+23J2+23J2=11J6=16J6=10J1+27J5+28J5=4J5=18J2+39J2+39J2+33J6+34J6+35J6+27J8+28J5+38J5=16J2+48J2=18J7=11J4+34J6+35J6=10J8+54J8=11J5+48J2+50J7+50J7+51J4+51J4=14J6=20J8+63J8+56J8=10J7+66J7=18J7+68J4+68J4+62J4+63J8+63J8+56J8=9J7+75J7+68J7=16J4+78J4+62J4
Solution:
+-------+-------+-------+
| 5 4 9 | 1 2 3 | 7 6 8 |
| 2 3 6 | 8 7 9 | 5 1 4 |
| 7 1 8 | 6 5 2 | 4 3 9 |
+-------+-------+-------+
| 9 6 7 | 4 1 8 | 3 5 2 |
| 4 5 1 | 9 3 6 | 8 2 7 |
| 6 8 3 | 7 4 5 | 2 9 1 |
+-------+-------+-------+
| 3 7 2 | 5 9 4 | 1 8 6 |
| 1 9 5 | 2 8 7 | 6 4 3 |
| 8 2 4 | 3 6 1 | 9 7 5 |
+-------+-------+-------+

Quote:
SSscore: 0.85

Ruud: A new variant in exotic sudoku: Jigsaw Killer.
This puzzle is like a regular killer, but it has irregular nonets. You can use killer solving techniques (Innies, outies) and jigsaw solving techniques (LoL) to solve this puzzle.

Andrew (in 2014): I found this one in enxio27's post Ruud's Specialty Puzzles. Ruud posted it at http://www.setbb.com/sudoku/viewtopic.p ... udoku#5903 on the Sudoku Programmers website.
It was posted there about a month before Ruud posted TJK1, so it's another easy jigsaw killer. If I'd known about this puzzle sooner, it would have been the first entry in this archive.

Andrew's walkthrough:
Prelims

a) R1C23 = {49/58/67}, no 1,2,3
b) R1C56 = {14/23}
c) R2C56 = {79}
d) R3C89 = {39/48/57}, no 1,2,6
e) R45C7 = {29/38/47/56}, no 1
f) R56C3 = {13}
g) R7C12 = {19/28/37/46}, no 5
h) R8C45 = {19/28/37/46}, no 5
i) R9C45 = {18/27/36/45}, no 9
j) R9C78 = {79}
k) 11(3) cage at R3C6 = {128/137/146/235/245}, no 9
l) 19(3) cage at R4C1 = {289/379/469/478/568}, no 1
m) 19(3) cage at R4C2 = {289/379/469/478/568}, no 1
n) 10(3) cage at R4C9 = {127/136/145/235}, no 8,9
o) 11(3) cage at R6C7 = {128/137/146/236/245}, no 9
p) 11(3) cage at R7C3 = {128/137/146/236/245}, no 9

1. Naked pair {13} in R56C3, locked for C3 and NR3C1
1a. 11(3) cage at R7C3 = {245} (only remaining combination), locked for C3, clean-up: no 8,9 in R1C2
1b. 11(3) cage at R7C3 = {245}, CPE no 2,4 in R7C12, clean-up: no 6,8 in R7C12
1c. Naked quad {6789} in R1234C3, locked for NR1C1, clean-up: no 6,7 in R1C3

2. Naked pair {79} in R2C56, locked for R2 and NR1C4
2a. 7 in C3 only in 16(3) cage at R2C2 = {178/367}, no 9 -> R2C2 = {13}

3. R1C3 = 9 (hidden single in C3) -> R1C2 = 4, placed for NR1C1, clean-up: no 1 in R1C56
3a. Naked pair {23} in R1C56, locked for R1 and NR1C4
3b. 7 in R1 only in R1C789, locked for NR1C7, clean-up: no 5 in R3C89

4. 2 in NR1C1 only in R2C12, locked for R2
4a. R4C9 = 2 (hidden single in NR1C7), clean-up: no 9 in R5C7
4b. R4C9 = 2 -> R56C9 = 8 = {17/35}, no 4,6

5. 45 rule on NR1C7 1 outie R3C7 = 4, placed for NR3C7, clean-up: no 8 in R3C89
5a. R3C7 = 4 -> R12C7 = 12 = [75], 5 placed for NR1C7, R9C78 = [97], both placed for NR6C7, clean-up: no 2 in R9C45
5b. R45C7 = {38} (only remaining combination), locked for C7 and NR3C7, clean-up: no 5 in R56C9 (step 4b)
5c. Naked pair {17} in R45C9, locked for C9 and NR3C7
5d. Naked triple {126} in R678C7, locked for NR6C7

6. 14(3) cage at R7C9 = {356} (only remaining combination) -> R7C9 = 6, placed for NR3C7, R89C9 = {35}, locked for C9 and NR6C7 -> R12C9 = [84], R3C89 = [39]

7. R3C2 = 1, placed for NR1C1 -> R1C1 = 5, clean-up: no 9 in R7C1
7a. R3C2 = 1 -> R34C3 = 15 = {78}, locked for C2 -> R2C2 = 6, R12C4 = [18], placed for NR1C4, R2C2 = 3 (cage sum), R2C1 = 2, R3C1 = 7 (cage sum), placed for NR3C1, R34C3 = [87], clean-up: no 2,9 in R8C5, no 1,8 in R9C5

8. Naked pair {56} in R3C45, locked for R3 and 15(3) cage -> R3C6 = 2, placed for NR3C6, R4C4 = 4, R1C56 = [23], clean-up: no 6 in R8C5, no 5 in R9C5
8a. 1 in R4 only in R4C56
8b. R3C6 = 2 -> R4C56 = 9 = {18}, locked for R4 and NR3C6 -> R45C7 = [38]

9. R6C2 = 8 (hidden single in NR3C1) -> R45C2 = {29/56}, no 9 in R5C2
9a. Killer pair 6,9 in R4C1 and R45C2, locked for NR3C1 -> R5C1 = 4, placed for NR3C1
9b. R5C1 = 4 -> R46C1 = 15 = {69}, locked for C1

10. R6C5 = 4 (hidden single in NR3C6), R67C4 = 12 = {39/57}, clean-up: no 6 in R8C4, no 5 in R9C4
10a. Naked pair {36} in R9C45, locked for R9 and NR6C6 -> R89C9 = [35], R9C2 = 2, placed for NR6C1, R89C3 = [54]

11. R8C4 = 2 (hidden single in NR6C6) -> R8C5 = 8, placed for NR6C6, R89C1 = [18], R8C2 = 9 (cage total), placed for NR6C1

and the rest is naked singles, without using the nonets.


