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PostPosted: Sun May 08, 2011 11:53 pm 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Old SSv3.2.1 scores:
Score = SudokuSolver v3.2.1 Score, rounded to nearest 0.05
E = Easy
H = Hard
In these tables, Rating is the lowest of the ratings given by Afmob,
Andrew and Mike, including estimates for puzzles by Afmob and Mike

+--------------------------+--------------------------+--------------------------+
| Puzzle Rating Score | Puzzle Rating Score | Puzzle Rating Score |
+--------------------------+--------------------------+--------------------------+
| A104HBN! 1.00 1.20 | A106V2 H1.50 2.15 | A107V2 E1.50 1.25 |
| A105 E1.50 1.35 | MO#3 1.00 0.95 | A108 1.25 0.90 |
| A106 1.25 1.35 | A107 H1.00 1.05 | A108V2 1.75 1.60 |
+--------------------------+--------------------------+--------------------------+

Puzzle rating table, with links to archive entries; each of these has a link to the puzzle thread.

Abbreviations used in Rating Table on this page:
Est = Estimated rating by puzzle maker
E = Easy
H = Hard
G = gary w
Go = goooders
M = Mike (mhparker)
Score = SudokuSolver v3.3.0 score, rounded to nearest 0.05
! indicates that the Score has changed at least 0.10 from the SS v3.2.1 score
R indicates lowest score out of 4 rotations of puzzle, for some harder puzzles
** in the Afmob column indicates that these puzzles were made by him,
for these ones the estimate is his rating.
+------------------------+-----------+-------+-------+-------+-----------------------+-------+
| Puzzle | Made By | Est | Afmob | Andrew| Other Raters | Score |
+------------------------+-----------+-------+-------+-------+-----------------------+-------+
| Assassin 104HBN! | Ed | 1.00 | 1.00 | 1.00 |(M)1.0 | 1.20 |
| Assassin 105 | Nasenbaer | H1.25 | E1.50 | 1.50 |(Go)1.0(M)1.50(G)1.0 | 1.35 |
| Assassin 106 | Afmob | 1.25 | ** | 1.00 |(M)1.25(Ed)1.25(G)1.25 | !1.20 |
| Assassin 106V2 | Afmob | H1.50 | ** | H1.50 | |!R2.45 |
| Messy One #3 | Nasenbaer | | 1.00 | H1.00 |(G)1.0 | 0.95 |
| Assassin 107 | Mike | 1.25 | H1.00 | H1.25 |(Ed)E1.0 | 1.05 |
| Assassin 107V2 | Mike | 1.75 | E1.50 | 1.50 | | !1.45 |
| Assassin 108 | Nasenbaer | | 1.25 | H1.25 |(Ed)H1.25(G)1.50 | !1.10 |
| Assassin 108V2 | Nasenbaer | | 1.75 | 1.75 | | !1.75 |
+------------------------+-----------+-------+-------+-------+-----------------------+-------+


Some of the selected quotes in the puzzle entries have been edited to remove "spoilers"; the full rating comments are included with the walkthroughs. In some cases the puzzle makers gave hints; these are included in tiny text in the selected quotes.


Last edited by Andrew on Thu Aug 04, 2011 1:15 am, edited 5 times in total.

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PostPosted: Mon May 09, 2011 2:16 am 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 104 HBN! by Ed (May 2008) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:1537:1537:5378:5635:5635:5635:1540:2821:2821:2310:2310:5378:5378:5635:1540:1540:3847:3847:5640:5640:2569:2569:4874:3595:3595:5132:5132:5389:5640:2569:4874:4874:4874:3595:5132:3342:5389:5389:9231:9231:9231:9231:9231:3342:3342:5389:3600:4369:9231:2066:9231:4883:3092:3342:3600:3600:4369:4369:2066:4883:4883:3092:3092:3349:3349:2326:2326:7447:3864:3864:3609:3609:1306:1306:2326:7447:7447:7447:3864:1307:1307:
Solution:
+-------+-------+-------+
| 1 5 8 | 7 4 6 | 3 2 9 |
| 6 3 4 | 9 5 2 | 1 8 7 |
| 7 9 2 | 3 1 8 | 4 5 6 |
+-------+-------+-------+
| 4 6 5 | 8 3 7 | 2 9 1 |
| 2 7 3 | 6 9 1 | 8 4 5 |
| 8 1 9 | 4 2 5 | 6 7 3 |
+-------+-------+-------+
| 5 8 7 | 1 6 4 | 9 3 2 |
| 9 4 1 | 2 7 3 | 5 6 8 |
| 3 2 6 | 5 8 9 | 7 1 4 |
+-------+-------+-------+
Quote:
Ed: Happy Birthday Nasenbaer :cheers: :cheers: :cheers: ! Bit late but had any excuse to party!. Just can't seem to get my timing right. Too early for 2nd June Anniversary of the start of Assassins.
SS(v3.1)score = 1.19 Seems a bit high - a 1.0 trick puzzle. Though may have gotten lucky and spotted the key quickly for once.

Mike (mhparker): Thanks for the relaxing puzzle, Ed!
Is it only me, or was this puzzle particularly easy? Just needed to squeeze out an important candidate in step 4 (which was pretty easy to see) and the rest came very quickly, without a single "45" test being necessary anywhere! :shock:
Rating: 0.75? Upgraded to 1.0 in a later post after forum discussion.
Last but not least, as Ed said: Happy (belated) birthday, Nasenbaer!
P.S. Nice photo you've got of Nasenbaer there (taking a break from Sudoku, I see)! ;)

Afmob: Happy Birthday, Nasenbaer! (even though it's a bit late) And also thanks Ed for providing another Assassin!
I used the same breakthrough move as Mike though with a different reasoning.
Rating: 1.0

gary w: Yep,same "step" as with Mike and Afmob.Then pretty easy.Very enjoyable!!

Andrew: Happy Birthday Nasenbaer! I hope you enjoyed solving your birthday present.
Interesting thoughts from Mike about the rating of a puzzle where there is one key move that is significantly harder than all the other moves. I think that some software solvers rate puzzles on the hardest move while others apply some form of weighting to get an average.
I'll rate A104 at 1.0 because, as Mike says, the key move isn't too difficult to find and because I finished it in one session. However one could argue for a higher rating based only on the hardest move.

Nasenbaer: Belated thanks for the late birthday wishes! :cheesey:
Just one question: WHO GAVE YOU MY PASSPORT PHOTO?!? :cheesey:
The puzzle was very nice. It took me some time to find the key move, I should have seen it coming earlier. Oh well, my age is finally showing... :doh: :roll:
Walkthrough by Mike:
Thanks for the relaxing puzzle, Ed!

Is it only me, or was this puzzle particularly easy? Just needed to squeeze out an important candidate in step 4 (which was pretty easy to see) and the rest came very quickly, without a single "45" test being necessary anywhere! :shock:

Rating: 0.75? Upgraded to 1.0 in a later post after forum discussion. (Added to Mike's original post, while compiling this archive entry)

Last but not least, as Ed said: Happy (belated) birthday, Nasenbaer!

Assassin 104 HBN Walkthrough (16 steps):

Prelims

a) 6(2) at R1C1 = {15/24}
b) 21(3) at R1C3 = {489/579/678} (no 1..3)
c) 6(3) at R1C7 = {123}
d) 11(2) at R1C8 = {29/38/47/56} (no 1)
e) 9(2) at R2C1 = {18/27/36} (no 4,5,9) (Note: {45} combo blocked by R1C12 (prelim a))
f) 15(2) at R2C8 = {69/78}
g) 22(3) at R3C1 = {589/679} (no 1..4); 9 locked
h) 10(3) at R3C3 = {127/136/145/235} (no 8,9)
i) 20(3) at R3C8 = {389/479/569/578} (no 1,2)
j) 13(4) at R4C9 = {1237/1246/1345} (no 8,9); 1 locked for N6
k) 8(2) at R6C5 = {17/26/35} (no 4,8,9)
l) 19(3) at R6C7 = {289/379/469/478/568} (no 1)
m) 13(2) at R8C1 = {49/58/67} (no 1..3)
n) 9(3) at R8C3 = {126/135/234} (no 7..9)
o) 29(4) at R8C5 = {5789}, locked for N8; cleanup: no 1,3 in R6C5
p) 14(2) at R8C8 = {59/68}
q) 5(2) at R9C1 and R9C8 = {14/23}

1. Naked quad at R9C1289 = {1234}, locked for R9
1a. -> 9(3) at R8C3 (prelim n) = {126/135}
1b. -> R8C34 = {12/13} (no 4..6)
1c. 1 locked in R8C34 for R8

2. {58} combo for 13(2) at R8C1 blocked by 14(2) at R8C8 (prelim p)
2a. -> no 5,8 in R8C12

3. 13(2) at R8C1 and 14(2) at R8C8 form killer pair on {69}
3a. -> no 6,9 elsewhere in R8
3b. 9 in 29(4) locked in R9C456 for R9

4. [456] permutation for 15(3) at R8C6 blocked by 14(2) at R8C8
4a! Also, 15(3) at R8C6 cannot be [276] (blocked by combined cage 27(4) at R8C1289)
4b. -> no 6 in R9C7

5. Hidden single (HS) in R9 at R9C3 = 6
5a. -> split 3(2) at R8C34 = {12}, locked for R8
5b. cleanup: no 7 in R8C12

6. Naked pair (NP) at R8C1 = {49}, locked for R8 and N7
6a. cleanup: no 1 in R9C12, no 5 in R8C89

7. Naked single (NS) at R8C6 = 3
7a. -> split 12(2) at R89C7 = {57}, locked for C7 and N9
7b. cleanup: no 5 in R6C5

8. NP at R9C12 = {23}, locked for R9 and N7
8a. -> R8C34 = [12]
8b. cleanup: no 6 in R6C5

9. NP at R8C89 = {68} locked for R8 and N9

10. NP at R9C89 = {14}, locked for N9

11. 1 unavailable for 12(3) at R6C8
11a. -> {129} combo blocked
11b. -> R7C89 = {23}, locked for R7
11c. -> R7C7 = 9, R6C8 = 7 (cage sum)
11d. cleanup: no 4 in R1C9, no 8 in R2C9

12. R67C5 = [26]
12a. -> R7C46 = [14]
12b. -> R6C7 = 6 (cage sum)

13. Split 16(2) at R67C3 = [97] (only possible permutation)

14. NP at R7C12 = {58}
14a. -> R6C2 = 1 (cage sum)
14b. cleanup: no 5 in R1C1, no 8 in R2C1

15. HS in N6 at R4C8 = 9
15a. cleanup: no 2 in R1C9, no 6 in R2C9

16. 21(3) at R1C3 = {489} (last combo)
16a. -> R2C4 = 9; R12C3 = {48}, locked for C3 and N1
16b. -> R1C12 = [15], R2C89 = [87]
16c. -> R12C3 = [84], R7C12 = [58], R8C89 = [68] ...

Rest is pretty trivial now.
P.S. Nice photo you've got of Nasenbaer there (taking a break from Sudoku, I see)! ;)
Walkthrough by Afmob:
Happy Birthday, Nasenbaer! (even though it's a bit late) And also thanks Ed for providing another Assassin!

I used the same breakthrough move as Mike though with a different reasoning. I think my step 2a and Mike's 4a is too difficult for a 0.75 Killer, so I can understand SS rating. The size of one's walkthrough doesn't always relate to the difficulty of the Killer and just because a complicated step is expressed in a short way doesn't mean that it's less difficult as you can (also) see in the thread about HATMAN's Human Solvable.

A 104 Walkthrough:

1. R789
a) Naked quad (1234) locked in R9C1289 for R9
b) 9(3) = 1{26/35} because R9C3 = (56) -> 1 locked for R8 and R8C45 <> 5,6
c) 29(4) = {5789} locked for N8
d) 13(2) <> {58} since it's a Killer pair 14(2)
e) Killer pair (69) locked in 13(2) + 14(2) for R8
f) 8 locked in R7C123 @ N7 for R7
g) 15(3) <> 5{28/46} because they are blocked by Killer pairs (56,58) of 14(2)

2. R789 !
a) ! R9C7 <> 6 because 15(3) = [276] leaves no placement for Innies N9 = 26(5)
since R7C789 <> 8 and {13679} blocked by Killer pair (13) of 5(2)
b) Hidden Single: R9C3 = 6 @ R9
c) 13(2) = {49} locked for R8+N7
d) 5(2) @ N7 = {23} locked R9+N7
e) R8C3 = 1 -> R8C4 = 2
f) 15(3) = {357} -> R8C6 = 3; {57} locked for C7+N9
g) Naked triple (146) locked in R7C456 for R7
h) 19(3) = {469} because R7C6 = (46) -> R7C7 = 9
i) 12(3) = {237} because R7C89 = (23) -> R6C8 = 7
j) 8(2) = {26} -> R7C5 = 6, R6C5 = 2

3. N78
a) R7C6 = 4 -> R6C7 = 6
b) R7C4 = 1 -> R67C3 = 16(2) = [97] -> R6C3 = 9, R7C3 = 7
c) 14(3) = {158} because R7C12 = {58} -> R6C2 = 1
d) Hidden Single: R4C8 = 9 @ N6

4. N36
a) 13(4) = {1345} locked for N6
b) Naked pair (28) locked in R45C7 for C7
c) 6(3) = {123} -> R2C6 = 2, {13} locked for N3
d) R3C7 = 4 -> R3C6+R4C7 = 10(2) = [82] -> R3C6 = 8, R4C7 = 2
e) 20(3) = {569} -> {56} locked for R3+N3
f) R2C8 = 8 -> R2C9 = 7

5. N15
a) 21(3) = {489} -> R2C3 = 4, R1C3 = 8, R2C4 = 9
b) 10(3) = {235} -> R3C4 = 3, R3C3 = 2, R4C3 = 5
c) 36(7) = {1345689} -> R5C3 = 3, R5C7 = 8, R6C6 = 5, R6C4 = 4, R5C4 = 6
-> {19} locked for R4+N5

6. Rest is singles.

Rating: 1.0
Walkthrough by Andrew:
Happy Birthday Nasenbaer! I hope you enjoyed solving your birthday present.

I must admit that I did originally start using some 45s, even through there aren't as many as usual because of the way R12, R89, C12 and C89 don't have any innies-outies with the other rows and columns. Innies-outies on R9 made me realise that 6 in R9 was important so then I looked again for a shortcut move and found it quickly. Then I reworked my walkthrough from the beginning and, like Mike, had a solving path without any 45s.

Interesting thoughts from Mike about the rating of a puzzle where there is one key move that is significantly harder than all the other moves. I think that some software solvers rate puzzles on the hardest move while others apply some form of weighting to get an average.

I'll rate A104 at 1.0 because, as Mike says, the key move isn't too difficult to find and because I finished it in one session. However one could argue for a higher rating based only on the hardest move.

Here is my walkthrough.

Prelims

a) R1C12 = {15/24}
b) R1C89 = {29/38/47/56}, no 1
c) R2C12 = {18/27/36} (cannot be {45} which clashes with R1C12), no 4,5,9
d) R2C89 = {69/78}
e) R56C5 = {17/26/35}, no 4,8,9
f) R8C89 = {59/68}
g) R8C12 = {49/67} (cannot be {58} which clashes with R8C89)
h) R9C12 = {14/23}
i) R9C89 = {14/23}
j) 21(3) cage at R1C3 = {489/579/678}, no 1,2,3
k) 6(3) cage at R1C7 = {123}
l) 22(3) cage at R3C1 = {589/679}
m) 10(3) cage at R3C3 = {127/136/145/235}, no 8,9
n) 20(3) cage at R3C8 = {389/479/569/578}, no 1,2
o) 19(3) cage at R6C7 = {289/379/469/478/568}, no 1
p) 9(3) cage at R8C3 = {126/135/234}, no 7,8,9
q) 13(4) cage in N6 = {1237/1246/1345}, no 8,9 , 1 locked for N6
r) 29(4) cage in N8 = {5789}, locked for N8, clean-up: no 1,3 in R6C5
s) 36(7) cage at R5C3 must contain 9

1. Naked quad {1234} in R9C1289, locked for R9

2. Killer pair 6,9 in R8C12 and R8C69, locked for R8
2a. 9 in N8 locked in R8C456, locked for R8
2b. 8 in N7 locked in R7C123, locked for R7

3. 9(3) cage at R8C3 = {126/135} (cannot be {234} because R9C3 only contains 5,6), no 4, 1 locked for R8
3a. R9C3 = {56} -> no 5 in R8C3

4. Hidden killer pair 2,3 in R8C34 and R8C67 for R8 -> R8C67 must contain 2 or 3
4a. 15(3) cage at R8C6 = {348/357} (cannot be {258} which clashes with R8C89, cannot be {267} because 8 cannot be placed in N9, cannot be {456} which doesn’t contain 2 or 3), no 2,6, 3 locked for R8
4b. 8 of {348} must be in R9C7 -> no 8 in R8C7
[The hidden killer pair was how I first saw this step. As others have shown it isn't needed, {456} can be eliminated because it clashes with R8C89. The rating should therefore take account of the elimination of {267} but ignore my hidden killer pair.]

5. R9C3 = 6 (hidden single in R9), clean-up: no 7 in R8C12
5a. Naked pair {49} in R8C12, locked for R8 and N7 -> R8C6 = 3, clean-up: no 5 in R6C5, no 5 in R8C89, no 1 in R9C12
5b. Naked pair {23} in R9C12 , locked for R9 and N7 -> R8C34 = [12], clean-up: no 6 in R6C5
5c. Naked pair {68} in R8C89, locked in R8 and N9
5d. Naked pair {57} in R89C7, locked for C7 and N9
5e. Naked pair {14} in R9C89, locked for N9

6. 6(3) cage at R1C7 = {123}
6a. 3 locked in R12C7, locked for C7 and N3, clean-up: no 8 in R1C89

7. 19(3) cage at R6C7 = {469} (only remaining combination, cannot be {289} because R7C6 only contains 4,6), no 2,8 -> R7C7 = 9
7a. R7C89 = {23} -> R6C8 = 7 (cage sum), R6C5 = 2, R7C5 = 6, R7C46 = [14], R6C7 = 6, clean-up: no 4 in R1C9, no 8 in R2C9

8. 13(4) cage in N6 = {1345} (only remaining combination), locked for N6

9. Naked pair {28} in R45C7, locked for C7 and N6 -> R4C8 = 9, clean-up: no 2 in R1C9, no 6 in R2C9
9a. Naked pair {13} in R12C7, locked for C7 and 6(3) cage -> R2C6 = 2, R3C7 = 4, clean-up: no 7 in R1C9, no 7 in R2C12

10. R7C4 = 1 -> R67C3 = 16 = [97]

11. R4C8 = 9 -> R3C89 = 11 = {56}, locked for R3 and N3 -> R1C89 = [29], R2C89 = [87], R7C89 = [32], R8C89 = [68], R3C89 = [56], clean-up: no 4 in R1C12, no 1 in R2C12

12. Naked pair {15} in R1C12, locked for R1 and N1 -> R12C3 = [84], R2C4 = 9 (cage sum), R12C7 = [31]

13. Naked pair {36} in R2C12, locked for R2 and N1 -> R3C3 = 2, R2C5 = 5, R8C5 = 7, R89C7 = [57], R1C5 = 4

14. Naked pair {67} in R1C46, locked for N2 -> R3C4 = 3, R45C3 = [53]

15. R3C12 = {79} -> R4C2 = 6 (cage sum), R2C12 = [63], R9C12 = [32]

16. R7C12 = {58} -> R6C2 = 1 (cage sum), R1C12 = [15], R7C12 = [58]

17. R3C7 = 4 -> R3C6 + R4C7 = 10, R3C6 = 8, R4C7 = 2

and the rest is naked singles.


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PostPosted: Mon May 09, 2011 3:05 am 
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Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 105 by Nasenbaer (May 2008) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:2816:4097:4097:4097:1796:4613:4613:4613:5128:2816:2816:3595:3595:1796:3598:3598:5128:5128:4114:2579:2579:3595:8726:3598:3352:3352:4122:4114:4114:2579:8726:8726:8726:3352:4122:4122:2852:2852:4646:4646:8726:4393:4393:2859:2859:3629:2852:3887:4646:8726:4393:2867:2859:1333:3629:2359:3887:5177:5177:5177:2867:2109:1333:3903:2359:2113:2113:5177:4932:4932:2109:4423:3903:3903:2113:4683:4683:4683:4932:4423:4423:
Solution:
+-------+-------+-------+
| 1 9 4 | 3 2 6 | 5 7 8 |
| 2 8 6 | 7 5 4 | 1 9 3 |
| 7 5 3 | 1 8 9 | 4 6 2 |
+-------+-------+-------+
| 5 4 2 | 9 1 7 | 3 8 6 |
| 3 1 9 | 4 6 8 | 7 2 5 |
| 6 7 8 | 5 3 2 | 9 4 1 |
+-------+-------+-------+
| 8 3 7 | 6 9 1 | 2 5 4 |
| 9 6 1 | 2 4 5 | 8 3 7 |
| 4 2 5 | 8 7 3 | 6 1 9 |
+-------+-------+-------+
Quote:
Nasenbaer: We have to work on that Assassin schedule. I can't believe it, still no volunteers on Thursday evening!?! :cry: That's why you get a nasty one from me. :twisted:
It took me several hours to create this one. Either the score was too low (0.7) or way too high (1.9+). Finally this beauty came up. SSolver(3.1)score is 1.37, so you might have to kick your brain cells a little bit. :lol:

Mike (mhparker): Thanks for doing this, and working into the small hours of the morning to get it finished. You are a hero! :cheers:
As to the "nasty" puzzle, I actually quite enjoyed doing it! :shock: (maybe because I am a masochist? :) ). Will post my WT later, after I have a chance to work through my rough notes. :study:

goooders: Not particularly scientific but I rate assassins on how long it takes me to do them.This was less than an hour so I would give it a 1.0

Afmob: This was quite a Killer. I took me quite some time to make progress (hence the high rating) and there is probably an easier way to solve it. But it was still interesting because I discovered ...
Rating: (Easy) 1.5.

Mike(mhparker): As promised, here my walkthrough for the Assassin 105. Very nice puzzle, Nasenbaer. Many thanks! :-D
Even though it didn't take me so long to do (more like 2 hours for me), as far as techniques are concerned, it's probably in the 1.5 rating category.
Hopefully, others can get involved in posting Assassins in the future. Variants are nice, but the priority has to remain on keeping to the Assassin schedule as closely as possible, so that people get used to checking out this site every Friday.

udosuk: Mike, great walkthrough, with step 4 being the thoughtful critical move that cracks it all open. :thumbs:

Then various discussions and suggested alternatives to a couple of Mike's steps; these posts are included in the walkthroughs below.

gary w: A nice puzzle..
about 1.0-1.25?

Andrew: Thanks for a challenging puzzle Nasenbaer!
I found it difficult because I didn't spot Mike's step 4. Nice one! I also liked the way that Afmob used ...
I decided to post my walkthrough to show there is another way to break this puzzle open; mine was steps 28 and 29. It's good when some puzzles have multiple ways to solve them yet are also challenging.
I'll rate A105 at 1.5 the way I solved it.

