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PostPosted: Wed May 25, 2011 3:21 am 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
This is Part F of the Assassin Forum Archive, continuing to include puzzles posted on this site and provide easy links to the puzzles and threads. Please read the first part of the Archive Index to get the background to this archive including Mike (mhparker)'s original post about ratings.

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

Abbreviations used in Rating Table:
Est = Estimated rating by puzzle maker
E = Easy
H = Hard
U = udosuk
Score = SudokuSolver v3.3 score, rounded to nearest 0.05
** in the Afmob column indicates that these puzzles were made by him,
for these ones the estimate is his rating.
+--------------------------------+-----------+-------+-------+-----------------------+-------+
| Puzzle Est | Made By | Afmob | Andrew| Other Raters | Score |
+--------------------------------+-----------+-------+-------+-----------------------+-------+
| Assassin 151 E1.25 | Ed | E1.00 | H1.00 | | 1.35 |
| Assassin 151V2 | Frank | H1.50 | H1.50 | | 1.75 |
| Assassin 152 1.25 | Afmob | ** | H1.25 | (Ed) E1.50 | 1.25 |
| JF"F"K6 | manu | H1.25 | 1.50 | | 2.75 |
| Assassin 153 | manu | H1.00 | 1.25 | | 1.25 |
| Assassin 153V2 | manu | | 1.50 | | 2.35 |
| Assassin 154 | Ronnie G | H1.00 | E1.25 | | 1.35 |
| Assassin 154V1.5 | Ronnie G | 1.50 | 1.50 | | 1.60 |
| Assassin 155 H1.00 | Afmob | ** | H1.25 | | 1.10 |
+--------------------------------+-----------+-------+-------+-----------------------+-------+
Page #1

+--------------------------------+-----------+-------+-------+-----------------------+-------+
| Puzzle Est | Made By | Afmob | Andrew| Other Raters | Score |
+--------------------------------+-----------+-------+-------+-----------------------+-------+
| Assassin 156 | manu | H1.00 | 1.00 | | 1.30 |
| Assassin 156V2 | manu | H1.00 | E1.25 | | 2.35 |
| Assassin 157 | Ed | 1.25 | E1.25 | | 1.30 |
| Assassin 158 H1.25 | Afmob | ** | 1.25 | | 1.50 |
| Assassin 159 | manu | H1.00 | 1.25 | | 1.35 |
| Human Solvable 4X | HATMAN | | 1.50 | | 2.00 |
| Assassin 160 | Ronnie G | H1.25 | H1.25 | | 1.25 |
| Assassin 161X | HATMAN | H1.00 | 1.25 | (U) 0.75 | 1.70 |
| JFFK7 1.50 | manu | | 1.50 | Solved as a "tag" | 2.10 |
+--------------------------------+-----------+-------+-------+-----------------------+-------+
Page #2

+--------------------------------+-----------+-------+-------+-----------------------+-------+
| Puzzle Est | Made By | Afmob | Andrew| Other Raters | Score |
+--------------------------------+-----------+-------+-------+-----------------------+-------+
| Assassin 162 | Ed | H1.00 | 1.25 | | 1.30 |
| Human Solvable 5X | HATMAN | | 1.25 | | DNF |
| Assassin 163 1.00 | Afmob | ** | E1.25 | (U) 1.00 | 1.10 |
| Assassin 163V2 1.75 | Afmob | ** | 1.25 | (U) 1.25 | 2.00 |
| Assassin 164 | manu | H1.00 | H1.00 | | 1.30 |
| Assassin 164V1.5 | manu | H1.00 | E1.25 | | 1.60 |
| Assassin 165 | Ronnie G | H1.00 | 1.25 | | 1.30 |
| Assassin 165V2 | Ronnie G | E1.25 | E1.25 | | 1.75 |
| Assassin 166 1.25 | Afmob | ** | 1.25 | | 1.50 |
+--------------------------------+-----------+-------+-------+-----------------------+-------+
Page #3

+--------------------------------+-----------+-------+-------+-----------------------+-------+
| Puzzle | Made By | Afmob | Andrew| Other Raters | Score |
+--------------------------------+-----------+-------+-------+-----------------------+-------+
| Assassin 167 | Ed | | E1.25 | | 0.95 |
| JFFK8 | manu | | H1.25 | | 1.45
| Assassin 168 | manu | 1.25 | 1.25 | | 1.60 |
| Assassin 169 | Ronnie G | H1.25 | H1.25 | | 1.45 |
| Assassin 170 | Ed | | H1.00 | | 1.30 |
| Assassin 170V1.5 | Ed | 1.50 | 1.50 | | 1.60 |
| Assassin 171 | manu | H1.00 | 1.25 | | 1.25 |
| Assassin 171 Alt | manu | | H1.25 | | 1.35 |
| Assassin 171V2.5 | manu | H1.00 | H1.25 | | 2.50 |
+--------------------------------+-----------+-------+-------+-----------------------+-------+
Page #4

+--------------------------------+-----------+-------+-------+-----------------------+-------+
| Puzzle | Made By | Afmob | Andrew| Other Raters | Score |
+--------------------------------+-----------+-------+-------+-----------------------+-------+
| Assassin 172 | tarek | | 1.25 | | 1.45 |
| Assassin 173 | Ronnie G | E1.25 | 1.25 | | 1.45 |
| Assassin 174 | manu | 1.25 | 1.25 | | 1.50 |
| Assassin 175 | udosuk | H1.50 | 1.50 | | 2.80 |
| Assassin 175V0.75 | udosuk | | E1.50 | | 3.35 |
| Assassin 175V0.25 | udosuk | | 1.25 | | 1.35 |
| Assassin 175V0.5 | udosuk | | E1.50 | | 2.05 |
| Assassin 175V2 | udosuk | H1.50 | 1.50 | | 3.20 |
| Assassin 175V3 | udosuk | E1.75 | E1.75 | | 3.45 |
+--------------------------------+-----------+-------+-------+-----------------------+-------+
Assassin 175 V2 and Assassin 175 V3 are in the same archive entry.
Other Images in the Assassin 175 thread and (often light-hearted) discussion about them are here
Page #5

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.

Many thanks to Ed for providing the format for the rating tables, including links to the puzzle threads; I think it was a great idea that in the rating tables he provided separate columns for Afmob and myself, the most regular posters of walkthroughs. Thanks also to Børge for generating so many diagrams, which are also in the Images with "udosuk Style Killer Cages" thread, and for providing links from diagrams in that thread to the archive entries.

If you solve any of these puzzles and decide that there aren't enough walkthroughs for that puzzle or, better still, that you've found an interesting way to solve it which hasn't been posted, please feel free to post your walkthrough in the puzzle thread. Your comments and walkthrough will then be added to this archive.


Last edited by Andrew on Mon Jan 14, 2013 9:15 pm, edited 9 times in total.

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PostPosted: Thu Aug 04, 2011 5:02 am 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 151 by Ed (April 2009) here
Puzzle Diagram:
Image
NOTE: r5c5 is intentionally blank.
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:4097:4097:4610:2563:2563:2563:5892:4869:4869:4097:4614:4610:4610:5383:5892:5892:4616:4869:4617:4617:4614:4610:5383:5892:4616:3338:4869:3339:4617:4617:4614:5383:4616:3338:3338:3852:3339:4621:4621:4621:0000:2830:2830:2830:3852:3339:5903:5903:4368:2577:2322:3347:3347:3852:1812:5903:4368:3093:2577:5910:2322:3347:6935:1812:4368:3093:3093:2577:5910:5910:2322:6935:3608:3608:3608:3097:3097:3097:6935:6935:6935:
Solution:
+-------+-------+-------+
| 8 2 7 | 5 1 4 | 9 6 3 |
| 6 1 3 | 2 9 7 | 4 5 8 |
| 4 5 9 | 6 8 3 | 7 1 2 |
+-------+-------+-------+
| 1 7 2 | 8 4 6 | 3 9 5 |
| 3 4 5 | 9 7 2 | 1 8 6 |
| 9 6 8 | 3 5 1 | 2 7 4 |
+-------+-------+-------+
| 2 9 6 | 7 3 8 | 5 4 1 |
| 5 8 1 | 4 2 9 | 6 3 7 |
| 7 3 4 | 1 6 5 | 8 2 9 |
+-------+-------+-------+
Quote:
Ed: This one seems straightforward after recent Assassins but hopefully still passes the interesting test. I'll give it an Easy 1.25 rating. This cage pattern started out as an X, but this puzzle is not. I'm not planning a V2 - though started one with a couple of cages joined together. Someone else feel free!
SS(3.3.0)score = 1.36.
An alternate score - could be a spoiler or hint With r5c5 as a single cell cage the SSscore = 0.93.

Andrew: Thanks Ed for a straightforward Assassin. It was fun. It will be a good one for newbies to do once they have solved the puzzles recommended in my Advice "sticky".
Nice cage pattern! I like the way R5C5 was left blank to maintain the central symmetry.
This was the quickest I've solved an Assassin since the first dozen of Ruud's ones. After finding fairly early breakthroughs it flowed nicely. I'm sure there must be a lot of ways to solve this one; Afmob will probably find a more direct solution.
I'll rate A151 as Hard 1.0.

Afmob: Don't put so much pressure on me, Andrew! :lol:
Thanks for this nice Killer, Ed! It was fun to solve ...
Rating: Easy 1.0.

Joe Casey: For me the biggest difficulty was remembering that it wasn't a diagonal. You get into an awful mess if you think it is.

(Further comments by Andrew after the walkthroughs)

Walkthrough by Andrew:
Thanks Ed for a straightforward Assassin. It was fun. It will be a good one for newbies to do once they have solved the puzzles recommended in my Advice "sticky".

Nice cage pattern! I like the way R5C5 was left blank to maintain the central symmetry.

This was the quickest I've solved an Assassin since the first dozen of Ruud's ones. After finding fairly early breakthroughs it flowed nicely. I'm sure there must be a lot of ways to solve this one; Afmob will probably find a more direct solution.

I'll rate A151 as Hard 1.0 because that's the rating that I give for easy killer triples; others would rate it Easy 1.25 because they never rate killer triples lower than that.

Here is my walkthrough. Looks like I was careless with my manual eliminations first time through. I've done minor editing, moving R78C4 to step 24.

Prelims

a) R78C1 = {16/25/34}, no 7,8,9
b) R1C456 = {127/136/145/235}, no 8,9
c) R234C5 = {489/579/678}, no 1,2,3
d) R5C678 = {128/137/146/236/245}, no 9
e) 23(3) cage at R6C2 = {689}, CPE no 6,8,9 in R45C2
f) R678C5 = {127/136/145/235}, no 8,9
g) 9(3) cage at R6C6 = {126/135/234}, no 7,8,9
h) 23(3) cage at R7C6 = {689}, CPE no 6,8,9 in R8C45

1. 45 rule on complete grid 1 innie R5C5 = 7

2. 45 rule on C5 2 remaining innies R19C5 = 7 = {16/25/34}, no 8,9

3. 8,9 in C5 locked in R234C5 = {489}, locked for C5, clean-up: no 3 in R19C5 (step 2)

4. 45 rule on R1234 2 innies R4C19 = 6 = {15/24}

5. 45 rule on R6789 2 innies R6C19 = 13 = {49/58/67}, no 1,2,3
5a. Killer triple 6,8,9 in R6C19 and R6C23, locked for R6

6. 45 rule on R5 2 remaining innies R5C19 = 9 = {18/36/45}, no 2,9
6a. 9 in R5 locked in R5C234 = {189/369/459}, no 2
6b. 1,3 of {189/369} must be in R5C2 -> no 1,3 in R5C34

7. 45 rule on R9 2 outies R78C9 = 8 = {17/26/35}
7a. 45 rule on R9 3 innies R9C789 = 19 = {289/379/469/478/568}, no 1

8. 45 rule on C1234 2 innies R19C4 = 6 = {15/24}

9. 45 rule on C6789 2 innies R19C6 = 9 = [18]/{27/36/45}, no 1,9 in R9C6

10. R9C456 = {147/156/246/345} (cannot be {138/237} because 3,7,8 only in R9C6), no 8, clean-up: no 1 in R1C6 (step 9)

11. 45 rule on N6 3 outies R37C8 + R5C6 = 7
11a. Min R37C8 = 3 -> max R5C6 = 4
11b. Max R37C8 = 6, no 6,7,8,9

12. 13(3) cage at R6C7 = {157/247}, no 3, 7 locked in R6C78, locked for R6 and N6, clean-up: no 6 in R6C19 (step 5)
12a. 6 in R6 locked in R6C23, locked for N4 and 23(3) cage at R6C2, clean-up: no 3 in R5C9 (step 6)
12b. 7 in R4 locked in R4C23, locked for 18(4) cage at R3C1

13. R456C1 = {139/148/238}, no 5, clean-up: no 1 in R4C9 (step 4), no 4 in R5C9 (step 6), no 8 in R6C9 (step 5)
13a. Killer pair 8,9 in R456C1 and R6C23, locked for N4

14. R5C234 (step 6a) = {459} (only remaining combination, cannot be {189/369} because R5C3 only contains 4,5) -> R5C4 = 9, R5C23 = {45}, locked for R5 and N4, clean-up: no 2 in R4C9 (step 4), no 9 in R6C9 (step 5)

15. Naked pair {45} in R46C9, locked for C9 and N6, R5C9 = 6 (cage sum), R5C1 = 3 (step 6), clean-up: no 4 in R78C1, no 2,3 in R78C9 (step 7)
15a. Killer pair 1,2 in R4C1 and R78C1, locked for C1
15b. Naked pair {17} in R78C9, locked for C9 and N9, clean-up: no 3 in R9C789 (step 7a)
15c. 3 in C9 locked in R123C9, locked for N3

16. R4C78 = {39} (hidden pair in N6), locked for R4, R3C8 = 1 (cage sum)
16a. Naked triple {127} in R4C123, locked for R4

17. 3 locked in R123C9 = {238/239} (R123C9 = {389} is more than 19), 2 locked for C9 and N3, R1C8 = {56} (cage sum)

18. 3 in N9 locked in R7C7 + R8C8, locked for 9(3) cage at R6C6
18a. 9(3) cage at R6C6 = {135/234}, no 6
18b. 1 of {135} must be in R6C6 -> no 5 in R6C6

19. 9 in N8 locked in R78C6, locked for C6 and 23(3) cage at R7C6
19a. 9 in N9 locked in R9C789, locked for R9
19b. R9C789 (step 7a) = {289/469}, no 5

20. R9C456 (step 10) = {147/156/345} (cannot be {246} which clashes with R9C789), no 2, clean-up: no 4 in R1C4 (step 8), no 5 in R1C5 (step 2), no 7 in R1C6 (step 9)
20a. 3,7 of {147/345} must be in R9C6 -> no 4 in R9C6, clean-up: no 5 in R1C6 (step 9)

21. R1C456 = {136/145/235}
21a. Killer pair 5,6 in R1C456 and R1C8, locked for R1
21b. 3,4 only in R1C6 -> R1C6 = {34}, clean-up: no 3,7 in R9C6 (step 9)
21c. R1C456 = [163/514/523], no 2 in R1C4, clean-up: no 4 in R9C4 (step 8)

22. Naked pair {15} in R19C4, locked for C4
22a. Naked triple {156} in R9C456, locked for R9 and N8, clean-up: no 4 in R9C78 (step 19b)
22b. Naked triple {289} in R9C789, locked for R9 and N9 -> R8C7 = 6, clean-up: no 1 in R7C1
22c. Naked triple {347} in R9C123, locked for N7

23. Naked pair {23} in R78C5, locked for C5 and N8, R6C5 = 5 (prelim f), R6C9 = 4, R4C9 = 5, R4C1 = 1 (step 4), R6C1 = 9 (step 5), clean-up: no 6 in R7C1
23a. Naked pair {25} in R78C1, locked for C1 and N7

24. R6C4 = 3 (hidden single in R6), R7C3 = 6 (hidden single in R7), R8C2 = 8 (cage sum), R6C23 = [68], R7C2 = 9, R78C6 = [89], R8C3 = 1, R78C9 = [17], R78C4 = [74]

25. Naked pair {12} in R56C6, locked for C6

26. 7 in C6 locked in R23C6, locked for 23(4) cage at R1C7
26a. 7 in N3 locked in 18(3) cage at R2C8 = {567} (only remaining combination) -> R4C6 = 6, R4C45 = [84], R9C456 = [165], R1C45 = [51], R1C6 = 4 (step 9), R1C8 = 6
26b. Naked pair {57} in R2C8 + R3C7, locked for N3

27. R23C6 = {37} = 10 -> R12C7 = 13 = [94], R4C78 = [39], R7C7 = 5, R78C8 = [43], R78C1 = [25], R78C5 = [32], R3C7 = 7, R2C8 = 5, R23C6 = [73], R6C6 = 1 (step 18a)

28. R23C4 = {26} = 8 -> R12C3 = 10 = [73]

and the rest is naked singles.
Walkthrough by Afmob:
Don't put so much pressure on me, Andrew! :lol:

Thanks for this nice Killer, Ed! It was fun to solve and I didn't use any Killer pairs. I was tempted to also use the Killer triple Andrew found but after having a closer look at the grid I found that it wasn't needed.