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PostPosted: Mon Apr 28, 2014 3:54 am 
Offline
Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1893
Location: Lethbridge, Alberta, Canada
Texas Jigsaw Killer 34 by Para (July 2008) here
Puzzle Diagram:
Image
Moderator note: The link to Para's diagram stopped working. I've replaced it by a diagram from SudokuSolver.
Børge's images with udosuk Style Killer Cages and Ruud TJK#33 inspiration:
Image



Image



Image
Code: Select, Copy & Paste into solver:
SumoCueV1=17J0=34J0+1J0+1J1=11J1+4J1=12J2+6J2=13J2+0J3+0J0+1J1+1J1+4J1=15J1+14J4+6J4+8J2=20J3+18J0+18J5=19J5+4J4+14J1=13J1+24J4+8J2+18J3=10J0+18J5+21J6=21J4=13J4+32J4+32J4=14J2=11J3+28J0+28J5+21J6+31J4=15J6=21J2+42J2+35J2+36J3=13J0+46J5+46J6+31J6+41J7=22J7+42J7+51J7=11J3=9J0+55J5=12J6=21J8+41J6+51J7+51J7+51J7+54J3=19J5+57J5+57J6+58J8=20J8+68J6=19J8+70J7+54J3+64J3+64J5+58J8+58J8+68J8+68J8+68J8+70J7
Solution:
+-------+-------+-------+
| 9 8 7 | 6 5 3 | 4 1 2 |
| 3 5 9 | 4 1 2 | 6 7 8 |
| 4 1 6 | 9 2 7 | 8 5 3 |
+-------+-------+-------+
| 7 6 2 | 8 4 9 | 1 3 5 |
| 5 3 1 | 2 8 4 | 7 6 9 |
| 6 2 4 | 7 9 5 | 3 8 1 |
+-------+-------+-------+
| 8 4 5 | 1 3 6 | 9 2 7 |
| 2 7 8 | 3 6 1 | 5 9 4 |
| 1 9 3 | 5 7 8 | 2 4 6 |
+-------+-------+-------+

Quote:
SSscore: 0.88 (v3.2.1), 1.25 (v3.6.3)

Para: Just continueing one more of Ruud's legacies. I always enjoyed them. Don't know if ayone else is interested. But just giving it a try. If anyone else feels like making one some time, go ahead. Otherwise I'll try posting one in some random time interval.
First an easy one to get started again.
Have fun!

Glyn: Nice one Para. I kept losing track of the 'K' in TJK, lots of fun things happening there.

Ed: Thanks for getting these started again Para. :applause: No way was it easy for me. Must be out of practice. I'd have to give it a 1.25 rating with my step 13 and 14.

SudokuSolver V3.2.1 gives it a score of 0.88 so I've missed something important. [edit: v3.6.3 gives it a score of 1.25 so a massive increase.]

Andrew (in 2013): When I was discussing a later TJK with Ed, he suggested that I should post my walkthrough on the forum, rather than just in the archive entries, to give the TJKs more prominence. I'm therefore posting my walkthroughs for all the TJKs from TJK 34 onward; the ones which were first posted on this site - I solved them earlier this year, but am only now looking at how others solved them.
Ed commented on the SS score at the time. The current SS score (1.25 from SS V3.6.2) agrees with Ed’s evaluation of this puzzle. SS scoring for killers has evolved a lot since this puzzle was posted, thanks to hard work by Ed and Richard; it’s very likely that it’s also improved for jigsaw killers.

Ed's walkthrough:
Para wrote:
First an easy one to get started again.
Thanks for getting these started again Para. :applause: No way was it easy for me. Must be out of practice. I'd have to give it a 1.25 rating with my step 13 and 14.

SudokuSolver V3.2.1 gives it a score of 0.88 so I've missed something important. [edit: v3.6.3 gives it a score of 1.25 so a massive increase.] [Thanks to Andrew for helping with some inaccuracies in this walkthrough]

Thanks again.
Ed

TJK 34 Walkthrough

Prelims
i. 34(5)r1c2 = {46789}
ii. 11(4)r1c5 = {1235}
iii. 13(2)r3c7: no 1..3
iv. 10(3)r4c2: no 8 or 9
v. 21(3)r4c5: no 1,2 or 3
vi. 14(2)r4c9 = {59/68}
vii. 11(2)r5c1: no 1
viii. 21(3)r5c7: no 1,2 or 3
ix. 11(3)r7c1: no 9
x. 9(2)r7c2: no 9
xi. 19(3)r8c2: no 1

1. "45" on c1234: 1 innie r9c4 = 5

2. "45" on c6789: 1 innie r1c6 = 3

3. Naked triple {125} in r123c5: all locked for c5 & no 1, 2 or 5 in r3c67
3a. no 8 in r3c8

4. 21(3)r4c5 = 8{49/67}
4a. 8 locked for c5

5. 21(4)r7c5 must have 3 for c5 = 35{49/67}
5a. 3 locked for Nr7c5

6. "45" Nr1c7: 2 outies r26c8 = h15(2) = {69/78}
6a. min. r2c8 = 6 -> max r1c78 = 6 (no 6..9)
6b. 12(3)r1c7 = {129/147/156/246}(no 8)
6c. no 7 r6c8 (h15(2))

7. 3 in Nr1c7 only in 13(3)r1c9 = 3{19/28/46}(no 5 or 7)

8. 7 in Nr1c7 only in r5c78
8a. 7 locked for r5
8b. no 4 r6c1

9. 21(3)r5c7 must have 7 = 7{59/68}
9a. 9 in {579} must be in r6c8 -> no 9 in r5c78