Nasenbaer posted "If you are interested here is the scoring output of the puzzle. WARNING. Might contain spoiler.":
SudokuSolver V3.1.0 by Richard Broughton
Scoring Puzzle:

Puzzle Completed. Step Analysis:
7 Find Hidden Cages
68 Naked Singles
3 Naked Pairs
1 Naked Quads
2 Hidden Singles
1 Locked Cages 2
3 Cage Blockers
2 Cage Placement 2
5 Cage Placement 1
2 Cage Placement
4 Cage Combinations
2 Cage Combinations Extended
8 Cage Blockers Extended 9
1 Cage Placement Extended
5 45 Rule Two Innies/Outies 2
3 45 Rule Single House 4
1 Common Peer Elimination Extended 3
55 Cage Cleanup
1 Locked Candidates (House/Cage)
1 Forced Cage Candidates - Extended
6 45 Rule Simple Single House 3
1 Forced Killer Pairs
5 45 Rule Simple Single House Innies&Outies 3
2 45 Rule Single House Innies&Outies 4
3 45 Rule Extended Single House Innies&Outies 5
Total solving time (seconds): 17,90
Calculated score: 1,37
Walkthrough by Afmob:
Afmob wrote:
This was quite a Killer. I took me quite some time to make progress (hence the high rating) and there is probably an easier way to solve it. But it was still interesting because I discovered 3 Killer quads but one was omitted since it was redundant after some less difficult moves.

A105 Walkthrough:

1. C1234
a) Innies C12 = 14(2) = [86/95]
b) Innies = 23(3) = {689} locked for C4
c) Innies+Outies N7: -13 = R8C4 - R7C13 -> R8C4 <> 5
d) 10(3): R34C3 <> 5,6,7 because R3C2 >= 5;
e) 16(3) @ R1: R1C3 <> 8,9 because R1C2 >= 8
f) 8(3) = 1{25/34} -> 1 locked between N7+R8 -> R8C12 <> 1
g) 9(2): R7C2 <> 8
h) 8(3): R8C4 <> 4 because (13) is a Killer pair of 10(3)
i) 8(3): R89C3 <> 3 because (34) is a Killer pair of 10(3)
j) 9(2) <> 4,5 because (45) is a Killer pair of 8(3)
k) 18(3): R5C3 = (6789) because R56C4 <= 12; 18(3) <> 1 because R56C4 <> 8,9
l) 18(3): R5C3 <> 7 because (689) only possible there

2. R1234
a) Innies R1 = 11(3) <> 9; R1C1 <> 8 because R1C9 >= 3
b) Outies R12 = 10(2) <> 5; R3C6 <> 1,2,4
c) 14(3) @ N1: R2C3 <> 1 because R23C4 <= 12
d) 14(3) @ N3: R2C67 <> 8 because R3C6 <> 1,2,4,5
e) 18(3) <> 1 because R1C2 = (89)

3. R789
a) Innies N8 = 7(2) -> R8C6 = (456)
b) 19(3) <> 2,3 because R8C6 = (456)
c) Innies+Outies N9: -1 = R8C6 - R7C79; R8C6 = (456)
-> R7C7 <> 7,8,9
d) 11(2): R6C7 <> 2,3,4

4. N5+C6789 !
a) Outies N5 = 24(2+1) <> 1,2,3,4,5; R3C5 = (789)
b) Innies C89 = 13(2) <> 1,2,3
c) Innies C6789 = 11(3) <> 9
d) Using Innies C89: Innies N3 = 12(4) <> 7,8,9
e) ! 20(3) <> 6 because (569) is a Killer triple of Innies C89

5. N5+C3456 !
a) ! Hidden Killer quad (1234) in 18(3) for N5 because 17(3) can only have one
and 34(6) must have two of (1234) -> 18(3) <> {567}
b) 18(3) must have 8 xor 9 -> R5C3 <> 6
c) Killer pair (89) locked in R5C3+15(2) for C3
d) Both 14(3): R2C37 <> 4 because 10(2) @ R23C4 or R23C6 clashes with Outies R12
e) 18(3) @ N2: R1C6 <> 5 because 13(2) @ R1C78 clashes with Innies C89
f) 16(3): R1C4 <> 2 because 14(2) @ R1C23 clahes with Innies C12

6. R123+C34 !
a) Using Innies C12: Innies N1 = 20(4): R3C1 <> 1,2,3 because R123C3 <= 16
since {467} is blocked by Killer pair (67) of 15(2)
b) 14(3) @ N1: R2C3 <> 2,3 because (47,57) are Killer pairs of 18(3) @ N5
c) Innies N1 = {1379/1478/2378/2459/2468/3467} since other combos blocked by Innies C12
d) Innies N1: R3C1 = (6789) since 8,9 can only be placed there and only
one of (67) can be placed @ R12C3 because of Killer pair (67) of 15(2)
e) ! Hidden Killer quad (1234) in R1C3 for N1 since 11(3) must have two of (1234) -> R1C3 = (1234)
f) 16(3) @ R1: R1C4 <> 1 because R123 <= 13

7. N14 !
a) ! Innies+Outies N14: -26 = R1C4 - (R256C3+R6C1)
-> R1C4 <> 5,7 because R256C3+R6C1 <= 30
b) 16(3) @ R1 = {349} -> R1C2 = 9 -> {34} locked for R1
c) Innies C12 = 14(2) = [95] -> R3C2 = 5

8. C789
a) 18(3) = {567} locked for R1, 5 also locked for N3
b) R1C9 = 8 -> R2C89 = {39} locked for R2+N3
c) Innies C89 = 13(2) = {67} locked for C8+N3
d) 8(2) = {35} locked for C8+N9
e) Hidden Single: R4C7 = 3 @ C7 -> R3C78 = 10(2) = [46] -> R3C7 = 4, R3C8 = 6
f) R1C7 = 5
g) 11(2) = {29} -> R7C7 = 2, R6C7 = 9
h) 17(3) @ N9 = 7{19/46} -> 7 locked for C9+N9
i) 16(3) = {268} -> R3C9 = 2, R4C9 = 6, R4C8 = 8
j) 19(3) = {568} -> R8C6 = 5

9. R789
a) Innies N8 = 7(2) = {25} -> R8C4 = 2
b) 8(3) = {125} -> R8C3 = 1, R9C3 = 5
c) 9(2) = {36} locked for C2+N7
d) 3 locked in 18(3) @ R9 for N8 -> 18(3) = 3{69/78}
e) 17(3) = {179} locked for N9

10. Rest is singles.

Rating: (Easy) 1.5. I used two Killer quads and Innies+Outies analysis.
Walkthrough by Mike:
As promised, here my walkthrough for the Assassin 105. Very nice puzzle, Nasenbaer. Many thanks! :-D

Even though it didn't take me so long to do (more like 2 hours for me), as far as techniques are concerned, it's probably in the 1.5 rating category.

Hopefully, others can get involved in posting Assassins in the future. Variants are nice, but the priority has to remain on keeping to the Assassin schedule as closely as possible, so that people get used to checking out this site every Friday.

Edit: Incorporated udosuk's optimization (steps 10 and 11), which proved to be a slot-in replacement for my original steps 10 - 13, shortening the WT by 2 steps. Many thanks!

Edit2: More minor corrections due to missed common peers in step 4 (thanks again udosuk!)

Assassin 105 Walkthrough (16 steps)

Prelims

a) 11(3) at R1C1, R5C1 and R5C8 = {128/137/146/236/245} (no 9)
b) 7(2) at R1C5 = {16/25/34} (no 7..9)
c) 20(3) at R1C9 = {389/479/569/578} (no 1,2)
d) 10(3) at R3C2 = {127/136/145/235} (no 8,9)
e) 14(2) at R6C1 = {59/68} (no 1..4,7)
f) 15(2) at R6C3 = {69/78} (no 1..5)
g) 11(2) at R6C7 = {29/38/47/56} (no 1)
h) 5(2) at R6C9 = {14/23} (no 5..9)
i) 9(2) at R7C2 = {18/27/36/45} (no 9)
j) 8(2) at R7C8 = {17/26/35} (no 4,8,9)
k) 8(3) at R8C3 = {125/134}; 1 locked -> no 1 in R8C12; cleanup: no 8 in R7C2
l) 19(3) at R8C6 = {289/379/469/478/568} (no 1)

1. Innies C12: R13C2 = 14(2) = [86/95]
1a. 16(3) at R1C2 can only have 1 of {89}
1b. -> no 8,9 in R1C34
1c. R34C3 must sum to 4 or 5 = {13/14/23} (no 5..7)

2. Outies N5: R3C5+R5C37 = 24(1+2) = {7+89/8+79/9+69/9+78} (no 1..5, no 6 in R3C5)
(Note: thanks to Ruud's killer calculator for that one! :-))

3. Innie/Outie (I/O diff. N7): R7C13 = R8C4 + 13
3a. max. R7C13 = 17 -> max. R8C4 = 4
3b. -> no 5 in R8C4
3c. R7C13 = 14..17 = {(8/9)..}

4. Chain (AIC): (5=6)r3c2,(13)r34c3-((1|3)=5)r89c3
4a. -> 5 locked in R3C2+R89C3
4b. -> no 5 in R12C3+R789C2 (common peers)
4c. cleanup: no 4 in R78C2

(Note: This can be expressed in more verbose form as follows:
(i) Either R3C2 = 5, or...
(ii) R3C2 = 6 -> R34C3 = {13}
(iii) -> {134} combo for 8(3)n78 blocked -> 8(3) = {125}
(iv) -> R89C3 must contain a 5
(v) => from (i) and (iv), either R3C2 or R89C3 (or both) must contain a 5)

5. 5 in C3 locked in R89C3 for N7
5a. -> 8(3) at R8C3 = {125} (no 3,4)
5b. no 2 in R8C12 (CPE)
5c. cleanup: no 9 in R6C1, no 7 in R7C2

6. 4 in N7 locked in 15(3) at R8C1
6a. furthermore: 15(3), 9(2) and R89C3 form hidden killer triple on {123} within N7
6b. -> 15(3) = {249/348} (no 1,6,7) = {(8/9)..}

7. 15(3) at R8C1 (step 6b) and R7C13 (step 3c) form killer pair on {89} within N7
7a. -> no 8 in R8C2
7b. cleanup: no 1 in R7C2

8. 1 in N7 locked in R89C7 for C3 and 8(3)
8a. -> R8C4 = 2
8b. cleanup: no 6 in R7C8

9. Innie N8: R8C6 = 5
9a. -> R89C3 = [15]
9b. no 3,7 in R7C8

10. Split 14(2) at R89C7 = {68} (last combo), locked for C7 and N9
10a. cleanup: no 2 in R8C8; no 3,5 in R67C7

11. Innies R89: R8C258 = 13(3) = [643] (last permutation)
11a. -> R8C28 = [35], R89C7 = [86], R8C19 = [97], R3C2 = 5
11b. -> R67C1 = [68]
11c. -> R67C3 = [87]
11d. -> R5C37 = [97]
11e. -> R3C5 = 8 (step 2)
11f. cleanup: no 4 in R34C3, no 4 in R67C7, no 2 in R6C9, no 3 in R12C5


12. Innies N9: R7C79 = 6(2) = [24] (last permutation)
12a. -> R6C79 = [91]
12b. -> R9C89 = [19]

13. R12C2 = [98] (both HS@C2)
13a. -> split 3(2) at R12C1 = {12}, locked for C1 and N1
13b. -> R34C3 = [32], R9C12 = [42]
13c. -> R3C1 = 7
13d. -> split 9(2) at R4C12 = [54] (last permutation)
13e. -> 11(3) at R5C1 = [317]

14. Outies R1234: R56C5 = 9(2) = [63] (last permutation)
14a. -> R9C5 = 7
14b. cleanup: no 1 in R12C5

15. 11(3) at R5C8 = {245} (last combo)
15a. -> R5C9 = 5; R56C8 = {24}, locked for C8

16. 20(3) at R1C9+R2C89 = [893] (last permutation)

Rest is just singles and simple cage sums now.
udosuk's alternative step 9:
Mike, great walkthrough, with step 4 being the thoughtful critical move that cracks it all open. :thumbs:

However, after step 9, I tried another route which seemed to make the rest of the puzzle dissolve a little bit faster:

...
9. Innie N8: R8C6 = 5
9a. -> R89C3 = [15]
9b. no 3,7 in R7C8

19/3 @ r8c67+r9c7: r89c7=19-5=14={68} (NP @ c7,n9)
Innies @ r89: r8c258=13=[643]
... :geek:
Mike's further thoughts on step 4:
udosuk wrote:
However, after step 9, I tried another route which seemed to make the rest of the puzzle dissolve a little bit faster:

Thanks for the suggestion :-D. Very welcome! I've updated my WT to incorporate your alternative path, and it does indeed shorten the WT. Interestingly, my endgame is unaffected by your optimization.

udosuk wrote:
...with step 4 being the thoughtful critical move...

Some more thoughts on this one:
Alternative view of my step 4
Although I expressed it as a chain, it's interesting to note that the chain only involves 2 cages (edit: and nothing else). Therefore, an automated solver could in theory find this move by performing a permutational analysis of the combined cage 18(6) at r3c2+r3489c3+r8c4, which would likewise also show that 5 is locked in r3c2+r89c3, removing 5 from their common peers. SudokuSolver already handles combined cages, so might already be able to find this, if it weren't for easy moves available earlier in the pecking order. (However, I'm not sure that SS handles combined cages that aren't completely contained within a single house.)

Although permutational analysis of combined cages is not something a human would do sitting in the armchair at home (!), the results can usually be re-expressed in simpler (WT-friendly) form, as in this case.

Note: a permutational analysis on a combined cage is not to be confused with J-C's cages grouping solver, which performs a partial recursive solve on 2 cages.
And udosuk's further thoughts on step 4:
More thought's on Mike's step 4:
At that point r34c3={13|14|23}, r89c3={13|14|15|25|34}.

If r34c3={13} => r89c3 can't be {13|14|34}, must be {15|25}
If r34c3={14|23} => r3c2=5

Therefore r123c3+r789c2, seeing r3c2+r89c3, can't have 5.

Another way is writing r3c2+r3489c3+r8c4 as

[ 5{14}{25}1 | 5{23}{14}3 | 5{23}{15}2 | 6{13}{25}1 ]

and observing that 5 always appears in r3c2+r89c3. :geek:
gary w's solving outline:
A nice puzzle..

r13c2=14 so r3c2=5/6
outies n5 24(3) so r5c3<>5
thus if r3c2 is 5 OR 6 a 5 is forced into r89c3
so in 15(3) cage n7 4 is placed so 15(3) <>7
i-o n7=13 and r8c4=1/2 but if =1 cannot place a 7 in n7.So r8c4=2
so r89c3=15 so r3c2 must be a 5 and r1c2=9
extended mop-up now.

about 1.0-1.25?
udosuk "Here is a walkthrough following gary w's efficient approach":
Here is a walkthrough following gary w's efficient approach:
8/3 @ r89={1(25|34)}
Innies @ c12: r13c2=14=[86|95]
10/3 @ r34=[5{14|23}|6{13}]
Outies @ n5: r3c5+r5c37=[7{89}|8{79}|9{69|78}]

Now r3c2+r3489c3+r8c4=[5{14}{25}1|5{23}{14}3|5{23}{15}2|6{13}{25}1]
=> at least one of r3c2+r89c3 must have 5
=> r12c3+r789c2, seeing r3c2+r89c3, can't have 5
=> 5 @ c3,n7 locked @ r89c3
=> 8/3 @ r89=[{15}2|{25}1], 9/2 @ n7={18|27|36}
=> 4 @ n7 locked @ 15/3 @ n7={4(29|38)}

Innies/outies @ n7: r7c13=r8c4+13=14|15={68|69|78}
=> r7c13+15/3 form Killer NP {89} @ n7
=> 9/2 @ n7 can't have 8, must be {27|36}
=> 1 @ c3,n7 locked @ r89c3={15} => r8c4=2
=> r34c3={23} (NP @ c3), r3c2=5

Innies @ n8: r8c6=5 => r89c3=[15]
=> r89c7=19-5=14={68} (NP @ c7,n9)
=> 11/2 @ c7={29|47} => HS @ n9: 8/2 @ n9=[53]

Innies @ n9: r7c79=6={24} (NP @ r7,n9)
=> 9/2 @ n7=[36] => r89c7=[86]
=> r7c13=2+13=15=[87] => r6c13=[68]
=> 20/4 @ n8=[{169}4] => r8c19=[97]

r3c5+r5c37=[897] => 11/2 @ c7=[92]
=> 5/2 @ c9=[14] => r56c6=17-7=10=[64|82]
=> r6c68={24} (NP @ r6) => 11/3 @ n4=[317]
=> r34c3=[32] => 16/3 @ r34=[754]
=> HS @ n6: r5c9=5 => r56c8=11-5=6={24} (NP @ c8)

20/3 @ n3 from {36789} must be {389} (NT @ n3)
=> HS @ n3: r1c8=7 => r1c67=18-7=11=[65]
=> 13/3 @ r34=[463] => 14/3 @ r23=[419]

All naked singles from here. :geek:

Added later: Just realise the major steps above is almost identical to Mike's ones. So nothing too brand new there. gary w just used the same logic, but presenting them in a more concise manner with a heavier T&E feel. :ugeek:
Walkthrough by Andrew:
Thanks for a challenging puzzle Nasenbaer!

I found it difficult because I didn't spot Mike's step 4. Nice one! I also liked the way that Afmob used Innies on Innies in steps 4d and 6a.

I decided to post my walkthrough to show there is another way to break this puzzle open; mine was steps 28 and 29. It's good when some puzzles have multiple ways to solve them yet are also challenging.

I'll rate A105 at 1.5 the way I solved it.

Here is my walkthrough. Like Afmob I spotted three different hidden killer quads but I only used one of mine; the other ones were in N6 and R4.

Prelims

a) R12C5 = {16/25/34}, no 7,8,9
b) R67C1 = {59/68}
c) R67C3 = {69/78}
d) R67C7 = {29/38/47/56}, no 1
e) R67C9 = {14/23}
f) R78C2 = {18/27/36/45}, no 9
g) R78C8 = {17/26/35}, no 4,8,9
h) 11(3) cage in N1 = {128/137/146/236/245}, no 9
i) 20(3) cage in N3 = {389/479/569/578}, no 1,2
j) 10(3) cage at R3C2 = {127/136/145/235}, no 8,9
k) 11(3) cage in N4 = {128/137/146/236/245}, no 9
l) 11(3) cage in N6 = {128/137/146/236/245}, no 9
m) 8(3) cage at R8C3 = {125/134}, CPE no 1 in R8C12, clean-up: no 8 in R7C2
n) 19(3) cage at R8C6 = {289/379/469/478/568}, no 1
o) 34(6) cage at R3C5 must contain 9

1. 45 rule on R1 3 innies R1C159 = 11 = {128/137/146/236/245}, no 9

2. 45 rule on R12 2 outies R3C46 = 10 = {19/28/37/46}, no 5

3. 45 rule on R1234 2 outies R56C5 = 9 = {18/27/36/45}, no 9

4. 45 rule on C12 2 innies R13C2 = 14 = [86/95]

5. R1C234 = {169/178/259/268/349/358} (cannot be {367/457} because R1C2 only contains 8,9)
5a. R1C2 = {89} -> no 8,9 in R1C34

6. 18(3) cage at R1C6 = {279/369/378/459/468/567} (cannot be {189} which clashes with R1C2), no 1

7. 10(3) cage at R3C2 = {136/145/235} (cannot be {127} because R3C2 only contains 5,6), no 7
7a. R3C2 = {56} -> no 5,6 in R34C3
7b. R34C3 = {13/14/23}

8. 45 rule on C89 2 innies R13C8 = 13 = {49/58/67}, no 1,2,3

9. 45 rule on C1234 3 innies R479C4 = 23 = {689}, locked for C4, clean-up: no 1,2,4 in R3C6 (step 2)

10. 45 rule on C6789 3 innies R479C6 = 11 = {128/137/146/236/245}, no 9

11. 18(3) cage at R5C3 = {279/378/459/567} (cannot be {189/369/468} because 6,8,9 only in R5C3), no 1
11a. 6,8,9 can only be in R5C3 -> R5C3 = {689}

12. 45 rule on N5 3 (1+2) outies R3C5 + R5C37 = 24
12a. Max R5C37 = 17 -> min R3C5 = 7
12b. R3C5 = {789} -> R5C37 = 15,16,17 = {69/78/79/89}, no 1,2,3,4,5 in R5C7
12c. If R3C5 = 9 => 9 in N5 in R56C6 => no 9 in R5C7
12d. -> no 6 in R5C3

13. Killer pair 8,9 in R5C3 and R67C3, locked for C3

14. 45 rule on R789 4 outies R6C1379 = 24 = {1689/2589/2679/3579/3678/4569/4578} (cannot be {3489} which clashes with R5C3)
14a. 1,2,3,4 must be in R6C9 -> no 2,3,4 in R6C7, clean-up: no 7,8,9 in R7C7

15. 45 rule on N7 2 innies R7C13 = 1 outie R8C4 + 13, max R7C13 = 17 -> max R8C4 = 4

16. 45 rule on N8 2 innies R8C46 = 7 = [16/25/34/43], no 2,7,8,9 in R8C6

17. 19(3) cage at R8C6 = {379/469/478/568} (cannot be {289} because 2,8,9 only in R89C7), no 2
17a. 3 of {379} must be in R8C6 -> no 3 in R89C7

18. 45 rule on C123 2 innies R25C3 = 2 outies R18C4 + 10
18a. Min R18C4 = 3 -> min R25C3 = 13, no 1,2,3 in R2C3
18b. Max R25C3 = 16 -> max R18C4 = 6, no 7 in R1C4

19. R1C234 (step 5) = {169/178/259/268/349/358}
19a. 1 of {169/178} must be in R1C4 -> no 1 in R1C3

20. 45 rule on C789 2 outies R18C6 = 2 innies R25C7 + 3
20a. Min R25C7 = 7 -> min R18C6 = 10, no 2,3 in R1C6
20b. Max R18C6 = 15 -> max R25C7 = 12, max R2C7 = 5 (R25C7 cannot be [66])

21. 13(3) cage at R3C7 = {139/148/157/238/247/256/346}
21a. 9 of {139} must be in R3C8 -> no 9 in R34C7

22. 14(3) cage at R2C6 = {149/158/167/239/248/257/347/356}
22a. 8 of {158/248} must be in R3C6 -> no 8 in R2C6

23. 45 rule on N1 4 innies R2C3 + R3C123 = 1 outie R1C4 + 18
23a. Min R2C3 + R3C123 = 19, max R2C3 + R3C23 = 17 -> min R3C1 = 2

24. 8(3) cage at R8C3 = {125/134}
24a. 1,4 of {134} must be in R89C3 (R89C3 cannot be {13/34} which clash with R34C3) -> no 3 in R89C3, no 4 in R8C4, clean-up: no 3 in R8C6 (step 16)
24b. Killer pair 1,2 in R34C3 and R89C3, locked for C3

25. 45 rule on R12 2 innies R2C34 = 1 outie R3C6 + 4, IOU no 4 in R2C3

26. 45 rule on R12 2 innies R2C67 = 1 outie R3C4 + 4, IOU no 4 in R2C7

27. 45 rule on N47 2 innies R45C3 = 2 outies R3C1 + R8C4 + 2
27a. Min R45C3 = 9 -> min R3C1 + R8C4 = 7, min R3C1 = 4

28. 45 rule on N7 3 outies R6C13 + R8C4 = 16
28a. R8C4 = {123} -> R6C13 = 13,14,15 = [58/59/68/86/69/87/96] (cannot be [67] because that would make R7C13 = [88])
[Alternatively, and better,
R7C13 = R8C4 + 13 (step 15),
R8C4 = {123} -> R7C13 = 14,15,16 = [59/68/86/69/87/96/97]
-> R6C13 = [96/87/69/86/68/59/58] from cage sums for R67C1 and R67C3]

29. Killer pair 8,9 in R5C3 and R6C13, locked for N4

30. Hidden killer quad 1,2,3,4 in R4C12, R4C3 and 11(3) cage in N4 for N4, 11(3) cage must contain two of 1,2,3,4 -> R4C12 must contain one of 1,2,3,4
30a. 16(3) cage at R3C1 = {169/178/259/268/358/367/457} (cannot be {349} because R4C12 cannot contain both of 3,4)
30b. 4 of {457} must be in R4C12 (step 30) -> no 4 in R3C1

31. 45 rule on N7 4 innies R7C13 + R89C3 = 21 = {1479/1569/1578/2568} (cannot be {2469/2478} because 8(3) cage cannot be {24}2)
31a. 5 of {1569/1578/2568} must be in R89C3 -> no 5 in R7C1, clean-up: no 9 in R6C1

32. 15(3) cage in N7 = {249/258/348} (cannot be {159/168/267/357/456} which clash with R7C13 + R89C3), no 1,6

33. Killer pair 8,9 in R7C13 and 15(3) cage for N7, locked for N7, clean-up: no 1 in R7C2

34. 1 in N7 locked in R89C3, locked for C3 and 8(3) cage -> no 1 in R8C4, clean-up: no 4 in R34C3 (step 7b), no 6 in R8C6 (step 16)
34a. Naked pair {23} in R34C3, locked for C3, R3C2 = 5 (step 7), clean-up: no 4 in R78C2
[I could now have used step 4 to fix R1C2 but missed it; it comes in step 37 so I didn’t rework to put it here.]