A151 Walkthrough:

1. C456
a) Innie of grid = R5C5 = 7
b) 21(3) = {489} locked for C5
c) Innies C1234 = 6(2) = {15/24}
d) Innies C5 = 7(2) <> 3
e) 10(3) @ N2: R1C6 <> 2,6 since (37) only possible there
f) 12(3) @ R9 <> 8 since R9C45 <> 3,8
g) Innies C6789 = 9(2) <> 1,9; R9C6 <> 3,7
h) 12(3) @ R9 = 6{15/24} -> 6 locked for R9+N8
i) 23(3) = {689} -> R8C7 = 6; 8,9 locked for C6+N8

2. N69 !
a) ! Outies N6 = 7(2+1) <> 6,7,8,9; R5C6 <> 5
b) Innies+Outies N9: -3 = R6C6 - R7C8 -> R7C8 = (45), R6C6 = (12)
c) 9(3) = 3{15/24} -> 3 locked for N9
d) Outies R9 = 8(2) = {17} locked for C9+N9
e) Innies R9 = 19(3) = {289} locked for R9+N9

3. N78
a) 14(3) = {347} locked for R9+N7
b) 12(3) @ R9 = {156} locked for N8
c) 10(3) = {235} since R78C5 = (23) -> R6C5 = 5; 2,3 locked for C5+N8
d) 12(3) @ N7 = {147} because R78C4 = (47) -> R8C3 = 1; 4,7 locked for C4
e) 7(2) = {25} locked for C1+N7
f) 17(3) = 6{29/38} -> R6C4 = (23), R7C3 = 6
g) 23(3) @ N7 = {689} -> R6C2 = 6

4. R456
a) Innies R6789 = 13(2) = {49} locked for R6
b) R6C3 = 8, R7C2 = 9, R8C2 = 8 -> R6C4 = 3
c) Innies R1234 = 6(2) = [15/42]
d) 13(3) @ N4 = {139} -> R4C1 = 1, R6C1 = 9, R5C1 = 3
e) 18(3) @ R5 = {459} since R5C23 = (245) -> R5C4 = 9; 4,5 locked for R5+N4
f) 9 locked in 13(3) @ N6 = {139} -> R3C8 = 1

5. C456
a) Innies C1234 = 6(2) = {15} locked for C4
b) 10(3) = 1{36/45} -> 1 locked for R1+N2; R1C6 = (34)
c) Hidden Single: R2C2 = 1 @ N1
d) 18(3) @ N1 = {189} -> R3C3 = 9, R4C4 = 8
e) 18(4) = {2367} since R23C4 = (26) -> 3,7 locked for C3+N1; 2,6 locked for N2

6. R123
a) 18(4) @ R3C1 = {2457} -> R3C1 = 4, R4C3 = 2, R4C2 = 7, R3C2 = 5
b) R4C5 = 4, R4C6 = 6, R3C5 = 8, R2C5 = 9
c) 18(3) @ N3 = {567} because R3C7 <> 4,8 -> R3C7 = 7, R2C8 = 5
d) R4C9 = 5, R6C9 = 4 -> R5C9 = 6, R7C8 = 4, R8C8 = 3, R7C7 = 5 -> R6C6 = 1

7. Rest is singles.

Rating: Easy 1.0. I used IOD and hidden cages.
Further comments by Andrew:
Returning briefly to A151; I've only recently noticed that Frank's V2 was posted so haven't tried it yet.

Afmob wrote:
I was tempted to also use the Killer triple Andrew found but after having a closer look at the grid I found that it wasn't needed.
Nice walkthrough! It's not often we get one without any killer pairs, etc. and still only using simple combo work.

You probably weren't surprised that I used that killer triple. It was so obvious and consistent with the SS score and Ed's rating that I didn't think twice about looking for something simpler.

I guess that's one difference between our solving approaches. You, Ed, Mike and udosuk (there may be others) like to optimise WTs. I've never been interested in doing that. If I can find a solving path at the expected level of difficulty, or occasionally better, I'm happy with that; I've usually spent quite a long time solving the puzzle.


Andrew wrote:
Nice cage pattern! I like the way R5C5 was left blank to maintain the central symmetry.
Although I thought that at the time, I've now realised it would be better for me with a single cell cage at R5C5. When I set up a killer on my Excel worksheet the first check I do is that the cage totals sum to 405, which obviously can't be done when there are blank cell(s). It's only after that check that I set up the cage patterns by colouring the cells.


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PostPosted: Thu Aug 04, 2011 5:21 am 
Offline
Grand Master
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 151 V2 by Frank (April 2009) here
Puzzle Diagram:
Image
NOTE: r5c5 is intentionally blank. See hint in Frank's quote below :)
Images with "udosuk style Killer Cages" by Børge:
pre]Image Image[/pre]
Code: Select, Copy & Paste into solver:
3x3:d:k:5889:5889:5378:2307:2307:2307:7172:3845:3845:5889:3590:5378:5378:4615:7172:7172:3080:3845:5889:5385:3590:5378:4615:7172:3080:3338:3845:2315:5385:5385:3590:4615:3080:3338:3338:4364:2315:3085:3085:3085:26:4110:4110:4110:4364:2315:4367:4367:2320:3857:3090:3091:3091:4364:5908:4367:2320:5653:3857:5910:3090:3091:6423:5908:2320:5653:5653:3857:5910:5910:3090:6423:5908:5908:5653:2585:2585:2585:5910:6423:6423:
Solution:
+-------+-------+-------+
| 6 1 8 | 3 2 4 | 9 5 7 |
| 9 5 3 | 1 7 6 | 8 4 2 |
| 7 4 2 | 9 8 5 | 6 3 1 |
+-------+-------+-------+
| 1 8 9 | 7 3 2 | 4 6 5 |
| 5 2 6 | 4 9 1 | 7 8 3 |
| 3 7 4 | 5 6 8 | 1 2 9 |
+-------+-------+-------+
| 4 6 1 | 2 5 7 | 3 9 8 |
| 2 3 7 | 8 4 9 | 5 1 6 |
| 8 9 5 | 6 1 3 | 2 7 4 |
+-------+-------+-------+
Quote:
Frank: I am not sure of the protocol for creating a version 2. So if I have committed procedural trangressions, please disregard the rest of this post :).
JSudoku, Perfect Sudoku, SudokuSolver and I have together conspired to create the following. We decided to make it diagonal and symmetric. It measures 1.74 on the Richter scale. To help you get started I will tell you 1.2345679% of the solution. The centre cell is 9. Enjoy :).
SS(3.3.0)score = 1.74.

Ed: Thanks for making a harder version, this puzzle certainly needs one. Did you solve it yourself first? I'd rather solve puzzles that I know have been hand solved first.
I wish I could say something witty to match your wonderful way with words Frank...you add some much needed humour to this forum!
Thanks

Afmob: Thanks for giving us a V2, Frank!
I usually try to avoid ...
Rating: Hard 1.5.

manu: Thanks for this V2 !
The main difficulty was ...

Andrew (in 2010): A151 V2 was then next one of my backlog of unfinished puzzles that I went back to.
In Afmob's walkthrough steps ... were neat while in manu's walkthrough the key step ... was similar to my final breakthrough; I liked the way manu expressed it ...
Even though my solving path is similar to manu's, I decided to post my walkthrough because of the interesting step ...
I'll also rate A151 V2 at Hard 1.5. I think my hardest steps were of the same difficulty level as Afmob's hardest steps.

Walkthrough by Afmob:
Thanks for giving us a V2, Frank!

I usually try to avoid contradiction chains but I found it easier to express my hardest step (6b) this way than by explaining it via a forcing chain.

A151 V2 Walkthrough:

1. D\/
a) Innie of grid = R5C5 = 9
b) Innies D/ = 15(2) = {78} locked for D/; CPE: R1C1+R9C9 <> 7,8
c) Innies D\ = 10(2) = {46} locked for D\
d) 14(3) = 5{18/27} -> 5 locked for D\
e) 15(4) = 1{248/257/347} <> 6,9 since R1C9 = (78); 1 locked for N3; R1C8+R23C9 <> 7,8

2. C456
a) Innies C5 = 3(2) = {12} locked for C5
b) Innies C1234 = 9(2) <> 1; R9C4 <> 2
c) Innies C6789 = 7(2) <> 7

3. R456
a) Innies R1234 = 6(2) = {15/24}
b) Innies R6789 = 12(2) = [39/48/57]
c) 9(3) = 3{15/24} since R6C1 = (345) -> 3 locked for C1+N4

4. C789
a) Outies C9 = 12(2) = [39/48/57]
b) 15(4) = 1{248/257/347} -> 1 locked for C9
c) 17(3) <> 7 because 7{46/28} blocked by R1C9 = (78) and R9C9 = (46)

5. R456
a) Innies R6789 = 12(2) <> 5
b) Innies R5 = 8(2) <> 1,4,8; R5C9 <> 2
c) Innies R1234 = 6(2): R4C1 <> 5

6. R456+D\ !
a) 17(3) @ N6 <> {368} since R4C9 = (245)
b) ! 17(3) @ N6 <> 8 since it sees all 4 of D\ because 17(3) = {458} -> R9C9 <> 4 and R6C9 = 8 -> R6C1 = 4 (Innie R6) -> R1C1 <> 4
c) Innies R6 = 12(2) = [39] -> R6C1 = 3, R6C9 = 9
d) 17(3) @ N6 = [26/53]9
e) Innies R1234 = 6(2): R4C1 <> 2
f) 12(3) @ D\ = 3{18/27} -> 3 locked for N9
g) 9 locked in 21(3) @ N4 for 21(3); 21(3) <> 6

7. R1234 !
a) 12(3): R4C6 <> 3,6 because 1 only possible there and (24,45) are Killer pairs of 15(4)
b) ! Hidden Killer pair (36) in R4C5 + 13(3) for R4 since R4C78 = {36} blocked by R5C9 = (36)
-> R4C5 = (36) and 13(3) = {139/238/256/346} <> 7
c) 18(3): R23C5 <> 3,6 since R4C5 = (36)
d) ! 3 in R23 must be in 21(4) + N3 -> CPE: R1C38 <> 3 (Grouped/Caged X-Wing?)

8. C6789
a) Outies C9 = 12(2) <> 9
b) 25(4) = {4678} locked for N9; 4,6 also locked for C9
c) R5C9 = 3 -> R4C9 = 5
d) 13(3) <> 9 because R4C78 <> 3,9
e) Hidden Single: R7C8 = 9 @ C8, R4C5 = 3 @ R4
f) 12(3) @ N6 = {129} -> 1,2 locked for R6+N6
g) 23(4) = 59{18/27} because R89C7 = (125) -> R8C6 = 9; R7C6 = (78); 5 locked for C7
h) Naked pair (78) locked in R67C6 for C6
i) 28(4) = {5689} since R23C6 = (456) -> R23C6 = {56} locked for C6+N2; R12C7 = {89} locked for C7+N3
j) 16(3) = 8{17/26} -> R5C8 = 8
k) 13(3) = {346} since R4C78 = (46) -> R3C8 = 3; 4,6 locked for R4+N6

9. R1234
a) 9(3) @ N2 = {234} -> R1C5 = 2; 3,4 locked for R1+N2
b) R1C1 = 6, R9C9 = 4
c) Naked pair (78) locked in R23C5 for C5+N2
d) Hidden Single: R2C3 = 3 @ N1
e) 21(4) = {1389} since R23C4 = (19) -> R1C3 = 8; 1 locked for C4
f) 23(4) = {1679} locked for N1
g) Innie R1234 = R4C1 = 1
h) Cage sum: R5C1 = 5

10. Rest is singles.

Rating: Hard 1.5. I used a small contradiction chain and an interesting CPE move.
Walkthrough by manu:
Thanks for this V2 !

The main difficulty was working both on cage combinations and Innies for the four 3-cages at the puzzle border.
The killer pair found at step 5g then enables to crack this puzzle.


Walkthrough A151 V2

1) Frank's hint : r5c5=9 (too tired for counting the cages total ! )

2)a) Innies for D/ : r9c1+r1c9=15 → r9c1=(78), r1c9=(78) locked for D/ and for r1c1, r9c9
b) Innies for D\ : r1c1+r9c9=10 → r1c1=(46), r9c9=(46) locked for D\

3)a) Innies for r1234 : r4c1+r4c9=6 : no 36789
b) Innies for r5 : r5c1+r5c9=8 : no 489
c) Innies for r6789 : r6c1+r6c9=12 : no 126
d) Deduce from c) that 9(2) at n4 ={234/135} : combinations : [423/234/135/153/513]

Some combinations are not possible:

e) Combination [135] is impossible since from a)b), it would imply that both cells r4c9 and r5c9 are equal to 5
f) From step 2a), both cages 9(2) and 17(2) at n4 and n6 cannot contain digit 4 ; combination [234] is not possible since it would force r4c9 to be 4.
g) Conclusion of step 3 : r6c1=3 and r6c9=9, r45c1=[42/15/51], r45c9=[26/53/17]

4)a) Outies of c9 : r19c8 total 12 : no 126
b) Combinations of cage 15(4) at n3 : {1248/1257/1347} → r1c8=(345), r9c8=(789).
c) 1 is locked for n3 and c9 at r23c9 → 17(3) at n6 = [269/539], 9(3) at n4 = [153/423]

5)a) Innies for c1234 : r 19c4 total 9
b) Innies for c5 : r19c5 total 3 : {12}
c) Innies for c6789 : r19c7 total 7

Focus on combinations of cage 9(3) at n2

d) [513] is not possible since from steps a)c), it would force both cells r9c4 and r9c6 to be
equal to 4
e) [315] is not possible since from steps b)c) it would force both cells r9c5 and r9c6 to be
equal to 2.
f) Combination {135} is impossible : 9(3) = {126/234}
g) Killer pair {46} locked for r1 at cells r1c1 and cage 9(3) : r1c8=(35) and r9c8=(79)

6)a) Combination of 15(4) at n3 : {1248} is no longer possible since r1c8=(35)
→ (step 4b) r1c9=7 and r9c1=8
→ r23c9=(124) : no 3
b) Combinations of 25(4) at n9 using r9c8=(79) and r9c9=(46) : {2689/4678} : no 3 and
8 is locked for n9/c9 at r78c9
c) Hidden single for c9 : r5c9=3 → r4c9=5 (cage combination) and
r45c1=[15] (steps 3)ab)

7)a) r3c8<>6,8, 9 since combinations {139/148/238/346} are blocked by cells r4c1 and r5c9.
b) Hidden pair {89} for n3 locked for c7 and r23c6 at cells r12c7.
c) 6 locked at cells r2c8 and r3c7 for D/
d) Last combination for cage 12(3) at D/ : {246} locked for D/ → 9(3) at D/ = {135}

8)a) Combinations of cage 14(3) at D\ : 5{18/27} → 5 is locked at r2c2 and r3c3 for n1 and D\ (r4c4<>5 since r4c9=5)
b) From step a), cage 12(3) at D\ =3{18/27} with 3 locked at r7c7 and r8c8 for D\ and n9.
c) No 5 for cage 21(3) at n1
d) 9 is locked for r4 at r4c23.
e) We deduce from steps c) and d) that 21(3)={489}
f) Hidden single for r4 : r4c5=3

9)a) Last combination : r23c5={78} locked for c5 and n2
b) Last combination for cage 28(4) : r23c6={56} locked for c6 and n2
c) Last combination for cage 9(2) : r1c456={234}locked for r1/n2, with r1c5=2.
d) Using steps 5)abc, r9c5=2, r9c4=(56) and r9c6=(34)
e) Naked pair {34} locked for c6 at r19c6
f) Naked triple {456} at cells r78c5 and r9c4 locked for n8 → r9c6=3, r9c4=6 and r6c5=6
g) Hidden singles for r1 : r1c2=1, r1c7=9
i) r1c1=6 (naked single) and 9 is locked for n1 at cge 23(4) : last combination {1679}
→ r23c1={79} locked for n1/c1
j) last combination for cage 14(3) at D\ (step 8a) : {257} with r4c4=7, r23c23={25}
locked for n1 and D\ → 12(3) at D\ = {138} with r6c6=8
k) Naked single : r6c4=5, r5c4=4 r5c6=1
l)Last combinations for cages at r5 : 12(3)={246} with {26} locked for r5/n4 and 16(3)={178} with {78} locked for n6

10)Rest is singles
Walkthrough by Andrew (in 2010):
A151 V2 was then next one of my backlog of unfinished puzzles that I went back to.

In Afmob's walkthrough steps 6b and 7d were neat while in manu's walkthrough the key step 3f was similar to my final breakthrough; I liked the way manu expressed it that the 3-cell cages in C1 and C9 cannot both contain 4.

Even though my solving path is similar to manu's, I decided to post my walkthrough because of the interesting step 31.

Rating Comment. I'll also rate A151 V2 at Hard 1.5. I think my hardest steps were of the same difficulty level as Afmob's hardest steps.

Here is my walkthrough for A151 V2. At the time this puzzle was originally active I'd got as far as step 24. When I resumed I did a bit of nibbling and then made real progress from step 28 onward.