10. 9 in Nr1c7 is only in c9: 9 locked for c9

11. "45" on Nr7c5: 1 outie r8c7 + 4 = 1 innie r8c8
11a. = [26/37/48/59]
11b. r8c7 = (2..5), r8c8 = (6..9)

12. "45" on c9: 1 outie r8c8 - 1 = 2 innies r67c9
12a. -> max r67c9 = 8 (no 8)

13. no 7 in r8c8 because of 7's in c9. Like this.
13a. 7 in r89c9 -> no 7 in r8c8 (same cage)
13b. 7 in r67c9, r8c8 must be greater than 7 (step 12)
13c. -> no 7 in r8c8
13d. -> no 3 in r8c7 (i/o Nr7c5 = -4)

14. "45" on Nr1c7 + Nr6c6: 2 outies r2c8 + r8c8 - 11 = 1 innie r6c6
14a. r2c8 + r8c8 = [79] and r6c6 = 5! All others blocked.
14b. ([76] blocked by 21(3)r5c7 = {67}8 using h15(2)r26c8;[68] blocked by no 3 in innie r6c6; {69/[78]} clashes with h15(2)r26c8; [98] = 17 blocked by 6 forced into both r6c68 in innie r6c6 and h15(2))
14c. no 4 or 6 in r3c7
14d. no 6 in r5c1

15. r6c8 = 8 (h15(2))
15a. no 3 r5c1

16. r8c7 = 5 (i/o Nr7c5 = -4)

17. r5c78 = [76] (last permutation)

18. 14(2)r4c9 = {59}: both locked for c1 and Nr1c7

19. r1c78 = 5 = {14}: both locked for r1 & Nr1c7

20. Naked triple {238} in r123c9: locked for c9

21. r89c9 = 10 = {46}: both locked for Nr6c6

22. Naked pair {17} in r67c9: both locked for Nr6c6

23. 9 in Nr6c6 only in c7: 9 locked for c7
23a. r3c78 = [85]

24. LoL c1: r1c1 = r9c2 -> no 5 in r1c1, no 3 or 4 in r9c2

25. r1c5 = 5 (hsingle r1)

26. 7 & 8 which must be in 34(5) are only in r1: both locked for r1 and 8 locked for Nr1c1
26a. r1c9 = 2
26b. no 2, 7 or 8 in r9c2 (lol c1)
26c. no 1 r7c3

27. r23c9 = [83]

28. 15(3)r2c6 cannot have 1 since no 5 or 8 is available
28a. r2c5 = 1 (hsingle Nr1c4)
28b. r3c5 = 2
28c. r2c6 = 2 (hsingle Nr1c4)
28d. 15(3)r2c6 = [2][49/67]: r2c7 = (46), r3c6 = (79)

29. Naked triple {469} r2c347: all locked for r2 and 9 locked for Nr1c4 & 34(5)r1c2
29a. r3c6 = 7, r2c7 = 6

30. r1c1 = 9 (hsingle r1)
30a. r9c2 = 9 (LoL c1)

31. r789c5 = 16 = {367}: locked for c5 and Nr7c5

32. r7c6 = 6 (hsingle c6)
32a. no 3 in 9(2)r7c2
32b. r5c6 = 4 (cage sum)
32c. no 7 r6c1

Rest unfolds from here.
Andrew's walkthrough:
When I was discussing a later TJK with Ed, he suggested that I should post my walkthrough on the forum, rather than just in the archive entries, to give the TJKs more prominence. I'm therefore posting my walkthroughs for all the TJKs from TJK 34 onward; the ones which were first posted on this site - I solved them earlier this year, but am only now looking at how others solved them.

Ed commented on the SS score at the time. The current SS score (1.25 from SS V3.6.2) agrees with Ed’s evaluation of this puzzle. SS scoring for killers has evolved a lot since this puzzle was posted, thanks to hard work by Ed and Richard; it’s very likely that it’s also improved for jigsaw killers.

Thanks Ed for correcting some typos.

Prelims

a) R3C78 = {49/58/67}, no 1,2,3
b) R45C9 = {59/68}
c) R56C1 = {29/38/47/56}, no 1
d) R7C23 = {18/27/36/45}, no 9
e) 19(3) cage at R3C4 = {289/379/469/478/568}, no 1
f) 10(3) cage at R4C2 = {127/136/145/235}, no 8,9
g) 21(3) cage at R4C5 = {489/579/678}, no 1,2,3
h) 21(3) cage at R5C7 = {489/579/678}, no 1,2,3
i) 11(3) cage at R7C1 = {128/137/146/236/245}, no 9
j) 19(3) cage at R8C2 = {289/379/469/478/568}, no 1
k) 19(3) cage at R8C8 = {289/379/469/478/568}, no 1
l) 11(4) cage at R1C5 = {1235}
m) 34(5) cage at R1C2 = {46789}

1. 45 rule on C1234 1 innie R9C4 = 5, placed for NR7C5
1a. 1 in C4 only in R678C4, locked for disjoint NR4C4, no 1 in R57C6 + R8C7
1b. 1 in R5 only in R5C23, locked for 10(3) cage at R4C2, no 1 in R4C2

2. 45 rule on C6789 1 innie R1C6 = 3
2a. Naked triple {125} in R123C5, locked for C5
2b. Naked triple {125} in R123C5, CPE no 1,2,5 in R3C67, clean-up: no 8 in R3C8
2c. R12C5 + R2C6 = {125} (hidden triple in NR1C4)

3. 3 in NR1C7 only in R23C9, locked for C9
3a. 13(3) cage at R1C9 = {139/238/346}, no 5,7
3b. 7 in C9 only in R6789C9, locked for NR6C6

4. 21(3) cage at R4C5 = {489/678}, 8 locked for C5
4a. 3 in C5 only in R789C5, locked for NR7C5

5. 45 rule on NR1C7 2 outies R26C8 = 15 = [69/78/96]
5a. Min R2C8 = 6 -> max R1C78 = 6, no 6,7,8,9 in R1C78