35. Killer pair 6,7 in R2C3 and R67C3, locked for C3 -> R1C3 = 4, clean-up: no 3 in R2C5

36. Naked pair {15} in R89C3, locked for N7, R8C4 = 2 (step 24), R8C6 = 5 (step 16), R89C3 = [15], clean-up: no 7 in R7C2, no 3,6,7 in R7C8

37. R1C234 (step 5) = [943] (only remaining permutation), clean-up: no 4 in R2C5

38. 18(3) cage at R5C3 (step 11) = {459} (only remaining combination) -> R5C3 = 9, R56C4 = {45}, locked for C4 and N5, clean-up: no 6 in R67C3

39. Naked pair {17} in R23C4, locked for N2 and 14(3) cage -> R2C3 = 6, clean-up: no 6 in R1C5

40. R2C6 = 4 (hidden single in N2), R2C7 + R3C6 = [19/28] (step 22)

41. Naked pair {89} in R3C56, locked for R3 and N2 -> R1C6 = 6, R3C1 = 7, R23C4 = [71]

42. 11(3) cage in N1 = {128} (only remaining combination), locked for N1 -> R34C3 = [32]

43. 16(3) cage at R3C1 (step 30a) = {367/457}, no 1
43a. 5 of {457} must be in R4C2 -> no 4 in R4C1

44. 1 in N4 locked in 11(3) cage = {137/146}, no 5

45. R7C13 = R8C4 + 13 (step 15)
45a. R8C4 = 2 -> R7C13 = 15 = [87], R6C13 = [68], clean-up: no 3 in R4C12 (step 43), no 4 in 11(3) cage in N4 (step 44), no 1,3 in R5C5 (step 3), no 2 in R7C2, no 3,5 in R7C7
45b. R4C12 = [54]

46. Naked pair {36} in R78C2, locked for C2 and N7 -> R9C2 = 2

47. Naked pair {17} in R56C2, locked for C2 and N4 -> R2C2 = 8, R5C1 = 3

48. 9 in R2 locked in R2C89
48a. 20(3) cage in N3 = {389} (only remaining combination) -> R1C9 = 8, R2C89 = {39}

49. R1C6 = 6 -> R1C78 = 12 = {57}, locked for R1 -> R12C5 = [25], R12C1 = [12], R2C7 = 1, R3C6 = 9 (step 40), R3C5 = 8, R5C7 = 7 (step 12b), R56C2 = [17], R1C78 = [57], R5C5 = 6, R6C5 = 3 (step 3), R4C4 = 9, R79C4 = [68], R78C2 = [36], R6C7 = 9, R7C7 = 2

and the rest is naked singles and a cage sum.


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PostPosted: Mon May 09, 2011 4:03 am 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 106 by Afmob (June 2008) here
Puzzle Diagram:
Image

Note the two "remote" cages at R1C19 = 10(2) and R9C19 = 9(2).
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:2560:2817:3842:3842:6148:2309:2309:2309:2560:4873:2817:2817:3842:6148:6148:6415:6415:6415:4873:2579:3092:3092:6148:4119:4119:2841:6415:4873:2579:3092:3102:6687:6687:2841:2841:5667:4132:4132:2854:3102:6687:1833:2841:5667:5667:4132:2854:2854:6687:6687:1833:3891:3124:3893:4406:2854:2616:2616:5690:3891:3891:3124:3893:4406:4406:4406:5690:5690:4932:3909:3909:3893:2376:3913:3913:3913:5690:4932:4932:3909:2376:
Solution:
+-------+-------+-------+
| 9 7 8 | 5 6 4 | 2 3 1 |
| 5 1 3 | 2 9 8 | 7 6 4 |
| 6 4 2 | 3 1 7 | 9 5 8 |
+-------+-------+-------+
| 8 6 7 | 4 5 3 | 1 2 9 |
| 4 9 5 | 8 2 1 | 3 7 6 |
| 3 2 1 | 9 7 6 | 4 8 5 |
+-------+-------+-------+
| 2 3 9 | 1 8 5 | 6 4 7 |
| 1 8 6 | 7 4 2 | 5 9 3 |
| 7 5 4 | 6 3 9 | 8 1 2 |
+-------+-------+-------+
Quote:
Afmob: The funny thing about this Assassin is that it was quite hard to find a fitting V1 since I created the forthcoming V2 first (in about 15 minutes) but it took me nearly 1 and a half hour to find a suitable V1. I created this Assassin around the two "remote" cages (not really remote if you look around the corner), the center Nonet and those T-shaped cages at N2 and N8. I hope you recognize the symmetry of this Killer.
V2 will be posted when someone has volunteered for A107.
SS Score (v3.2): 1.34. Estimated rating: 1.25.

Andrew: Thanks Afmob for a fun puzzle!
It's good to be able to post the first walkthrough, which I rarely manage to do on the day that a puzzle appears. Puzzles by Afmob obviously give me a better chance to be first. ;) It's also helpful to be in North America for easier puzzles.
I'll rate A106 at 1.0.

Para: I agree with Andrew. I did about the same as Andrew. And yeah, you just need to spot Andrews first step (although i used 4 innies) and it's not really hard from there on.

Mike (mhparker): Many thanks to Afmob for another enjoyable puzzle. :-D
Thought my step 2 would be Andrew's step 1 that he and others were talking about, which proved not to be the case. Unfortunately, my step 2 didn't get me as far as I would have liked. :-(
I'll rate this puzzle, as Afmob did, as 1.25.
Looking forward (with some trepidation...) to the V2.
P.S. Good to see you back around, Para! :sun:

Ed: As usual, Para gets the 'just', just right! So, congrats to Andrew for getting this key move so quickly and making it look very pedestrian in an excellent Walkthrough ;clapclap; . I didn't get it so quickly and missed (my path's equivalent to) Andrew's step 4 ( :oops: ). So, I'll invoke the "narrow" clause in the ratings and call this one a 1.25. It seemed way harder than A104 or Messy One #1 for example, both of which got a 1.0 rating from respondents.
Will be interested if others found the same narrowness.
I really enjoyed the remote cages in this one. Thanks for another fun innovation Afmob!

gary w: Yes,V1 was a good puzzle.I too used ...
Didn't take long to do...< 1 hour ..but it seems quite a tricky move to find.1.25??

goooders: I'm hopeless at doing walkthroughs I tried and failed twice on sudocue(tiny text). Anyway I thought that was about 1.25 to 1.5.
The question I have is this .I like killersudokuonlines weekly offering because it can always be solved by logic (ie not attrition) The cage sizes are much bigger . I think only Gary has referred to them .Also some of ND,s killers have similarly large cage sizes and fall by logic not attrition.Probably this is the wrong place to raise the issue(logic v attrition) anyway I did

There was some discussion on goooders question; see hidden section after the walkthroughs.

Andrew replied: Thanks goooders for mentioning nd's puzzles. I found them shortly before I found Ruud's Assassins. There are still some of nd's puzzles that I've never solved although I never tried them using elimination solving; I must try them again that way.
They can be found at http://www.ndorward.com/blog/?page_id=71 and there are WTs for each puzzle. Have fun with them!
nd#10 was first posted on Ruud's Assassin forum in December 2006. It's included in Ed's Assassin Archive & Ratings Update, page 3, as A New One. nd's other puzzles are only on his website. I think the hardest ones are reckoned to be #6alt, #9 and #10.
(Note added for archive)Ed later posted nd's #9; see Archive, Part C.
Walkthrough by Andrew:
Thanks Afmob for a fun puzzle!

It's good to be able to post the first walkthrough, which I rarely manage to do on the day that a puzzle appears. Puzzles by Afmob obviously give me a better chance to be first. ;) It's also helpful to be in North America for easier puzzles.

I'll rate A106 at 1.0. I think that's a valid rating for step 1.

Here is my walkthrough. I only used one 45, discarding a couple of routine ones after I found step 1 fairly quickly. My mop-up was fairly routine; I expect there are quicker ways to do it but I didn't try to optimise it. Thanks Afmob for pointing out what I missed at the beginning. I've added them in blue but not amended my walkthrough since they wouldn't make it significantly quicker.

Prelims

a) R1C19 wrap-around cage = {19/28/37/46}, no 5
b) R34C2 = {19/28/37/46}, no 5
c) R3C67 = {79}, locked for R3, clean-up: no 1,3 in R4C2
d) R45C4 = {39/48/57}, no 1,2,6
e) R56C6 = {16/25/34}, no 7,8,9
f) R67C8 = {39/48/57}, no 1,2,6
g) R7C34 = {19/28/37/46}, no 5
h) R9C19 wrap-around cage = {18/27/36/45}, no 9
i) 11(3) cage at R1C2 = {128/137/146/236/245}, no 9
j) R1C678 = {126/135/234}, no 7,8,9
k) R234C1 = {289/379/469/478/568}, no 1
l) 22(3) cage in N6 = {589/679}, 9 locked for N6, clean-up: no 3 in R7C8
m) 19(3) cage at R8C6 = {289/379/469/478/568}, no 1
n) 11(4) cage at R3C8 = {1235}, CPE no 5 in R5C8
o) 11(4) cage at R5C3 = {1235}, CPE no 1,2,3,5 in R5C2
I missed no 3,5 in R6C8 in n) and no 2 in R4C2 in o); these would both have had clean-ups if I'd included them.

1. 45 rule on C1234 2 innies R68C4 = 2 wrap-round outies R19C9 + 13
1a. Min R19C9 = 3 -> minR68C4 = 16 -> R68C4 = {79/89}, 9 locked for C4, clean-up: no 3 in R45C4, no 1 in R7C3
1b. Max R68C4 = 17 -> max R19C9 = 4 -> R19C9 = {12/13}, 1 locked for C9, R1C1 = {789}, R9C1 = {678}
1c. R19C9 = [12/13/31] (cannot be [21] because R19C1 cannot be [88]), R19C1 = [97/96/78], no 8 in R1C1, no 2 in R1C9

2. R234C1 = {469/478/568} (cannot be {289} which clashes with R19C1, cannot be {379} which clashes with R1C1), no 2,3

3. Killer pair 6,7 in R19C1 and R234C1, locked for C1
3a. Killer pair 8,9 in R19C1 and R234C1, locked for C1

4. Naked quint {12345} in R5C13 and R6C123, locked for N4, clean-up: no 6,8 in R3C2
[If I’d spotted 4 in N6 locked in R6C789, locked for R6, this would have fixed R5C1 = 4 here.]

5. 4 in N4 locked in R56C1, locked for C1
5a. R234C1 = {568} (only remaining combination), locked for C1, 5 locked for N1 -> R19C1 = [97], R19C9 = [12], clean-up: no 3 in R7C4

6. 16(3) cage in N4 = {349} (only remaining combination containing 4) -> R5C2 = 9, naked pair {34} in R56C1, locked for C1 and N4, clean-up: no 1 in R3C2
6a. Naked triple {678} in R4C123, locked for R4, clean-up: no 4,5 in R5C4

7. R7C2 = 3 (hidden single in 11(4) cage at R5C3), clean-up: no 7 in R4C2, no 7 in R7C4

8. R4C3 = 7 (hidden single in R4)
8a. R4C9 = 9 (hidden single in N6)

9. R1C678 = {234} (only remaining combination), locked for R1

10. 7 in N1 locked in R12C2 -> 11(3) cage = {137} -> R1C2 = 7, R2C23 = [13]

11. R3C23 = {24} (hidden pair in N1), locked for R3

12. 12(3) cage at R3C3 = {147/237}
12a. R3C3 = {24} -> R3C4 = {13}

13. 15(3) cage at R1C3 = {258/456} (cannot be {267} because 2,7 only in R2C4), no 7
13a. 2,4 only in R2C4 -> R2C4 = {24}, R1C4 = 5, R4C4 = 4, R5C4 = 8, R2C4 = 2, R1C3 = 8 (step 13), R1C5 = 6, clean-up: no 3 in R56C6, no 2,6 in R7C3

14. 22(3) cage in N6 = {679} (only remaining combination), no 5, naked pair {67} in R5C89, locked for R5 and N6, clean-up: no 1 in R6C6, no 5 in R7C8

15. Naked pair {12} in R78C1, locked for N7 and 17(4) cage -> R8C23 = 14 = [59/86], no 4, no 6 in R8C2, no 5 in R8C3

16. 6,8 in N2 locked in 24(4) cage = {1689/3678}, no 4
16a. 1,3 only in R3C5 -> R3C5 = {13}
16b. Naked pair {13} in R3C45, locked for R3 and N2 -> R1C6 = 4, R3C8 = 5, R23C1 = [56], R3C9 = 8, R4C12 = [86], R3C2 = 4, R3C3 = 2, R3C4 = 3 (step 12), R3C5 = 1, clean-up: no 7 in R2C56 (step 16), no 7 in R7C8

17. R3C6 = 7 (hidden single in N2), R3C7 = 9

18. 5 in N6 locked in R6C79, locked for R6 -> R6C23 = [21], R5C3 = 5, R6C6 = 6, R5C6 = 1

19. 3 in C9 locked in R678C9 = {357} (only remaining combination), locked for C9, 7 locked for N9 -> R5C89 = [76], R2C789 = [764]

20. Naked pair {23} in R5C57, locked for R5 -> R56C1 = [43], R6C9 = 5, R78C9 = [73], clean-up: no 9 in R7C8

21. Naked pair {23} in R15C7, locked for C7 -> R4C7 = 1
21a. Naked pair {48} in R67C8, locked for C8
21b. R89C8 = {19} -> R8C7 = 5 (cage sum), R8C2 = 8, R8C3 = 6 (step 15)
21c. R9C2 = 5, R9C34 = 10 = [46] (cannot be [91] which clashes with R9C8)

and the rest is naked singles and a cage sum.
Walkthrough by Mike:
Many thanks to Afmob for another enjoyable puzzle. :-D

Thought my step 2 would be Andrew's step 1 that he and others were talking about, which proved not to be the case. Unfortunately, my step 2 didn't get me as far as I would have liked. :-(

I'll rate this puzzle, as Afmob did, as 1.25. A rating of 1.0 seems a bit low for a puzzle requiring(?) killer triples and/or quads.

Looking forward (with some trepidation...) to the V2. In the meantime, here's my V1 WT:

Edit: Small typo fixed, as pointed out by Andrew (thanks!).

Assassin 106 Walkthrough (23 steps)

Prelims

a) 10(2) at R1C19, R3C2 and R7C3 = {19/28/37/46} (no 5)
b) 11(3) at R1C2 = {128/137/146/236/245} (no 9)
c) 9(3) at R1C6 = {126/135/234} (no 7..9)
d) 19(3) at R2C1 and R8C6 = {289/379/469/478/568} (no 1)
e) 16(2) at R3C6 = {79}, locked for R3; cleanup: no 1,3 in R4C2
f) 11(4) at R3C8 and R5C3 = {1235} (no 4,6..9); no 1,2,3,5 in R45C2+R56C8 (CPE); cleanup: no 8 in R3C2
g) 12(2) at R4C4 = {39/48/57} (no 1,2,6)
h) 22(3) at R4C9 = {589/679} (no 1..4), 9 locked for N6
i) 7(2) at R56C6 = {16/25/34} (no 7..9)
j) 12(2) at R6C8 = [48/75/84] (no 6 in R6C8; no 1..3,6,7,9 in R7C8)
k) 9(2) at R9C19 = {18/27/36/45} (no 9)

1. 4 in N6 locked in R6C789 for R6
1a. cleanup: no 3 in R5C6

2. Either R3C8 = 5, OR...
2a. ...R4C78+R5C7 must contain a 5
2b. -> 22(3) at R4C9 (prelim h) must contain a 7
2c. either way, 12(2) at R6C8 cannot be [75]
2d. -> 12(2) at R6C8 = {48}, locked for C8

3. Innie/Outie (I/O) diff. N1234: R3C8 = R7C2 + 2
3a. -> R3C8 = {35}, R7C2 = {13}
3b. no 3 in R3C2 (CPE); cleanup: no 7 in R4C2

4. {25} in 11(4) at R5C3 locked in R5C3+R6C23 for N4

5. {12} in 11(4) at R3C8 locked in R4C78+R5C7 for N6

6. I/O diff C56789: R68C4 = R19C9 + 13
6a. min. R19C9 = 3 -> min. R68C4 = 16
6b. -> R68C4 = {79/89} (no 1..6)
6c. 9 locked in R68C4 for C4
6d. -> R19C9 = 3 or 4 = {12/13} (no 4..9)
6e. 1 locked in R19C9 for C9
6f. cleanup: no 3 in R45C4, no 1 in R7C3; no 1..4,6 in R1C1; no 1..5 in R9C1

7. R45C4 and R68C4 (step 6b) form killer triple on {789} within C4
7a. -> no 7..9 elsewhere in C4
7b. cleanup: no 2,3 in R7C3

8. 19(3) at R2C1 = {289/379/469/478/568}
8a. 3 of {379} must go in R3C1
8b. -> no 3 in R24C1

9. R1C1, R1C9 and 19(3) at R2C1 form killer quad on {6789} within C1
9a. -> no 6..9 elsewhere in C1 (R5..8C1)

10. 16(3) at R5C12+R6C1 = [493] (last permutation)
10a. cleanup: no 1,6 in R3C2; no 8 in R4C4

11. Naked triple on {125} at R5C3+R6C23, locked for N4 and 11(4)
11a. -> R3C8+R7C2 (step 3) = [53]
11b. cleanup: no 7 in R7C3

12. 22(3) at R4C9 = {679} (last combo)
12a. -> R4C9 = 9; R5C89 = {67}, locked for R5
12b. cleanup: no 5 in R4C4, no 1 in R6C6

13. 19(3) at R2C1 = [928/568/586] (no 7) (last permutations)
13a. -> R2C1 = {59}, no 7 in R4C1
13b. 8 locked in R34C1 for C1
13c. cleanup: no 2 in R1C9, no 1 in R9C9

14. from step 6d, R19C9 = [12/13]
14a. -> R1C9 = 1
14b. -> R1C1 = 9 (remote 10(2) cage sum)
14c. -> R2C1 = 5
14d. -> split 14(2) at R34C1 = {68}, locked for C1
14e. -> R9C1 = 7
14f. -> R9C9 = 2 (remote 9(2) cage sum)

15. Naked pair (NP) at R4C12 = {68}, locked for R4
15a. -> R4C3 = 7
15b. -> 12(2) at R45C4 = [48]
15c. cleanup: no 6 in R7C3; no 6 in R3C34

16. 9(3) at R1C6 = {234} (no 5,6) (last combo), locked for R1
16a. -> 15(3) at R1C3 = [852/861] (last permutations)
16b. -> R1C3 = 8; no 3,6 in R2C4
16c. cleanup: no 2 in R7C4

17. R34C1 = [68]
17a. -> R4C2 = 6
17b. -> R3C2 = 4 (cage sum)

18. 11(3) at R1C2+R2C23 = [713] (last permutation)
18a. -> split 7(2) at R12C4 = [52]
18b. -> split 5(2) at R3C34 = [23]
18c. -> R1C56 = [64]
18d. -> R3C59 = [18]

19. NP at R56C3 = {15}, locked for C3 and 11(4)
19a. -> R6C2 = 2

20. HS in C9 at R8C9 = 3
20a. -> split 12(2) at R67C9 = [57] (last permutation)
20b. -> R5C89 = [76], R56C3 = [51], R56C6 = [16]

21. Split 17(3) at R2C789 = [764] (last permutation)
21a. -> R3C67 = [79]

22. NP at R89C8 = {19}, 1 locked for C8 and N9
22a. -> R8C7 = 5 (cage split)
22b. -> R89C2 = [85]

23. 19(3) at R8C6+R9C67 = [298] (last permutation)

Rest is naked singles.
P.S. Good to see you back around, Para! :sun:
gary w's solving outline:
Yes,V1 was a good puzzle.I too used
c1234 outies > r68c4=r19c9+13 > r19c9=3/4 > r19c1=15/16>r5678c1=10/11=123 4/5
with the 11(4) cage in n4 and the 16(3) cage in n4 there's then only the possibility that the 1 in c1 is at r78 and that r7c2=3 and it's cracked now.

Didn't take long to do...< 1 hour ..but it seems quite a tricky move to find.1.25??
Discussion of goooders question:
goooders: The question I have is this .I like killersudokuonlines weekly offering because it can always be solved by logic (ie not attrition) The cage sizes are much bigger . I think only Gary has referred to them .Also some of ND,s killers have similarly large cage sizes and fall by logic not attrition.Probably this is the wrong place to raise the issue(logic v attrition) anyway I did

Afmob: Goooders, are you talking about V1 or V2? Both can be solved using logic only (whatever that means) and even V2 didn't require any contradiction moves or chains.

goooders: OK I accept that intellectually killers are solved by logic if each cage has all numbers 1 to 9 enumerated and bit by bit they get chipped away .I think all I am trying to say is that there is much more satisfaction if killers can be solved by "putting numbers in rather than taking them away" I couldnt do this with your V2.Row 1 column 9 had three options .Thereafter there is a "tree diagram" of options.I think thats all I am saying By the way thanks for producing such brilliant puzzles

Andrew: Thanks goooders for mentioning nd's puzzles. I found them shortly before I found Ruud's Assassins. There are still some of nd's puzzles that I've never solved although I never tried them using elimination solving; I must try them again that way.
They can be found at http://www.ndorward.com/blog/?page_id=71 and there are WTs for each puzzle. Have fun with them!
nd#10 was first posted on Ruud's Assassin forum in December 2006. It's included in Ed's Assassin Archive & Ratings Update, page 3, as A New One. nd's other puzzles are only on his website. I think the hardest ones are reckoned to be #6alt, #9 and #10.