Prelims

a) R1C456 = {126/135/234}, no 7,8,9
b) 21(3) cage at R3C2 = {489/579/678}, no 1,2,3
c) R456C1 = {126/135/234}, no 7,8,9
d) 9(3) cage at R6C4 = {126/135/234}, no 7,8,9
e) R9C456 = {127/136/145/235}, no 8,9
f) 28(4) cage at R1C7 = {4789/5689}, no 1,2,3

1. 45 rule on complete grid 1 innie R5C5 = 9, placed for D/ and D\

2. 45 rule on R1234 2 innies R4C19 = 6 = {15/24}

3. 45 rule on R5 2 remaining innies R5C19 = 8 = [17]/{26/35}, no 4,8, no 1 in R5C9

4. 45 rule on R6789 2 innies R6C19 = 12 = [39/48/57]

5. R456C1 = {135/234} (cannot be {126} because R6C1 only contains 3,4,5}, no 6, 3 locked in R56C1, locked for C1 and N4, clean-up: no 2 in R5C9 (step 3)

6. 45 rule on C1 2 outies R19C2 = 10 = {19/28/37/46}, no 5

7. 45 rule on C9 2 outies R19C8 = 12 = {39/48/57}, no 1,2,6

8. 45 rule on C1234 2 innies R19C4 = 9 = [27]/{36/45}, no 1, no 2 in R9C4

9. 45 rule on C5 2 remaining innies R19C5 = 3 = {12}, locked for C5

10. 45 rule on C6789 2 innies R19C6 = 7 = {16/25/34}, no 7

11. 45 rule on D/ 2 remaining innies R1C9 + R9C1 = 15 = {78}, locked for D/, CPE no 7,8 in R1C1 + R9C9

12. 45 rule on D\ 2 remaining innies R1C1 + R9C9 = 10 = {46}, locked for D\

13. R1C9 = {78} -> 15(4) cage in N3 = {1248/1257/1347} (other combinations don’t contain 7 or 8), no 6,9, 1 locked in R23C9, locked for C9 and N3, clean-up: no 5 in R4C1 (step 2), no 3 in R9C8 (step 7)
13a. R1C9 = {78} -> no 7,8 in R1C8 + R23C9, clean-up: no 4,5 in R9C8 (step 7)
13b 5 of {1257} must be in R1C8 -> no 5 in R23C9

14. R456C9 = {269/359/458} (cannot be {278} which clashes with R1C9, cannot be {368} because R4C9 only contains 2,4,5, cannot be {467} which clashes with R9C9), no 7, clean-up: no 1 in R5C1 (step 3), no 5 in R6C1 (step 4)

15. 12(3) cage at R6C6 = {138/237}, no 5, 3 locked for D\

16. R5C234 = {138/147/246} (cannot be {156/237} which clash with R5C19, cannot be {345} because 3 only in R5C4 and R5C23 = {45} clashes with R456C1), no 5
16a. 3 of {138} must be in R5C4 -> no 8 in R5C4

17. 45 rule on N4 4 innies R46C23 = 1 outie R5C4 + 24
17a. Max R46C23 = 30 -> max R5C4 = 6
17b. Min R46C23 = 25 (when R5C4 = 1) but cannot be {1789} which clashes with R5C234 = {47}1 -> no 1 in R6C23

18. Interactions between R1C456 and R9C456 using steps 8,9,10
18a. R1C456 cannot be [315/513] because R9C456 cannot be [622/424] -> R1C456 = {126/234}, no 5, 2 locked for R1 and N2, clean-up: no 8 in R9C2 (step 6), no 4 in R9C4 (step 8), no 2 in R9C6 (step 10)
18b. Killer pair 4,6 in R1C1 and R1C46, locked for R1, clean-up: no 4,6 in R9C2 (step 6), no 8 in R9C8 (step 7)
[Step 18a can be expressed more logically as
R19C4 = 9, R19C6 = 7 -> R1C4 cannot be 2 more than R1C6 -> R1C456 cannot be [513]
R19C5 = 3, R19C6 = 7 -> R1C6 cannot be 4 more than R1C5 -> R1C456 cannot be [315]
-> R1C456 cannot be [315/513] -> R1C456 = {126/234} …]

19. 15(4) cage in N3 (step 13) = {1257/1347} (cannot be {1248} because R1C8 only contains 3,5), no 8 -> R1C9 = 7, placed for D/ -> R9C1 = 8, clean-up: no 3 in R9C2 (step 6)
19a. R1C8 = {35} -> no 3 in R23C9

20. R9C456 = {136/145/235} (cannot be {127} which clashes with R9C28, ALS block), no 7, clean-up: no 2 in R1C4 (step 8)
20a. R1C456 (step 18a) = {126/234}
20b. 6 of {126} must be in R1C4 -> no 6 in R1C6, clean-up: no 1 in R9C6 (step 10)

21. 25(4) cage in N9 = {2689/3679/4678} (cannot be {3589} because R9C8 only contains 4,6, cannot be {4579} because 5,9 in R78C9 clashes with R456C9), no 5, 6 only in R789C9, locked for C9 and N9, clean-up: no 2 in R4C9 (step 14), no 4 in R4C1 (step 2), no 2 in R5C1 (step 3)
21a. 5 in C9 only in R45C9, locked for N6

22. Naked pair {35} in R5C19, locked for R5
22a. R5C234 (step 16) = {147/246}, no 8, 4 locked for R5

23. 23(4) cage in N1 = {1679/2489/2678/3479/3569} (cannot be {1589/2579/3578} because R1C1 only contains 4,6, cannot be {4568} which clashes with R456C1)
23a. R1C1 = {46} -> no 4,6 in R23C1

24. 45 rule on N3 3 innies R12C7 + R3C8 = 1 outie R4C6 + 18
24a. Max R12C7 + R3C8 = 23 -> max R4C6 = 5

25. 12(3) cage at R2C8 = {156/246/345}
25a. 3 of {345} must be in R2C8 + R3C7 (R2C8 + R3C7 cannot be {45} which clashes with 15(4) cage in N3), no 3 in R4C6

26. 12(3) cage at R6C6 (step 15) = {138/237}
26a. 1 of {138} must be in R7C7 + R8C8 (R7C7 + R8C8 cannot be {38} which clashes with 25(4) cage), no 1 in R6C6

27. 15(3) cage at R6C5 = {348/357/456}
27a. 7,8 of {348/357} must be in R78C5 (R78C5 cannot be {34/35} which clash with R9C456), no 7,8 in R6C5

28. 23(4) cage in N7 = {1589/2489/2678} (cannot be {4568} because R9C2 only contains 1,2,7,9)
28a. Killer triple 4,5,6 in R1C1, R56C1 and R78C1, locked for C1

29. R9C456 (step 20) = {136/145/235}
29a. R9C456 cannot be [514], here’s how
R9C456 = [514] => R1C4 = 4 (step 8), R9C9 = 6 => R1C1 = 4 clashes with R1C4
29b. -> R9C456 = {136/235}, no 4, 3 locked for R9 and N8, clean-up: no 3 in R1C6 (step 10)

30. R1C456 (step 18a) = {126/234}
30a. 3,6 only in R1C4 -> R1C4 = {36}, clean-up: no 5 in R9C4 (step 8)
30b. Naked pair {36} in R19C4, locked for C4

31. 6 in R1 only in R1C14, 6 in C4 only in R19C4, 6 on D\ only in R1C1 + R9C9 -> R9C49 must contain 6, locked for R9, clean-up: no 1 in R1C6 (step 10)
[I guess this is a sort of X-Wing using the diagonal, possibly XY-Wing, XYZ-Wing or some strangely named Fish.]
31a. 1 in N2 only in R1C5 + R23C4, CPE no 1 in R1C3
[This CPE has been there since step 9 but I’ve only just spotted it.]

32. 15(3) cage at R6C5 (step 27) = {348/357/456}
32a. 3 of {348} must be in R6C5, 4 of {456} must be in R78C5 (R78C5 cannot be {56} which clashes with R9C456), no 4 in R6C5

[Steps 29 and 31 gave me the idea how to continue.]
33. R456C1 (step 5) = {135/234}, R456C9 (step 14) = {359/458}
33a. R456C1 cannot be {234} and R456C9 cannot be {458}, here’s how
R456C1 = [234] => R456C9 = [458] (steps 2,3,4) => R1C1 = 6, R9C9 = 6 clash on D\
33b. -> R456C1 = [153], R456C9 = [539]

34. R4C5 = 3 (hidden single in R4), R23C5 = 15 = {78}, locked for C5 and N2

35. R9C6 = 3 (hidden single in C6), R9C4 = 6, R9C5 = 1 (step 29b) R1C4 = 3, R1C5 = 2, R1C6 = 4, R1C1 = 6, R9C9 = 4, R1C8 = 5, R9C8 = 7 (step 7)

36. R4C6 = 2, placed for D/, R2C8 + R3C7 = 10 = {46}, locked for N3 and D/

37. Naked pair {78} in R4C4 + R6C6, locked for N5 and D\

38. Naked pair {45} in R78C5, locked for C5 and N8 -> R6C5 = 6, R5C6 = 1, R5C4 = 4, R6C4 = 5, placed for D/

39. Naked pair {19} in R23C4, locked for C4, N2 and 21(4) cage at R1C3, no 1,9 in R12C3
39a. R23C4 = {19} = 10 -> R12C3 = 11 = [83], R7C3 = 1, R8C2 = 3, R12C7 = [98], R23C5 = [78], R1C2 = 1, R9C2 = 9 (step 6)

40. Naked pair {12} in R23C9, locked for C9 and N3 -> R3C8 = 3

41. Naked pair {25} in R2C2 + R3C3, locked for N1 and D\, R4C4 = 7 (cage sum)

42. Naked pair {28} in R78C4, locked for N8 and 22(4) cage at R7C4 -> R9C3 = 5

and the rest is naked singles.


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PostPosted: Thu Nov 03, 2011 8:59 pm 
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Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 152 by Afmob (April 2009) here
Puzzle Diagram:
Image
Note that this is an X-Killer and R2C8+R3C7+R5C5+R7C3+R8C2 is a 29(5) remote cage.
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3:d:k:4608:4608:4608:4608:3844:3844:3846:3846:2568:3849:3849:3851:3596:4877:3844:3846:7440:2568:3849:3851:3596:3596:3606:4877:7440:3097:3097:3355:5148:5148:3596:3606:3606:4877:4386:3097:3355:5148:4134:6183:7440:3606:4877:4386:3116:4653:5148:4134:6183:6183:6450:4386:4386:3116:4653:4653:7440:4134:6183:6450:6450:2621:2622:2367:7440:5697:2114:4134:6450:2621:2622:2622:2367:5697:5697:2114:2114:6477:6477:6477:6477:
Solution:
+-------+-------+-------+
| 1 2 7 | 8 9 4 | 5 6 3 |
| 3 8 6 | 1 5 2 | 4 9 7 |
| 4 9 5 | 6 3 7 | 8 1 2 |
+-------+-------+-------+
| 8 7 3 | 2 4 6 | 1 5 9 |
| 5 4 2 | 9 7 1 | 6 3 8 |
| 9 6 1 | 5 8 3 | 7 2 4 |
+-------+-------+-------+
| 6 3 4 | 7 2 5 | 9 8 1 |
| 7 1 9 | 3 6 8 | 2 4 5 |
| 2 5 8 | 4 1 9 | 3 7 6 |
+-------+-------+-------+
Quote:
Afmob: This cage pattern created some very hard Killers and I had to adjust the cage sums several times to get the difficulty right, but I think it was worth it. Here you can find lots of I/O moves, so have fun spotting and using them! :dance:
SS Score: 1.26. Estimated rating: 1.25.
Edit: No further version planned.

manu: Thanks for this interesting Assassin Afmob. There is nothing too much difficult (my walkthrough uses ..., and the steps flow nicely. There are many opportunities which makes this Assassin very enjoyable ; I guess there should be others different solving paths.
Are you planning a V2 Afmob ?

Ed:
Afmob wrote:
had to adjust the cage sums several times to get the difficulty right, but I think it was worth it.
Perfect difficulty thanks Afmob. No need for a V2!
manu wrote:
There is nothing too much difficult
Was difficult for me! But I missed manu's really neat step 3d. A very elegant trick which I'd give an Easy 1.5 rating since I think it is very hard to find. Here is a more conventional (though much longer) way to solve this puzzle which....I'll rate at Easy 1.50 [but see edit following] I took sooo many tangents before getting to this optimised, simple to follow WT. Also, I like to rate puzzles that have remote cages and diagonal cages a bit higher than standard patterns. [edit: I'll up that rating to 1.50 since ...]. It's cracked by step 17.

Andrew: Thanks Afmob for a challenging puzzle!
It took me quite a long time to find the key breakthrough in step 22, which is the same as manu's step 3d. With hindsight it's the same sort of move as my step 2 but there's something I can't explain which makes it a much harder move to spot.
I'll rate my walkthrough for A152 at Hard 1.25 because although step 22 isn't technically difficult IMHO it's hard to spot.
Ed wrote:
Also, I like to rate puzzles that have remote cages and diagonal cages a bit higher than standard patterns.
I've probably also factored that into my rating.

Afmob: Good walkthroughs from all of you! I used ... to solve it.

Walkthrough by manu:
Thanks for this interesting Assassin Afmob. There is nothing too much difficult (my walkthrough uses some cage blockers and a min-max argument to open the puzzle), and the steps flow nicely. There are many opportunities which makes this Assassin very enjoyable ; I guess there should be others different solving paths.

Are you planning a V2 Afmob ?

Walkthrough A152

0) Cage 15(2) n1 - cells only uses 6789
Cage 12(2) n6 - cells do not use 126
Cage 13(2) n4 - cells do not use 123
Cage 9(2) n7 - cells do not use 9
Cage 10(2) n9 - cells do not use 5
Cage 10(2) n3 - cells do not use 5
Cage 8(3) n8 - cells do not use 6789
Cage 22(3) n7 - cells do not use 1234
Cage 10(3) n9 - cells do not use 89
Cage 14(4) n25 - cells do not use 9
Cage 14(4) n125 - cells do not use 9

1)a) Innies for n47 : r5c3+r6c3+r7c3+r8c2=8
b) Max r2c8+r3c7+r5c5=24 → (using cage 29(5) at D/) Min r7c3+r8c2=5
→ Max r5c3+r6c3=3 : r56c3={12} locked for n4/c3 and cells r7c4/r8c5
c) hcage 5(2) at r7c3 and r8c2 : combinations : [41/32]
hcage 24(3) at r2c8, r3c7 and r5c5 : combination {789} locked for D/

2)a) Outies for r89 : r8c134 total 19
b) We deduce there is no 1 at r8c134 → 1 locked at r9c45 for n8/r9 since the cage 8(3) must contain 1, and r8c1<>8 since r9c1<>1
c) r8c134=[793/694/784/685] → r8c1=(67), r8c3=(89) and r8c4=(345)
d) r9c1=(23) (combination of cage 9(2)). We deduce that r9c1 blocks combination {23} for
the hcage 5(2) at n 7 : 5(2)=[41]
e) Remaining innie for n4 : r6c1=9, so cage 13(2) at n4 is {58} locked for r1/n4 since combination {67} is blocked by cell r8c1=(67)
f) cage 22(3) at n7 contains at least one of {58} since r789c1, r7c3,r8c2<>5,8 : combination {589} all locked for n7
g) Using D/, we deduce that cage 10(2) at n3 is [28/37/64]

3)a) 9 is locked for n6/r4 at r4c789, so cage 12(2) at n6 is {48/57}
b) Outies for n3 (using hcage 24(3)) : r5c5+r4c9=16 → r4c9=(789)
c) ! 7 is locked for c9 in cages 10(2) and 12(2), since from steps 2)g) and 3)a), 12(2)={48}
→ 10(2)={37}. We deduce r4c9=(89) so (step b) r5c5=(78)
d) ! Cage 13(2) at n4 contains 8 so both cells r4c9 and r5c5 cannot be 8. We deduce from
steps b)c) that r4c9=9 and r5c5=7.
e) r3c89={12} (using cage 12(3)) locked for n3/r3, and there is a naked pair {89} at r2c8
and r3c7 locked for n3. Cage 10(2) at n3 is [37/64]

4)a) Hidden single for r5 : r5c4=9
b) Innies and outies for n1 : r1c4=3+r3c3
c) We deduce that r3c3=(35) and r1c4=(68)
d) Let us focus on cage 14(4) at c4 : combination {1247} is not possible since r3c3=(35)
→ there is no 7 in this cage because {1247} is the only combination of 14(4) that contains
digit 7, so r7c4=7 (hidden single for c4) and r8c5=6 (cage combination for 16(4))

5)a) r89c1=[72], r7c12={36} locked for n7/r7
b) There is no 2 at cage 8(3) at n8 : combination {134} locked for n8
c) Innies-outies for n126 : r3c5=r4c4+1.
d) Innies-Outies for n89 : r6c6=r7c5+1 : we deduce r6c6=(36) and r7c5=(25)
e) Naked triple {356} locked for n5 at r4c6 and r6c46.
f) We have r4c4=(1248). But from step c), since r3c5 – 1= r4c4 and since r3c5<>9 and r3c5<>2 (step 3)e)), we deduce that r4c4<>1,8. We thus have r4c4=(24) and r3c5=(35)

6)a) ! Naked pair {35} at r3c35 locked for r3 and r2c4
b) Hidden single for c4 : r6c4=5
c) 3 is locked for c4 at cage 8(3) : r9c5=(14)
d) Naked single : r7c5=2
cage combination : r6c5=8
step 5)d) → r6c6=3
naked single : r4c6=6, r12c9=[37]

7)a) Last combination : 12(2) at n6 = [84], 13(2) at n4 = [85]
b) Naked single : r3c3=5 → (step 4)b)) r1c4=8
c) Naked single : r3c5=3 → (step 5)c)) r4c4=2
d) Naked single : r8c8=4 → r78c9=[15], and hidden single for D\ : r1c1=1
e) Cage 18(4) at n1 : {1278} with r1c23=[27]
f) Cage 15(2) at n1 : {69} locked for n1
g) Hidden single for n9 : r8c7=2 → r7c8=8
h) Innies for n6 : r4c7+r5c7=7 → r45c7=[16]

8)The rest is naked and hidden singles.