6. 7 in NR1C7 only in R5C78, locked for R5, clean-up: no 4 in R6C1
6a. 21(3) cage at R5C7 = {579/678}, no 4
6b. Hidden killer triple 1,2,4 in R1C78 + 13(3) cage at R1C9 for NR1C7, 13(3) cage (step 3a) contains one of 1,2,4 -> R1C78 = {124}

7. 45 rule on NR7C5 1 innie R8C8 = 1 outie R8C7 + 4, R8C8 = {6789}, R8C7 = {2345}

8. 1,2 in NR7C5 only in R8C6 + R9C678, locked for 20(5) cage at R8C6, clean-up: no 6 in R8C8 (step 7)
8a. 20(5) cage = {12368/12458/12467} (cannot be {12359} because 3,5 only in R8C7), no 9
8b. R26C8 = 15 (step 5) -> min R268C8 = 22, must contain 9, locked for C8, clean-up: no 4 in R3C7

9. 19(3) cage at R8C8 = {289/469/478} (cannot be {568} which clashes with R45C9), no 5

10. 45 rule on R12 3 outies R3C569 = 12
10a. Min R3C56 = 5 -> no 8,9 in R3C9

11. 45 rule on R6789 3 innies R6C158 = 1 outie R5C6 + 19
11a. Max R6C158 = 24 -> max R5C6 = 5
11b. Min R5C6 = 2 -> min R6C158 = 21, no 2,3 in R6C1, clean-up: no 8,9 in R5C1

12. 45 rule on R123 2 outies R4C13 = 1 innie R3C4
12a. Min R4C13 = 3 -> min R3C4 = 3
12b. Max R4C13 = 9, no 9 in R4C13

13. 45 rule on R789 1 innie R7C6 = 2 outies R6C79 + 2
13a. Min R6C79 = 3 -> min R7C6 = 5
13b. Max R6C79 = 7, no 7,8,9 in R6C79

14. 2 in C4 only in R45678C4, locked for disjoint NR4C4, no 2 in R5C6
14a. R6C158 = R5C6 + 19 (step 11)
14b. R5C6 = {45} -> R6C158 = 23,24 = {689/789}, no 4,5, 8,9 locked for R6, clean-up: no 6 in R5C1
14c. 13(3) cage at R6C2 = {157/247/256/346}
14d. Killer pair 6,7 in R6C158 and 13(3) cage, locked for R6

15. 15(3) cage at R5C6 = {159/249/258/456} (cannot be {168/267} because R5C6 only contains 4,5), no 7
15a. Max R56C6 = 9 -> min R7C6 = 6

16. 13(3) cage at R1C9 (step 3a) = {139/238/346}, R45C9 = {59/68} -> 13(3) cage + R45C9 must contain 9, locked for C9 and NR1C7

17. 3 in NR6C6 only in 22(5) cage at R6C7 = {12379/13459/13468/13567} (cannot be {23458/23467} which clash with 19(3) cage at R8C9), 1 locked for NR6C6

18. 45 rule on NR6C6 2 innies R6C68 = 1 outie R8C8 + 4, IOU no 4 in R6C6

19. Hidden killer triple 2,5,8 in R6C6, 22(5) cage at R6C7, R6C8 and 19(3) cage at R8C8 for NR6C6, R6C6 = {25}, 22(5) cage contains one of 2,5,8 -> 19(3) cage cannot contain more than one of 2,8 in R89C9
19a. 19(3) cage at R8C8 (step 9) = {469/478} (cannot be {289} which contains both of 2,8 in R89C9), no 2, 4 locked for C9 and NR6C6
19b. 13(3) cage at R1C9 (step 3a) = {139/238}, no 6

20. 4 in NR1C7 only in R1C78, locked for R1
20a. R1C78 = {14/24} -> R2C8 = {67}, clean-up: no 6 in R6C8 (step 5)
20b. 21(3) cage at R5C7 (step 6a) = {579/678}
20c. R6C8 = {89} -> no 8 in R5C78
20d. 34(5) cage at R1C2 = {46789}, 4 locked for R2 and NR1C4

21. 8 in NR1C7 only in R1245C9, locked for C9
21a. 19(3) cage at R8C8 (step 9) = {469/478}
21b. 8,9 only in R8C8 -> R8C8 = {89}, clean-up: no 3 in R8C7 (step 7)
[This elimination could have been obtained a bit earlier using Law of Leftovers for C6789, but I’m avoiding using it for this puzzle which has a fairly low SS score.]
21c. Naked pair {89} in R68C8, locked for C8

22. 22(5) cage at R6C7 (step 17) = {12379} (only remaining combination, cannot be {13567} which clashes with R89C9) -> R7C9 = 7, R7C7 = 9, R6C8 = 8, R2C8 = 7 (step 5), placed for NR2C7, R8C8 = 9, clean-up: no 6 in R3C7, no 4 in R3C8, no 3 in R5C1, no 2 in R7C23
22a. R6C6 = 5 (hidden single in NR6C6), R5C6 = 4, R7C6 = 6 (cage sum), both placed for disjoint nonet at R4C4, R8C7 = 5, clean-up: no 7 in R6C1, no 3 in R7C23

23. R6C158 = R5C6 + 19 (step 11)
23a. R5C6 = 4 -> R6C158 = 23 = {689} -> R6C5 = 9, placed for NR4C4, R6C1 = 6, placed for NR2C1, R5C1 = 5, R5C78 = [76], R5C5 = 8, placed for NR2C7, R4C5 = 4 (cage sum), R5C9 = 9, R4C9 = 5, R3C7 = 8, placed for NR1C4, R3C8 = 5, R7C5 = 3

24. 13(3) cage at R1C9 (step 3a) = {238} (only remaining combination), locked for C9 and NR1C7 -> R6C9 = 1, R7C8 = 2, R6C7 = 3
24a. Naked pair {14} in R1C78, locked for R1

25. R7C23 = {45} (only remaining combination, cannot be {18} which clashes with R7C4), locked for R7
25a. Naked pair {45} in R7C23, CPE no 4 in R8C2