Andrew, again:
goooders wrote:
OK I accept that intellectually killers are solved by logic if each cage has all numbers 1 to 9 enumerated and bit by bit they get chipped away .I think all I am trying to say is that there is much more satisfaction if killers can be solved by "putting numbers in rather than taking them away"

I ought to have also replied earlier to this point.

When I first started doing Assassins, having done easier Killers on a different site which I still do, I also used insertion solving. For some time I resisted changing over to elimination solving, although others started using it before I did as can be seen from posted walkthroughs. I was surprised to see, from Ed's excellent Assassin Archive & Rating Update thread, that I was still using insertion solving as late as A27.

It's certainly more satisfying if one can make steady progress rather than chipping away one candidate or one combination at a time. That sort of progress can happen with many puzzles, whether one is using insertion or elimination solving.


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PostPosted: Mon May 09, 2011 4:26 am 
Offline
Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 106 V2 by Afmob (June 2008) here
Puzzle Diagram:
Image

Note the two "remote" cages at R19C1 = 10(2) and R19C9 = 13(2).
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:2560:7169:2818:2818:6660:2565:2565:2565:3336:7169:7169:7169:6660:6660:2318:2318:3856:3856:3090:7169:4628:4628:6660:2318:3608:3608:3856:3090:3090:4628:2846:2846:7712:3608:3362:3362:4132:2341:2341:7712:7712:7712:1322:1322:3362:4132:4132:3375:7712:1073:1073:4403:4148:4148:5174:3375:3375:3897:4154:4403:4403:7229:4148:5174:5174:3897:3897:4154:4154:7229:7229:7229:2560:2377:2377:2377:4154:4429:4429:7229:3336:
Solution:
+-------+-------+-------+
| 4 6 8 | 3 5 1 | 7 2 9 |
| 5 1 9 | 7 8 2 | 3 4 6 |
| 2 7 3 | 9 6 4 | 1 8 5 |
+-------+-------+-------+
| 1 9 6 | 4 7 8 | 5 3 2 |
| 3 2 7 | 6 9 5 | 4 1 8 |
| 8 5 4 | 2 1 3 | 9 6 7 |
+-------+-------+-------+
| 7 8 1 | 5 4 6 | 2 9 3 |
| 9 4 2 | 8 3 7 | 6 5 1 |
| 6 3 5 | 1 2 9 | 8 7 4 |
+-------+-------+-------+
Quote:
Afmob: Here is the original or first Assassin I created for this week but judging by SS score it was too hard, so I created V1 afterwards. I checked through a lot of Innies+Outies and this one shouldn't fall so fast and should provide a good practice for your creativity.
SS Score (v3.2): 2.15. Estimated rating: 1.5 - (Hard) 1.5.

udosuk: If just judging by difficulty of techniques involved, this one is not particularly hard. But it's quite tricky with the ... :ugeek:

Andrew, in 2010: I jumped back a bit to my earliest unfinished puzzle on this site, as a change from working backward through my unfinished puzzles.
Thanks Afmob for a challenging variant! It deserves more than one walkthrough.
At the time when this puzzle appeared I'd got stuck after step 15. When I started again last week I got stuck again after step 32 so I took a break and solved another of my backlog, one from Ruud's site, before trying again and finding how I could use ... I'm glad that I persevered as I enjoyed finding the somewhat unusual step 46.
I'll rate my walkthrough for A106 V2 at least Hard 1.5.
Walkthrough by udosuk:
Here is a complete walkthrough for Assassin 106 v2:

1. 9/3 @ r9c2 from {123456}
20/3 @ r7c1 from {3456789}
13/2 @ r19c9={49|58|67}
4/2 @ r6c5={13} (NP @ r6,c5)
17/2 r9c6={89} (NP @ r9)
=> 10/2 @ r19c1=[37|46|64|73|82|91]

2. Innies @ n9: r79c79=17
Min r9c7=8 => max r7c79+r9c9=17-8=9
=> r9c9 from {456}, r7c79 from {123}
=> 13/2 @ r19c9=[76|85|94]

3. Outies @ n3: r123c6+r4c7+r9c9=16
r6c6 from {13} => r123c6 can't be {123}
=> Min r123c6=1+2+4=7, min r9c9=4
=> Max r4c7=16-7-4=5

4. Innies @ n6: r4c7+r6c789=27 from {3..9}
=> r4c7 from {345} => r6c789=22..24
=> r4c7+r6c789=[3{789}|4{689}|5{679}]
=> 9 @ r6,n6 locked @ r6c789

5. Outies @ c789: r12379c6=22
Min r123c6=7 => max r79c6=22-7=15
Min r9c6=8 => max r7c6=15-8=7
r7c7 from {123} => r6c7+r7c6=17-r7c7=14..16
=> r6c7 from {89}, r7c6 from {567}

6. r69c7={89}=17 (NP @ c7) => r6c7=r9c6=17-r9c7
=> r79c6=r6c7+r7c6=14..16
=> r123c6=22-r79c6=7|8 (min r123c6=7)
=> 1 @ c6,n2 locked @ r123c6 => r6c56=[13]
=> r123c6=7|8={124|125} (2 @ c6,n2 locked)
=> r12379c6=22={12469|12478|12568}
=> r6c7+r7c6=r79c6={68|69|78}=14|15
=> r7c7=17-r6c7-r7c6=2|3

7. 16/4 @ r7c5 from {2..9}={2347|2356}
=> {23} @ c5,n8 locked @ r789c5
=> 16/4 @ r7c5 & r7c6 form CNP {67} @ n8
=> r78c4 from {14589} can't sum to 7|8|11
=> r8c3=15-r78c4 can't be 4|7|8
9/3 @ r9c2 can't contain {14|45}
r9c4 from {145} => r9c23 can't have 4

8. Outies @ n78: r1c1+r6c3+r7c7=27-17=10
r7c7 from {23} => r1c1+r6c3=7|8
=> r6c3 from {245}, r19c1=[37|46|64]
=> 13/3 @ r6c3 can't be {139}
=> r7c23 can't have 9

9. Outies @ n8: r7c7+r8c3+r9c23=29-17=12
r7c7 from {23} => r8c3+r9c23=12-r7c7=9|10
=> r8c3+r9c23={126|135|136|235} has 5|6
=> 9 @ n7 locked @ 20/3 @ r7c1={389|479}
=> 13/3 @ r6c3 can't be [1{57}|2{56}|6{25}|7{15}]
=> r7c23 can't have 5
=> 5 @ n7 locked @ r8c3+r9c23={135|235}

10. 9/3 @ r9c2 from {12345}={135|234}
=> 3 @ r9,n7 locked @ r9c23
=> 20/3 @ r7c1={479} (NT @ n7)
=> r19c1=[46] => HS @ n7: r8c2=4
=> 8 @ n7 locked @ 13/3 @ r6c3 from {12458}
=> r6c3=4, r7c23={18} (NP @ r7,n7)
=> r7c79={23}=5 (NP @ r7,n9)
=> r9c79=17-5=12=[84] => r6c7=r9c6=r1c9=9
=> r7c67=17-9=8=[62] => r7c9=3

11. r123c6=22-r79c6=22-6-9=7={124}
=> 4 @ c6,n2,9/2 locked @ r23c6
=> r23c6={24}, r2c7=3, r1c6=1
=> r4c7=16-r123c6-r9c9=16-7-4=5
=> r1c78=10-1=9=[72] => r3c78=14-5=9=[18]
=> 5/2 @ r5c7=[41], r8c7=6
=> 15/3 @ r7c4 from {12458} must be [528]
Innies @ n2: r13c4=19-7=12=[39] => r1c3=11-3=8
Innies @ n1: r3c13=17-4-8=5=[23]

All naked singles from here.

If just judging by difficulty of techniques involved, this one is not particularly hard. But it's quite tricky with the wrap-around cages and all to deal with the big outies. Note I didn't use a single innie-outie. :ugeek:

(Edited: Complete walkthrough replaced.)
Walkthrough by Andrew in 2010:
I jumped back a bit to my earliest unfinished puzzle on this site, as a change from working backward through my unfinished puzzles.

Thanks Afmob for a challenging variant! It deserves more than one walkthrough.

udosuk wrote:
If just judging by difficulty of techniques involved, this one is not particularly hard. But it's quite tricky with the wrap-around cages and all to deal with the big outies. Note I didn't use a single innie-outie.
You forgot to mention that you also used a "clone" and made very good use of it. It took me a long time until I spotted the "clone". Later you had clearly reached a stage where you were avoiding innies-outies; I felt that one step might have been simpler using one.

At the time when this puzzle appeared I'd got stuck after step 15. When I started again last week I got stuck again after step 32 so I took a break and solved another of my backlog, one from Ruud's site, before trying again and finding how I could use the "clone" in step 33. I'm glad that I persevered as I enjoyed finding the somewhat unusual step 46.

I'll rate my walkthrough for A106 V2 at least Hard 1.5. It would be hard to rate udosuk's walkthrough, because I don't think there's a consensus on the rating of large outies, but steps 8 and 9 which use to "clone" to redefine large outies were close to the difficulty level of my hardest steps.

Here is my walkthrough for A106 V2.

Prelims

a) R19C1 wrap-around cage = {19/28/37/46}, no 5
b) R1C34 = {29/38/47/56}, no 1
c) R19C9 wrap-around cage = {49/58/67}, no 1,2,3
d) R4C45 = {29/38/47/56}, no 1
e) R5C23 = {18/27/36/45}, no 9
f) R5C78 = {14/23}
g) R6C56 = {13}
h) R9C67 = {89}
i) 9(3) cage at R2C6 = {126/135/234}, no 7,8,9
j) 10(3) cage at R1C6 = {127/136/145/235}, no 8,9
k) 20(3) cage in N7 = {389/479/569/578}, no 1,2
l) 9(3) cage at R9C2 = {126/135/234}, no 7
m) 26(4) cage in N2 = {2789/3689/4589/4679/5678}, no 1

Steps resulting from Prelims
1a. Naked pair R6C56 = {13}, locked for R6 and N5, clean-up: no 8 in R4C45
1b. Naked pair {89} in R9C67, locked for R9, clean-up: no 1,2 in R1C1, no 4,5 in R1C9

2. 45 rule on C12 3 innies R579C2 = 1 outie R2C3 + 4
2a. Min R579C2 = 6 -> min R2C3 = 2

3. 45 rule on N6 4 innies R4C7 + R6C789 = 27 = {3789/4689/5679}, no 1,2, 9 locked for N6

4. 45 rule on C6789 4 innies R4568C6 = 23 = {1679/3479/3569/3578} (cannot be {1589/2489} which clash with R9C6, cannot be {2579/2678/4568} which don’t contain 1,3), no 2
4a. R6C6 = {13} -> no 1,3 in R8C6
4b. Killer pair 8,9 in R4568C6 and R9C6, locked for C6

5. 9(3) cage at R2C6 = {126/135/234}
5a. 5 of {135} must be in R23C6 (R23C6 cannot be {13} which clashes with R6C6) -> no 5 in R2C7

6. 45 rule on N9 4 innies R79C79 = 17 = {1259/1268/1349/1358/2348} (cannot be {1367/1457/2357/2456} which don’t contain 8,9), no 7, clean-up: no 6 in R1C9
6a. R9C7 = {89} -> no 8,9 in R7C79
6b. R9C9 = {456} -> no 4,5,6 in R7C79

7. 17(3) cage at R6C7 = {179/269/278/359/368} (cannot be {458/467} because R7C7 only contains 1,2,3), no 4
7a. R7C7 = {123} -> no 1,2,3 in R7C6
7b. 8,9 can only be in R6C7 -> R6C7 = {89}
7c. Killer single 7 in R4568C6 and R7C6, locked for C6
7d. 2 in C6 only in R123C6, locked for N2, clean-up: no 9 in R1C3

8. Naked pair {89} in R69C7, locked for C7
8a. 9 in N6 locked in R6C789, locked for R6

9. 13(3) cage at R6C3 = {148/157/238/247/256/346} (cannot be {139} because R6C3 doesn’t contain any of 1,3,9), no 9

10. 45 rule on R789 2 outies R6C37 = 3 innies R7C9 + R9C19
10a. Min R79C7 = 9 -> max R79C9 = 8 (step 6) -> max R7C9 + R9C19 = 15 -> max R6C37 = 15 -> no 8 in R6C3

11. 45 rule on N3 4 outies R123C6 + R4C7 = 1 innie R1C9 + 3
11a. R1C9 = {789} -> R123C6 + R4C7 = 10,11,12
11b. Min R123C6 = 7 (cannot be 6 because {123} clashes with R6C6) -> max R4C7 = 5

12. 14(3) cage at R3C7 = {149/158/239/248/257/347/356} (cannot be {167} because R4C7 only contains 3,4,5)
12a. 8,9 of {149/158} must be in R3C8 -> no 1 in R3C8

13. R4C7 + R6C789 (step 3) = {3789/4689/5679}
13a. R4C7 = {345} -> no 4,5 in R6C89

14. 16(3) cage in N4 = {259/268/358/457} (cannot be {169/349} because 1,3,9 only in R5C1, cannot be {178/367} which clash with R6C789), no 1
14a. 9 of {259} must be in R5C1, 2 of {268} must be in R6C12 (R6C12 cannot be {68} which clashes with R6C789) -> no 2 in R5C1

15. 45 rule on N4 2 innies R46C3 = 1 outie R3C1 + 8
15a. Max R46C3 = 16 -> max R3C1 = 8
15b. Min R46C3 = 9 -> min R4C3 = 2

16. 45 rule on C1234 4 innies R2456C4 = 19 = {2359/2368/2458/2467/3457}
16a. 3 of {2359} must be in R2C4 -> no 9 in R2C4
16b. 5 of {2359} must be in R4C4 (R4C4 cannot be 2 or 9 because R456C4 would clash with R4C45, CCC) -> no 9 in R4C4, clean-up: no 2 in R4C5

17. 16(4) cage in N8 = {1249/1258/1267/1456/2347/2356} (cannot be {1348/1357} which clash with R6C5)
17a. Killer pair 1,3 in R6C5 and R789C5, locked for C5

18. 45 rule on N23 3(2+1) innies R1C49 + R3C4 = 1 outie R4C7 + 16
18a. Min R4C7 = 3 -> min R1C49 + R3C4 = 19 -> no 1 in R3C4 (because max R1C49 = 17)
18b. 1 in N2 only in R123C6, locked for C6 -> R6C56 = [13]
18c. 9(3) cage at R2C6 = {126/135/234}
18d. 3 of {234} must be in R2C7 -> no 4 in R2C7

19. 16(4) cage in N8 (step 17) = {2347/2356}, no 8,9, 2,3 locked for N8, 2 also locked for C5
19a. 9(3) cage at R9C2 = {126/135/234}
19b. 4 of {234} must be in R9C4 -> no 4 in R9C23

20. Hidden killer pair 8,9 in R78C4 and 16(4) cage for N8 -> 15(3) cage at R7C4 must contain one of 8,9 in R78C4
20a. 15(3) cage at R7C4 = {159/168/249/258/348} (cannot be {267/357/456} which don’t contain 8 or 9), no 7
20b. 8,9 must be in R78C4 -> no 8,9 in R8C3
20c. 2,3 of {249/348} must be in R8C3 -> no 4 in R8C3

21. 45 rule on R789 4 outies R6C3789 = 2 innies R9C19 + 16
21a. Max R9C19 = 13 -> R6C3789 cannot total 30 -> no 6,7 in R6C3
21b. Min R6C3789 = [29]{67} (because R6C789 must contain 9) = 24 -> min R9C19 = 8, no 1 in R9C1, clean-up: no 9 in R1C1

22. 16(3) cage in N4 (step 14) = {259/268/358/457}
22a. 16(3) cage cannot be {268} => R5C23 = {45} clashes with R6C3
22b. -> 16(3) cage = {259/358/457}, no 6, 5 locked for N4, clean-up: no 4 in R5C23
22c. 3 of {358} must be in R5C1 -> no 8 in R5C1

23. Hidden killer quad 6,7,8,9 in R4C12, R4C3, 16(3) cage and R5C23 for N4, R4C12 cannot contain more than one of 6,7,8,9, 16(3) cage contains one of 7,8,9, R5C23 contains one of 6,7,8 -> R4C3 = {6789} and R4C12 must contain one of 6,7,8,9
23a. 12(3) cage at R3C1 cannot contain more than one of 6,7,8,9 which must be in R4C12 -> no 6,7,8 in R3C1

24. Max R7C1 + R8C12 + R9C1 = 27 -> min R7C23 + R8C3 + R9C34 = 18 must contain at least one of 4,7,8 -> R7C23 must contain at least one of 4,7,8
24a. 13(3) cage at R6C3 (step 9) = {148/238/247} (cannot be {157} because R6C3 only contains 2,4, cannot be {256/346} which don’t contain any of 4,7,8 in R7C23), no 5,6
24b. R7C23 = {18/27/38/47}

25. 9 in N7 only in 20(3) cage = {389/479/569}
25a. R7C23 + R8C3 + R9C34 cannot be 18 = {12348/12357/12456} (cannot be {12348/12456} because 4 must be in R7C23 and neither combination contains both of 4,7, cannot be {12357} which clashes with 20(3) cage)
25b. Min R7C23 + R8C3 + R9C34 = 19 -> max R7C1 + R8C12 + R9C1 = 26 -> max R9C1 = 6, clean-up: no 3 in R1C1

26. Hidden killer pair 7,8 in R7C23 and 20(3) cage for N7, R7C23 contains one of 7,8 -> 20(3) cage must contain one of 7,8 -> 20(3) cage (step 25) = {389/479} (cannot be {569} which doesn’t contain 7 or 8), no 5,6

27. 3 in N2 only in R123C4
27a. Hidden killer triple 7,8,9 in R13C4 and 26(4) cage for N2, 26(4) cage contains two of 7,8,9 -> R13C4 must contain one of 7,8,9
27b. 26(4) cage in N2 = {3689/4589/4679/5678}
27c. 26(4) cage = {3689} or R13C4 contains 3 and one of 7,8,9 -> no 6 in R13C4, clean-up: no 5 in R1C3

28. 45 rule on N78 3 innies R9C1 + R79C6 = 1 outie R6C3 + 17
28a. Min R6C3 = 2 -> min R9C1 + R79C6 = 19, max R79C6 = 16 -> min R9C1 = 3, clean-up: no 8 in R1C1

29. 7 in R9 only in R9C58
29a. 45 rule on R9 4 innies R9C1589 = 19 = {1567/2467/3457}
29b. 5,6 of {1567} must be in R9C19, 4,6 of {2467} must be in R9C19 -> no 6 in R9C58

30. 45 rule on R1 4 innies R1C1259 = 24 = {1689/2679/3489/3678/4569/4578} (cannot be {2589} because R1C1 only contains 4,6,7, cannot be {3579} which clashes with R1C34)
30a. 1,2,3 of {1689/2679/3489/3678} must be in R1C2, 9 of {4569} must be in R1C9 -> no 9 in R1C2

31. 45 rule on R1234 3 innies R1C19 + R4C6 = 1 outie R5C9 + 13
31a. Min R1C19 + R4C6 = 15 -> min R5C9 = 2

32. 16(3) cage in N4 (step 22b) = {259/358/457}
32a. {259} => R6C3 = 4
{358} => R5C23 = {27} => R6C3 = 4
{457} => R6C3 = 2
32b. -> no 2,4 in R4C12

33. R4568C6 (step 4) = {3479/3569/3578} -> R458C6 = {479/569/578}
33a. R69C7 = {89}, R9C67 = {89} -> R6C7 = R9C6
33b. 17(3) cage at R6C7 (step 7) = {179/269/278/359/368} -> R7C67 + R9C6 = {179/269/278/359/368}
33c. R7C67 + R9C6 = [629/638/728] (cannot be [539/719] which clash with R458C6) -> R7C6 = {67}, R7C7 = {23}
33d. Killer pair 6,7 in R458C6 and R7C6, locked for C6
33e. Killer pair 6,7 in 16(4) cage and R7C6, locked for N8

34. 9(3) cage at R2C6 = {126/135/234}
34a. 3,6 only in R2C7 -> R2C7 = {36}

35. 14(3) cage at R3C7 (step 12) = {149/158/248/257/347} (cannot be {239} which clashes with R7C7, cannot be {356} because R2C7 “sees” all cells of the 14(3) cage), no 6
35a. 5 of {158/257} must be in R4C7 -> no 5 in R3C78

36. 10(3) cage at R1C6 = {127/145/235} (cannot be {136} which clashes with R2C7), no 6
36a. R1C1259 (step 30) = {1689/2679/3489/3678/4569} (cannot be {4578} which clashes with 10(3) cage at R1C6)
36b. 1,2,3 of {1689/2679/3489/3678} must be in R1C2 -> no 7,8 in R1C2

37. R79C79 (step 6) = {1259/1268/1349/1358/2348}
37a. 1 of {1259/1268/1349/1358} must be in R7C9, 2 of {2348} must be in R7C7 (R79C7 cannot be [38] which clashes with R7C67 + R9C6) -> no 2 in R7C9

38. Killer triple 3,4,6 in 20(3) cage, R9C1 and R9C23 (because 3,6 of 9(3) cage must be in R9C23), locked for N7

39. 13(3) cage at R6C3 (step 24a) = {148/247} -> R6C3 = 4, clean-up: no 7 in R1C4
39a. 16(3) cage in N4 (step 32) = {259/358}, no 7
39b. 3,9 only in R5C1 -> R5C1 = {39}, 5 locked in R6C12, locked for R6

40. Killer triple 1,2,3 in R7C23, R7C7 and R7C9, locked for R7, 3 also locked for N9
40a. R79C79 (step 6) = {1349/1358/2348}, no 6, clean-up: no 7 in R1C9

41. R6C37 = R7C9 + R9C19 (step 10), R6C3 = 4, R6C7 = {89} -> R7C9 + R9C19 = 12,13
41a. Max R79C9 = 8 -> min R9C1 = 4, clean-up: no 7 in R1C1
41b. Naked pair {46} in R19C1, locked for C1
41c. R9C1589 (step 29a) = {1567/2467/3457}
41d. 1,2,3,7 only in R9C58 -> no 4,5 in R9C58

42. 16(4) cage in N8 (step 19) = {2347/2356}
42a. 2,3 only in R89C5 -> R89C5 = {23}

43. R9C8 = 7 (hidden single in R9)
43a. 1 in R9 locked in 9(3) cage at R9C2 = {126/135}, no 4

44. 45 rule on N4 1 remaining innie R4C3 = 1 outie R3C1 + 4, no 1 in R3C1, no 8 in R4C3

45. 20(3) cage in N9 (step 26) = {389/479}
45a. 4 of {479} must be in R8C2 -> no 7 in R8C2
45b. 8 of {389} must be in R78C1 (R78C1 cannot be {39} which clashes with R5C1) -> no 8 in R8C2

46. 45 rule on R6789 3 innies R6C4 + R9C19 = 1 outie R5C1 + 9
46a. 4 in R9 only in R9C19 = [45/64] = 9,10
46b. R5C1 = {39} cannot be the same as R6C4 = {2678} -> R9C19 cannot total 9
46c. R9C19 = 10 = [64], R1C1 = 4, R1C9 = 9, R9C5 = 2 (step 41c), R8C5 = 3, clean-up: no 2,7 in R1C3

47. R8C2 = 4 (hidden single in N7), R78C1 (step 45) = {79}, locked for C1 and N7, R5C1 = 3, clean-up: no 6 in R5C23, no 2 in R5C78, no 2 in R6C12 (step 39a), no 2 in R7C23 (step 39)

48. Naked pair {58} in R6C12, locked for R6 and N4 -> R4C1 = 1, R6C7 = 9, R6C89 = [67], R6C4 = 2, clean-up: no 9 in R4C5

49. Naked pair {18} in R7C23, locked for R7 and N7 -> R7C9 = 3, R7C7 = 2, R9C67 = [98], R7C6 = 6 (step 33c), clean-up: no 5 in 16(4) cage in N8 (step 19)
49a. R8C6 = 7, R7C5 = 4, R7C4 = 5, R9C4 = 1, R8C34 = [28], R1C4 = 3, R1C3 = 8, R5C23 = [27], R7C23 = [81], R6C12 = [85], R9C23 = [35]

50. 13(3) cage in N6 = {238} (only remaining combination) -> R45C9 = [28], R4C8 = 3
50a. Naked pair {14} in R5C78, locked for R5 and N6 -> R4C7 = 5, R5C6 = 5

51. 45 rule on N1 2 remaining innies R3C13 = 5 -> R3C3 = 3, R3C1 = 2, R4C2 = 9 (cage sum), R4C3 = 6, R3C4 = 9 (cage sum), R2C13 = [59], R5C45 = [69]

52. R4C7 = 5 -> R3C78 = 9 = [18]

and the rest is naked singles.