Edit : Ed has pointed out some steps that needed further explanations that have been added to my WT.
Walkthrough by Ed:
Afmob wrote:
had to adjust the cage sums several times to get the difficulty right, but I think it was worth it.
Perfect difficulty thanks Afmob. No need for a V2!
manu wrote:
There is nothing too much difficult
Was difficult for me! But I missed manu's really neat step 3d. A very elegant trick which I'd give an Easy 1.5 rating since I think it is very hard to find. Here is a more conventional (though much longer) way to solve this puzzle which....I'll rate at Easy 1.50 [but see edit following] I took sooo many tangents before getting to this optimised, simple to follow WT. Also, I like to rate puzzles that have remote cages and diagonal cages a bit higher than standard patterns. [edit: I'll up that rating to 1.50 since step 13c is chainy then a bit extra for the non-standard pattern]. It's cracked by step 17.

Assassin 152 (39 steps)
This is an optimised solution so I have occasionally missed some obvious eliminations since they aren't essential. However, I try and do clean-up as I go. Let me know of any corrections or clarifications.

Prelims
i. 10(2)n3: no 5
ii. 15(2)n1 = {69/78}
iii. two 14(4)n2 cages: no 9
iv. 13(2)n4: no 1,2,3
v. 12(2)n6: no 1,2,6
vi. 10(2)n9: no 5
vii. 10(3)n9: no 8,9
viii. 9(2)n7: no 9
ix. 22(3)n7: no 1,2,3,4
x. 8(3)n8: no 6,7,8,9

1. 29(5)n357: max. any three cells = {789} = 24 -> min. any two cells = 5 (important for next step)

2. "45" n7: 1 outie r6c1 - 4 = 2 innies r7c3 + r8c2
2a. min. 2 innies = 5 (step 1) -> r6c1 = 9
2b. r7c3 + r8c2 = h5(2) = {14/23}
2c. no 3 in r5c9
2d. no 4 in 13(2)n4

3. split cage 24(3)r2c8 + r3c7 + r5c5 = 24 = {789}
3a. all locked for D/
3b. no 1,2 in r8c1
3c. no 1,2,3 in r2c9

4. "45" n4: 2 remaining innies r56c3 = 3 = {12}
4a. both locked for c3 & n4 & 16(4)n4
4b. no 3,4 in r8c2 (h5(2)n7)

5. "45" r9: 3 outies r8c134 = 19 (no 1)

6. 8(3)n8 = 1{25/34}
6a. 1 locked for r9 & n8
6b. no 8 in r8c1

7. h19(3)r8c134 (step 5) = {379/469/478/568}(no 2) ({289} blocked by no candidates in r8c1)
7a. = one of 3/4/5 which must go in r8c4 -> no 3,4,5 in r8c13
7b. and = one of 8/9 -> r8c3 = (89)
7c. r9c1 = (23)

8. [32] blocked from h5(2)n7 by r9c1 (missed this first two times through - originally got it from hidden single 4 in n7)
8a. -> r7c3 = 4 & r8c2 = 1 (both placed for D/)
8b. no 6,9 in r2c9
8c. no 6 in r8c7
8d. no 9 in r7c8

9. r7c12 = 9 (cage sum) = {27/36}(no 5,8)

10. 5 locked in n7 in 22(3) = {589}: 5 locked for r9

11. 13(2)n4 = {58} ({67} blocked by r8c1)
11a. both locked for c1 & n4

12. 9 in n5 only in r5: 9 locked for r5
12a. no 3 in r6c9

13. 12(2)n6 = {48/57}
13b. 10(2)n3 = [28/37/46]
13c. -> 7 locked in these cages (since if 12(2) is not 7 it is {48} -> 10(2) = [37])
13d. 7 locked for c9

14. "45" n3: 1 outie r4c9 + 8 = 2 innies r2c8 + r3c7
14a. min. 2 innies = {78} = 15 -> min. r4c9 = 7
14b. -> r4c9 = (89)
14c. 2 innies = 16/17 = {79/89}
14d. 9 locked for n3 & D/

15. r5c4 = 9 (hsingle n5)

16. "45" n1: 1 outie r1c4 - 3 = r3c3
16a. r1c4 = (68)
16b. r3c3 = (35)

17. 14(4)n2 must have 3/5 for r3c3 = {1238/1256/1346/2345}(no 7)

This cracks it. Now trying to get to singles ASAP so cleanup missing.
18. r7c4 = 7 (hsingle c4)

19. r8c5 = 6 (cage sum)

20. r89c1 = [72] (2 placed for D/)

21. 8(3)n8 = {134}: 3 & 4 locked for n8

22. "45" c1234: 3 innies r689c4 = 12 and must have 3/4 for r8c4 = {345} only.
22a. r6c4 = 5
22b. r89c4 = {34}: both locked for c4 & n8
22c. r9c5 = 1

23. r67c5 = 10 (cage sum) = {28}: both locked for c5

24. r5c5 = 7 (placed for both D\ & D/)

25. naked pair {89} in n3: 8 locked for n3

26. "45" n3: 1 remaining outie r4c9 = 9 (finally caught up with manu!)

27. r3c89 = 3 (cage sum) = {12}: both locked for r3 & n3

28. naked pair {68} in r13c4: both locked for c3 & n2

29. 14(4)r3c5 must have 1/2 to keep below the cage sum
29a. r5c6 = (12)

30. naked pair {12} in n5 and r5c36: both locked for n5 & r5

31. r67c5 = [82]

32. "45" n89: 1 remaining outie r6c6 = 3 (placed for D\)

33. r3c3 = 5 (placed for D\)

34. "45" n1: 1 remaining outie r1c4 = 8
34a. r3c4 = 6

35. r34c5 = [34]

36. "45" n6: 2 remaining outies r2c5 + r3c6 = 12 = [57]

37. r1c9 + r4c6 = [36]
37a. r2c9 = 7 (cage sum)

38. r1c5 = 9
38a. r12c6 = 6 (cage sum) = {24} both locked for n2 & c6

39. r3c1 = 4
39a. r2c12 = 11 (cage sum) = [38]

rest are naked singles
Walkthrough by Andrew:
Thanks Afmob for a challenging puzzle!

It took me quite a long time to find the key breakthrough in step 22, which is the same as manu's step 3d. With hindsight it's the same sort of move as my step 2 but there's something I can't explain which makes it a much harder move to spot.

I'll rate my walkthrough for A152 at Hard 1.25 because although step 22 isn't technically difficult IMHO it's hard to spot.

Ed wrote:
Also, I like to rate puzzles that have remote cages and diagonal cages a bit higher than standard patterns.
I've probably also factored that into my rating.

Here is my walkthrough for A152.

Prelims

a) R2C3 + R3C2 = {69/78}
b) R12C9 = {19/28/37/46}, no 5
c) R45C1 = {49/58/67}, no 1,2,3
d) R56C9 = {39/48/57}, no 1,2,6
e) R7C8 + R8C7 = {19/28/37/46}, no 5
f) R89C1 = {18/27/36/45}, no 9
g) 22(3) cage in N7 = {589/679}, 9 locked for N7
h) 8(3) cage in N8 = {125/134}, 1 locked for N8
i) 10(3) cage in N9 = {127/136/145/235}, no 8,9
j) 14(4) cage at R2C4 = {1238/1247/1256/1346/2345}, no 9
k) 14(4) cage at R3C5 = {1238/1247/1256/1346/2345}, no 9

1. 45 rule on R9 2 outies R8C34 = 1 innie R9C1 + 10
1a. Max R8C34 = 14 -> max R9C1 = 4, clean-up: min R8C1 = 5
1b. Min R8C34 = 11, no 5 in R8C3, no 1 in R8C4
1c. 1 in N8 locked in R9C45, locked for R9, clean-up: no 8 in R8C1
1d. Min R9C1 = 2 -> min R8C34 = 12, no 6 in R8C3, no 2 in R8C4
[I enjoy recursive steps!]

2. 45 rule on N7 1 outie R6C1 = 2 innies R7C3 + R8C2 + 4
2a. Min R7C3 + R8C2 = 5 (because max R2C8 + R3C7 + R5C5 = 24) -> R6C1 = 9, clean-up: no 4 in R45C1, no 3 in R5C9
2b. R7C3 + R8C2 = 5 = {14/23}, R2C8 + R3C7 + R5C5 = 24 = {789}, locked for D/, clean-up: no 1,2,3 in R2C9
2c. R7C12 = 9 = {18/27/36/45}
2d. 9 in N5 locked in R5C45, locked for R5, CPE no 9 in R7C5, clean-up: no 3 in R6C9
[With hindsight 45 rule on N7 4(3+1) outies R2C8 + R3C7 + R5C5 + R6C1 = 33 -> R6C1 = 9, R2C8 + R3C7 + R5C5 = 24 = {789}, locked for D/ … is more direct. I’m not sure how I’d rate that step; I’m tending to rate 2+1, 3+1 etc. outies higher than I used to do.]

3. 45 rule on N4 2 remaining innies R56C3 = 3 = {12}, locked for C3, N4 and 16(4) cage at R5C3, clean-up: no 3,4 in R8C2
3a. R56C3 = 3 -> R7C4 + R8C5 = 13 = {49/58/67}, no 3

4. 45 rule on R9 3 outies R8C134 = 19 = {379/469/478/568}
4a. 8,9 of {379/478} must be in R8C3 -> no 7 in R8C3
4b. 6 of {568} must be in R8C1 -> no 5 in R8C1, clean-up: no 4 in R9C1

5. R7C3 + R8C2 (step 2b) = [41] (cannot be {23} which clashes with R9C1), placed for D/, clean-up: no 6,9 in R2C9, no 5,8 in R7C12 (both step 2c), no 9 in R7C8, no 9 in R8C5 (step 3a), no 6 in R8C7
5a. 4 in N4 locked in R456C2, locked for C2
5b. 5 on D/ locked in R4C6 + R6C4, locked for N5

6. Naked quad {2367} in R7C12 + R89C1, locked for N7
6a. 5 in N7 locked in R9C23, locked for R9

7. R45C1 = {58} (cannot be {67} which clashes with R9C1), locked for C1 and N4

8. 45 rule on N3 2 innies R2C8 + R3C7 = 1 outie R4C9 + 8
8a. Min R2C8 + R3C7 = 15 -> min R4C9 = 7
8b. Min R4C9 = 7 -> max R3C89 = 5 -> R3C89 = {1234}

9. 45 rule on C1 2 outies R27C2 = 1 innie R1C1 + 10
9a. Max R27C2 = 16 -> max R1C1 = 6
9b. Min R27C2 = 11, no 2,3 in R2C2

10. Hidden killer pair 1,4 in R1C1 and R23C1 for C1, R23C1 cannot contain both of 1,4 -> R1C4 = {14}, R23C1 must contain one of 1,4
10a. 15(3) cage in N1 = {168/249/348/456} (cannot be {258/267/357} which don’t contain 1 or 4, cannot be {159} because 5,9 only in R2C2), no 7
10b. 5,8 of {168/456} must be in R2C2 -> no 6 in R2C2
10c. 7 in C1 locked in R78C1, locked for N7, clean-up: no 2 in R7C1 (step 2c)

11. 45 rule on N1 1 outie R1C4 = 1 innie R3C3 + 3, R1C4 = {689}, R3C3 = {356}

12. 45 rule on N9 1 innie R7C7 = 1 outie R9C6, no 1,5 in R7C7, no 4 in R9C6

13. 6,7 in R9 locked in R9C6789 = {3679/4678}, no 2, clean-up: no 2 in R7C7 (step 12)

14. 45 rule on N9 4 innies R7C7 + R9C789 = 25 = {3679/4678}, 6,7 locked for N9, clean-up: no 3,4 in R7C8 + R8C7

15. 45 rule on N1 4 innies R1C123 + R3C3 = 15 = {1239/1257/1356} (cannot be {1248} because R3C3 only contains 3,5,6, cannot be {1347} because 1,4 only in R1C1, cannot be {2346} which clashes with 15(3) cage in N1), no 4,8 -> R1C1 = 1, placed for D\
15a. 2 of {1239/1257} must be in R1C2 -> no 7,9 in R1C2

16. 45 rule on N6 3 innies R45C7 + R4C9 = 16 = {169/259/268/349/367} (cannot be {178/358/457} which clash with R56C9)
16a. R4C9 = {789} -> no 7,8,9 in R45C7

17. 14(4) cage at R2C4 = {1238/1256/1346/2345} (cannot be {1247} because R3C3 only contains 3,5,6), no 7

18. 45 rule on R1 3 outies R2C679 = 13 = {148/157/238/247/346} (cannot be {139/256} because R2C9 only contains 4,7,8), no 9

19. 45 rule on C1234 2 remaining innies R56C4 = 2 outies R89C5 + 7
19a. Max R56C4 = 15 -> max R89C5 = 8, no 8 in R8C5, no 4 in R9C5, clean-up: no 5 in R7C4 (step 3a)
19b. Min R89C5 = 5 -> min R56C4 = 12, no 1,2,3,4,6 in R5C4, no 2 in R6C4
19c. 1 in N5 locked in R4C5 + R5C6 + R6C5, CPE no 1 in R3C5
19d. 7 in C4 locked in R57C4, CPE no 7 in R7C5

20. 45 rule on N89 1 outie R6C6 = 1 remaining innie R7C5 + 1, no 2,8 in R6C6, no 8 in R7C5

21. 24(4) cage at R5C4 = {2589/2679/3579/3678/4569} (cannot be {1689/3489/4578} because 1,4,7,8,9 only in R5C4 + R6C5), no 1
21a. 4,7,8,9 only in R5C4 + R6C5 -> no 2,3,6 in R6C5

22. 45 rule on N3 2(1+1) remaining outies R4C9 + R5C5 = 16 = {79} (cannot be [88] which clashes with R45C1), no 8, CPE no 7 in R5C9, clean-up: no 5 in R6C9
22a. 8 on D/ locked in R2C8 + R3C7, locked for N3, clean-up: no 2 in R1C9
22b. Killer pair 4,7 in R2C9 and R56C9, locked for C9 -> R4C9 = 9, R5C5 = 7 (step 22), placed for D/ and D\, clean-up: no 6 in R7C4 (step 3a), no 6 in R7C5 (step 20), no 7 in R9C6 (step 12)
22c. 9 on D/ locked in R2C8 + R3C7, locked for N3

23. R4C9 = 9 -> R3C89 = 3 = {12}, locked for R3 and N3

24. R5C4 = 9 (hidden single in R5), clean-up: no 6 in R3C3 (step 11), no 4 in R8C5 (step 3a)
24a. 24(4) cage at R5C4 (step 21) = {2589/4569}, no 3, clean-up: no 4 in R6C6 (step 20)

25. 45 rule on N8 4 remaining innies R7C5 + R789C6 = 24
= {2589/2679} (cannot be {3489/3678} because R7C5 only contains 2,5, cannot be {3579/4569/4578} which clash with 8(3) cage), no 3,4, 2 locked for N8, clean-up: no 3 in R7C7 (step 12), no 5 in R8C4 (prelim h)

26. R9C1 = 2 (hidden single in R9), placed for D/, R8C1 = 7, clean-up: no 9 in R2C2 (step 10a)
26a. Naked pair {36} in R7C12, locked for R7, clean-up: no 6 in R9C6 (step 12)

27. R1C2 = 2 (hidden single in C2)
27a. R1C12 = [12] -> R1C123 + R3C3 (step 15) = {1239/1257}
27b. 7,9 only in R1C3 -> R1C3 = {79}

28. R7C7 = 9 (hidden single on D\), R3C7 = 8, R2C8 = 9, R8C7 = 2, R7C8 = 8, R7C4 = 7, R8C5 = 6 (step 3a), R9C6 = 9 (step 12), clean-up: no 7 in R2C3, no 6 in R3C2

29. Naked pair {25} in R7C56, locked for R7 and N8 -> R7C9 = 1, R3C89 = [12], R8C6 = 8, R8C3 = 9, R1C3 = 7, R3C2 = 9, R2C3 = 6, R4C3 = 3, R3C3 = 5, placed for D\, R2C2 = 8, R9C23 = [58]
29a. R7C9 = 1 -> R8C89 = 9 = [45], clean-up: no 7 in R6C9
29b. R8C4 = 3, R9C45 = [41], R3C4 = 6, R1C4 = 8, R24C4 = [12], R6C4 = 5, R4C6 = 6, placed for D/, R6C6 = 3, placed for D\, R7C56 = [25], R6C5 = 8 (step 25)

and the rest is naked singles
Walkthrough by Afmob:
Good walkthroughs from all of you! I used a Killer quad (step 3f) and an unusual IOD using a nonet and a column (step 4e) to solve it.