26. 19(3) cage at R3C4 = {289/379} (cannot be {469/478} because R5C4 only contains 2,3) -> R3C4 = 9, placed for NR3C3, R4C4 = {78}
26a. R3C6 = 7, placed for NR1C4, R2C67 = 8 = [26], 2 placed for NR1C4, R12C5 = [51] , R1C4 = 6, R2C34 = [94]

27. R3C5 = 2, placed for NR2C7, 13(3) cage at R4C6 = [913]

28. R2C2 = 5 (hidden single in R2), R12C1 = 12 = [93], 3 placed for NR2C1, R7C23 = [45]
28a. 11(3) cage at R7C1 = {128} (only remaining combination), locked for C1 and NR2C1 -> R45C1 = [47], 7 placed for NR2C1, R9C2 = 9, R4C4 = 8, R5C4 = 2 (cage sum)

29. R6C4 = 7, R78C4 = [13], R8C3 = 8 (cage sum)

and the rest is naked singles, without using the nonets.
JSudoku formats:
Ed's walkthrough ended with

u9 u8 u7 u6 u5 u3 u4 u1 u2
u3 u5 u9 u4 u1 u2 u6 u7 u8
u4 u1 u6 u9 u2 u7 u8 u5 u3
u7 u6 u2 u8 u4 u9 u1 u3 u5
u5 u3 u1 u2 u8 u4 u7 u6 u9
u6 u2 u4 u7 u9 u5 u3 u8 u1
u8 u4 u5 u1 u3 u6 u9 u2 u7
u2 u7 u8 u3 u6 u1 u5 u9 u4
u1 u9 u3 u5 u7 u8 u2 u4 u6

I wonder what Jsudoku is trying to tell me ;)

Jean-Christophe:
Ruud wrote:
Sudoku Puzzle Progress (.sdx)
This format contains a line for each row in the grid. A blank separates the cells. For unsolved cells, the candidates are listed without separating space. Solved cells are preceded by u when they are placed by the user, the givens have no prefix.

That's the format used for "Copy Candidates". You could "Copy Solution" instead.


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PostPosted: Tue Apr 29, 2014 1:39 am 
Offline
Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1893
Location: Lethbridge, Alberta, Canada
Texas Jigsaw Killer 35 by Para (November 2008) here
Puzzle Diagrams:
Image   Image
Børge's images with udosuk Style Killer Cages and Ruud TJK#33 inspiration:
Image



Image



Image
Cages with cells in 3 jigsaw nonets:  pink
Cages with cells in 2 jigsaw nonets: green, yellow and brown
Cages with cells in 1 jigsaw nonet: red, blue and grey
Code: Select, Copy & Paste into solver:
SumoCueV1=21J0=27J1+1J1+1J1=12J2+4J2+4J2=15J2+7J2+0J0+0J0+1J0+1J1=15J2+13J3=16J2+15J3+7J2+0J0=9J0=15J1+20J1+13J4=15J3+23J3+15J3+15J2=25J0+19J0+19J0=12J1+30J4=14J4+23J4=29J3+34J3+27J5+27J5=3J1+38J1+32J4+32J6=13J6+42J3+34J3+27J5=19J5+46J4=11J4+48J4+32J6=19J7+34J7+34J7=18J8+46J5=8J5+56J5=18J4+51J6+51J6=21J7+61J7+54J8+54J5=24J8+65J5+58J8+58J6+61J7+61J7=13J7+54J8+65J8+65J8+65J8=13J8+76J6+76J6+71J6+71J7
Solution:
+-------+-------+-------+
| 6 3 4 | 8 2 1 | 9 7 5 |
| 9 2 7 | 5 6 8 | 4 1 3 |
| 4 5 6 | 9 1 7 | 2 3 8 |
+-------+-------+-------+
| 8 1 3 | 7 5 2 | 6 4 9 |
| 7 9 2 | 1 3 4 | 8 5 6 |
| 1 6 9 | 4 7 5 | 3 8 2 |
+-------+-------+-------+
| 2 4 5 | 3 8 9 | 7 6 1 |
| 3 8 1 | 2 4 6 | 5 9 7 |
| 5 7 8 | 6 9 3 | 1 2 4 |
+-------+-------+-------+

Quote:
Para: Here's the second one by my hand. It's not that hard either, but harder than the previous one. But I enjoyed solving it. Hope you do too.

Ed: Yes it is! Absolutely perfect hard. Had to use many of the fun jigsaw techniques that I find hard to see; LoL, weird CPE and cage blocks. But nothing outrageously technically difficult. Really enjoyed it. I'll give it a rating of Hard 1.25 since it took me a long, long time to finish. Didn't find the cracking move till step 31a.

Para: If someone else feels like making one too, go ahead. ;) Can't be the only one keeping this up.

Ed: I really appreciate you keeping this fine tradition alive. Thanks! I'll make one before the NY.

Andrew (in 2013): Loved the jigsaw pattern for this puzzle!

Ed's walkthrough:
Para wrote:
It's not that hard
Yes it is! Absolutely perfect hard. Had to use many of the fun jigsaw techniques that I find hard to see; LoL, weird CPE and cage blocks. But nothing outrageously technically difficult. Really enjoyed it. I'll give it a rating of Hard 1.25 since it took me a long, long time to finish. Didn't find the cracking move till step 31a.

Para wrote:
If someone else feels like making one too, go ahead. ;) Can't be the only one keeping this up.
I really appreciate you keeping this fine tradition alive. Thanks! I'll make one before the NY.

Walkthrough for TJK 35
(please let me know of any mistakes or things that could be clearer; [edit: thanks Andrew!])