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PostPosted: Mon May 09, 2011 4:56 am 
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Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Messy One #3 by Nasenbaer (June 2008) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:2560:2560:5634:5634:5634:5634:5382:5382:5382:2560:6410:6410:6410:3853:3342:1807:1807:5382:6418:6410:3860:3853:3853:3342:4376:4376:3098:6418:6410:3860:5150:5150:4128:4128:4376:3098:6418:6418:2086:5150:5150:5150:4128:3115:3115:5421:5421:2095:2086:5681:2866:3635:3115:4405:4918:5421:2095:5681:5681:2866:3635:3635:4405:4918:5421:2369:2369:4675:7236:7236:3635:4405:4918:4918:2369:4675:4675:4675:7236:7236:4405:
Solution:
+-------+-------+-------+
| 7 1 6 | 9 4 3 | 2 5 8 |
| 2 5 9 | 1 8 7 | 4 3 6 |
| 3 4 8 | 5 2 6 | 1 7 9 |
+-------+-------+-------+
| 8 6 7 | 2 1 4 | 5 9 3 |
| 5 9 1 | 6 3 8 | 7 4 2 |
| 4 2 3 | 7 5 9 | 8 6 1 |
+-------+-------+-------+
| 6 7 5 | 8 9 2 | 3 1 4 |
| 1 8 4 | 3 6 5 | 9 2 7 |
| 9 3 2 | 4 7 1 | 6 8 5 |
+-------+-------+-------+
Quote:
Nasenbaer: Well, the next 3 Assassins are booked, so it's time for a new Messy One! :cheers:
Have fun!
SScore(3.2.0): 0.97

Afmob: Thanks Nasenbaer for providing a fun killer! I think of all Messies posted so far, this one was the easiest.
Rating: 1.0.

gary w: Not messy at all !!
Agree with Afmob.1.0 at most.

Nasenbaer: pointed out that Messy One #3 meets all of Ed's rules for Messy Ones.

Andrew: After doing A108 I moved on to Messy One #3, two puzzles in a row by Nasenbaer.
I think I found this one a bit harder than Messy One #1, possibly because I took some time to find the key step ...
Maybe I'm going a bit high but I'll rate it as High 1.0.
I won't post a full walkthrough. My one was fairly similar to Afmob's apart from the order of some steps.

Walkthrough by Afmob:
Thanks Nasenbaer for providing a fun killer! I think of all Messies posted so far, this one was the easiest.

Messy One #3 Walkthrough:

1. R456
a) Outies R6789 = 7(3) = {124} locked for R4; 4 also locked for N6
b) 8(2) @ R5C3 = [17/26]
c) Innies R6789 = 13(2) = {67} locked for R6

2. R123
a) Outies R1 = 8(2) <> 4,8,9
b) Outies N3 = 12(2) = {39/57}
c) 12(2) <> 4,8
d) Innies+Outies N2: 5 = R1C3 - R2C4 -> R1C3 = (6789), R2C4 = (1234)

3. C123
a) Outies C12 = 10(2) = [64/73/82/91]
b) Naked quad (6789) locked in R1234C3 for C3
c) 8(2) @ R6C3 = {35} locked for C3
d) 4 locked in 9(3) @ C3 = {234} -> R8C4 = 3, {24} locked for C3+N7
e) R5C3 = 1 -> R6C4 = 7, R6C8 = 6
f) Outies N7 = 9(3) = {234} -> R6C3 = 3, {24} locked for R6+N4
g) 21(4) = 24{69/78} because R6C12 = {24}

4. R456
a) Hidden triple (124) in R4C456 @ N5 locked for R4
b) R7C3 = 5
c) 11(2) = [56/92]
d) 3,6 locked in 20(5) @ N5 = 236{18/45} -> 2 locked for N5; 3,6 locked for R5

5. R6789
a) Outies N9 = 14(2+1): R8C6 <> 9 because R6C79 >= 6
b) 18(4) <> 9 because R7C6 = (26) blocks {1269}
c) 9 locked in R7C45 @ N8 for R7+22(3)
d) Hidden Single: R6C6 = 9 @ N5 -> R7C6 = 2
e) Outies N9: 1 locked in R6C79 @ N6 -> R6C79 = 1{5/8} -> R8C6 = (58)
f) 4 locked in R7C789 @ R7 for N9
g) 28(4) = {5689} -> 6,9 locked for N9; 6 also locked for C7
h) 14(4) = 2{138/147/345} -> R8C8 = 2
i) 17(4) = 15{38/47} -> 1,5 locked for C9

6. C789
a) 12(2) = {39} locked for C9
b) 17(4) = {1457} locked for C9, {47} locked for N9
c) R5C8 = 4, R5C9 = 2, R2C9 = 6, R1C9 = 8
d) 21(4) = 68{25/34} -> R1C7 = (24)
e) 7(2) = [25/43]
f) Naked pair (35) locked in R12C8 for N3+C8
g) R3C9 = 9, R4C9 = 3
h) 17(3) = {179} -> R4C8 = 9, {17} locked for R3
i) R9C8 = 8, R7C8 = 1 -> R67C7 = 11(2) = [83] -> R6C7 = 8, R7C7 = 3

7. R123+N5
a) R6C5 = 5 -> R7C45 = 17(2) = {89} locked for R7+N8
b) 20(5) = {12368} locked for N5, 8 locked for R5
c) R8C6 = 5
d) 13(2) = {67} -> R2C6 = 7, R3C6 = 6
e) R3C3 = 8 -> R4C3 = 7
f) 7 locked in 10(3) @ N1 = {127} locked for N1

8. Rest is singles.

Rating: 1.0
Solving Outline by gary w:
Not messy at all !!

r2c34=10 and r1c3=r2c4+5 so if r2c4=x -> r1c3=x+5 and r2c3=10-x ->r12c3=15
so r12c3 and r34c3 naked quad 6789
so in c3 4 at r89 so in c3 naked single r5c3=1 so r6c4=7 -> r6c8=6 and r67c3={35}
r89c3={24} so r8c4=3


Still some work to do but it's cracked now.

Agree with Afmob.1.0 at most.
Andrew's post, including "spoilers":
After doing A108 I moved on to Messy One #3, two puzzles in a row by Nasenbaer.

I think I found this one a bit harder than Messy One #1, possibly because I took some time to find the key step, the innie-outie on N2 which leads directly to the naked quad in C3. My methodical approach must have been upset by this being a Messy One! ;) Maybe 45s on rows, columns and corner nonets are easier to spot. Just a thought!

The key step is part of a narrow path to start this puzzle. Maybe I'm going a bit high but I'll rate it as High 1.0.

I won't post a full walkthrough. My one was fairly similar to Afmob's apart from the order of some steps.


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PostPosted: Mon May 09, 2011 8:13 pm 
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Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 107 by Mike(mhparker) (June 2008) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:3584:3584:2050:2050:5892:4101:4101:4871:4871:5385:3584:2050:5892:5892:5892:4101:4871:2065:5385:5385:6676:6676:6676:6676:6676:2065:2065:2843:2843:3357:3357:4895:800:800:3106:3106:2852:2852:1830:1830:4895:1321:1321:4395:4395:813:813:3631:3631:4895:2866:2866:2356:2356:3126:3126:7736:7736:7736:7736:7736:2621:2621:3126:4672:3393:4418:4418:4418:5189:3910:2621:4672:4672:3393:3393:4418:5189:5189:3910:3910:
Solution:
+-------+-------+-------+
| 6 3 2 | 5 4 1 | 9 7 8 |
| 8 5 1 | 7 3 9 | 6 4 2 |
| 4 9 7 | 8 2 6 | 3 1 5 |
+-------+-------+-------+
| 3 8 9 | 4 6 2 | 1 5 7 |
| 7 4 6 | 1 5 3 | 2 8 9 |
| 2 1 5 | 9 8 7 | 4 6 3 |
+-------+-------+-------+
| 9 2 4 | 6 7 5 | 8 3 1 |
| 1 7 8 | 3 9 4 | 5 2 6 |
| 5 6 3 | 2 1 8 | 7 9 4 |
+-------+-------+-------+
Quote:
Mike: Time for the next Assassin and I still haven't had the chance to take a look at Afmob's A106V2 :( .
Here's an interesting and unusual cage pattern that will drive you crazy (well, at least it almost drove me crazy!). Like trying to break out of a fortress guarded by a huge moat...
SudokuSolver v3.2.0 score: 1.05. Est. rating: 1.25

Afmob: Thank you Mike for a relaxing killer! I had to use one small trick (step 3f) to crack it which I think deserves to be of rating 1.0. By the way, this is my shortest wt so far (farewell, A102X :salute:).
Rating: (Easy) 1.0. Edit: I think my initial rating is too low like Andrew mentioned, so following my wt A107 would be of rating (Hard?) 1.0.

gary w: V1 solving path same as Afmob's..including the "blocking" move.I don't think I'll ever beat him to the tape though !!!

Ed: Ditto & ditto. Easy 1.0 rating seems generous but correct. For once I agree with Afmob ;) .
With no word from Ruud, seems just a matter of when the forums go down again as well as sudocue.net.

Andrew: Congratulations to Afmob, Ed, Gary and anyone else who found Afmob's step 3f.
I think the key to solving this puzzle was ...
I'll rate my walkthrough as solid to hard 1.25.

Walkthrough by Afmob:
Thank you Mike for a relaxing killer! I had to use one small trick (step 3f) to crack it which I think deserves to be of rating 1.0. By the way, this is my shortest wt so far (farewell, A102X :salute:).

Edit: I think my initial rating is too low like Andrew mentioned, so following my wt A107 would be of rating (Hard?) 1.0.

A107 Walkthrough:

1. R456
a) Innies R4 = 6 = R4C5
b) Innies R5 = 5 = R5C5
c) Innies R6 = 8 = R6C5
d) R6C4 = 9 -> R6C3 = 5
e) 13(2) = [94] -> R4C3 = 9, R4C4 = 4
f) Outies N6 = 12(3) = {237} -> R4C6 = 2, R5C6 = 3, R6C6 = 7
g) Cage sums: R4C7 = 1, R5C7 = 2, R6C7 = 4
h) R5C4 = 1 -> R5C3 = 6
i) 11(2) @ R4 = {38} locked for R4+N5

2. C123
a) Innies N7 = 15(3) = {348} locked for C3+N7
b) 8(3) = {125} -> R1C4 = 5, {12} locked for N1
c) R3C3 = 7
d) 14(3) = {356} -> R2C2 = 5, {36} locked for R1+N1
e) 12(3) = 1{29/56} -> 1 locked for N7

3. C789 !
a) Innies+Outies N3: -2 = R1C6 - R3C7 -> R1C6 <> 8,9; R3C7 = (36)
b) 5 locked in 8(3) @ N3 = {125} locked for N3
c) 5 locked in Innies N9 @ C7 = 20(3) = 5{69/78} -> 5 locked for N9
d) 10(3) = 1{27/36} -> 1 locked for N9
e) 4 locked in 15(3) @ N9 = 4{29/38}
f) ! 10(3) = {136} locked for N9 because R234C9 = (1257) blocks {127}
g) Innies N7 = 20(3) = {578} locked for C7+N9
h) 10(3): R7C8 <> 1 because 1{36} blocked by R6C9 = (36)
i) Hidden Single: R3C8 = 1 @ C8, R9C5 = 1 @ R9

4. R789
a) Innies+Outies N7: -2 = R9C4 - R7C3 -> R7C3 <> 3 and R9C4 = (26)
b) Innies+Outies N9: R7C7 = R9C6 = (58)
c) 4,7,8 locked in 30(5) @ R7 = 478{29/56} -> 7 locked for N8
d) 17(4) = 19{25/34} because R9C4 = (26) and R9C6 = (58) block 18{26/35}
-> 9 locked for R8+N8
e) 30(5) = {45678} locked for R7, 6 locked for N8
f) 13(3) = {238} -> R9C4 = 2, {38} locked for N7

5. Rest is singles.

Rating: (Easy) 1.0. I used a combo blocker.
Walkthrough by Andrew:
Congratulations to Afmob, Ed, Gary and anyone else who found Afmob's step 3f.

I think the key to solving this puzzle was interactions between L-shaped cages in R23 or R78 and adjacent 2-cell cages in R4 or R6. That's what Afmob's step 3f was except that it used two L-shaped cages. Having not spotted that one and done a lot of nibbling I eventually found one of those interactions in step 38.

It's interesting to note that Afmob's walkthrough could have been a bit shorter by using a UR after step 3f but we know that Afmob prefers not to use UR. I've given a note in my walkthrough where I could have used the UR.

Afmob wrote:
Rating: (Easy) 1.0. I used a combo blocker.
I'm surprised at that rating. IMHO step 3f belongs in a 1.25 rating, probably an Easy 1.25 since it's the only tricky move in Afmob's walkthrough. If I'd found it, I would definitely have rated it a 1.25.

Here is my rather longer walkthrough. I'll rate it as solid to hard 1.25. Thanks Afmob and Ed for pointing out the minor error in step 21 and to Ed for pointing out that I'd missed a pair of permutations in step 32d. Steps 39 to 41 have been deleted and steps 44 to 47 have been reworked to take account of these extra permutations.

Prelims

a) R4C12 = {29/38/47/56}, no 1
b) R4C34 = {49/58/67}, no 1,2,3
c) R4C67 = {12}, locked for R4, clean-up: no 9 in R4C12
d) R4C89 = {39/48/57}
e) R5C12 = {29/38/47/56}, no 1
f) R5C34 = {16/25/34}, no 7,8,9
g) R5C67 = {14/23}
h) R5C89 = {89}, locked for R5 and N6, clean-up: no 3,4 in R4C89, no 2,3 in R5C12
i) R6C12 = {12}, locked for R6 and N4, clean-up: no 5,6 in R5C4
j) R6C34 = {59/68}
k) R6C67 = {47/56}/[83], no 9, no 3 in R6C6
l) R6C89 = {36/45}, no 7
m) 8(3) cage at R1C3 = {125/134}, CPE no 1 in R1C12
n) 21(3) cage in N1 = {489/579/678}, no 1,2,3
o) 19(3) cage in N3 = {289/379/469/478/568}, no 1
p) 8(3) cage in N3 = {125/134}, 1 locked for N3
q) 10(3) cage in N9 = {127/136/145/235}, no 8,9
r) 20(3) cage at R8C7 = {389/479/569/578}, no 1,2

1. Naked pair {57} in R4C89, locked for R4 and N6, clean-up: no 4,6 in R4C12, no 6,8 in R4C34, no 4,6 in R6C6, no 4 in R6C89

2. Naked pair {38} in R4C12, locked for R4 and N4, clean-up: no 4 in R5C4, no 6 in R6C4

3. Naked pair {36} in R6C89, locked for R6 and N6, clean-up: no 2 in R5C6, no 8 in R6C4, no 5,8 in R6C6

4. Naked pair [74] in R6C67, clean-up: no 1 in R5C6

5. Naked pair {12} in R45C7, locked for C7

6. R46C5 = [68] (hidden singles in R4 and R6), R5C5 = 5 (cage sum), R6C34 = [59], R4C34 = [94], R5C6 = 3, R5C7 = 2, R5C4 = 1, R5C3 = 6, R4C67 = [21]

7. 8(3) cage at R1C3 = {125/134}
7a. 1 locked in R12C3, locked for C3 and N1
7b. 5 of {125} must be in R1C4 -> no 2 in R1C4
7c. 3 of {134} must be in R1C4 -> no 3 in R12C3

8. 45 rule on N1 3 innies R123C3 = 10 = {127} (only remaining combination) -> R3C3 = 7, R12C3 = {12}, locked for C3 and N1, R1C4 = 5 (step 7)

9. Naked triple {348} in R789C3, locked for N7

10. 18(3) cage in N7 = {279/567}, no 1, 7 locked for N7

11. 45 rule on N3 1 innie R3C7 = 1 outie R1C6 + 2, no 8,9 in R1C6, no 5,9 in R3C7

12. 45 rule on N7 1 innie R7C3 = 1 outie R9C4 + 2, no 3 in R7C3, no 3,7,8 in R9C4

13. 45 rule on N9 1 innie R7C7 = 1 outie R9C6, no 3,7 in R7C7, no 4 in R9C6

14. 45 rule on R3 2 outies R2C19 = 10 = [64/82/91], no 4,5 in R2C1, no 3,5 in R2C9

15. 45 rule on R7 2 outies R8C19 = 7 = {16/25}, no 9 in R8C1, no 3,4,7 in R8C9

16. 14(3) cage in N1 = {356} (only remaining combination) -> R2C2 = 5, R1C12 = {36}, locked for R1 and N1, clean-up: no 4 in R2C9 (step 14), no 8 in R3C7 (step 11)
16a. 4 in N1 locked in R3C12, locked for R3

17. 8(3) cage in N3 = {125} (only remaining combination), no 3, locked for N3

18. Naked pair {12} in R2C39, locked for R2

19. 5 in C7 locked in R789C7, locked for N9, clean-up: no 2 in R8C1 (step 15)
19a. 45 rule on N9 3 innies R789C7 = 20 = {569/578}, no 3
19b. 3 in C7 locked in R23C7, locked for N3

20. 10(3) cage in N9 = {127/136}, no 4, 1 locked for N9
[I could have used Afmob’s combo blocker here if I’d spotted it.]

21. R9C5 = 1 (hidden single in R9)

22. 1 in R8 locked in R8C19 = {16} (step 15), locked for R8

23. 16(3) cage at R1C6 = {169/178/349} (cannot be {367} because R1C6 only contains 1,4)
23a. 3,6 of {169/349} must be in R2C7 -> no 9 in R2C7

24. 3,6,7 in R3 locked in R3C34567 = {13679/23678}
24a. 1 of {13679} must be in R3C6 -> no 9 in R3C6

25. 17(4) cage in N8 = {1259/1349/1358/1457}
25a. 2 of {1259} must be in R8C4 -> no 2 in R8C5
25b. 5 of {1358} must be in R8C6 -> no 8 in R8C6
25c. 7 of {1457} must be in R8C4 -> no 7 in R8C5

26. 4 in N9 locked in 15(3) cage = {249/348}, no 6,7

27. 18(3) cage in N7 (step 10) = {279/567}
27a. 5 of {567} must be in R9C1 -> no 6 in R9C1

28. 5 in R8 locked in R8C67, CPE no 5 in R9C6, clean-up: no 5 in R7C7 (step 13)

29. 20(3) cage at R8C7 = {569/578}
29a. 8 of {578} must be in R9C6 -> no 8 in R89C7

30. 8 in R1 locked in R1C789, locked for N3

31. 16(3) cage at R1C6 (step 23) = {169/178/349}
31a. 8 of {178} must be in R1C7 -> no 7 in R1C7

32. 1 in R7 locked in R7C1289
32a. 45 rule on R7 4 innies R7C1289 = 15 = {1239/1257/1356}
32b. {1239} must be {29}{13}
32c. {1257} must be [51]{27} (other permutations not consistent with the 12(3) and 10(3) cages)
32d. {1356} must be [51]{36}/[56]{13}
32e. -> no 1,6 in R7C1 (no eliminations from R7C2)
[Thanks Ed for pointing out the extra permutations in step 32d. Ed also pointed out that I could have got the eliminations from R7C1 from the permutations of the 12(3) cage.]


33. 45 rule on R9 4 outies R8C2378 = 22 = {2389/2479/2578} (cannot be {3478} because 3,4,8 only in R8C38), 2 locked for R8
33a. 4 of {2479} must be in R8C3 -> no 4 in R8C8

34. 4 in N9 locked in R9C89, locked for R9

35. 4 in N3 locked in 19(3) cage = {469/478}
35a. 6 of {469} must be in R2C8 -> no 9 in R2C8

36. Killer pair 8,9 in R1C7 and R1C89, locked for R1

37. 1 in N2 locked in R13C6
37a. 45 rule on N2 4 remaining innies R1C6 + R3C456 = 17 = {1268/1349}
37b. 2 of {1268} must be in R3C5 -> no 2 in R3C4
37c. 9 of {1349} must be in R3C5 -> no 3 in R3C5
37d. 2 in N2 locked in R13C5, locked for C5

38. 12(3) cage in N7 = {129/156}
38a. {129} must be [921] (R7C12 cannot be [29] because R78C1 clashes with R6C1), no 2 in R7C1, no 9 in R7C2

39 to 41. Deleted

42. 10(3) cage in N9 (step 20) = {127/136}
42a. 1 must be in R78C9 (R78C9 cannot be {36} which clashes with R6C9) -> no 1 in R7C8
[A similar interaction between R6C89 and 10(3) cage in N9 could be used to eliminate 6 from R7C89 but that would involve using a UR, which I don’t use. Fortunately the finish is quick without using it.]

43. 1 in N9 locked in R78C9, locked for C9 -> R23C9 = [25], R3C8 = 1, R12C3 = [21], R4C12 = [57]

44. R1C6 = 1, R3C5 = 2 (hidden singles in N2), R3C46 = {68} (step 37a), locked for R3 and N2 -> R3C7 = 3

Rest of step 44 deleted, steps 45 to 47 reworked.

45. R2C1 = 8 (hidden single in R2), R4C12 = [38], R1C12 = [63], R8C1 = 1, R6C12 = [21], R8C9 = 6, R6C89 = [63], R7C9 = 1, R7C8 = 3 (step 20), clean-up: no 9 in 19(3) cage in N3 (step 35), no 6 in R9C6 (step 13), no 8 in 15(3) cage in N9 (step 26)

46. Naked triple {478} in 19(3) cage in N3, locked for N3 -> R12C7 = [96], R7C7 = 8, R9C6 = 8 (step 13), R3C46 = [86], R7C3 = 4, R89C3 = [83], R9C4 = 2 (cage sum)

47. Naked pair {59} in R7C16, locked for R7 -> R7C45 = [67]


and the rest is naked singles.