A152 Walkthrough:

1. R789
a) Outies R9 = 19(3) <> 1; R8C13 <> 2,3,4,5 since R8C4 <= 5
b) 8(3) = 1{25/34} -> 1 locked for R9+N8
c) 9(2) = [63/72]
d) 22(3) = {589} locked for N7 since {679} blocked by R8C1 = (67); 5 also locked for R9
e) 18(3) <> 1 since R7C12 <> 8,9
f) 1 locked in 29(5) @ N7 for D/ -> 29(5) = 189{47/56} -> 8,9 locked for D/
g) Innies+Outies N7: 4 = R6C1 - (R7C3+R8C2) -> R6C1 = 9 because R7C3+R8C2 >= 5
-> R7C3+R8C2 = 5 = {14} locked for N7+D/
h) 4 locked in 29(5) @ N7 = {14789} -> 7 locked for D/

2. C123
a) Innies N4 = 3(2) = {12} locked for C4+N4+16(4)
b) R7C3 = 4, R8C2 = 1
c) 13(2) = {58} locked for C1+N4 because {67} blocked by R8C1 = (67)
d) 15(3) <> 1 because R23C1 <> 5,9 and {168} blocked by Killer pair (68) of 15(2)
e) Hidden Single: R1C1 = 1 @ C1
f) 4 locked in 15(3) @ C1 for N1 -> 15(3) = 4{29/38/56} <> 7; R2C2 = (589)
g) Innies+Outies N1: 3 = R1C4 - R3C3 -> R3C3 <> 7,8; R1C4 = (689)

3. N356 !
a) 9 locked in R5C45 @ N5 for R5
b) 12(2) <> 3
c) Innies+Outies N3: -8 = R4C9 - (R2C8+R3C7): R4C9 = (789) since R2C8+R3C7 >= 15
d) 12(3): R3C89 = (1234) since R4C9 >= 7
e) 10(2) = [28/37/64]
f) ! Killer quad (4789) locked in 15(3) + R2C89+R3C7 for N3
g) 10(2) <> {28} since it's a Killer pair of 12(3)
h) Killer pair (47) locked in R2C9 + 12(2) for C9
i) 12(3) = 1{29/38} -> 1 locked for R3+N3
j) 5 locked in 15(3) @ N3 <> 9
k) 9 locked in R2C8+R3C7 @ N3 for D/

4. C456 !
a) 14(4) @ N1 <> 7 because R3C3 = (356)
b) Hidden Single: R5C4 = 9 @ N5, R7C4 = 7 @ C4
c) 16(4) = {1267} -> R8C5 = 6
d) R8C1 = 7 -> R9C1 = 2
e) ! Innies+Outies N1+C4: -4 = R9C5 - R6C4 -> R9C5 = 1, R6C4 = 5
f) 8(3) = {134} -> 3,4 locked for C4+N8
g) 24(4) = {2589} -> 2,8 locked for C5
h) R5C5 = 7
i) Innies+Outies N9: R7C7 = R9C6 = (89)
j) 25(4) = 89{26/35} -> R6C6 = (36)
k) Naked pair (36) locked in R46C6 for C6+N5

5. N59
a) Naked pair (28) locked in R4C4+R6C5 for N5
b) 14(4) = {1346} @ R3C5 -> R5C6 = 1, R4C5 = 4, R3C5 = 3. R4C6 = 6
c) R1C9 = 3 -> R2C9 = 7
d) R6C6 = 3
e) 10(3) = {145} -> R8C9 = 5, R8C8 = 4, R7C9 = 1
f) 10(2) = {28} locked for N9
g) Innies+Outies N9: R7C7 = R9C6 = 9

6. Rest is singles.


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PostPosted: Thu Nov 03, 2011 9:30 pm 
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Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
JF"F"K6 by manu (April 2009) here
Puzzle Diagram:
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Images with "udosuk style Killer Cages" by Børge:
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Code: Select, Copy & Paste into solver:
3x3:d:k:3840:1793:1793:6147:6147:6147:3590:3590:2824:3840:3840:2315:2315:4109:6147:3590:2824:2577:5906:5906:2324:4109:4109:3607:3607:2577:794:2587:5906:5906:2324:4127:3607:3873:3873:794:2587:4901:4901:4901:4127:3113:3113:3113:3116:2093:2094:2094:2352:4127:2610:4915:4915:3116:2093:3383:2352:2352:4666:4666:2610:4915:4915:3383:3136:3137:5186:4666:2116:2116:4934:4934:3136:3137:3137:5186:5186:5186:2638:2638:4934:
Solution:
+-------+-------+-------+
| 1 3 4 | 7 6 9 | 8 2 5 |
| 9 5 8 | 1 3 2 | 4 6 7 |
| 2 7 6 | 5 8 4 | 9 3 1 |
+-------+-------+-------+
| 6 9 5 | 3 4 1 | 7 8 2 |
| 4 8 2 | 9 7 6 | 1 5 3 |
| 3 1 7 | 2 5 8 | 6 4 9 |
+-------+-------+-------+
| 5 6 3 | 4 9 7 | 2 1 8 |
| 7 4 1 | 8 2 3 | 5 9 6 |
| 8 2 9 | 6 1 5 | 3 7 4 |
+-------+-------+-------+
Quote:
manu: Here is something quite hard, so this one might be '' Just For a Frightful Killer!'' ;) for some people, which explains the "F" :evil: .
However, I have tried to keep a similar concept as Assassin 152 : a X-killer with many Innies and outies.
It is still solvable without using too heavy moves, but the cracking step I have used for this one was not easy to find for me.
On the other hand, it then could be solved quite (?) straightforwardly if you find the good trick ! See by yourself ...
SSscore : 2.75
Edit : There was a trouble (thanks Afmob) with the PS-code (there was a "broken" cage). I apologize for that problem, everything is OK now !

Afmob: I tried to tackle this Killer a week ago but I didn't come far, so I stopped and deleted my partial wt since I thought vast T & E was needed because of SudokuSolver's score. Today I tried it again and it fell quite fast. :D
My rating is not "just" 1.25 because ... Rating: Hard 1.25.

manu: Thanks Afmob for having taken time with this puzzle.
This is not so much difficult (by the way, all the puzzles I post on this forum are solvable using logic, without vast T_E steps whereas I am not completely against some short contradiction move : trust me,I solve each of them several times).
PS : A strange fact : without the diagonal condition, the puzzle is still solvable and SSscore is 2.23 < 2.75 ! However, I have kept this condition since it seems to me more interesting to solve.

Andrew (in 2010): Another from my backlog of unfinished puzzles.
Afmob wrote:
I tried to tackle this Killer a week ago but I didn't come far, so I stopped ... Today I tried it again and it fell quite fast. :D
I also came back to it but, in my case, just over a year and a half later. ;)
Thanks manu for an interesting Killer. Nice solving paths by both Afmob and manu! I also used ... thanks manu for deciding to post this puzzle as a Killer-X. :)
As can be seen from my walkthrough, when I came back to this puzzle I found that I had been very close to solving it at the time; my key breakthrough was only two steps later ...
I'll rate my walkthrough for JF"F"K at 1.5.

Walkthrough by Afmob:
I tried to tackle this Killer a week ago but I didn't come far, so I stopped and deleted my partial wt since I thought vast T & E was needed because of SudokuSolver's score. Today I tried it again and it fell quite fast. :D

My rating is not "just" 1.25 because the Hidden Cage I used seems useless at first and one normally looks for other Hidden Cages first.

JFF 6 Walkthrough:

1. R1234
a) Innies N3 = 10(2) = [91/82]
b) 11(2) <> {29} because it's a Killer pair of Innies N3
c) Outies N3 = 7(2+1) <> 6,7,8,9; R3C6 <> 3 since it sees R4C6
d) Innies N2 = 5(2) = [14/32/41]
e) 9(2) = [54/63/81]
f) Outies R12 = 16(3) must have one of (1234)

2. R1234 !
a) ! Innies N1 = 23(4) <> 49{28/37} since R3C123 can only have one of (1234) because of Killer quad (1234) in Outies R12 + R3C69
b) ! Innies N1 = 23(4) <> 56{39/48} because they're blocked by Killer triples (356,456) of 7(2)
-> Innies N1 = 23(4) = {1589/1679/2579/2678/3578} <> 4
c) ! Hidden Killer pair (56) locked in Innies N1 + Outies R12 for R3 since none of them can have both
-> Innies N1 = 23(4): R2C3 <> 5,6
d) R2C3 = 8 -> R2C4 = 1
e) Innie N2 = R3C6 = 4
f) 14(3) = 4[82/91]
g) Innies R12 = 10(2) = [37/64/73]
h) 10(2) @ N3 = [37/46/73]
i) 11(2) <> {47} since it's a Killer pair of 11(2)

3. R456
a) Naked pair (12) locked in R4C69 for R4
b) Innies R1234 = 10(2) = {37/46}
c) 9(2): R4C4 <> 5
d) 5 locked in R4C23 @ R4 for N4+23(4)
e) 10(2) @ N4 = {37/46}
f) 8 locked in R45C2 @ N4 for C2
g) 8(2) @ R6C2 <> 3
h) Killer pair (67) locked in 10(2) + 8(2) @ R6C2 for N4
i) 8(2) @ C1: R7C1 = (567)

4. R789
a) Outies R89 = 22(3) = 9{58/67} -> 9 locked for R7
b) Innies N7 = 8(2): R7C3 = (123)
c) 4 locked in Innies N89 = 15(4) @ R7 = 4{128/137/236} <> 5
d) 9(3): R7C4 <> 6 because [216] blocked by R4C6 = (12)
e) Innies N8 = 7(2) = [25/43]
f) 8(2) = {35} locked for R8
g) Innies R89 = 9(2) = [72/81]
h) 13(2) = [58/67]
i) 12(2) = [48/93] since (57) is a Killer pair of 12(2)

5. D\/
a) 11(2) = {56} locked for N3+D/ because {38} blocked by R9C1 = (38)
b) 9(3) = {234} -> 4 locked for C4
c) 9(2) <> 5

6. C456
a) 5 locked in 16(3) @ R3 = {358} because R2C5 = (367) -> R2C5 = 3; 5,8 locked for R3+N2
b) R3C7 = 9 -> R4C6 = 1, R8C2 = 4 -> R9C1 = 8, R8C1 = 7 -> R7C2 = 6
c) R7C1 = 5 -> R6C1 = 3, R6C4 = 2, R7C3 = 3
d) 16(3) @ N5 = {457} -> R5C5 = 7, R4C5 = 4, R6C5 = 5

7. Rest is singles.

Rating: Hard 1.25. I used a Killer quad and a Hidden Killer pair.
Walkthrough by manu:
Thanks Afmob for having taken time with this puzzle.
This is not so much difficult (by the way, all the puzzles I post on this forum are solvable using logic, without vast T_E steps whereas I am not completely against some short contradiction move : trust me,I solve each of them several times).

Here is the way I solve this killer :

Walkthough JFFK 6

The main cracking move is the killer triple at step 1)e) which was hard to see for me (Afmob obtains the same result in a different way) : it enables to have a starting point for this puzzle.
The other important move is step 2)e) which will enable to state step 3)b) and fix cell r5c5 : from this point, the puzzle is cracked : everything then follow straightforwardly.

1)a) Innies for n3 : r3c7+r3c9=10 : r3c79=[91/82]
b) Innies for n2 : r2c4+r3c6=5 : r2c4=(1234), r3c6=(1234)
c) Cage combinations for 9(2) : r2c3=(5678)
d) Outies for r12 : r3c4+r3c5+r3c8 = 16 : r3c458 cannot contain both {56} → r3c123 contain at least one of {56}
e) Killer triple {456} locked for n1 at cages 15(3), 7(2) and cells r3c123 (each of these three
sets contain at least one of {456} (*)) → r2c3 <> 56 : r2c3=(78), r2c4=(12)
f) Using step b), r3c6=(34)
g) Min r3c6+r3c7=3+8=11 → max r4c6=3 : r4c6=(123).
h) r4c6 <> 3 since r3c67<>[47] (no 7 at r3c7) → r4c6 = (12)
i) Naked pair {12} locked for r4 at r4c69
j) Outies for n3 : r3c6+r4c6+r4c9=7 and r4c69={12)}→ r3c6=4, r2c4=1 (step b) ) and
r2c3=8 (cage 9(2))

2)a) Innies for n8 : r7c4+r8c6=7 : r7c4=(12456), r8c6=(12356)
b) r7c4<>3 since r8c6<>4
c) r7c4<>6 since r4c6=(12) would block combinations of 9(3)
d) r7c4=(245), r8c6=(235) → r8c7=(356)
e) We deduce that either r8c67={35}, either r8c6=2 and r7c4=5 : r8c5 <>5 since 5 is locked
at r7c4 or r8c67
f) Innies for r89 : r8c1+r8c5=9 : r8c15=[81/72/63/54] → r8c1=(5678). We deduce that 13(2)
at n7 is {58/67}.
g) 12(2) at n7 is {39/48} since {57} blocks combinations of 13(2).
i) Killer pair {89} locked for D/ at cage 12(3) and cell r3c7
j) Innies-Outies for D/ : r7c4+r3c6=1+r5c5 → r5c5=3+r7c4 since r3c6=4
→ r5c5<>8, so r7c4<>5
→ r7c4=(24), so r5c5=(57)
k) From step a), r8c6=(35) : cage 8(2) at n8 is {35} locked for r8

3)a) Innies for r6789 : r6c5+r6c9=14 → r6c59=[59/68/95] since r6c9<>6, and using
cage 12(2) at n6, we get r5c9=(347).
b) Combinations of 16(3) at n5 : [475/376] ([259] is not possible since r4c5<>2) → r5c5=7
c) Step 2)j) → r7c4=4, step 2)a) → r8c6=3, r8c7=5.
d) Hidden single for c5 : r4c5=4 → r6c5=5
e) Step a) → r6c9=9 → r5c9=3
f) One remaining innie for r5 : r5c1=4 → r4c1=6
g) Last combination : 15(2) at n6 is {78} locked for n6/r4

4)a) Naked single : r4c4=3 → r3c3=6
b) Last combination for cage 9(3) at D/ : r6c4=2, r7c3=3
c) Remaining innie for n7 : r7c1=5 → r6c1=3
d) Last combination for 12(3) at D/ : {48} → r9c1=8, r8c2=4
e) Naked single : r4c6=1 → r34c9=[12]. R3c7=9.
f) Naked single : r9c9=4
g) Naked pair : r4c23={59} locked for n4 and r3c12
h) Last combination : r6c23={17} locked for n4/c6 → r5c3=2, r5c2=8 → r5c4=9
i) Last combination : 13(2) at n7 is {67} : r8c1=7, r7c2=6
j) Naked singles : r56c6=[68] → last combo : r5c78=[15] and r7c7=2
k) Hidden single for c9 : r1c9=5 → r2c8=6

The rest is singles ….

(*) To convince yourself that 15(3) contains one of {456}, check there is no combo only with {123789}. You needn't try all of them : since two of {123} total at most 5 and two of {789} total at least 15, there is no combo only with {123789}.

PS : A strange fact : without the diagonal condition, the puzzle is still solvable and SSscore is 2.23 < 2.75 ! However, I have kept this condition since it seems to me more interesting to solve.
Walkthrough by Andrew (in 2010):
Another from my backlog of unfinished puzzles.

Afmob wrote:
I tried to tackle this Killer a week ago but I didn't come far, so I stopped ... Today I tried it again and it fell quite fast. :D
I also came back to it but, in my case, just over a year and a half later. ;)

Thanks manu for an interesting Killer. Nice solving paths by both Afmob and manu! I also used a hidden cage for my breakthrough but a very different one, the 22(5) one on D/, so thanks manu for deciding to post this puzzle as a Killer-X. :)

As can be seen from my walkthrough, when I came back to this puzzle I found that I had been very close to solving it at the time; my key breakthrough was only two steps later, immediately after finding the hidden cage which Afmob used for his breakthrough.

I seem to have been good at spotting innies for this puzzle but not as good at seeing outies. I either missed several of them or saw them but didn't realise their importance. I don't know now which of those categories 3 outies for R12 R3C458 = 16 fits into. That was a key step in both Afmob's and manu's walkthroughs.

Rating Comment. I'll rate my walkthrough for JF"F"K at 1.5. I used a manu-style step and later a very short forcing chain.

Here is my walkthrough for JF"F"K6.