Prelims
i. 9(3)r3c2: no 7,8,9
ii. 15(2)r3c3 = {69/78}
iii. 12(2)r4c4: no 1,2,6
iv. 14(4)r4c6: no 9
v. 3(2)r5c3 = {12}
vi. 13(2)r5c7: no 1,2,3
vii. 19(3)r6c2: no 1
viii. 11(2)r6c4: no 1
ix. 19(3)r6c7: no 1
x. 8(2)r7c3: no 4,8,9

1. 3(2)r5c3 = {12}: both locked for r5 and Nr1c2

2. LOL r1234: 4 outies r5c3489 = 4 innies r3c5 & r4c567
2a. 4 outies must have 1 & 2 -> 4 innies must have 1 & 2
2b. 1,2 locked for Nr3c5
2c. no 9 in r6c45

3. "45" Nr1c1: 2 innies r2c3 & r4c1 = h15(2) = {69/78}

4. 27(5)r1c2 = {34569/34578} = 345{69/78}
4a. 3,4,5 all locked for Nr1c2
4b. r4c5 = (345)

5. "45" Nr1c1 + Nr1c2: 2 innies r4c14 = h15(2) = [69]{78} (no 9 r4c1)
5a. no 6 r2c3 (h15(2)Nr1c1)

6. "45" c1234: r46c4 = h11(2) = [74/83]
6a. r4c5 = (45)
6b. r6c5 = (78)
6c. r4c1 = (78) (h15(2))
6d. r2c3 = (78) (h15(2))

7. naked pair {78} in r2c3+r4c1: both locked for Nr1c1

8. naked pair {78} in r4c14: both locked for r4

9. 15(2)r3c3 = {69} ({78} blocked by r4c4)
9a. both locked for Nr1c2 & r3

10. "45" Nr1c5: r23c8 + 2 = r2c5
10a. min r23c8 = {12} = 3 -> min r2c5 = 5
10b. max. r23c8 = 7: no 7,8,9

11. LOL c6789: 2 innies r4c67 = 2 outies r12c5
11a. 2 innies must have at least 1 of 1 or 2 for LOL r1234 (step 2)
11b. -> the only place for 1 or 2 in 2 outies r12c5 is in r1c5 = (12)
11c. NOTE: this also means that r4c67 has exactly 1 of (12)
11d. no 3,4 in r12c5 -> no 3,4 in r4c67
11e. no 7,8 in r4c67 -> no 7,8 in r2c5

12. hidden pair (12) for Nr3c5 (from step 11c) -> r3c5 = (12)

13. naked pair (12) in r13c5: both locked for c5
13a. no 1,2 in r3c9 (CPE sees both r13c5)

14. LOL r1234: 4 outies r5c3489 = 4 innies r3c5 & r4c567
14a. 4 innies don't have 3,7,8 -> 4 outies cannot have them in r5c89
14b. no (56) in r5c7

15. hidden killer pair (78) in Nr2c6: r2c6 & r3c67 must have both of 7 & 8 for Nr2c6
15a. r3c67 can only have 1 of 7 or 8 because of 15(3) -> r2c6 = (78)
15b. & 15(3)r3c6 must have 7 or 8 = {168/258/267/357}(no 4,9) (can't place {348} because of r4c7)
15c. 6 in {168} must be in r4c7 -> no 1 in r4c7

16. 9 in r4 only in Nr2c6: 9 locked for that N & for 29(5)r4c8
16a. no 4 in r5c7

17. naked pair {78} in r2c36: both locked for r2

18. hidden killer pair (78) in r3: r3c67 has at most one of 7/8 -> r3c9 = (78)

19. 15(3)r2c5: min. r2c6 + r3c5 = [71] = 8 -> max. r2c5 = 7 (no 9)

20. "45" Nr1c5: r23c8 + 2 = r2c5
20a. -> r23c8 = 3/4 = 1{2/3}
20b. 1 locked for c8 & Nr2c6 & 16(4)r2c7
20c. r23c8 & r3c9 = 10/11/12 -> r2c7 = (456)

21. "45" Nr1c5: 3 innies r2c57 & r3c9 = h18(3) = {468/567} (only valid combinations)
21. 6 locked for r2 & Nr1c5

22. r1c1 = 6 (hsingle r1)

23. 9 in Nr1c1 only in r2: 9 locked for r2
23a. 21(4)r1c1 = 69{15/24}(no 3)

24. "45" r123: r3c2 + 1 = r4c7
24a. NOTE: i/o [12] blocked by r3c5 = (12) which sees both r3c2 & r4c7!
24b. r3c2 = (45)
24c. r4c7 = (56)

25. 3 in Nr1c1 only in r4: 3 locked for r4

26. 3 in r3 only in Nr2c6: 3 locked for that N

27. 1 & 2 in r1 only in Nr1c5: no 1 or 2 in r2c9

28. hidden killer pair (12) in r2: r2c12 must have 1 of 1/2 for r2
28a. 21(4)r1c1 = 69{15/24} -> no 1 or 2 in r3c1

29. naked pair (45) in r3c12: both locked for r3 & Nr1c1

30. hidden pair (12) in Nr3c5 -> r4c6 = (12)
30a. CPE -> no 2 in r3c6

31. 15(3)r2c5 = {168/258/267} = [2/8..]
31a. 15(3)r3c6 = {267/357}(no 8) (NOTE: {258} = {28}[5] clashes with 15(3)r2c5 (step 31))
31b. 7 locked for r3 & Nr2c6

32. r3c9 = 8
32a. 16(4)r2c7 = 8{125/134} (no 6)

33. r2c6 = 8

34. r2c5 = 6 (hsingle r2)
34a. r3c5 = 1 (cage sum)

35. r1c5 & r4c6 = 2

36. naked pair {13} in r4c23: 1 locked for Nr1c1
36a. r3c2 = 5 (cage sum)
36b. r4c7 = 6 (i/o r1234)
36c. r3c67 = 9 = [72]
36d. r23c8 = [13] -> r2c7 = 4

37. r2c3 = 7
37a. r34c1 = [48]
37b. r4c45 = [75]

38. r6c4 = 4 (h11(2)r46c4)
38a. r6c5 = 7

39. 5&6 in Nr2c6 only in r5: 5&6 locked for r5
39a. CPE: no 5,6 in r6c8

40. r1c5 = 2
40a. r1c67 = 10 = {19/37) (no 5)

41. r4c89 = {49}: 4 locked for Nr2c6
41a. no 9 in r5c7
41b. r5c9 + r6c89 = 16
41c. max. r56c9 = {56} = 11 -> min. r6c8 = 7
41d. r6c8 = 8
41e. r67c9 = 8 = [53/62] (r6c9 = (23)