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PostPosted: Mon May 09, 2011 8:27 pm 
Offline
Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 107 V2 by Mike(mhparker) (June 2008) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:4864:4864:2050:2050:5124:4101:4101:4103:4103:4361:4864:2050:5124:5124:5124:4101:4103:4369:4361:4361:5652:5652:5652:5652:5652:4369:4369:1307:1307:3613:3613:5407:1824:1824:3106:3106:1572:1572:2598:2598:5407:2857:2857:3115:3115:2861:2861:3887:3887:5407:1074:1074:1844:1844:3894:3894:6712:6712:6712:6712:6712:3389:3389:3894:4416:2625:6210:6210:6210:4421:3398:3389:4416:4416:2625:2625:6210:4421:4421:3398:3398:
Solution:
+-------+-------+-------+
| 9 6 3 | 4 2 8 | 5 1 7 |
| 2 4 1 | 7 5 6 | 3 8 9 |
| 7 8 5 | 3 1 9 | 4 2 6 |
+-------+-------+-------+
| 3 2 9 | 5 7 1 | 6 4 8 |
| 1 5 8 | 2 6 4 | 7 9 3 |
| 4 7 6 | 9 8 3 | 1 5 2 |
+-------+-------+-------+
| 6 1 4 | 8 3 2 | 9 7 5 |
| 8 9 7 | 6 4 5 | 2 3 1 |
| 5 3 2 | 1 9 7 | 8 6 4 |
+-------+-------+-------+
Quote:
Mike: And just so you don't have to wait in suspense, here's a V2:
SudokuSolver v3.2.0 score: 1.26. Est. rating: 1.75.

Afmob: This one was certainly tougher than V1. I wonder whether my chain in step 8a can somehow be replaced by an easier move.
Rating: (Easy) 1.5.

Andrew, in 2010: Continuing working through my backlog, the next variant I tried was A107 V2 which I found that I hadn't started.
Thanks Mike for a challenging variant!
My solving path is very similar to Afmob's and we used the same breakthrough move but our reasoning for steps was sometimes different so I'm posting my walkthrough.
I could have gone with Afmob's rating but I decided to go for a solid 1.5.

Walkthrough by Afmob:
This one was certainly tougher than V1. I wonder whether my chain in step 8a can somehow be replaced by an easier move.

A107 V2 Walkthrough:

1. R456
a) Innies R4 = 7 = R4C5
b) Innies R5 = 6 = R5C5
c) Innies R6 = 8 = R6C5
d) R6C4 = 9 -> R6C3 = 6, R4C4 = 5 -> R4C3 = 9
e) 12(2) @ R4 = {48} locked for R4+N6
f) 7(2) @ R4 = {16} -> R4C6 = 1, R4C7 = 6
g) Outies N6 = 8(3) = {134} -> R6C6 = 3, R6C5 = 4
h) Cage sum: R5C7 = 7, R6C7 = 1, R5C4 = 2 -> R5C3 = 8
i) 7(2) @ R6 = {25} locked for R6+N6

2. C1234
a) Innies N1 = 9(3) = 3{15/24} -> 3 locked for C3+N1
b) 7 locked in R789C3 @ C3 for N7
c) Innies+Outies N1: -1 = R1C4 - R3C3 -> R3C3 <> 1,3
d) 3 locked in 8(3) @ C3 = {134}; R1C4 <> 3
e) Innies+Outies N1: -1 = R1C4 - R3C3 -> R3C3 <> 4
f) Innies+Outies N7: -3 = R9C4 - R7C3 -> R9C4 = (14), R7C3 = (47)
g) Naked pair (14) locked in R19C4 for C4
h) 17(3) @ N1 <> 5 because {458} blocked by Killer pair (45) of Innies N1

3. N39
a) Innies N3 = 12(3) = {345} locked for C7+N3
b) Naked triple (289) locked in R789C7 for N9
c) Innies+Outies N3: 4 = R1C6 - R3C7 -> R1C6 = (789)
d) Innies+Outies N9: -2 = R9C6 - R7C7 -> R9C6 = (67), R7C7 <> 2

4. R123
a) Innies R12 = 11(2) <> 1,6,8; R2C1 <> 7
b) 17(3) @ N1: R3C12 <> 4 because R2C1 <> 6,7
c) 3,4,5 locked in 22(5) @ R3 = 345{19/28}

5. R789 !
a) Innies R89 = 9(2): R8C1 <> 1,9
b) Innies N7 = 7{15/24}
c) 17(3) @ N7 <> 4 because (458) blocked by Killer pair (45) of Innies N7
d) 15(3) <> 5{28/46} because they are blocked by Killer pairs (25,45) of Innies N7
e) Outies R89 = 19(4) <> 47{26/35} because R7C3 = (47)
f) ! Outies R89 = 1{279/369/378/459/468/567} because:
- 2 can only be @ 15(3) which must be {29}4 and R7C89 @ 13(3) cannot be {35}
-> 1 locked for R7
g) 24(4) <> 2 because {2589} blocked by Killer pair (25) of 26(5) and {2679}
blocked by R9C6 = (67)
h) 2 locked in 26(5) @ N8 for R7

6. R789
a) 15(3): R7C12 <> 5 because R8C1 <> 1,9
b) 13(3) @ R7: R8C9 <> 6 because Outies R89 have no combo with (34)
c) Innies R89 = 9(2): R8C1 <> 3
d) Innies+Outies N9: -2 = R9C6 - R7C7 -> R7C7+R9C6 = [86/97]
e) 26(5) <> {23678} because 8 in R7C7 forces R9C6 = 6
f) 15(3) <> 2 because {49}2 blocked by Killer pair (49) of 26(5)
g) Innies R89 = 9(2): R8C9 <> 7

7. C4+N8
a) Hidden Killer pair (67) in R78C4 for C4 since only other place possible is R2C4
b) Killer pair (67) locked in R78C4+R9C6 for N8

8. R789 !
a) ! 15(3): R8C1 <> 4 because it forces Outies R89 = 19(4) = {1378} but at
the same time R8C1 = 4 -> R7C3 = 7
b) Innies N7 = 13(3) = 7{15/24}: R89C3 <> 4 because 10(3) cannot have (24)
c) 4 locked in R7C123 @ N7 for R7
d) 13(3) @ R7: R8C9 <> 3 because R7C89 <> 4
e) Innies R89 = 9(2) = [54/81]
f) 13(3) @ R7 must have 1 xor 4 and R8C9 = (14) -> R7C89 <> 1
g) 1 locked in 15(3) @ R7 for N7 -> 15(3) = 1{59/68}
h) Innies N7 = {247} locked C3+N7 -> R7C3 = 4
i) 10(3) = {127} -> R9C4 = 1

9. R123
a) R1C4 = 4
b) 22(5) = 345{18/29} -> R3C3 = 5, R3C7 = 4; 3 locked for N2
c) 20(4) = {2567} because 18{29/56} blocked by Killer pair (18) of 22(5)
-> R12C5 = {25} locked for C5+N2; R2C46 = {67} locked for R2
d) Innies R12 = 11(2) = {29} locked for R2
e) 16(3) @ N2 = {358} -> R1C6 = 8
f) R3C6 = 9, R8C6 = 5, R8C1 = 8
g) Hidden Single: R2C2 = 4 @ R2 -> R1C12 = 15(2) = {69} locked for R1+N1

10. R789
a) Innies R89 = 9(2) = [81] -> R8C9 = 1

11. Rest is singles.

Rating: (Easy) 1.5. I used some combo analysis and a small contradiction chain.
Walkthrough by Andrew, in 2010:
Continuing working through my backlog, the next variant I tried was A107 V2 which I found that I hadn't started.

Thanks Mike for a challenging variant!

My solving path is very similar to Afmob's and we used the same breakthrough move but our reasoning for steps was sometimes different so I'm posting my walkthrough.

Rating Comment. I could have gone with Afmob's rating but I decided to go for a solid 1.5. That's partly because there appears to be a narrow solving path, based on the similarity of Afmob's walkthrough and my one, and because several times I took a long time to find the next step.

Here is my walkthrough for A107 V2.

Prelims

a) R4C12 = {14/23}
b) R4C34 = {59/68}
c) R4C67 = {16/25/34}, no 7,8,9
d) R4C89 = {39/48/57}, no 1,2,6
e) R5C12 = {15/24}
f) R5C34 = {19/28/37/46}, no 5
g) R5C67 = {29/38/47/56}, no 1
h) R5C89 = {39/48/57}, no 1,2,6
i) R6C12 = {29/38/47/56}, no 1
j) R6C34 = {69/78}
k) R6C67 = {13}
l) R6C89 = {16/25/34}, no 7,8,9
m) 19(3) cage in N1 = {289/379/469/478/568}, no 1
n) 8(3) cage at R1C3 = {125/134}
o) 21(3) cage at R4C5 = {489/579/678}, no 1,2,3
p) 10(3) cage at R8C3 = {127/136/145/235}, no 8,9

Steps resulting from Prelims
1a. Naked pair {13} in R6C67, locked for R6, clean-up: no 8 in R6C12, no 4,6 on R6C89
1b. Naked pair {25} in R6C89, locked for R6 and N6, clean-up: no 2,5 in R4C6, no 7 in R4C89, no 6,9 in R5C6, no 7 in R5C89, no 6,9 in R6C12
1c. Naked pair {47} in R6C12, locked for R6 and N4, clean-up: no 1 in R4C12, no 2 in R5C12, no 3,6 in R5C4, no 8 in R6C34
1d. Naked pair {23} in R4C12, locked for R4 and N4, clean-up: no 4 in R4C67, no 9 in R4C89, no 7,8 in R5C4
1e. Naked pair {16} in R4C67, locked for R4, clean-up: no 8 in R4C34
1f. Naked pair {15} in R5C12, locked for R5 and N4 -> R4C34 = [95], R6C34 = [69], R5C3 = 8, R5C4 = 2, R6C5 = 8, clean-up: no 3 in R5C6, no 3,6,9 in R5C7, no 4 in R5C89
1g. Naked pair {48} in R4C89, locked for R4 and N6 -> R4C5 = 7, R5C67 = [47], R5C5 = 6, R4C67 = [16], R6C67 = [31]
What an incredible start to an Assassin variant! The hard work will come later.

2. 45 rule on R12 2 innies R2C19 = 11 = {29/38/47/56}, no 1

3. 45 rule on R89 2 innies R8C19 = 9 = {18/27/36/45}, no 9

4. 45 rule on N1 1 innie R3C3 = 1 outie R1C4 + 1, no 1,3,7 in R3C3
4a. 45 rule on N1 3 innies R123C3 = 9 = {135/234}, 3 locked for C3, N1 and 8(3) cage at R1C3, no 3 in R1C4, clean-up: no 8 in R2C9 (step 2), no 4 in R3C3
4b. R3C3 = {25} -> no 2,5 in R12C3
4c. 8(3) cage at R1C3 = {134}, CPE no 4 in R1C12

5. 45 rule on N3 3 innies R123C7 = 12 = {345} (only remaining combination), locked for C7 and N3, clean-up: no 6,7,8 in R2C1 (step 2)
5a. 45 rule on N3 1 outie R1C6 = 1 innie R3C7 + 4, no 2,5,6 in R1C6

6. Naked triple {289} in R789C7, locked for N9, clean-up: no 1,7 in R8C1 (step 3)
6a. 45 rule on N9 1 innie R7C7 = 1 outie R9C6 + 2, no 2 in R7C7, no 2,5,8,9 in R9C6

7. 45 rule on N7 1 innie R7C3 = 1 outie R9C4 + 3, no 1,2,5 in R7C3, no 3,6,7 in R9C4
7a. R7C3 + R9C4 = [41/74], CPE no 4 in R7C45 + R89C3 + R9C12

8. 45 rule on N7 3 innies R789C3 = 13 = {157/247}, 7 locked for N7
8a. 10(3) cage at R8C3 = {127/145}, CPE no 1 in R9C12

9. Naked pair {14} in R19C4, locked for C4

10. 17(3) cage in N7 = {269/359/368} (cannot be {458} which clashes with R789C3), no 1,4

11. 17(3) cage in N1 = {179/269/278/467} (cannot be {458} which clashes with R123C3), no 5, clean-up: no 6 in R2C9 (step 2)
11a. 4 of {467} must be in R2C1 -> no 4 in R3C12

12. 3,4,5 in R3 only in 22(5) cage at R3C3 = {13459/23458}, no 6,7
12a. 1 of {13459} must be in R3C5 -> no 9 in R3C5

13. Hidden killer pair 6,7 in R2C4 and R78C4 for C4, R78C4 cannot be {67} which clashes with R9C6 -> R2C4 = {67}, R78C4 must contain one of 6,7
13a. Killer pair 6,7 in R78C4 and R9C6, locked for N8

14. 6 in N2 only in R2C46, locked for R2
14a. 20(4) cage = {1568/2369/2468/2567/3467} (cannot be {1469} which clashes with R1C4)
14b. 6,7,8 of {1568/2567} must be in R2C46 -> no 5 in R2C6

15. 19(3) cage in N1 = {289/469/478/568}
15a. 7 of {478} must be in R1C1 (R12C2 cannot be [74] which clashes with R6C2) -> no 7 in R12C2

16. 15(3) cage in N7 = {159/168/249/348} (cannot be {258/456} which clash with R789C3)
16a. 5 of {159} must be in R8C1 -> no 5 in R7C12

17. 13(3) cage at R7C8 = {157/346}
17a. 45 rule on R7 4 innies R7C1289 = 19 = {1279/1369/1378/1459/1468/1567} (cannot be {2467/3457} which clash with R7C3, cannot be {2359/2458} because {35/45} in R7C89 aren’t consistent with combinations for 13(3) cage at R7C8, cannot be {2368} because no {258} combination in 15(3) cage in N7), 1 locked for R7
17b. No combination for R7C1289 contains both of 3,4 -> R7C89 cannot be {34} -> no 6 in R8C9, clean-up: no 3 in R8C1 (step 3)

18. Hidden killer pair 1,4 in 24(4) cage and R9C4 for N8, R9C4 = {14} -> 24(4) cage must contain one of 1,4 -> {1689/3489/4569/4578} (only combinations containing one of 1,4), no 2
18a. 2 in N8 only in R7C56, locked for R7

19. 20(4) cage in N2 (step 14a) = {2369/2468/2567/3467} (cannot be {1568} which clashes with 22(5) cage at R3C3 which must have 1 or 8 in R3C456), no 1

20. 45 rule on N2 5 innies R1C46 + R3C456 = 25 = {12589/13489/13579/14578} (from combinations for 20(4) cage, step 19)
20a. R1C46 + R3C456 cannot be {12589} which clashes with R3C3
R1C46 + R3C456 cannot be {13579} = [17]+[359] which clashes with 22(5) cage at R3C3 because {13459} must have 5 in R3C3)
-> R1C46 + R3C456 = {13489/14578}, 4,8 locked for N2

21. 20(4) cage in N2 (step 19) = {2369/2567}, 2 locked for N2
21a. 5 of {2567} must be in R2C5 (R2C456 cannot be {267} which clashes with R2C19), no 5 in R1C5
21b. R1C4 + R3C5 = {14} (hidden pair in N2)

22. Hidden grouped X-Wing for 1 in 8(3) cage at R1C3 and 16(3) cage at R1C8 for R12, 8(3) cage at R1C3 contains 1 -> 16(3) cage at R1C8 must contain 1 = {169/178}, no 2, 1 locked for N3

23. 45 rule on R3 4 innies R3C1289 = 23 = {1679/2678}
23a. 1 of {1679} only in R3C12, R3C12 cannot be {19} (because no 7 in R2C1) -> no 9 in R3C12

24. 2 in R7 only in 26(5) cage at R7C3 = {23489/23579/24569/24578} (cannot be {23678} which clashes with 24(4) cage in N8 or 24(4) cage + R9C6 in the case of the {4578} combination)
[Alternatively {23678} can be eliminated by looking at the permutations for 5 innies R7C456 + R9C69 in N8.]

25. R7C1289 (step 17a) = {1369/1378/1468/1567} (cannot be {1459} which clashes with 26(5) cage)
25a. 4,6 of {1468} must be in R7C89 (R7C12 cannot contain both of 4,8 because no 3 in R8C1), no 4 in R7C12
25b. 15(3) cage in N7 (step 16) = {159/168/348} (cannot be {249} because 2,4 only in R8C1), no 2, clean-up: no 7 in R8C9 (step 3)
25c. 9 of {159} must be in R7C1 (R78C1 cannot be [15] which clashes with R5C1) -> no 9 in R7C2

26. 15(3) cage in N7 (step 25b) = {159/168/348}
26a. Cannot be {348}
R8C1 = 4 => R7C12 = {38} => R7C89 (step 25) = {17} clashes with R7C3
26b. 15(3) cage = {159/168}, no 3,4, clean-up: no 5 in R8C9 (step 3)

27. R7C3 = 4 (hidden single in N7), R9C4 = 1 (step 7), R1C4 = 4, R3C5 = 1, R3C3 = 5 (step 4), clean-up: no 9 in R3C89 (step 23)

28. Naked quad {2678} in R3C1289, locked for R3 -> R3C46 = [39], R3C7 = 4, R1C6 = 8 (step 5a), R12C5 = [25], R78C6 = [25]

29. Naked pair {67} in R2C46, locked for R2, clean-up: no 4 in R2C1 (step 2)
29a. Naked pair {29} in R2C19, locked for R2

30. R2C2 = 4 (hidden single in N1), R6C12 = [47]
30a. R2C2 = 4 -> R1C12 = 15 = {69}, locked for R1 and N1, R2C1 = 2, R2C9 = 9, R3C12 = [78], R4C12 = [32], R5C89 = [93]

31. Naked pair {17} in R1C89, locked for R1 and N3 -> R2C8 = 8, R4C89 = [48]

32. R8C19 = 9 (step 3) = [81], R1C89 = [17]

33. R8C9 = 1 -> R7C89 = 12 = [75]

34. R7C4 = 8 (hidden single in C4), R7C7 = 9, R9C6 = 7 (step 6a)

and the rest is naked singles.


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PostPosted: Mon May 09, 2011 9:11 pm 
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Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 108 by Nasenbaer (June 2008) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:5120:5120:3330:3330:2308:2308:6662:6662:6662:5120:4362:3330:3340:2308:1294:1294:7952:6662:2066:4362:4362:3340:3094:7952:7952:2585:6662:2066:3868:3868:7198:3094:7952:7952:2585:2851:3108:3108:3868:7198:7198:3369:3369:2851:2851:4141:4910:4910:7198:7198:7198:2611:2611:3637:4141:4141:4910:4910:3898:3898:3388:3637:3637:3903:3903:4161:4910:3651:3898:3388:3388:4679:3903:3903:4161:4161:3651:3149:3149:4679:4679:
Solution:
+-------+-------+-------+
| 9 4 1 | 7 2 6 | 8 5 3 |
| 7 6 5 | 8 1 3 | 2 4 9 |
| 2 3 8 | 5 4 9 | 6 7 1 |
+-------+-------+-------+
| 6 9 4 | 1 8 5 | 7 3 2 |
| 5 7 2 | 6 3 4 | 9 1 8 |
| 8 1 3 | 9 7 2 | 4 6 5 |
+-------+-------+-------+
| 3 5 9 | 4 6 8 | 1 2 7 |
| 4 8 7 | 2 5 1 | 3 9 6 |
| 1 2 6 | 3 9 7 | 5 8 4 |
+-------+-------+-------+
Quote:
Nasenbaer: Dear Assassin friends,
due to the European Soccer Championship quarter final game Germany vs. Portugal there will be a slight delay of 1-2 hours for Assassin 108. As soon as I'm back home I'll start posting (even if we lost). ;)

Børge: Since Norway is not participating in the European Soccer Championship, Drücke ich heute Abend Deutschland die Daumen! Image

Nasenbaer: :cheers: :applause: 3-2 victory!!! Hurray!!! :applause: :cheers:
Well, that was a great game. And because of that I'm in a kind mood. :cheesey: Here's an easy v1 with a rating of 0.9. I know, it's too easy for you ;) , but I liked the symmetry, and v2 (which will be posted tomorrow) will destroy that.
Have fun!

udosuk: Congrats on the German victory! :cheers:
I've stated elsewhere before the start of the tournament that I tipped Germany to win it all, but even I wasn't confident enough to predict them to beat Portugal based on the group matches performance. Glad to see Ballack finally get some payback from C-Ron after the Premier and Champions Leauge rivalries. :applause:
Will try your v2 later. :cheesey: (see next archive entry for the v2 and udosuk's walkthrough)

Afmob: Thanks Nasenbaer for an interesting Killer ... It took me some time to find the key moves and it didn't fall so fast, so I rate it higher than SS. I have to see if I missed something obvious.
Rating: 1.25.

Ed: Yes, it was a really good challenge. I (finally) found a way to make big inroads quickly so by step 8 I was thinking, "yep - hard 1.0 rating". But it didn't give in!
I'll say this one is a hard 1.25 rating. Back to my usual half a rating higher than Afmob! For quite a few puzzles recently, the SSscore is coming in a bit low. Might be time to tweak it up half a rating.
So thanks Nasenbaer for a fine challenge!
BTW - Mike hasn't logged in for over a week. Hope he's OK. [edit: just noticed he went on holidays this time last year. So, hopefully, lots of warm sun now :) ]

gary w: Wow!! v1 was MUCH harder than I anticipated given the 0.9 rating.It wasn't until I saw ...
I'ld rate it about 1.5..I really struggled for a long time!

Nasenbaer: I'm glad you appreciated this Assassin! Truth is that the rating of 0.9 also seemed to be a little off for me, but hey, I'm no rating expert. ;)
Now I'll have to look at all your nice walkthroughs. I'm almost sure that one of you found an easier way than I did...

Andrew: Thanks Nasenbaer for an enjoyable and challenging Assassin. :)
I'm slowly catching up with them after our holiday.
Ed's step 4 was a clever move. Maybe Mike can tell us what sort of step this is?
I'll also rate A108 as Hard 1.25.
I'm not sure whether I'll try the V2, certainly not until I've caught up on the V1s and other forum puzzles. (I only tried it 2 years later, see next archive entry)

Walkthrough by Afmob:
Thanks Nasenbaer for an interesting Killer which had lots of Innies/Outies of multiple Nonets. It took me some time to find the key moves and it didn't fall so fast, so I rate it higher than SS. I have to see if I missed something obvious.