Prelims

a) R1C23 = {16/25/34}, no 7,8,9
b) 11(2) cage in N3 = {29/38/47/56}, no 1
c) R2C34 = {18/27/36/45}, no 9
d) 10(2) cage in N3 = {19/28/37/46}, no 5
e) 9(2) cage at R3C3 = {18/27/36/45}, no 9
f) R34C9 = {12}
g) R45C1 = {19/28/37/46}, no 5
h) R4C78 = {69/78}
i) R56C9 = {39/48/57}, no 1,2,6
j) R67C1 = {17/26/35}, no 4,8,9
k) R6C23 = {17/26/35}, no 4,8,9
l) 10(2) cage at R6C6 = {19/28/37/46}, no 5
m) 13(2) cage in N7 = {49/58/67}, no 1,2,3
n) 12(2) cage in N7 = {39/48/57}, no 1,2,6
o) R8C67 = {17/26/35}, no 4,8,9
p) R9C78 = {19/28/37/46}, no 5
q) 19(3) cage at R5C2 = {289/379/469/478/568}, no 1
r) 9(3) cage at R6C4 = {126/135/234}, no 7,8,9
s) 19(3) cage in N9 = {289/379/469/478/568}, no 1

1. Naked pair {12} in R34C9, locked for C9, clean-up: no 9 in R2C8, no 8,9 in R3C8

2. 45 rule on N2 2 innies R2C4 + R3C6 = 5 = {14/23}, clean-up: no 1,2,3,4 in R2C3

3. 45 rule on N3 2 innies R3C79 = 10 = [82/91]
3a. Min R3C67 = 9 -> max R4C6 = 5
3b. 11(2) cage in N3 = {38/47/56} (cannot be [92] which clashes with R3C79), no 2,9
3c. R3C79 = 10 -> R3C67 cannot be 10 (CCC) -> no 4 in R4C6

4. 45 rule on N7 2 innies R7C13 = 8 = {26/35}/[71], no 1 in R7C1, no 4 in R7C3, clean-up: no 7 in R6C1
4a. R7C13 = 8 -> R7C34 cannot be 8 (CCC) -> no 1 in R6C4

5. 45 rule on N8 2 innies R7C4 + R8C6 = 7 = {16/25}/[43], no 3 in R7C4, no 7 in R8C6, clean-up: no 1 in R8C7

6. 45 rule on R12 2 innies R2C59 = 10 = [19/28]/{37/46}, no 5,8,9 in R2C5

7. 45 rule on R89 2 innies R8C15 = 9 = [45/54/63/72/81], no 9 in R8C1, no 6,7,8,9 in R8C5, clean-up: no 4 in R7C2
7a. Max R8C5 = 5 -> min R7C56 = 13, no 1,2,3 in R7C56

8. 45 rule on R1234 2 innies R4C15 = 10 = {19/28/37/46}, no 5 in R4C5

9. 45 rule on R6789 2 innies R6C59 = 14 = [59/68/95], clean-up: no 5,8,9 in R5C9

10. 45 rule on C1234 4 innies R1389C4 = 26 = {2789/3689/4589/4679/5678}, no 1

11. 45 rule on D/ 5 innies R3C7 + R4C6 + R5C5 + R6C4 + R7C3 = 22
11a. R3C7 = {89} -> no 8,9 in R5C5

12. 12(3) cage in N7 = {129/138/147/246} (cannot be {156/237} which clash with R7C13), cannot be {345} which clashes with 12(2) cage), no 5

13. 45 rule on N3 3(1+2) outies R3C6 + R4C69 = 7 = [151/232/412/421] (cannot be [322/331]), no 3 in R3C6, clean-up: no 2 in R2C4 (step 2), no 7 in R2C3

14. 45 rule on N6 5 innies R4C9 + R56C78 = 18 = {12348/12357/12456}, no 9
14a. 12(3) cage at R5C6 = {138/147/156/237/246/345} (cannot be {129} which clashes with R4C9), no 9

15. R45C1 = {19/28/46} (cannot be {37} because R67C1 and R6C23 cannot both be {26}), no 3,7, clean-up: no 3,7 in R4C5 (step 8)
15a. When R45C1 = {28/46} either R67C1 must be [17] or R6C23 must be {17}
-> 1 in N4 locked in R45C1 + R6C123, locked for N4

[The next step is in the spirit of many of manu’s puzzles and walkthroughs. When I first spotted it I couldn’t see how to use but when I came back to the puzzle I found that I was able to use it for a later step so I’ve numbered it, even though there aren’t any immediate eliminations.]

16. R4567C1 + R6C23 is effectively a 26(6) cage because, although R7C1 cannot “see” R6C23, R7C1 cannot be the same as one of R6C23 which would cause a clash between R6C1 and one of R6C23 because R67C1 and R6C23 are both 8(2) cages.
The only valid combinations for this 26(6) are {123569/123578/134567} (cannot be {123479/124568} because 4,8,9 are only in R45C1 and cannot have two of them in a 10(2) cage) -> only of the 8(2) cages must be {17/26} and the other {35}.

17. 45 rule on R5 3 innies R5C159 = 14 = {149/167/239/248/257/347/356} (cannot be {158} because R5C9 only contains 3,4,7)
17a. 4 of {149/248} must be in R5C9, 4 of {347} must be in R5C1 -> no 4 in R5C5

18. 1,2 on D/ only in R3C7 + R4C6 + R5C5 + R6C4 + R7C3 = 22 (step 11) = {12379/12469/12478/12568}
18a. 7 of {12379} must be in R5C5 -> no 3 in R5C5

19. 3 in R4 locked in R4C2346
19a. 45 rule on R123 5 outies R4C23469 = 20 = {12359/13457/23456} (cannot be {12368} which clashes with R4C78), no 8, clean-up: no 1 in R3C3

[This is how far I got when this puzzle first appeared.]

20. 45 rule on N1 4 innies R2C3 + R3C123 = 23 = {1589/1679/2678/3578} (cannot be {3479} because R2C3 only contains 5,6,8, cannot be {2489/2579} which clash with R3C79, cannot be {3569/4568} which clash with R1C23), no 4, clean-up: no 5 in R4C4

21. R3C7 + R4C6 + R5C5 + R6C4 + R7C3 (step 18) = {12379/12568} (cannot be {12469/12478} because 9(3) cage at R6C4 cannot contain both of 1,4 or both of 4,6 and R6C4 + R7C3 cannot be [42] because there’s no 3 in R7C4), no 4
[This is what I missed earlier.]
21a. 5 of {12568} must be in R4C6 + R5C5 (because 9(3) cage at R6C4 cannot contain both of 2,5 or both of 5,6 and R6C4 + R7C3 cannot be [51] because there’s no 3 in R7C4) -> no 5 in R6C4 + R7C3, clean-up: no 3 in R7C1 (step 4), no 5 in R6C1
21b. {12379} can only be [91723/91732/93721] (R3C7 + R4C6 cannot be [92] because no 3 in R3C6)
21c. 9(3) cage at R6C4 = {126/234} (cannot be {135} because R6C4 + R7C3 cannot be [31], step 21b), no 5 in R7C4, clean-up: no 2 in R8C6 (step 5), no 6 in R8C7

22. 11(2) cage in N3 (step 3b) = {47/56} (cannot be {38} which clashes with R3C7 + R4C6 + R5C5 + R6C4 + R7C3), no 3,8
22a. 10(2) cage in N3 = {37}/[82/91] (cannot be {46} which clashes with 11(2) cage), no 4,6, clean-up: no 4,6 in R2C5 (step 6)

23. 12(2) cage in N7 = {39/48} (cannot be {57} which clashes with R3C7 + R4C6 + R5C5 + R6C4 + R7C3), no 5,7
23a. 13(2) cage in N7 = {58/67} (cannot be [94] which clashes with 12(2) cage), no 4,9, clean-up: no 5 in R8C5 (step 7)
23b. 12(3) cage in N7 (step 12) = {129/147/246} (cannot be {138} which clashes with 12(2) cage), no 3,8

24. R4567C1 + R6C23 (step 16) = {123569/123578/134567}, 3 locked for R6 and N4, clean-up: no 7 in R7C7
24a. R6C23 = {17/35} (cannot be {26} which clashes with R6C4), no 2,6

25. 16(3) cage at R4C5 = {169/259/457} (cannot be {178} because R6C5 only contains 5,6,9, cannot be {268} which clashes with R6C4), no 8, clean-up: no 2 in R4C1 (step 8), no 8 in R5C1

26. Consider the placement for 3 in R6
R6C1 = 3
or R6C23 = {35} => R6C59 (step 9) = [68] => R6C4 = 2 => R6C1 = 1
-> R6C1 = {13}, clean-up: no 2,6 in R7C1, no 2,6 in R7C3 (step 4)
25a. Killer pair 1,3 in R6C1 and R6C23, locked for R6 and N4, clean-up: no 9 in R45C1, no 1,9 in R4C5 (step 8), no 9 in R7C7
25b. Killer pair 5,7 in R7C1 and 13(2) cage, locked for N7

27. 9(3) cage at R6C4 (step 21c) = {126/234}, 2 locked for C4, clean-up: no 7 in R3C3
27a. R7C3 = {13} -> no 1 in R7C4, clean-up: no 6 in R8C6 (step 5), no 2 in R8C7

28. 45 rule on D/ 2 outies R3C6 + R7C3 = 1 innie R5C5 + 1
28a. Min R3C6 + R7C3 = 3 -> min R5C5 = 2

29. 16(3) cage at R4C5 (step 25) = {259/457}, no 6, 5 locked for C5 and N5, clean-up: no 4 in R4C1 (step 8), no 6 in R5C1, no 8 in R6C9 (step 9), no 4 in R5C9
29a. Naked pair {59} in R6C59, locked for R6, clean-up: no 3 in R6C23, no 1 in R7C7
29b. Naked pair {17} in R6C23, locked for R6 and N4 -> R6C1 = 3, R7C1 = 5, R7C3 = 3 (step 4), placed for D/, clean-up: no 4 in R1C2, no 6 in R4C4, no 8 in 13(2) cage in N7, no 9 in 12(2) cage in N7, no 1,4 in R8C5 (step 7)

30. 9(3) cage at R6C4 (step 21c) = {234} (only remaining combination) -> R7C4 = 4, R6C4 = 2, R4C6 = 1, R34C9 = [12], R3C7 = 9 (step 3), R3C6 = 4 (cage sum), R2C4 = 1 (step 2), R2C3 = 8, R8C6 = 3 (step 5), R8C7 = 5, R4C5 = 4, R56C5 = {57} (step 29) -> R6C5 = 5, R5C5 = 7, placed for both diagonals, R56C9 = [39], R4C1 = 6 (step 8), R5C1 = 4, 13(2) cage in N7 = [67], R9C1 = 8, R8C2 = 4, placed for D/, R3C1 = 2, R12C1 = [19], R2C2 = 5 (cage sum),R4C4 = 3, R3C3 = 6, placed for D\, R6C6 = 8, placed for D\

and the rest is naked singles without using diagonals.


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PostPosted: Thu Nov 03, 2011 10:03 pm 
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Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 153 by manu (April 2009) here
Puzzle Diagram:
Image
PS : r1c19 is a 10(2) cage
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:2560:6913:2306:2306:3332:4101:4101:7431:2560:6913:3850:6913:2306:3332:4101:7431:2320:7431:6913:3850:6913:3349:2070:1303:7431:2320:7431:3611:6913:3357:3349:2070:1303:4129:7431:3875:3611:3357:3357:6951:6951:6951:4129:4129:3875:2349:2862:2607:6951:7729:6951:2611:2612:3125:2349:2862:2607:7729:2106:7729:2611:2612:3125:3135:3135:5697:7729:2106:7729:4421:3910:3910:3135:5697:5697:5697:7729:4421:4421:4421:3910:
Solution:
+-------+-------+-------+
| 3 9 1 | 5 4 8 | 2 6 7 |
| 4 7 2 | 3 9 6 | 8 5 1 |
| 6 8 5 | 7 1 2 | 9 4 3 |
+-------+-------+-------+
| 5 1 8 | 6 7 3 | 4 2 9 |
| 9 2 3 | 1 8 4 | 5 7 6 |
| 7 6 4 | 9 2 5 | 3 1 8 |
+-------+-------+-------+
| 2 5 6 | 8 3 1 | 7 9 4 |
| 1 3 7 | 4 5 9 | 6 8 2 |
| 8 4 9 | 2 6 7 | 1 3 5 |
+-------+-------+-------+
Quote:
manu: Don't be afraid of these hexagonal cages because they could be very useful !
I have made different versions with the same cage pattern and hope that this one
has the right difficulty. It is the kind of killer I like : no early cage combination fixed, and the whole cage pattern must be used. I have an other version that could offer a nice V2 : it will be sent later
according to the activity on this V1 and JFFK6 as soon as A154 is claimed.

Afmob: I really liked the cage pattern of this Killer, thanks manu!
There were lots of moves I omitted to shorten my wt which would have probably led to a different solution path, so I guess there should be different ways to tackle it.
Rating: Hard 1.0.

Andrew: Thanks manu for a great puzzle with an interesting cage pattern!
As will be seen from my comment before steps 11, 12 and 13, I took some time to get going on this Assassin. I also ought to have spotted step 19 earlier. I had seen my other key move, step 24, much earlier but it wasn't useful until ...
I'll rate my walkthrough for A153 at least 1.25.

Ronnie G: manu,
I think this is your best puzzle yet! I loved all the interesting interactions you created: the hexagonal cages combined with the X(2) 's made so many CPE's.
It was a really fun puzzle. Thanks

manu: Thanks for all these kind words about this puzzle. I recall I have a V2 ready (with a similar cage pattern) for you but I wait for A154 to be claimed, so ... :scratch: :idea:

Ed: Here's a completely different way to get into this puzzle. No combo work just "45"s! More suitable for manu's V2 really, if he ever gets to post it.

Walkthrough by Afmob:
I really liked the cage pattern of this Killer, thanks manu!

There were lots of moves I omitted to shorten my wt which would have probably led to a different solution path, so I guess there should be different ways to tackle it.

A153 Walkthrough:

1. N689
a) 8(2) <> {17} since it's a Killer pair of 30(6)
b) 30(6) = 1489{26/35} since other combos blocked by Killer pairs (23,25,36) of 8(2)
c) 7 locked in R9C46 @ N8 for R9
d) Innies+Outies N69: 5 = R9C6 - R4C8 -> R9C6 = (6789), R4C8 = (1234)

2. N1
a) 27(6) = 1245{69/78} since other combos blocked by Killer pairs (67,68,79,89) of 15(2)
b) 3 locked in R1C13 @ N1 for R1

3. R789+N46
a) Outies R89 = 14(4) <> 9
b) 30(6) = 1489{26/35} -> 9 locked for N8
c) Innies+Outies N47: 1 = R9C4 - R4C2 -> R9C4 <> 1,4; R4C2 <> 8,9
d) 1,4 in N8 locked in 30(6) for 30(6)
e) Innies+Outies N69: 5 = R9C6 - R4C8: R4C8 <> 4

4. R123
a) Killer pair (89) locked in 27(6) + 15(2) for N1
b) 10(2): R1C1 <> 7; R1C9 <> 1,2
c) R4C8 <> 3 since it sees all 3 of N3
d) Innies+Outies N3: -7 = R4C8 - R1C79: R1C79 <> 9 since R4C8 <= 2
e) Innies+Outies N3: -7 = R4C8 - R1C79: R1C7 <> 4,6,7,8 since R1C9 >= 4, R1C9 <> 5 and R4C8 <= 2

5. C789 !
a) ! Hidden Killer pair (12) in R23C9 + R89C9 for C9 since 15(3) can only have one of (12) and R23C9 = (12) blocked by R4C8 = (12)
-> 15(3) = {159/168/249/258/267} <> 3 and Killer pair (12) locked in R23C9 + R4C8 for 29(6); CPE: R23C8 <> 1,2
b) 9(2) <> 7,8
c) 9 locked in 29(6) @ N3 = 1289{36/45} because {124679} blocked by Killer pair (46) of 9(2) -> 8 locked for N3
d) Hidden Single: R1C9 = 7 @ N3 -> R1C1 = 3
e) 15(2) = {69} locked for C9+N6
f) 12(2) = {48} locked for C9
g) 3 locked in 29(6) @ C9 = {123689} for N3
h) 9(2) = {45} locked for C8+N3
i) 15(3) = 5{19/28} because R89C9 = (125) -> 5 locked for N9; R8C8 = (89)
j) Innies+Outies N69: 5 = R9C6 - R4C8: R9C6 <> 8

6. C789
a) 17(4) = 16{28/37} because R9C6 = (67) -> 1 locked for N9
b) 15(3) = {258} -> R8C8 = 8; 2 locked for C9+N9
c) R7C9 = 4, R6C9 = 8
d) 16(3) @ N6 = {457} -> R5C8 = 7; 4 locked for C7
e) 10(2) @ C8 = {19} -> R6C8 = 1, R7C8 = 9
f) 10(2) @ C7 = {37} -> R6C7 = 3, R7C7 = 7
g) Innies+Outies N69: 5 = R9C6 - R4C8 -> R4C8 = 2, R9C6 = 7
h) 17(4) = {1367} -> R9C8 = 3; 1,6 locked for C7
i) R1C7 = 2, R1C8 = 6

7. C456
a) 13(2) @ C5 <> {67}
b) 16(3) @ N2 = {268} because {259} blocked by Killer pair (59) of 13(2) @ N2 -> R1C6 = 8, R2C6 = 6
c) 13(2) @ C5 = {49} locked for C5+N2
d) 9(3) = {135} because R1C34 <> 2,3 -> R2C4 = 3; 1,5 locked for R1
e) Innies R6789 = 14(2) = {59} locked for R6+N5
f) Both 8(2) <> {26} because it's blocked by R6C5 = (26)
g) Hidden Single: R3C6 = 2 @ N2 -> R4C6 = 3
h) 8(2) @ N2 = {17} locked for C5
i) 27(5) = {14589} -> R5C5 = 8; 1,4 locked for R5+N5
j) R5C7 = 5, R4C7 = 4, R4C5 = 7, R4C4 = 6, R4C9 = 9, R5C9 = 6, R5C1 = 9 -> R4C1 = 5

8. N14
a) 9(2) = {27} -> R7C1 = 2, R6C1 = 7
b) R4C2 = 1, R4C3 = 8
c) 10(2) = {46} -> R7C3 = 6, R6C3 = 4
d) 11(2) = {56} -> R6C2 = 6, R7C2 = 5
e) 15(2) = {78} locked for C2+N1

9. Rest is singles.

Rating: Hard 1.0. I used a Hidden Killer pair.
Walkthrough by Andrew:
Thanks manu for a great puzzle with an interesting cage pattern!