42. r9c8 = 2 (hsingle Nr5c6)

43. 4 in Nr5c1 only in c2: locked for c2
43a. r1c3 = 4 (hsingle r1)

44. 8 in Nr3c5 only in c5: locked for c5

45. CPE on 8s in Nr7c1: no 8 in r8c4

46. 8 in Nr5c1 only in c2: 8 locked for c2
46a. r1c35 = [38]
46b. r2c4 = 5
46c. no 3 in r7c3

47. r2c9 = 3
47a. r1c89 = 12 = {57}: 7 locked for r1

48. r6c9 = 2
48a. r5c9 = 6 (cage sum)

49. r5c78 = [85]
49a. r5c56 = [34]

50. naked pair (79) in r5c12: both locked for Nr5c1
50a. r6c1 = 1 (cage sum)

51. r6c23 = [69] -> r7c2 = 4 (cage sum)

52. "45" c1: 3 outies r258 = 19 = [298]

53. 8(2)r7c3 = [53] (last permutation)

54. r6c67 = [53]
54a. r7c67 = 16 = [97]
rest is singles and cage sums
Andrew's walkthrough:
Loved the jigsaw pattern for this puzzle!

Prelims

a) R3C34 = {69/78}
b) R4C45 = {39/48/57}, no 1,2,6
c) R5C34 = {12}
d) R5C78 = {49/58/67}, no 1,2,3
e) R6C45 = {29/38/47/56}, no 1
f) R7C34 = {17/26/35}, no 4,8,9
g) 9(3) cage at R3C2 = {126/135/234}, no 7,8,9
h) 19(3) cage at R6C2 = {289/379/469/478/568}, no 1
i) 19(3) cage at R6C7 = {289/379/469/478/568}, no 1
j) 14(4) cage at R4C6 = {1238/1247/1256/1346/2345}, no 9

1. Naked pair {12} in R5C34, locked for R5 and NR1C2

2. 45 rule on NR1C1 2 innies R2C3 + R4C1 = 15 = {69/78}
2a. 27(5) cage at R1C2 = {34569/34578}, 3,4,5 locked for NR1C2, clean-up: no 7,8,9 in R4C5

3. Min R3C45 + R4C4 = 22 must contain 9, locked for NR1C2
3a. 27(5) cage at R1C2 (step 2) = {34569/34578}
3b. 9 of {34569} must be in R2C3 -> no 6 in R2C3, clean-up: no 9 in R4C1 (step 2)

4. 45 rule on R123 1 outie R4C7 = 1 innie R3C2 + 1, no 1,8,9 in R4C7

5. 45 rule on R789 1 innie R7C2 = 1 outie R6C7 + 1, no 9 in R6C7, no 2 in R7C2
5a. 45 rule on R789 3 innies R7C267 = 20 = {389/479/569/578}, no 2

6. Min R5C56 = 7 -> max R46C6 = 7, no 7,8 in R46C6

7. 45 rule on C1234 2 outies R46C5 = 12 = [39/48/57], R6C4 = {234}

8. 45 rule on C1 3 outies R258C2 = 19, no 1 in R28C5

9. Law of Leftovers (LoL) for C1234 two outies R89C5 must exactly equal two innies R6C34, no 1 in R6C34 -> no 1 in R89C5

10. LoL for C6789 two outies R12C5 must exactly equal two innies R4C67, no 8,9 in R4C67 -> no 8,9 in R12C5

11. 27(5) cage at R1C2 = {34569/34578}, R2C3 + R4C1 = [78/87/96], R3C34 = {69/78} -> R2C3 + R4C1 and R3C34 must form naked quad {6789}, CPE no 6,7,8,9 in R3C12, clean-up: no 7 in R4C7 (step 4)
[It can be seen that R4C14 must be [69/78/87] which might be useful later, for example R4C14 contains both or neither of 7,8 -> R4C89 must contain both or neither of 7,8.]

12. 45 rule on R123 3 outies R4C237 = 10 = {136/145/235}
12a. R4C237 + R4C5 must contain 3, locked for R4

[I’m glad that I found the interesting step 11 before I spotted the important next step …]
13. LoL for R6789 four outies R5C1267 must exactly equal four innies R6C345 + R7C5, no 1,2 in R5C1267 -> no 1,2 in R6C34 + R7C5, clean-up: no 9 in R6C5, no 3 in R4C5 (step 7), no 9 in R4C5
13a. R3C34 = {69} (cannot be {78} which clashes with R4C4), locked for R3 and NR1C2
13b. 27(5) cage at R1C2 (step 2) = {34578} (only remaining combination), no 9, clean-up: no 6 in R4C1 (step 2)
13c. Naked pair {78} in R4C14, locked for R4
13d. Naked pair {78} in R2C3 + R4C1, locked for NR1C1
13e. 9 in R4 only in R4C89, locked for NR2C6 and 29(5) cage at R4C89, no 9 in R6C89, clean-up: no 4 in R5C7
13f. LoL (step 9), no 2 in R6C34 -> no 2 in R89C5

14. LoL (step 10), no 7 in R4C67 -> no 7 in R12C5
14a. 1,2 in NR3C5 only in R3C5 + R4C67, CPE no 1 in R2C6, no 1,2 in R3C6
14b. 9 in NR3C5 only in R6C3 + R7C5, CPE no 9 in R7C2, clean-up: no 8 in R6C7 (step 5)

15. R4C237 (step 12) = {136/235} (cannot be {145} which clashes with R4C5), no 4, clean-up: no 3 in R3C2 (step 4)

16. R46C5 (step 7) = [48/57], R6C45 = [38/47] -> R4C5 + R6C4 must contain 4 (locking cages), locked for NR3C5
16a. LoL (step 10), no 4 in R4C67 -> no 4 in R12C5

17. Hidden killer pair 1,2 in R1C5 and 15(3) cage at R2C5 for C5, 15(3) cage cannot contain both of 1,2 -> R1C5 = {12}, 15(3) cage must contain one of 1,2 = {168/258/267}, no 3,4
17a. 1,2 of 15(3) cage must be in R23C5 -> no 2 in R2C6
17b. LoL (step 9), no 3 in R12C5 -> no 3 in R4C7, clean-up: no 2 in R3C2 (step 4)

18. 3 in R4 only in R4C23, locked for NR1C1
18a. 9(3) cage at R3C2 = {135/234}, no 6

19. 14(4) cage at R4C6 = {1238/1247/1256/1346/2345}
19a. 7 of {1247} must be in R5C5 -> no 7 in R5C6

20. 15(3) cage at R3C6 = {168/258/267/357} (cannot be {348} because R4C7 only contains 2,5,6, cannot be {456} which clashes with R3C2 + R4C7 = [45/56], step 4, IOD blocker), no 4
[Thanks Ed for correcting this step and my terminology.]