A108 Walkthrough:

1. N1478
a) Outies N1 = 13(1+1) = [67/76/85]
b) 8(2): R3C1 <> 5,6,7
c) Innies N1 = 8(3) = 1{25/34} -> 1 locked for N1
d) Innies N478 = 13(1+1) = [58/67]
e) 8(2) <> 1
f) 1 locked in 13(3) @ N1 = 1{48/57} -> 1 locked for C3
g) 12(2) @ N8 = [75/84]

2. R456789 !
a) ! Innies+Outies: N4789 = -1 = R6C9 - R4C1; R4C1 = (56)
-> R6C9 = (45) -> R4C1+R6C9 = [54/65]
-> CPE: R4C79+R6C123 <> 5
b) Outies R6789 = 10(3) <> 8,9
c) Innies C1234 = 16(3): R6C4 = (789) because R45C4 <= 9
d) Outies C1234 = 12(3): R5C5 = (123) because R6C56 >= 9
e) 28(6) must have one of (89) and it's only possible @ R6C456
f) Innies R6789 = 18(3) <> 5 because R6C9 = (45) blocks {459} and
{567} has no 8,9
g) Hidden Single: R6C9 = 5 @ R6
h) Innies+Outies: N4789 = -1 = R6C9 - R4C1 -> R4C1 = 6
i) Outies N9 = 12(1+1) = [57] -> R9C6 = 7
j) Cage sums: R3C1 = 2, R9C7 = 5

3. R123
a) Innies N1 = 8(3) = {125} -> 5 locked for C3+N1
b) 13(3) = {157} -> R1C4 = 7
c) 5 locked in R3C456 @ R3 for N2
d) 13(2) <> 6 and R3C4 <> 8
e) 9(3) = 2{16/34} -> 2 locked for N2
f) 12(2): R4C5 <> 5

4. R456 !
a) 15(3) <> 1 because R45C3 <> 5
b) 1 locked in R6C12 @ N4 for R6
c) Using Innies R6789 = 18(3): Innies+Outies C1234: -6 = R5C5 - R6C4 -> R5C5 <> 1
d) 13(2): R5C6 <> 8, R5C7 <> 6
e) 10(2) <> 9
f) Outies C1234 = 12(3) <> 8,9
g) Outies R6789 = 10(3) <> 4 because R5C5 = (23)
h) ! Innies C1234 = 16(3) = {169/358} because {259} blocked by Killer pair (59)
of 13(2) @ C4 and {268} unplaceable since R45C4 @ 10(3) (Outies R6789) cannot be {26}

5. C456
a) 2 locked in Innies N8 @ C4 for N8 -> Innies N8 = 9(3) = 2{16/34}
b) Outies C1234 = 12(3) = 2{37/46} -> 2 locked for N5
c) Outies R6789 = 10(3) = 3{16/25} -> 3 locked for N5 and R5C4 <> 1 because
6 only possible there
d) Killer pair (36) locked in Innies C1234 + Outies C1234 = 28(6) for N5
e) 12(2): R3C5 <> 9
f) 13(2) @ R5 <> 7

6. R456
a) 12(2) <> 4,8 because (48) is a Killer pair of 13(2)
b) Killer pair (59) locked in 12(2) + 13(2) for R5
c) Innies C1234 = 16(3) must have 3 xor 6 and R5C4 = (36) -> R4C4 <> 3
d) 3 locked in R5C45 @ N5 for R5
e) 12(2) = {57} locked for R5+N4
f) 13(2) = {49} locked for R5
g) 15(3) = 4{29/38} -> 4 locked for R4+N4; R4C23 <> 2,8
h) 2 locked in R4C789 @ R4 for N6
i) 11(3) = {128} because R5C89 = (168) -> R4C9 = 2, {18} locked for R5+N6

7. R456
a) R5C3 = 2 -> R4C23 = 13(2) = {49} locked for R4+N4
b) R5C5 = 3, R5C4 = 6
c) Outies R6789 = 10(3) = {136} -> R4C4 = 1
d) 10(3) @ R3C8 = {37} locked for C8
e) 10(2) @ R6 = {46} locked for R6+N6
f) 19(5) = 123{49/67} -> R6C3 = 3, R6C2 = 1; 2 locked for C4
g) 19(5) = {12349} because R78C4 = (24) -> R7C3 = 9; 4 locked for C4+N8

8. R789
a) 16(3) @ R8C3 = {367} -> R9C4 = 3, R9C3 = 6, R8C3 = 7
b) 16(3) @ R6C1 = {358} -> R6C1 = 8, {35} locked for R7+N7
c) 15(3) = {168} locked for N8 because R7C56 <> 5,9
d) 14(3) = {257} because R7C56 = (168) blocks 5{18} -> R7C8 = 2, R7C9 = 7

9. N125
a) R6C6 = 2, R6C5 = 7, R4C5 = 8 -> R3C5 = 4
b) 9(3) = {126} locked for N2
c) R3C3 = 8 -> R23C2 = 9(2) = {36} locked for C2+N1

10. Rest is singles.

Rating: 1.25.
Walkthrough by Ed:
Yes, it was a really good challenge. I (finally) found a way to make big inroads quickly so by step 8 I was thinking, "yep - hard 1.0 rating". But it didn't give in!

I'll say this one is a hard 1.25 rating. Back to my usual half a rating higher than Afmob! For quite a few puzzles recently, the SSscore is coming in a bit low. Might be time to tweak it up half a rating.

So thanks Nasenbaer for a fine challenge!

BTW - Mike hasn't logged in for over a week. Hope he's OK. [edit: just noticed he went on holidays this time last year. So, hopefully, lots of warm sun now :) ]

Walkthrough for A108 (43 steps)
Please let me know if anything isn't clear or is wrong. Thanks Andrew for the corrections.

Prelims
i. 20(3)n1: no 1 or 2
ii. 8(2)n1: no 4, 8 or 9
iii. 9(3)n2: no 7, 8 or 9
iv. 13(2)n2: no 1, 2 or 3
v. 5(2)n2 = {14/23}
vi. 12(2)n2: no 1, 2 or 6
vii. 10(2)n3: no 5
viii. 12(2)n4: no 1, 2 or 6
ix. 13(2)n5: no 1, 2 or 3
x. 11(3)n6: no 9
xi. 10(2)n6: no 5
xii. 14(2)n8 = {59/68}
xiii. 12(2)n8: no 1, 2 or 6

1. "45" n478: r4c1 - 1 = r9c7
1a. = [54/65] = 5...
1b. no 5 in r4c7 or r9c1
1c. r4c1 = (56) -> r3c1 = (23)
1d. r9c7 = (45) -> r9c6 = (78)

2. "45" n9: r6c9 = r9c7 = (45)
Andrew has pointed out that combining step 1a and step 2 -> more CPEs for 5 in r4c9 and r6c123; of course the 5 in r4c9 is eliminated in step 3 and from r6c123 in step 5. Nice one Andrew! Wish I'd seen that.

3. 11(3)n6 = {128/137/146/236}(no 5) ({245} blocked by r6c9)

4. 12(2)n89 = [75/84] = [5/8..] -> [85] blocked from r5c67 in 13(2)n56

5. r6c9 = 5 (hsingle n6)
5a. r7c89 = 9 = {18/27/36}(no 4 or 9)

6. r9c7 = 5 (step 2)
6a. no 9 in r8c5

7. r4c1 = 6 (step 1)
7a. no 4 in r3c8
7b. r3c1 = 2
7c. no 8 in r4c8

8. "45" n1: r1c4 - 5 = r3c1 -> r1c4 = 7
8a. no 5 in r4c5
8b. no 6 in r23c4 (13(2)n2)
8c. r12c3 = 6 = {15}: both locked for n1 & c3

9. 20(3)n1 = {389/479}(no 6)
9a. = 9{..}: 9 locked for n1
9b. 17(3)n1 = 6{38/47}

10. r9c6 = 7 (cage sum)
10a. no 6 in r5c7

11. 5 in r3 only in n2: 5 locked for n2
11a. no 8 in r3c4
11b. 9(3)n2 = {126/234} = 2{16/34} = [1/3..]
11c. 2 locked for n2
11d. no 3 in r2c7
11e. if 1 is not in 9(3) it must be {234} -> r2c6 = 1
11f. -> 1 locked for n2 in r12c56

12. 1 in r3 only in n3: locked for n3
12a. no 4 in r2c6

13. Killer pair (13) in 9(2)n2 (step 11b) & r2c6: 3 locked for n2
13a. no 9 in r4c5

Looking around now for somewhere to make another big chunk
14. "45" n6: 3 innies r4c78 + r5c7 = h19(3): no 1
14a. no 9 in r3c8
14b. h19(3) = {289/379/478}

15. "45" n8: r789c4 = h9(3) = {126/135/234}(no 89) = [2/5..]
15a. 8 & 9 in n8 must be in 14(2) & 15(3) since they cannot both be in one
15b. 15(3) = {159/168/249/348} ({258} clashes with h9(3) step 15)

16. 15(3)n4 = {249/258/348/357}(no 1) ({159} not possible with 1 & 5 only in r4c2)

17. 1 in n4 only in r6: 1 locked for r6
17a. no 9 in 10(2)n6

18. 9 in n6 only in h19(3)(step 14) = 9{28/37}(no 4) = [2/3..]
18a. no 6 in r3c8
18b. no 9 in r5c6

19. "45" n5: 3 innies = h17(3)
19a. max. r4c5 + r5c6 = 14 -> min. r4c6 = 3
19b. h17(3) = {359/368/458/467}

20. "45" r1234: 4 innies r4c2349 = h16(4) & must have 1 for r4
20a. = 1{249/258/348/357} = [2/3..]

21. hidden Killer pair (23) in h16(4)(step 20) & r4c78 in h19(3)n6 (step 18)
[edit: Discussion with Andrew (by PM) about this one. I think of it as "hidden" because the 2 cages do not completely share a house.]
21a. 3 locked for r4
21b. no 9 in r3c5

22. h17(3)n5 = {458/467}(no 9) = [7/8..]
22a. = 4{58/67}: 4 locked for n5

23. "45" r6789: r6c456 = h18(3) = {279/369}(no 8) ({378} clashes with h17(3)n5 step 22)
23a. 9 locked for n5 & r6

24. split-cage 10(3) at r4c4 + r5c45 must have 1 = {127/136}(no 5, 8)

25. h17(3)n5 must have all of 4, 5 & 8 for n5: hidden triple (no 6, 7)
25a. no 7 in r5c7
25b. no 5 in r3c5
25c. 8 locked for r4
25d. 5 locked for c6

26. r3c4 = 5(hsingle r3)
26a. r2c4 = 8

27. r34c5 = [48]

28. r89c5 = [59] (last permutation)

29. r3c6 = 9 (hsingle n2) -> no 9 elsewhere in 31(5)n3

30. 9(3)n2 = {126} (last combination): 1 locked for n2

31. r2c67 = [32]

32. h19(3)n6 must have 3/7 for r4c7 = {379}
32a. r5c7 = 9, 3 & 7 locked for r4

33. r5c6 = 4 (cage sum) -> r4c6 = 5 (no 5 in r2c8)

34. 12(2)n4 = {57} last combination; 7 locked for r5 & n4

35. r6c5 = 7 (hsingle r6) -> r6c46 = [92] (h18(3)r6)

36. r4c4 = 1, r1c5 = 2 (hsingle n2), r1c8 = 5 (hsingle n3), r8c8 = 9 (hsingle c8)
36a. r78c7 = 4 = {13}: both locked for n9 & c7

37. 19(5)n478 must have 1: only in r6c2

38. 10(2)n6 = {46}: both locked for n6 & r6

39. r4c7 = 7 -> 10(2)n3 = [73]

40. 14(3)n69 = [5][27] (last permutation)

41. naked triple {468} in n3: all locked for n3

Looks like I lost the last few steps.
gary w's solving outline:
Wow!! v1 was MUCH harder than I anticipated given the 0.9 rating.It wasn't until I saw
that following the early placements at r6c9,r9c6,r4c1 etc the outies 17(3) in n5 cannot contain a 1..-> almost immediate contradiction in n6.Also in n4 1 must be in r6 so the 10(3) part of the 28(6) cage n5 must be {136} and the puzzle comes out,slowly,now.

I'ld rate it about 1.5..I really struggled for a long time!
Walkthrough by Andrew:
Thanks Nasenbaer for an enjoyable and challenging Assassin. :)

I'm slowly catching up with them after our holiday.

I found a different multiple nonet in my solution. It gives the same reductions to two-candidate cells as those used by Afmob and Ed but without the CPEs for 5s. However after going through both posted walkthroughs I don't think this made much difference. The key early (or fairly early) breakthrough is to fix R6C9. However after that there's still a lot of work to do but in my walkthrough I'd done a lot of that before I found that breakthrough so the remaining steps were easier.

Ed's step 4 was a clever move. Maybe Mike can tell us what sort of step this is?

I'll also rate A108 as Hard 1.25.

Here is my walkthrough for A108.

Prelims

a) R2C67 = {14/23}
b) R23C4 = {49/58/67}, no 1,2,3
c) R34C1 = {17/26/35}, no 4,8,9
d) R34C5 = {39/48/57}, no 1,2,6
e) R34C8 = {19/28/37/46}, no 5
f) R5C12 = {39/48/57}, no 1,2,6
g) R5C67 = {49/58/67}, no 1,2,3
h) R6C78 = {19/28/37/46}, no 5
i) R89C5 = {59/68}
j) R9C78 = {39/48/57}, no 1,2,6
k) 20(3) cage in N1 = {389/479/569/578}, no 1,2
l) 9(3) cage in N2 = {126/135/234}, no 7,8,9
m) 11(3) cage at R4C9 = {128/137/146/236/245}, no 9

1. 45 rule on N1 1 outie R1C4 = 1 innies R3C1 + 5, R3C1 = {123}, R1C4 = {678}, clean-up: no 1,2,3 in R4C1

2. 45 rule on N9 1 outie R6C9 = 1 innie R9C7, no 1,2,6 in R6C9

3. 45 rule on N2356 2 innies R1C4 + R6C9 = 12 = [75/84], no 6 in R1C4, no 3,7,8,9 in R6C9, clean-up: no 1 in R3C1 (step 1), no 7 in R4C1, R9C7 = {45} (step 2), R9C6 = {78}

4. 11(3) cage at R4C9 = {128/137/146/236} (cannot be {245} which clashes with R6C9), no 5

5. 45 rule on N1 3 innies R12C3 + R3C1 = 8 = {125/134}, 1 locked in R12C3, locked for C3 and N1
5a. R3C1 = {23} -> no 2,3 in R12C3

6. 14(3) cage at R6C9 = {149/158/248/257/347/356} (cannot be {167/239} because R6C9 only contains 4,5)
6a. R6C9 = {45} -> no 4,5 in R7C89

7. 45 rule on C123 4 outies R1789C4 = 16 = {1258/1267/1348/1357/2347} (cannot be {1249/1456/2356} because R1C4 only contains 7,8), no 9
7a. R1C4 = {78} -> no 7,8 in R789C4

8. 9(3) cage in N2 = {126/135/234}
8a. If {234} => R2C67 = [14] -> 1 locked in R12C56, locked for N2
8b. If {234} => R2C67 = [14] -> no 4 in R2C5
[I hope this doesn’t look like T&E, it seems to be the clearest way to look at the combination of the 9(3) and 5(2) cages. I see that Ed also used this in his WT.
If you don’t like this approach, my original move here was 45 rule on N2, 4 innies R1C4 + R2C6 + R3C56 = 23 (as in step 17) when combinations including 1 must have the 1 in R2C6 -> no 1 in R3C6. This doesn’t eliminate 4 from R2C5 but it doesn’t matter because step 26 eliminates 4 from the 9(3) cage.]

9. 45 rule on R6789 3 outies R4C4 + R5C45 = 10 = {127/136/145/235}, no 8,9
9a. 45 rule on C1234 3 innies R456C4 = 16
9b. From steps 9 and 9a, R6C4 = R5C5 + 6, R6C4 = {789}, R5C5 = {123}

10. Killer triple 7,8,9 in R1C4, R23C4 and R6C4, locked for C4

11. Grouped X-Wing on 1 in R12C3 and R12C56, locked for R12, clean-up: no 4 in R2C6
[Alternatively 1 in R3 locked in R3C789, locked for N3.]

12. Killer triple 1,2,3 in 9(3) cage and R2C6, locked for N2, clean-up: no 9 in R4C5

13. Hidden killer triple 4,5,6 in R1789C4, R23C4 and R45C4 -> R45C4 must contain one of 4,5,6
13a. R4C4 + R5C45 (step 9) = {136/235} (cannot be {145} which would contain both of 4,5 in R45C4), no 4, 3 locked for N5, clean-up: no 9 in R3C5

14. Killer pair 5,8 in R34C5 and R89C5, locked for C5

15. 45 rule on N5 3 innies R4C56 + R5C6 = 17 = {179/278/458/467} (cannot be {269} because R4C5 only contains 4,5,7,8)
15a. 1 of {179} must be in R4C6 -> no 9 in R4C6

16. 45 rule on R6789 3 innies R6C456 = 18 = {189/279/468} (cannot be {459} which clashes with R6C9, cannot be {567} which clashes with R4C4 + R5C45), no 5
16a. R4C56 + R5C6 (step 15) = {179/458/467} (cannot be {278} which clashes with R6C456), no 2

17. 45 rule on N2 4 innies R1C4 + R2C6 + R3C56 = 23 = {1589/2489/2678/3479/3578} (cannot be {1679} because 6,9 only in R3C6, cannot be {2579} which clashes with 9(3) cage, cannot be {3569} because R1C4 only contains 7,8, cannot be {4568} because R2C6 only contains 1,2,3)
17a. 9 of {2489/3479} must be in R3C6 -> no 4 in R3C6

18. 45 rule on N4 4 innies R4C1 + R6C123 = 18 = {1269/1359/1368/1458/1467/2358/2367/2457/3456} (cannot be {1278/2349} because R4C1 only contains 5,6)
18a. Only combination containing both of 5,6 is {3456} which must be 5{346} (cannot be 6{345} when R6C123 clashes with R6C9) -> no 5 in R6C123

19. R6C9 = 5 (hidden single in R6), R1C4 = 7 (step 3), R3C1 = 2 (step 1), R4C1 = 6, R9C7 = 5 (step 2), R9C6 = 7, clean-up: no 6 in R23C4, no 4 in R3C8, no 5 in R4C5, no 8 in R4C8, no 1 in R5C5 (step 9b), no 8 in R5C6, no 6 in R5C7, no 9 in R8C5
19a. R3C1 = 2 -> R12C3 = {15} (step 5), locked for C3 and N1

20. 20(3) cage in N1 = {389/479}, no 6, 9 locked for N1
20a. 7 of {479} must be in R2C1 -> no 4 in R2C1

21. R4C4 + R5C45 (step 13a) = {136/235}
21a. 6 of {136} must be in R5C4 -> no 1 in R5C4

22. 1 in R5 locked in R5C89, locked for N6, clean-up: no 9 in R3C8, no 9 in R6C78

23. 11(3) cage at R4C9 (step 4) = {128/137/146}
23a. 4 of {146} must be in R4C9 -> no 4 in R5C89

24. R1C4 + R2C6 + R3C56 (step 17) = {2678/3479/3578} (cannot be {1589/2489} which don’t contain 7), no 1, clean-up: no 4 in R2C7

25. Naked pair {23} in R2C67, locked for R2

26. 9(3) cage in N2 = {126/135} (cannot be {234} which clashes with R2C6), no 4
[Alternatively 1 in N2 locked in 9(3) cage = {126/135}]
26a. 5 of {135} must be in R1C6 -> 3 in R1C6

27. 9 in N6 locked in R4C78 + R5C7
27a. 45 rule on N6 3 remaining innies R4C78 + R5C7 = 19 = {289/379}, no 4, clean-up: no 6 in R3C8, no 9 in R5C6

28. 9 in N5 locked in R6C456, locked for R6
28a. R6C456 (step 16) = {189/279}, no 4,6

29. 4 in N5 locked in R4C56 + R5C6 (step 15) = {458/467}, no 1

30. 15(3) cage in N4 = {249/258/348/357} (cannot be {159} because 1,5 only in R4C2), no 1
30a. 5 of {357} must be in R4C2 -> no 7 in R4C2

31. 1 in N4 locked in R6C12, locked for R6, clean-up: no 8 in R6C46 (step 28a)
31a. R6C456 = [972], R2C67 = [32], R5C5 = 3, clean-up: no 4 in R23C4, no 5 in R45C4 (step 21), no 9 in R5C12, no 3,8 in R6C78
31b. R45C4 = [16], clean-up: no 4 in R4C9 (step 23)

32. Naked pair {58} in R23C4, locked for C4 and N2 -> R3C5 = 4, R4C5 = 8, clean-up: no 6 in R89C5
32a. R89C5 = [59]

33. 9 in N4 locked in 15(3) cage = {249} (only remaining combination, step 30), locked for N4, clean-up: no 8 in R5C12
33a. Naked pair {57} in R5C12, locked for R5 -> R5C6 = 4, R4C6 = 5, R5C7 = 9, R5C3 = 2, clean-up: no 1 in R3C8

34. 19(5) cage at R6C2 must contain 1 -> R6C2 = 1
34a. 19(5) cage = {12349/12367}, no 8 -> R6C3 = 3, R6C1 = 8
34b. R78C4 = {24} -> R7C3 = 9, R4C23 = [94]

35. R1C5 = 2, R3C6 = 9 (hidden singles in N2)
35a. R9C4 = 3 (hidden single in C4)
35b. R89C3 = 13 = [76], R3C3 = 8, R23C4 = [85], clean-up: no 3 in N1 (step 20), no 2 in R4C8
35c. R1C12 = [94], R2C1 = 7, R23C2 = [63], R5C12 = [57], R2C5 = 1, R1C6 = 6, R12C3 = [15], R2C89 = [49], R34C8 = [73], R4C79 = [72], R6C78 = [46], R7C5 = 6

36. 18(3) cage in N9 = {468} (only remaining combination) -> R9C89 = [84], R8C9 = 6

and the rest is naked singles

I'm not sure whether I'll try the V2, certainly not until I've caught up on the V1s and other forum puzzles. I can see that combining those two cages takes away the clean-ups in R6C78 which reduced that cage to a naked pair.


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PostPosted: Mon May 09, 2011 9:29 pm 
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Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 108 V2 by Nasenbaer (June 2008) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:5120:5120:3330:3330:2308:2308:6662:6662:6662:5120:4362:3330:3340:2308:1294:1294:7952:6662:2066:4362:4362:3340:3094:7952:7952:2585:6662:2066:3868:3868:7198:3094:7952:7952:2585:2851:3108:3108:3868:7198:7198:3369:3369:2851:2851:4141:4910:4910:7198:7198:7198:5939:5939:3637:4141:4141:4910:4910:3898:3898:5939:3637:3637:3903:3903:4161:4910:3651:3898:5939:5939:4679:3903:3903:4161:4161:3651:3149:3149:4679:4679:
Solution:
+-------+-------+-------+
| 9 4 1 | 7 2 6 | 8 5 3 |
| 7 6 5 | 8 1 3 | 2 4 9 |
| 2 3 8 | 5 4 9 | 6 7 1 |
+-------+-------+-------+
| 6 9 4 | 1 8 5 | 7 3 2 |
| 5 7 2 | 6 3 4 | 9 1 8 |
| 8 1 3 | 9 7 2 | 4 6 5 |
+-------+-------+-------+
| 3 5 9 | 4 6 8 | 1 2 7 |
| 4 8 7 | 2 5 1 | 3 9 6 |
| 1 2 6 | 3 9 7 | 5 8 4 |
+-------+-------+-------+
Quote:
Nasenbaer: I wanted to post v2 a little earlier but sadly the forum was down. Too much celebration, heh? ;)
For this version I combined the cages n6:10(2) and n9:13(3) to one cage n69:23(5) - small change, big difference (SSolver(3.2.0) rating: 1.6). The solution is the same.

udosuk: This is my walkthrough for v2: :king:
Had to employ some powerful moves to crack it up, but very enjoyable. :ugeek:
BTW congrats Ed for you 100th post. The bulk of your posts have been great archival work for this community. Keep up the great work! :thumbs:

Afmob: This was a true V2! Similar to V1, it took me quite some time to find the important moves ...
Rating: 1.75.

Andrew, in 2010: Another variant from my backlog of unfinished puzzles. Thanks Nasenbaer for a challenging variant.
A108 V2 was a very hard puzzle. However Afmob, udosuk and I managed to solve it with different solving paths ...
I found it hard to decide what rating to give my walkthrough. I'll rate it at 1.75, the same rating that Afmob gave for his walkthrough, because I think our solving paths were of similar difficulty.