As will be seen from my comment before steps 11, 12 and 13, I took some time to get going on this Assassin. I also ought to have spotted step 19 earlier. I had seen my other key move, step 24, much earlier but it wasn't useful until there was a cage sum limit on R3C456 and therefore on R1C37.

I'll rate my walkthrough for A153 at least 1.25 because I used several combined cages and the hidden killer pair in step 24 also feels like that rating.

Here is my walkthrough. Thanks Afmob for your comments about step 26; I hope it's clearer now. I've also added a note at the end of step 19.

Prelims

a) R1C19 = {19/28/37/46}, no 5
b) R12C5 = {49/58/67}, no 1,2,3
c) R23C2 = {69/78}
d) R23C8 = {18/27/36/45}, no 9
e) R34C4 = {49/58/67}, no 1,2,3
f) R34C5 = {17/26/35}, no 4,8,9
g) R34C6 = {14/23}
h) R45C1 = {59/68}
i) R45C9 = {69/78}
j) R67C1 = {18/27/36/45}, no 9
k) R67C2 = {29/38/47/56}, no 1
l) R67C3 = {19/28/37/46}, no 5
m) R67C7 = {19/28/37/46}, no 5
n) R67C8 = {19/28/37/46}, no 5
o) R67C9 = {39/48/57}, no 1,2,6
p) R78C5 = {17/26/35}, no 4,8,9
q) 9(3) cage at R1C3 = {126/135/234}, no 7,8,9

1. 45 rule on R6789 2 innies R6C46 = 14 = {59/68}

2. 45 rule on R1234 4 innies R4C1379 = 26 = {2789/3689/4589/4679/5678}, no 1

3. 45 rule on N6 4 innies R4C8 + R6C789 = 14 = {1238/1247/1256/1346/2345}, no 9, clean-up: no 1 in R7C78, no 3 in R7C9

4. 45 rule on N47 1 outie R9C4 = 1 innie R4C2 + 1, no 9 in R4C2, no 1 in R9C4

5. 45 rule on N69 1 outie R9C6 = 1 innie R4C8 + 5, R4C8 = {1234}, R9C6 = {6789}
5a. Min R9C6 = 6 -> max R8C7 + R9C78 = 11, no 9

6. 45 rule on N1 2 outies R1C9 + R4C2 = 1 innie R1C3 + 7, IOU no 7 in R4C2, clean-up: no 8 in R9C4 (step 4)

7. 45 rule on N3 2 outies R1C1 + R4C8 = 1 innie R1C7 + 3, IOU no 3 in R4C8, clean-up: no 8 in R9C6 (step 5)

8. Combined cage R4567C9 = 27 = {3789/4679/5679}, 9 locked for C9, clean-up: no 1 in R1C1

9. 45 rule on C9 3 innies R123C9 = 1 outie R8C8 + 3
9a. Min R123C9 = 6 -> min R8C8 = 3

10. 45 rule on N1235 2 outies R4C28 = 1 innie R6C5 + 1
10a. Min R4C28 = 3 -> min R6C5 = 2

I ought to have spotted the next three steps earlier; manu gave a clue about the hexagons. The central 2-cell cages “see” all the cells of the surrounding hexagons so each hexagon plus central cage forms a 8-cell cage.

11. 27(6) cage at R1C2 + R23C2 = 42 = {12456789}, no 3, clean-up: no 4 in R9C4 (step 4)
11a. 3 in N1 locked in R1C13, locked for R1, clean-up: no 7 in R1C1
11b. 45 rule on N1 2 innies R1C13 = 1 outie R4C2 + 3
11c. R4C2 = {124568} -> R1C13 {13/23/34/35/36/38} (because R1C13 must contain 3), no 9 in R1C1, clean-up: no 1 in R1C9

12. 29(6) cage at R1C8 + R23C8 = 38 = {12345689}, no 7, clean-up: no 2 in R23C8
12a. 7 in N3 locked in R1C79, locked for R1, clean-up: no 6 in R2C5
12b. 45 rule on N3 2 innies R1C79 = 1 outie R4C8 + 7
12c. R4C8 = {124} -> R1C79 = {17/27/47} (because R1C79 must contain 7), no 5,6,8,9, clean-up: no 2,4 in R1C1

13. 30(6) cage at R6C5 + R78C5 = 38 = {12345689}, no 7, clean-up: no 1 in R78C5
13a. 7 in N8 locked in R9C46, locked for R9
13b. 45 rule on N8 2 innies R9C46 = 1 outie R6C5 + 7
13c. R9C46 = {27/37/57/67/79} (because R9C46 must contain 7) -> no 4,8 in R6C5

14. Combined cage R3478C5 = 16 = {1267/1357/2356}
14a. 45 rule on C5 3 innies R569C5 = 16 = {268/349/358} (cannot be {169/259/457} which clash with R3478C5, cannot be {178} because no 1,7,8 in R6C5, cannot be {367} which clashes with R78C5), no 1,7

15. 1 in C5 locked in R34C5 = {17}, locked for C5, clean-up: no 6 in R1C5

16. R4C28 = R6C5 + 1 (step 10)
16a. R4C28 cannot total 4 -> no 3 in R6C5 -> no 3 in R9C4 (step 13c), clean-up: no 2 in R4C2 (step 4)
16b. R4C2 = {14568} -> R1C13 (step 11b) = {13/34/35/36/38}, no 2

17. R569C5 (step 14) = {268/349/358}
17a. 5,9 of {349/358} must be in R6C5 -> no 5,9 in R59C5

18. R6C46 = 14 (step 1) -> R5C456 = 13 = {139/148/238/247/346} (cannot be {157} which clashes with R4C5, cannot be {256} which clashes with R6C46), no 5

19. 45 rule on R89 4 outies R6C5 + R7C456 = 14 = {1238/1256/1346/2345}, no 9
[These are 4 outies, rather than 4(3+1) outies, because they can all “see” each other.]
19a. R6C5 = {256} -> R9C46 (step 13c) = {27/57/67}, no 9, clean-up: no 8 in R4C2 (step 4), no 4 in R4C8 (step 5)
[In hindsight, after seeing Ed’s partial walkthrough, I ought now to have continued with
19b. R4C2 = {1456} -> R1C13 (step 11b) = {13/34/35/36}, no 8, clean-up: no 2 in R1C9
19c. R4C8 = {12} -> R1C79 (step 12b) = {17/27}, no 4, clean-up: no 6 in R1C1
19d.R1C19 = [37]
This would have avoided the need for the hidden killer pair in step 24.
I can only think that I must have been distracted by spotting 9 locked in C5.]

20. 9 in C5 locked in R12C5 = {49}, locked for C5 and N2, clean-up: no 4,9 in R4C4, no 1 in R4C6

21. 45 rule on N5 4 innies R4C456 + R6C5 = 18 = {1368/1467/2367/2457} (cannot be {1278} because 1,7,8 only in R4C45, cannot be {1458/2358/3456} which clash with R6C46)
21a. 3,4 of {2367/2457} must be in R4C6 -> no 2 in R4C6, clean-up: no 3 in R3C6
21b. 7 of {2457} must be in R4C5 with 5 in R4C4 -> no 5 in R6C5
21c. R6C5 = {26} -> R9C46 (step 13c) = {27/67}, no 5, clean-up: no 4 in R4C2 (step 4)

22. R78C5 = {35} (cannot be {26} which clashes with R6C5), locked for C5 and N8

23. R5C456 (step 18) = {148/238/247} (cannot be {139} because R5C5 only contains 2,6,8, cannot be {346} which clashes with R4C6), no 6,9
23a. 9 in N5 locked in R6C46 = {59}, locked for R6 and N5, clean-up: no 8 in R3C4, no 4 in R7C1, no 2,6 in R7C2, no 1 in R7C3, no 7 in R7C9

24. Hidden killer pair 3,7 in R1C19 and R1C37 for R1 -> either R1C19 = [37] or R1C37 = [37] (R1C19 is a 10(2) cage so must either contain both of 3,7 or neither of them)
24a. 45 rule on N2 3 innies R3C456 = 2 outies R1C37 + 7
24b. Max R3C456 = 15 -> max R1C37 = 8 so cannot be [37]
24c. R1C19 = [37], clean-up: no 8 in R45C9, no 6 in R67C1, no 5 in R7C9
24d. R1C13 = R4C2 + 3 (step 11b), R1C1 = 3 -> R1C3 = R4C2, no 4 in R1C3
24e. R1C79 = R4C8 + 7 (step 12b), R1C9 = 7 -> R1C7 = R4C8, no 4 in R1C7

25. Naked pair {69} in R45C9, locked for C9 and N6, clean-up: no 3 in R6C9, no 4 in R7C78
25a. Naked pair {48} in R67C9, locked for C9

26. Hidden killer pair 1,2 in R23C9 and R89C9 for C9, R23C9 cannot contain both of 1,2 because they would clash with R1C7 (or R4C8), R89C9 cannot contain both of 1,2 -> R23C9 and R89C9 must each contain one of 1,2
26a. Killer pair 1,2 in R1C7 and R23C9, locked for N3, clean-up: no 8 in R23C8

27. 8 in N2 locked in R12C6, locked for C6
27a. 16(3) cage at R1C6 = {178/268}, no 3,5
27b. R1C7 = 1,2 -> no 1,2 in R12C6
27c. Naked triple {678} in R129C6, locked for C6

28. R6C6 = 5 (hidden single in C6), R6C4 = 9
28a. R8C6 = 9 (hidden single in C6)

29. R2C4 = 3 (hidden single in N2), R1C34 = 6 = {15}, locked for R1 -> R1C7 = 2, R4C8 = 2 (step 24e), R9C6 = 7 (step 5), clean-up: no 6 in R3C8, no 6 in R4C2 (step 24d), no 8 in R67C7, no 8 in R67C8
29a. Naked pair {68} in R12C6, locked for N2, clean-up: no 7 in R4C4

30. R3C6 = 2 (hidden single in N2), R4C6 = 3
30a. 4 in N5 locked in R5C46, locked for R5

31. 45 rule on N6 3 remaining innies R6C789 = 12 = {138/147}, 1 locked for R6 and N6, clean-up: no 8 in R7C1, no 9 in R7C3
31a. 4 of {147} must be in R6C9, no 4 in R6C78, clean-up: no 6 in R7C78

32. 1 in N3 locked in R23C9, locked for C9
32a. 2 in C9 locked in R89C9 -> 15(3) cage at R8C8 = {258} (only remaining combination, cannot be {267} because 6,7 only in R8C8) -> R8C8 = 8, R89C9 = {25}, locked for C9 and N9, R23C9 = [13], R67C9 = [84], R7C6 = 1, R5C6 = 4, clean-up: no 6 in R2C8, no 7 in R6C2, no 6 in R6C3, no 7 in R6C78 (step 31), no 3 in R7C2, no 2 in R7C3, no 3 in R7C78

33. Naked pair {45} in R23C8, locked for C8 and N3
33a. Naked pair {13} in R6C78, locked for R6 and N6 -> R5C78 = [57], R4C7 = 4, R7C8 = 9, R6C8 = 1, R67C7 = [37], R1C8 = 6, R12C6 = [86], R9C8 = 3, clean-up: no 9 in R3C2, no 9 in R4C1, no 2 in R6C1, no 2,4 in R6C2, no 8 in R7C2
33b. R67C2 = [65], R6C5 = 2, R5C45 = [18], R1C34 = [15], R34C4 = [76], R34C5 = [17], R45C9 = [96], R9C45 = [26], R78C4 = [84], R78C5 = [35], R89C9 = [25], R5C1 = 9, R4C1 = 5, R4C23 = [18], R89C7 = [61]

35. R7C1 = 2, R6C1 = 7

and the rest is naked singles.
Partial Walkthrough by Ed:
Here's a completely different way to get into this puzzle. No combo work just "45"s! More suitable for manu's V2 really, if he ever gets to post it.

Just to the first couple of placements. [edit: a couple of sub-steps shifted around for clarity; edit2: some important clarifications - thanks Andrew!]

Walk-partway for Assassin 153 [7 steps]

Prelims
i. 10(2)r1c19: no 5
ii. 9(3)n1: no 7,8,9
iii. 13(2)n2: no 1,2,3
iv. 15(2)n1 = {69/78}
v. 9(2)n3: no 9
vi. 13(2)n2: no 1,2,3
vii. 8(2)n2: no 4,8,9
viii. 5(2)n2 = {14/23}
ix. 14(2)n4 = {59/68}
x. 15(2)n6 = {69/78}
xi. 9(2)n4: no 9
xii. 11(2)n4: no 1
xiii. 10(2): no 5
xiv. 10(2) at r6c7 & r6c8: no 5
xv. 12(2)n6: no 1,2,6
xvi. 8(2)n8: no 4,8,9

1. "45" r89: 4 outies r6c5 + r7c456 = 14
1a. min. r7c456 = 6 -> max. r6c5 = 8

2. "45" n8: r6c5 + 7 = r9c46
2a. since the outie r6c5 sees all cells in n8 except the two innies -> the IOD (7) is locked in r9c46 (IOE: see here for a full explanation of this technique)
2b. 7 locked in r9c46 for r9 & n8
2c. no 7 in r6c5 (since r9c46 cannot be [77]
2d. no 9 in r9c46 since the other one of r9c46 = r6c5 (IOE)
2e. no 1 in 8(2)n8

3. "45" n3: 2 outies r1c1 + r4c8 - 3 = r1c7
3a. -> no 3 in r4c8 (IOU)

4. "45" n69: 1 outie r9c6 - 5 = 1 innie r4c8
4a. r9c6 = (67)
4b. r4c8 = (12)

5. "45" n3: 1 outie r4c8 + 7 = 2 innies r1c79
5a. since the one outie r4c8 see's all cells in n3 except the 2 innies -> IOD (7) locked in r1c79 (IOE)
5b. 7 locked for r1 & n3
5c. and r1c79 = 7{1/2} (since the other cell of r1c79 = r4c8 IOE)
5d. r1c1 = (389)
5e. no 2 in 9(2)n3

6. "45" n47: 1 outie r9c4 - 1 = 1 innie r4c2
6a. -> max r4c2 = 7, min. r9c4 = 2

7. "45" n1: 1 outie r4c2 + 3 = 2 innies r1c13
7a. and again! -> IOD (3) locked in r1c13 (IOE)
7b. 3 locked for r1 & n1
7c. no 8 or 9 in r1c1 (since the other cell of r1c13 = r4c2 IOE)
7d. r1c1 = 3
7e. r1c9 = 7 (cage sum)

On you go from there.


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PostPosted: Thu Nov 03, 2011 10:21 pm 
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Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 153 V2 by manu (May 2009) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:2048:7169:2818:2818:3076:4357:4357:7431:2048:7169:3594:7169:2818:3076:4357:7431:2832:7431:7169:3594:7169:2837:1558:1303:7431:2832:7431:3867:7169:5149:2837:1558:1303:4385:7431:3619:3867:5149:5149:7463:7463:7463:4385:4385:3619:3629:5149:2607:7463:7729:7463:2099:4385:5429:3629:3629:2607:7729:2618:7729:2099:5429:5429:3391:3391:4929:7729:2618:7729:3909:4678:4678:3391:4929:4929:4929:7729:3909:3909:3909:4678:
Solution:
+-------+-------+-------+
| 3 8 1 | 7 4 9 | 2 6 5 |
| 4 5 2 | 3 8 6 | 9 7 1 |
| 6 9 7 | 5 1 2 | 8 4 3 |
+-------+-------+-------+
| 7 1 9 | 6 5 3 | 4 2 8 |
| 8 2 3 | 1 9 4 | 7 5 6 |
| 5 6 4 | 8 2 7 | 3 1 9 |
+-------+-------+-------+
| 2 7 6 | 9 3 1 | 5 8 4 |
| 1 3 5 | 4 7 8 | 6 9 2 |
| 9 4 8 | 2 6 5 | 1 3 7 |
+-------+-------+-------+
Quote:
manu: Of course, the ideas you have used for the previous V1 will be useful, but you will certainly need other ones for cracking this one...
SSscore : 2.37

Ed: Got this V2 out way quicker than Ronnie's A154! So thank you manu! It just takes ... and no, it's nothing to do with IO anything :) . I felt reluctant to use it because of how many elements are in it but couldn't find a simpler way.