21. 3 in R3 only in R3C6789, CPE no 3 in R2C8 +R5C9
21a. 3 in NR2C6 only in R3C678, locked for R3

22. 45 rule on NR1C5 3 innies R2C57 + R3C9 = 18 = {189/279/459/468/567} (cannot be {369} because 3,9 only in R2C7, cannot be {378} because no 3,7,8 in R2C5), no 3
22a. 1,2 of {189/279} must be in R2C5 -> no 1,2 in R2C7 + R3C9

[First time through I made a careless, incorrect clean-up in an earlier step. Now I’ll use a harder step, and another one later (step 27), to get back on track.]
23. Consider placement for 3 in R3
3 in R3C67 => 15(3) cage at R3C6 (step 20) = {357} = {37}5 => R3C2 = 4 (step 4) => 16(4) cage at R2C7 cannot be {1249} = 9{12}4
or R3C8 = 3 => 16(4) cage cannot also contain 9
-> no 9 in R2C7
23a. R2C57 + R3C9 (step 22) = {468/567}, no 1,2, 6 locked for R2 and NR1C5
23b. R13C5 = {12} (hidden pair in C5)
23c. 15(3) cage at R2C5 (step 17) = {168/258/267}
23d. 7,8 only in R2C6 -> R2C6 = {78}
23e. Naked pair {78} in R2C36, locked for R2

24. R2C57 + R3C9 (step 23a) = {468/567}
24a. 7,8 only in R3C9 -> R3C9 = {78}
24b. R2C7 + R3C9 = [48/57/67] (from combinations for R2C57 + R3C9) = 12,13 -> R23C8 = 3,4 = {12}/[13], 1 locked for C8 and NR2C6
24c. 4 in R3 only in R3C12, locked for NR1C1

25. R1C1 = 6 (hidden single in R1)
25a. 9 in NR1C1 only in R2C12, locked for R2
25b. 21(4) cage at R1C1 = {1569/2469}
25c. 4 of {2469} must be in R3C1 -> no 2 in R3C1

26. 15(3) cage at R3C6 (step 20) = {258/267/357}
26a. Killer pair 7,8 in R2C6 + R3C67, locked for NR2C6, clean-up: no 5,6 in R5C7

27. Naked pair {78} in R2C36, R2C3 must equal R4C4 (because of naked pair {78} in NR1C1) -> R2C6 + R4C4 form naked pair {78}
27a. 15(3) cage at R3C6 (step 20) = {258/267} (cannot be {357} = {37}5 because R2C6 = 8, R4C45 = [75] clashes with R4C7), no 3, 2 locked for C7
27b. 2 in R34C7, CPE no 2 in R4C89

28. Naked quad {4569} in R45C89, locked for NR2C6
28a. Naked pair {78} in R23C6, locked for C6 and NR2C6 -> R3C7 = 2

29. R2C8 = 1, R3C8 = 3
29a. R23C8 = [13] = 4 -> R2C7 + R3C9 = 12 = [48/57], no 6
29b. R2C5 = 6 (hidden single in R2), R3C5 = 1, placed for NR3C5, R2C6 = 8 (cage sum), R3C6 = 7, R4C7 = 6 (cage sum), placed for NR3C5, R3C2 = 5 (step 4), placed for NR1C1, R4C23 = 4 = {13}

30. R2C3 = 7, R4C1 = 8, R4C4 = 7, R4C5 = 5, placed for NR3C5, R6C5 = 7 (step 7), R6C4 = 4, clean-up: no 3,7,8 in R7C2 (step 5), no 1 in R7C34

31. R3C9 = 8, R2C7 = 4 (cage sum), both placed for NR1C5
31a. R1C5 = 2, placed for NR1C5, R1C67 = 10 = [19/37/91], no 5, no 3 in R1C7

32. Naked pair {49} in R4C89, locked for NR2C6, clean-up: no 9 in R5C7
32a. Naked pair {56} in R5C89, locked for R5

33. R4C6 = 2 -> 14(4) cage at R4C6 = {1238/2345}, (cannot be {1256} because 5,6 only in R6C6)
33a. 1,5 only in R6C6 -> R6C6 = {15}
33b. 14(4) cage = {1238/2345}, 3 locked for R5

34. R4C1 = 8, 9 in R5 only in 24(4) cage at R4C1 = {1789} (only remaining combination because 2,3,5 only in R6C1) -> R6C1 = 1, placed for NR5C1, R5C12 = {79}, locked for R5 and NR5C1 -> R5C7 = 8, R5C8 = 5, R5C5 = 3, placed for NR3C5, R56C6 = [45], both placed for NR5C6

35. R6C7 = 3, placed for NR6C7, R7C2 = 4 (step 5), R5C9 = 6, R6C9 = 2, placed for NR6C7, R6C8 = 8, R6C2 = 6, placed for NR5C1, R6C3 = 9, placed for NR3C5, clean-up: no 2 in R7C34
35a. Naked pair {35} in R7C34, locked for R7 and NR5C1

36. R258C2 = 19 (step 8) = {289} (only remaining combination) -> R5C2 = 9, R2C2 = 2, R8C2 = 8

37. R6C7 = 3 -> R7C67 = 16 = [97], both placed for NR5C6 -> R9C7 = 1, R1C7 = 9, R1C6 = 1 (cage sum)

38. R7C5 = 8 -> R8C56 = 10 = [46]

and the rest is naked singles, without using the nonets.


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