Walkthrough by udosuk:
This is my walkthrough for v2: :king:

Assassin 108 v2 walkthrough: (9 major steps)


0. Preliminary work

5/2 @ r2c6={14|23}
8/2 @ r3c1={17|26|35}
12/2 @ r3c5,r5c1,r9c6={39|48|57}
13/2 @ r2c4,r5c6={49|58|67}
14/2 @ r8c5={59|68}
9/3 @ r1c5={126|135|234}


1. Gentle crackdown #1: n1, n478

Innies @ n478: r4c1+r9c6=13=[58|67]
=> r34c1=[26|35], r9c67=[75|84]
Outies @ n1: r1c4+r4c1=13=[85|76]
=> r1c4=r9c6=7|8
Innies @ n1: r12c3+r3c1=8=[{14}3|{15}2]
=> 1 @ c3,n1 locked @ r12c3


2. Gentle crackdown #2: n5, c4

Innies @ c1234: r456c4=16
Outies @ r6789: r45c4+r5c5=10
=> r6c4-r5c5=16-10=6 => r5c5+r6c6=[17|28|39]
=> r1c4,r23c4,r6c4 form killer NT {789} @ c4
=> r45c4+r5c5=10=[{16|25}3|{35}2|{36|45}1]
But 13/2 @ r2c4 has 4|5|7
=> r456c4=16 can't be [{45}7]
=> r45c4+r5c5 can't be [{45}1]
=> r45c4+r5c5={136|235} with 3 @ n5 locked
=> 12/2 @ r3c5=[39|48|57|75|84]


3. Gentle crackdown #3: n2, c5

Innies @ n2: r1c4+r2c6+r3c56=23
Max r1c4+r3c5=7+8=15 => Min r23c6=23-15=8
r2c6 from {1234} => r3c6 from {56789}
=> 1 @ r3,n3 locked @ r3c789
=> 5/2 @ r2c6=[14|23|32]
Max r1c4+r3c6=8+9=17 => Min r2c6+r3c5=6
=> r2c6 from {123} => r3c5 from {4578}
=> 12/2 @ r3c5={48|57}
=> r34c5,r89c5 form killer NP {58} @ c5


4. Powerful hammer #1: 28/6

Now consider 28/6 @ r4c4
r5c5+r6c4=[17|28|39], r45c4+r5c5=10
=> r6c456=28-10=18 => r45c4+r5c5+r6c456=
[{36}17{29}|{35}28{19|46}|{16}39{27|45}|{25}3918]
=> Either r45c4 has 6 or r6c46 has 8
=> Either r45c4 has 6 or r1c4=r9c6=7
=> 13/2 @ r2c4 can't be {67}
=> 13/2 @ r2c4 must be {49|58} with 5|9
=> r45c4+r5c5+r6c456 can't be [{25}3918], must be
[{36}17{29}|{35}28{19|46}|{16}39{27|45}]


5. Powerful hammer #2: c4

Now consider c4
r456c4=[{36}7|{35}8|{16}9] => r123456789c4=
[8{49}{36}7{125}|7{49}{35}8{126}|7{58}{16}9{234}]
=> Either r1c4=8 or r789c4={126} or r123c4=[7{58}]
=> Either r9c6=8 or r789c4={126} or 12/2 @ r3c5=[48]
=> 14/2 @ r8c5 can't be {68}, must be {59} (NP @ c5,n8)


6. Mop up #1: c5, n2

Now 12/2 @ r3c5={48} (NP @ c5)
=> r23c4,r3c5 form killer NP {48} @ n2
=> r1c4=7 => 12/2 @ r9c6=[75], 8/2 @ r3c1=[26]
=> 14/2 @ r8c5=[59], r12c3=13-7=6={15} (NP @ c3,n1)


7. Mop up #2: n5, n2

Innies @ n5: r4c56+r5c6=17 from {1245689}
r4c5 from {48} => r4c56+r5c6={458} (NT @ n5)
=> r6c4=9 => r5c5=3 => r4c45=[16], r6c56=[72]
=> 13/2 @ r2c4={58} (NP @ n2) => 12/2 @ r3c5=[48]
=> r23c6=23-7-4=12=[39] => r2c7=5-3=2


8. Mop up #3: n456

12/2 @ r5c1={48|57} has 5|8
=> 13/2 @ r5c6=[49] => r4c6=5
=> 12/2 @ r5c1={57} (NP @ r5,n4) => r5c89 from {128}
=> 11/3 @ r4c9=[2{18}] => r5c89={18} (NP @ r5,n6)
=> r5c3=2 => r4c23=15-2=13={49} (NP @ r4,n4)


9. Mop up #4: endgame

19/5 @ r6c2 from {12346789} can't be {12358}
=> it can't have 8 => r6c123=[813]
=> r78c4={24} (NP @ n8) => r7c3=19-1-3-2-4=9
=> r4c23=[94], r9c4=3 => r89c3=16-3=13=[76]
=> r3c3=8 => r23c2=17-8=9=[63] => r7c12=16-8=8=[35]
=> 10/2 @ r3c8=[73] => r4c7=7 => r2c8+r3c7=31-9-5-7=10=[46]


The rest is trivial.

Had to employ some powerful moves to crack it up, but very enjoyable. :ugeek:

BTW congrats Ed for you 100th post. The bulk of your posts have been great archival work for this community. Keep up the great work! :thumbs:
Walkthrough by Afmob:
This was a true V2! Similar to V1, it took me quite some time to find the important moves and I also had to use some chains to crack it.

A108 V2 Walkthrough:

1. N1478
a) Innies N478 = 13(1+1) = [58/67]
b) 8(2): R3C1 = (23)
c) Innies N1 = 8(3) = 1{25/34} -> 1 locked for C3+N1; R12C3 <> 2,3
d) Outies N1 = 13(2) = [76/85]
e) 12(2) @ N8: R9C7 = (45)
f) Innies+Outies N47: 3 = R789C4 - R4C1; R4C1 = (56)
-> R789C4 = 8/9(3) <> 7,8,9

2. C456 !
a) Outies R6789 = 10(3) <> 8,9
b) Using Outies R6789: Innies+Outies C1234: -6 = R5C5 - R6C4
-> R5C5 = (123), R6C4 = (789)
c) Killer triple (789) locked in R16C4+13(2) @ C4 for C4
d) ! Hidden Killer triple (456) in R45C4 for C4 because each of 13(2) @ C4
and R789C4 (step 1f) must have exactly one of (456)
e) Outies R6789 = 10(3) = 3{16/25} because R45C4 can only have one of (456)
-> 3 locked for N5
f) 12(2): R3C5 <> 9

3. N4
a) Hidden Killer triple (789) in R6C123 for N4 since 12(2) @ N4 + 15(3) can only have one of (789)
b) Innies N4 = 18(4) must have one of (789)
-> R4C1+R6C123 = 5+{139/148/238/247} / 6+{129/138/147/237}
-> R6C123 <> 5,6 and R6C123 must have two of (1234)

4. R123
a) Innies N2 = 23(4): R3C6 <> 1,2,3,4 because R1C4+R3C5 <= 15 and R2C6 <= 4
b) 1 locked in R3C789 @ R3 for N3
c) 5(2): R2C6 <> 4
d) Killer triple (123) locked in 9(3) + R2C6 for N2
e) 12(2) <> 9

5. C456
a) Killer pair (58) locked in 12(2) + 14(2) for C5
b) 28(6) must have one of (89) and it's only possible @ R6C456
c) Innies R6789 = 18(3) <> {567} because it has no 8,9

6. C456 !
a) ! Innies N8 = 16(4) <> 5 because:
-{125}+8 blocked by Killer pair (58) of 14(2)
-{135}+7 blocked by R45C4 since R45C4 @ 10(3) (Outies R6789) cannot be {26}
-> Innies N8 = {134}+8 / {126/234}+7
b) Innies C1234 = 16(3) <> 7 because {367} blocked by Killer pair (36) of Innies N8
c) Innies+Outies C1234: -6 = R5C5 - R6C4 -> R5C5 <> 1
d) 7 locked in R123C4 @ C4 for N2
e) 12(2): R4C5 <> 5
f) 15(3) = {159/168/267/348} because 5{28/46} blocked by Killer pairs (56,58) of 14(2)
and {249/357} blocked by Killer pairs (24,37) of Innies N8
g) ! 15(3): R7C5 <> 3 because it forces Innies C6789 = 17(4) = {2348} but at the
same time R5C5 = 2 -> R6C6 <> 2,3 -> no placement for Innies C6789
h) 15(3): R78C6 <> 4 because R7C5 <> 3,8
i) 9(3): R1C6 <> 4 because R5C5 = (23) blocks R12C5 = {23}

7. C456+N6 !
a) 4 locked in R456C6 @ C6 for N5
b) 12(2): R3C5 <> 8
c) Innies R6789 = 18(3) <> 5 because R6C45 <> 4,5; Innies R6789 = {189/279/468}
d) 5 locked in R6C789 @ R6 for N6
e) Innies N5 = 17(3) <> 2 because {278} blocked by Killer pair (78) of Innies R6789 and R4C5 = (78)
f) Killer pair (78) locked in R4C5+Innies R6789 for N5
g) 13(2) @ R5: R5C7 <> 6
h) ! Innies R6789 = 18(3) <> 4,6 because [864] forces R4C5 = 7 -> R3C5 = 5
but at the same time R6C5 = 6 -> 14(2) = {59} -> two 5's in C5
i) Innies R6789 = 18(3) = 9{18/27} -> 9 locked for R6+N5
j) 13(2) @ R5: R5C7 <> 4

8. R456+N9 !
a) 6 locked in R6C789 @ R6 for N6
b) 11(3) = 1{28/37} -> 1 locked for N6
c) Innies N5 = 17(3) <> 1
d) 1 locked in Innies R1234 = 16(4) = 1{249/258/267/348/357} because 1 locked
there @ R4 and R4C1 = (56) blocks {1456}
e) ! Killer triple (456) locked in R4C16+Innies R1234 for R4
f) Hidden triple (456) locked in R6C789 @ N6 for R6; R6C789 = (456)
g) Innies+Outies N9: 10 = R6C789 - R9C7 -> R9C7 = 5
h) Cage sum: R9C6 = 7

9. C123+N8
a) Innies N478 = 13(1+1) = [67] -> R4C1 = 6
b) Cage sum: R3C1 = 2
c) Innies N1 = 8(3) = {125} -> 5 locked for C3+N1
d) 13(2) = {157} -> R1C4 = 7
e) 15(3) @ N4 <> 1 because R45C3 <> 1,5
f) 1 locked in R6C12 @ N4 for R6

10. C456
a) Innies R6789 = 18(3) = {279} -> R6C4 = 9, R6C6 = 2, R6C5 = 7
b) 19(5) = 123{49/67} -> R6C3 = 3, R6C2 = 1
c) 28(6) = {123679} -> R5C5 = 3, R4C4 = 1, R5C4 = 6
d) 19(5) = {12349} because R78C4 = (24) -> R7C3 = 9, {24} locked for C4+N8
e) 13(2) = {58} @ N2 locked for N2
f) R3C5 = 4
g) 9(3) = {126} locked for N2
h) R9C4 = 3 -> R89C3 = 13(2) = {67} -> R9C3 = 6, R8C3 = 7

11. C123
a) 16(3) = {358} -> R6C1 = 8, {35} locked for R7+N7
b) R3C3 = 8 -> R23C2 = 9(2) = {36} locked for C2+N1
c) 15(3) = {249} because R45C3 = (24) -> R4C2 = 9, 4 locked for N4

12. N3
a) 10(2) = {37} locked for C8
b) R2C6 = 3 -> R2C7 = 2, R3C6 = 9

13. Rest is singles.

Rating: 1.75. I used combo analysis, Killer triples and two small contradiction chains.
Walkthrough by Andrew, in 2010:
Another variant from my backlog of unfinished puzzles. Thanks Nasenbaer for a challenging variant.

A108 V2 was a very hard puzzle. However Afmob, udosuk and I managed to solve it with different solving paths; udosuk used heavy combination analysis in conjunction with a "clone", Afmob and I both used short contradiction moves but in different parts of the grid.

Rating Comment. I found it hard to decide what rating to give my walkthrough. I'll rate it at 1.75, the same rating that Afmob gave for his walkthrough, because I think our solving paths were of similar difficulty. I used a short contradiction move, combination analysis and a "ruudiculous" combined cage.

Here is my walkthrough for A108 V2.

Prelims

a) R2C67 = {14/23}
b) R23C4 = {49/58/67}, no 1,2,3
c) R34C1 = {17/26/35}, no 4,8,9
d) R34C5 = {39/48/57}, no 1,2,6
e) R34C8 = {19/28/37/46}, no 5
f) R5C12 = {39/48/57}, no 1,2,6
g) R5C67 = {49/58/67}, no 1,2,3
h) R89C5 = {59/68}
i) R9C78 = {39/48/57}, no 1,2,6
j) 20(3) cage in N1 = {389/479/569/578}, no 1,2
k) 9(3) cage in N2 = {126/135/234}, no 7,8,9
l) 11(3) cage in N6 = {128/137/146/236/245}, no 9

1. 45 rule on N1 3 innies R12C3 + R3C1 = 8 = {125/134}, 1 locked for N1, clean-up: no 1,2 in R4C1
1a. 17(3) cage in N1 = {269/278/368/467} (cannot be {359/458} which clash with R12C3 + R3C1), no 5

2. 45 rule on N1 2 outies R1C4 + R4C1 = 13 = [67/76/85], clean-up: no 5 in R3C1

3. 45 rule on R6789 3 outies R45C4 + R5C5 = 10 = {127/136/145/235}, no 8,9
3a. 45 rule on C1234 3 innies R456C4 = 16
3b. -> R6C4 = R5C5 + 6, R6C4 = {789}, R5C5 = {123}

4. 45 rule on N478 1 innie R4C1 = 1 outie R9C7 + 1, no 7 in R4C1, R9C7 = {45}, clean-up: no 1 in R3C1, no 6 in R1C4 (step 2), R9C6 = {78}
4a. R4C1 + R9C7 = [54/65], CPE no 5 in R4C7 + R9C1
4b. Killer triple 7,8,9 in R1C4, R23C4 and R6C4, locked for C4

5. R12C3 + R3C1 (step 1) = {125/134}, 1 locked for C3
5a. R3C1 = {23} -> no 2,3 in R12C3
5b. 1 in N4 only in R4C2 + R6C12, CPE no 1 in R7C2

6. Max 9(3) cage in N2 + R2C6 = 13(4) must contain 1, locked for N2
6a. 1 in R3 only in R3C789, locked for N3, clean-up: no 4 in R2C6
6b. Killer triple 1,2,3 in 9(3) cage and R2C6, locked for N2, clean-up: no 9 in R4C5

7. 45 rule on C123 4 outies R1789C4 = 16 = {1258/1267/1348/1357/2347} (cannot be {1456/2356} because R1C4 must contain one of 7,8)
7a. Hidden killer triple 1,2,3 in R45C4 and R1789C4 for C4, R1789C4 contains two of 1,2,3 -> R45C4 must contain one of 1,2,3
7b. R45C4 + R5C5 (step 3) = {136/235} (cannot be {145} which doesn’t contain one of 1,2,3 in R45C4), no 4, 3 locked for N5, clean-up: no 9 in R3C5

8. Killer pair 5,8 in R34C5 and R89C5, locked for C5

9. 45 rule on N5 3 innies R4C56 + R5C6 = 17 = {179/278/458/467} (cannot be {269} because no 2,6,9 in R4C5)
9a. 1 of {179} must be in R4C6 -> no 9 in R4C6

10. 45 rule on N4 4 innies R4C1 + R6C123 = 18 = {1269/1359/1368/1458/1467/2367/2457} (cannot be {1278/2349} because R4C1 only contains 5,6, cannot be {2358/3456} which clash with R5C12)
10a. R4C1 = {56} -> no 5,6 in R6C123

11. 45 rule on N2 4 innies R1C4 + R2C6 + R3C56 = 23 = {1589/2489/2678/3479/3578} (cannot be {1679} because 6,9 only in R3C6, cannot be {2579/4568} which clash with R23C4, cannot be {3569} because R1C4 only contains 7,8)
11a. 9 of {2489/3479} must be in R3C6 -> no 4 in R3C6

12. 45 rule on R12 3 innies R2C248 = 1 outie R3C9 + 17
12a. Max R2C248 = 24 -> max R3C9 = 7

13. 13(3) cage at R1C3 = {148/157}
13a. 20(3) cage in N1 = {389/479/569/578}
13b. 7 of {479} must be in R2C1 (cannot be {47}9/{79}4 which clash with 13(3) cage) -> no 4 in R2C1

14. 45 rule on N69 4(3+1) innies R4C78 + R5C7 + R9C7 = 24
14a. Max R9C7 = 5 -> min R4C78 + R5C7 = 19, no 1 in R4C78, clean-up: no 9 in R3C8

15. R2C67 cannot be [14], here’s how
R2C67 = [14] => R2C3 = 5, R1C34 = [17] (step 13) is impossible because no combinations for R1C4 + R2C6 + R3C56 (step 11) contain both 1 and 7
15a. -> R2C67 = {23}, locked for R2
15b. 1 in N2 only in 9(3) cage at R1C5 = {126/135}, no 4
15c. 5 of {135} must be in R1C6 -> no 3 in R1C6
15d. 2 in N1 only in R3C123, locked for R3, clean-up: no 8 in R4C8

16. R12C3 + R3C1 (step 1) = {125/134} -> R12C3 + R34C1 = {14}[35]/{15}[26], CPE no 5 in R12C1 + R45C3
16a. 20(3) cage in N1 = {389/479/569/578}
16b. 5 of {569} must be in R1C2 -> no 6 in R1C2
16c. 7 of {479} must be in R2C1 (step 13b), 5 of {578} must be in R1C2 -> no 7 in R1C2

[At this stage I found two forcing chains but have omitted them after finding step 17a, which took me a long time to see. One chain has been replaced by steps 17a and 18; the other one by step 21.]

17. 45 rule on R1234 4 innies R4C2349 = 16 = {1249/1258/1267/1348/1357/2347} (cannot be {1456/2356} which clash with R4C1)
17a. R4C56 + R5C6 (step 9) = {278/458/467} (cannot be {179} = [719] which clashes with R4C2349), no 1,9, clean-up: no 4 in R5C7

18. 9 in N5 only in R6C456, locked for R6
18a. 45 rule on R6789 3 innies R6C456 = 18 = {189/279/459}, no 6
18b. 5 of {459} must be in R6C6 -> no 4 in R6C6
18c. 6 in R6 only in R6C789, locked for N6, clean-up: no 4 in R3C8, no 7 in R5C6

19. 15(3) cage in N4 = {168/249/258/267/348/357} (cannot be {159} because 1,5 only in R2C4, cannot be {456} which clashes with R4C1)
19a. R4C1 + R6C123 (step 10) = {1368/1458/2457} (cannot be {1467} which clashes with R6C456, cannot be {2367} which clashes with 15(4) cage)
19b. 15(3) cage = {168/249/267} (cannot be {258/348/357} which clash with R4C1 + R6C123), no 3,5
19c. 1 of {168} must be in R4C2 -> no 8 in R4C2

20. R5C12 = {39/57} (cannot be {48} which clashes with R4C1 + R6C123), no 4,8

21. R4C1 + R6C123 (step 19a) = {1368/1458/2457}, R6C456 (step18a) = {189/279/459} -> combined cage R4C1 + R6C123456 = 6{138}{279}/6{138}{459}/5{148}{279}/5{247}{189}, 1,8 locked for R6
[I wondered whether I could use a killer pair or hidden killer pair but decided that I couldn’t because 1,8 must be in R6C123 or in R6C456.]
21a. 1 in N6 only in 11(3) cage in N6 = {128/137}, no 4,5
21b. R6C789 (from combinations for combined cage R4C1 + R6C123456) = {356/456} (cannot be {267} which clashes with 11(3) cage) -> no 2,7 in R6C789, 5 locked for R6 and N6, clean-up: no 8 in R5C6, no 4 in R6C5 (step 18a)

22. 4 in N5 only in R4C56 + R5C6 (step 17a) = {458/467}, no 2

23. 4 in R5 only in R5C36 -> 15(3) cage in N4 = {249} or R5C67 = [49], CPE no 9 in R4C7

24. 9 in N6 only in R4C8 + R5C7
-> R4C78 + R5C7 + R9C7 = 24 (step 14) = {289/379}5/{389/479}4
24a. 4 of {479}4 must be in R4C8 + R9C7 -> no 4 in R4C7

25. 1 in R4 only in R4C249
25a. R4C2349 (step 17) = {1249/1258/1267} (cannot be {1348} because R4C23 in 15(3) cage cannot be {14/48}, cannot be {1357} because R4C23 in 15(3) cage cannot contain both of 1,7), no 3
25b. 8 of {1258} must be in R4C3 (R4C23 cannot contain both of 1,2) -> no 8 in R4C9

[The puzzle is now cracked.]

26. 3 in N5 only in R5C45, locked for R5, clean-up: no 7 in 11(3) cage in N6 (step 21a), no 9 in R5C12
26a. Naked pair {57} in R5C12, locked for R5 and N4 -> R4C1 = 6, R3C1 = 2, R9C7 = 5 (step 5), R9C6 = 7, R1C4 = 7 (step 2), clean-up: no 4 in R12C3 (step 13), no 6 in R23C4, no 5 in R4C5, no 6 in R5C6, no 8 in R5C7

27. R5C67 = [49], clean-up: no 8 in R3C5, no 1 in R3C8

28. Naked triple {128} in R5C389, locked for R5, 1 also locked for N6 -> R4C9 = 2, R5C45 = [63], R4C4 = 1 (step 3), clean-up: no 8 in R3C8
28a. Naked pair {18} in R5C89, locked for R5 and N6 -> R5C3 = 2

29. R6C456 (step 18a) = {279} (only remaining combination) -> R6C4 = 9, R6C6 = 2, R6C5 = 7, R4C5 = 8, R3C5 = 4, R4C6 = 5, R2C67 = [32], clean-up: no 6 in R89C5
29a. R89C5 = [59]

30. Naked pair {49} in R4C23, locked for R4 and N4, clean-up: no 6 in R3C8
30a. Naked pair {37} in R4C78, locked for N6
30b. Naked pair {37} in R34C8, locked for C8

31. R3C6 = 9 (hidden single in C6)
31a. R34C6 = [95] -> 31(5) cage at R2C8 = {35689/45679}, no 1, 6 locked for N3
31b. R4C7 = {37} -> no 3,7 in R3C7

32. R3C9 = 1 (hidden single in R3), R5C89 = [18]

33. Naked pair {15} in R12C3, locked for C3 and N1

34. 19(5) cage at R6C2 = {12349} (only remaining combination, cannot be {12367} because 6,7 only in R7C3) -> R6C2 = 1, R6C3 = 3, R7C3 = 9, R78C4 = {24}, locked for N8 -> R9C4 = 3, R4C23 = [94], R6C1 = 8

34. R9C4 = 3 -> R89C3 = 13 = [76], R9C9 = 4, R9C1 = 1

35. R6C1 = 8 -> R7C12 = 8 = {35}, locked for R7 and N7 -> R8C1 = 4, R78C4 = [42], R89C2 = [82]

36. 20(3) cage in N1 = {479} (only remaining combination) -> R1C1 = 9, R1C2 = 4

and the rest is naked singles.


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