Andrew:
Ed wrote:
Got this V2 out way quicker than Ronnie's A154!
In my case I was stuck on Afmob's A155, a fairly low rated puzzle that is clearly a lot harder than it rating, so I had a go at A153 V2.
manu wrote:
Of course, the ideas you have used for the previous V1 will be useful, but you will certainly need other ones for cracking this one...
Thanks manu for another excellent puzzle!
As you said the ideas used for the V1 were useful ...
My key breakthrough was step 15a, which is fairly similar to Ed's step 7a; I think spotting that step is almost certainly essential for a reasonable length solution. The SSscore is much higher than my rating so it's possible that it didn't find this step and used something more difficult.
I'll rank A153 V2 at 1.5 because of step 15a.

manu: I post the way I have solved that V2, especially for step 2 that offers another way for cracking this puzzle.

Walkthrough by Ed:
Got this V2 out way quicker than Ronnie's A154! So thank you manu! It just takes a single composite move to do crack it (step 7) and no, it's nothing to do with IO anything :) . I felt reluctant to use it because of how many elements are in it but couldn't find a simpler way.

Please let me know of any corrections or clarifications. Thanks.

Assassin 153 V2

Prelims
i. 8(2)r1c19: no 4,8,9
ii. 11(3)n1: no 9
iii. 12(2)n2: no 1,2,6
iv. 14(2)n1 = {59/68}
v. 11(2)n3: no 1
vi. 11(2)n2: no 1
vii. 6(2)n2 = {15/24}
viii. 5(2)n2 = {14/23}
ix. 15(2)n4 = {69/78}
x. 14(2)n6 = {59/68}
xi. 10(2)n4: no 5
xii. 8(2)n6: no 4,8,9
xiii. 21(3)n6: no 1,2,3
xiv. 10(2)n8: no 5

1. "45" n3: 2 outies r1c1 + r4c8 - 3 = 1 innie r1c7
1a. -> no 3 in r4c8 (IOU)

2. the 30(6)n5 and 10(2)n8 both "see" all of each other -> as a combined cage they make a 40(8) cage -> no 5

3. 5 in n8 only in r9c46: locked for r9.

4. "45" n69: 1 outie r9c6 - 3 = r4c8
4a. r9c6 no 1,2,3,6
4b. r4c8 no 7,8,9

5. "45" r123: 5 outies r4c24568 = 17 = 123{47/56}(no 8,9)
5a. 1,2 & 3 all locked for r4

6. "45" n47: 1 outie r9c4 - 1 = r4c2
6a. no 1,9 in r9c4

7. No 4 in r4c2. Like this.
7a. since 5 is locked at r9c46 and each of these two cells is linked to r4c28 -> r4c2 = 4 (step 6) OR r4c8 = 2 (step 4) and cannot be both
7b. for r4c2 to equal 4, the h17(5) in r5 can only be {12347} (step 5)
7c. and with 1 only in r4c8 (the only other available candidate is 2 but this is blocked see 7a.)
7c. -> the only permutation is [47231]
7d. but r4c45 as [72] is blocked by r3c45 = [44] (cage sums)
7e. -> no 4 in r4c2
7f. no 5 in r9c4 (i/o n47)

8. r9c6 = 5 (hsingle r9)
8a. other 3 cells in 15(4)n8 = 10 = 1{27/36}(no 4,8,9)
8b. 1 locked for n9
8c. no 7 in r6c7

The puzzle is cracked. I'll miss lots of clean-up since just getting it to singles ASAP.
9. "45" n69: 1 innie r4c8 = 2

10. "45" n6: 2 remaining innies r6c79 = 12 = [39/57] = [5/9...]

11. 14(2)n6 = {68} ({59} blocked by h12(2)n6 step 10)
11a. both locked for c9 and n6

12. 8(2)n6 = {35}: both locked for c7

13. "45" n3: 2 innnies r1c79 = 7 and must have 2 for n3 = [25]
13a. r1c1 = 3 (cage sum)

14. 15(4)n9 = {1356} only combination
14a. -> r9c8 = 3
14b. r89c7 = {16}: both locked for c7 & n9

15. r67c7 = [35]
15a. -> r6c9 = 9 (h12(2)n6)

16. h17(5)r4 = {1356}[2] ({1347}[2] blocked by r4c7)
16a. all locked for r4

17. r45c9 = [86]

18. r45c1 = [78] (last permutation)

19. "45" n1: 1 innie r1c3 = 1 outie r4c2 = (16)

20. "45" n4: 3 innies r4c2 + r6c13 = 10 = [1][54] (only permutation)
20a. -> r1c3 = 1 (step 19)

21. r7c89 = 12 = [84] (only permutation)

22. "45" n47: 1 outie r9c4 = 2

23. r7c3 = 6 (cage sum)

24. r7c12 = 9 = [27] (only permutation)

25. r4c34567 = [96534]
25a. r3c456 = [512] (cage sums)

26. r12c4 = 10 = [73] (last permutation)

27. r6c2 = 6 (hsingle n4)

28. r23c2 = [59] (last permutation)

29. r23c1 = {46}: both locked for n1 & c1
29a. r1c2 = 8

30. r12c5 = [48] (last permutation)

rest is singles.
Walkthrough by Andrew:
Ed wrote:
Got this V2 out way quicker than Ronnie's A154!
In my case I was stuck on Afmob's A155, a fairly low rated puzzle that is clearly a lot harder than it rating, so I had a go at A153 V2.

manu wrote:
Of course, the ideas you have used for the previous V1 will be useful, but you will certainly need other ones for cracking this one...
Thanks manu for another excellent puzzle!

As you said the ideas used for the V1 were useful, this time I started with the hexagons, but it needed more. My key breakthrough was step 15a, which is fairly similar to Ed's step 7a; I think spotting that step is almost certainly essential for a reasonable length solution. The SSscore is much higher than my rating so it's possible that it didn't find this step and used something more difficult.

Until I found that breakthrough move I looked several times at the hidden killer pair 3,5 in R1C19 and R1C37 for R1, either R1C19 = [35] or R1C37 = [35] but I was never able to find anything that stopped R1C37 being [35].

I'll rank A153 V2 at 1.5 because of step 15a.

Here is my walkthrough. I've deleted the original step 12 and rewritten some of the other steps for clarity.

Prelims

a) R1C19 = {17/26/35}, no 4,8,9
b) R12C5 = {39/48/57}, no 1,2,6
c) R23C2 = {59/68}
d) R23C8 = {29/38/47/56}, no 1
e) R34C4 = {29/38/47/56}, no 1
f) R34C5 = {15/24}
g) R34C6 = {14/23}
h) R45C1 = {69/78}
i) R45C9 = {59/68}
j) R67C3 = {19/28/37/46}, no 1
k) R67C7 = {17/26/35}, no 4,8,9
l) R78C5 = {19/28/37/46}, no 5
m) 11(3) cage at R1C3 = {128/137/146/236/245}, no 9
n) 21(3) cage at R6C9 = {489/579/678}, no 1,2,3

In the first 3 steps, the central 2-cell cages “see” all the cells of the surrounding hexagons so each hexagon plus central cage forms a 8-cell cage.

1. 28(6) cage at R1C2 + R23C2 = 42(8) = {12456789}, no 3
1a. 3 in N1 locked in R1C13, locked for R1, clean-up: no 5 in R1C1, no 9 in R2C5
1b. 45 rule on N1 2 innies R1C13 = 1 outie R4C2 + 3
1c. R1C13 contains 3 and one of {1245678} -> R4C2 = {1245678}, no 9
[Steps 1b and 1c, and the corresponding parts of steps 2 and 3, can also been seen as IOEs.]

2. 29(6) cage at R1C8 + R23C8 = 40(8) = {12346789}, no 5, clean-up: no 6 in R23C8
2a. 5 in N3 locked in R1C79, locked for R1, clean-up: no 7 in R2C5, no 5 in R4C2 (step 1c)
2b. 45 rule on N3 2 innies R1C79 = 1 outie R4C8 + 5
2c. R1C79 contains 5 and one of {1246789} -> R4C8 = {1246789}, no 3

3. 30(6) cage at R6C5 + R78C5 = 40(8) = {12346789}, no 5
3a. 5 in N8 locked in R9C46, locked for R9
3b. 45 rule on N8 2 innies R9C46 = 1 outie R6C5 + 5
3c. R9C46 = contains 5 and one of {12346789} -> R6C5 = {12346789}
[No eliminations at this stage but step 3c will be used for clean-ups later.]

4. 45 rule on N4 3 innies R4C2 + R6C13 = 10 = {127/136/145/235}, no 8,9, clean-up: no 8 in R1C3 (step 1c), no 1,2 in R7C3
4a. 5 of {145} must be in R6C1 -> no 4 in R6C1

5. 45 rule on N47 1 outie R9C4 = 1 innie R4C2 + 1, no 1,4,6,9 in R9C4

6. 45 rule on N69 1 outie R9C6 = 1 innie R4C8 + 3, no 7,8,9 in R4C8, no 1,2,3,6,8 in R9C6, clean-up: no 7,8,9 in R1C79 (step 2c), no 1 in R1C1, no 1,6 in R6C5 (step 3c)

7. 45 rule on R123 5 outies R4C24568 = 17 = {12347/12356}, no 8,9, 1,2,3 locked for R4, 3 locked for N5, clean-up: no 2,3 in R3C4
7a. R6C5 = {24789} -> R9C46 (step 3c) must contain 5 and one of {24789}, no 3, clean-up: no 2 in R4C2 (step 5)
7b. R4C2 = {1467} -> R1C13 (step 1c) must contain 3 and one of {1467}, no 2, clean-up: no 6 in R1C9

8. 3 in N5 locked in R4C46
8a. 45 rule on N5 4 innies R4C456 + R6C5 = 16 = {1348/1357/2347/2356}, no 9
8b. R6C5 = {2478} -> R9C46 (step 3c) must contain 5 and one of {2478}, no 9, clean-up: no 6 in R4C8 (step 6)
8c. R4C8 = {124} -> R1C79 (step 2c) = must contain 5 and one of {124}, no 6

9. R4C2 + R6C13 (step 4) = {127/136/145} (cannot be {235} because R4C2 only contains 1,4,6,7), 1 locked for N4

10. 45 rule on N6 3 innies R4C8 + R6C79 = 14 = {149/167/239/248/257/347} (cannot be {158} which clashes with R45C9, cannot be {356} because R4C8 only contains 1,2,4)
10a. 8,9 of {149/248} must be in R6C9, 4 of {347} must be in R4C8 -> no 4 in R6C9

11. 21(3) cage at R6C9 = {489/579/678}
11a. 4 of {489} must be in R7C9 (R67C9 cannot be {89} which clashes with R45C9), no 4 in R7C8

12. Deleted. This step, using 45 rule on C5 and then combined cage R34569C5, was unnecessarily complicated. Step 23b has been added to make the same eliminations.

13. Max R1C7 = 5 -> min R12C6 = 12, no 1,2
13a. Max R9C6 = 4 -> min R8C7 + R9C78 = 11, no 9

14. 45 rule on C9 2 outies R78C8 = 3 innies R123C9 + 8
14a. Min R123C9 = 6 -> min R78C8 = 14, no 1,2,3,4
14b. Max R78C8 = 17 -> max R123C9 = 9, no 7,8,9
14c. 7 in C9 locked in R6789C9, CPE no 7 in R7C8

15. 5 in N8 locked in R9C46
15a. Either R9C4 = 5 or R9C6 = 5 -> either R4C2 = 4 (step 5) or R4C8 = 2 (step 6) -> no 4 in R4C8, clean-up: no 7 in R9C6 (step 6)
15b. R4C8 = {12} -> R1C79 (step 2c) must contain 5 and one of {12}, no 4

16. R4C8 + R6C79 (step 10) = {167/239/257}, no 8
16a. R4C8 = {12} -> no 1,2 in R6C7, clean-up: no 6,7 in R7C7

17. 4 in N6 locked in 17(4) cage = {1349/1457/2348} (cannot be {2456} which clashes with R45C9), no 6

18. 15(4) cage at R8C7 = {1248/1257/1347/1356/2346}
18a. R9C6 = {45} -> no 4,5 in R8C7 + R9C78
18b. 4 in N9 locked in R789C9, locked for C9

19. Max R123C9 = 9 (step 14b) -> R123C9 = {123/125/126/135}, 1 locked for C9 and N3

20. R4C24568 (step 7) = {12347/12356}
20a. Either R4C2 = 4 or R4C8 = 2 (step 15a)
20aa. {12347} can only be [47231]
20ab. 2 of {12356} must be in R4C8
20b. -> no 7 in R4C2, no 2,4 in R4C46, no 4 in R4C5, clean-up: no 7,9 in R3C4, no 2 in R3C5, no 1,3 in R3C6, no 8 in R9C4 (step 5)
[I ought to have spotted this step when I did step 15a.
After step 20aa can eliminate [47231] because that would make R3C45 = [44], which is a multidirectional clash. However this elimination comes quickly by more normal steps so I haven’t used that clash.]

21. R9C46 contains 5 and one of {247} -> R6C5 = {247}, no 8 (step 3c)

22. R4C2 = {146} -> R1C13 (step 1c) must contain 3 and one of {146}, no 7, clean-up: no 1 in R1C9
22a. Naked pair {25} in R1C79, locked for R1 and N3, clean-up: no 9 in R23C8

23. 1 in C9 locked in R23C9, locked for 29(6) cage at R1C8 -> R4C8 = 2, R9C6 = 5 (step 6), clean-up: no 4 in R4C2 (step 15a), no 7 in R4C4 (step 20aa), no 4 in R3C4
23a. Naked quad {1356} locked in R4C2456, locked for R4, 5 locked for N5, clean-up: no 9 in R5C1, no 8,9 in R5C9
23b. Naked pair {15} in R34C5, locked for C5, clean-up: no 9 in R78C5

24. R4C2 = {16} -> R1C13 (step 1c) must contain 3 and one of {16}, no 4

25. R9C46 = [25/75] -> R6C5 (step 3c) = {27}, no 4

26. R9C6 = 5 -> 15(4) cage at R8C7 (step 18) = {1257/1356}, no 8, 1 locked for N9, clean-up: no 7 in R6C7

27. R4C8 = 2 -> R4C8 + R6C79 (step 16) = {239/257}, no 6, clean-up: no 2 in R7C7
27a. 5 of {257} must be in R6C7 -> no 5 in R6C9

28. Naked pair {35} in R67C7, locked for C7 -> R1C79 = [25], R1C1 = 3, R5C9 = 6, R4C9 = 8, clean-up: no 9 in R4C1, no 7 in R5C1
28a. R45C1 = [78], clean-up: no 3 in R7C3

29. 15(4) cage at R8C7 (step 26) = {1356} (only remaining combination) -> R9C8 = 3, R89C7 = {16}, locked for C7 and N9, R67C7 = [35], clean-up: no 8 in R23C8, no 7 in R7C3

30. 21(3) cage at R6C9 = {489} (only remaining combination) -> R6C9 = 9, R7C89 = [84], R45C7 = [47], R4C3 = 9, R7C3 = 6, R6C3 = 4, R1C3 = 1, clean-up: no 2 in R3C5, no 2,4,6 in R8C5

31. Naked pair {47} in R23C8, locked for C8 -> R8C8 = 9, R1C8 = 6

32. R1C13 = [31] -> R4C2 = 1 (step 1c), R34C5 = [15], R4C6 = 3, R3C6 = 2, R4C4 = 6, R3C4 = 5, R6C1 = 5 (step 4), R56C8 = [51], R6C5 = 2 (step 8a), R9C4 = 2 (step 5), R89C9 = [27], R9C3 = 8, R3C3 = 7, R23C8 = [74], R23C9 = [13], clean-up: no 7 in R1C5, no 9 in R2C2, no 8 in R8C5

33. R9C34 = [82] = 10 -> R8C3 + R9C2 = 9 = [54], R2C3 = 2, R5C23 = [23], R6C2 = 6, clean-up: no 8 in R23C2
33a. R23C2 = [59]

34. R6C1 = 5 -> R7C12 = 9 = [27]

and the rest is naked singles and a cage sum.
Walkthrough by manu:
I post the way I have solved that V2, especially for step 2 that offers another way for cracking this puzzle.

1)a) Cells of 28(6) and 14(2) at n1 see each other : 28(6)+14(2) = 42(7) = {12456789} : no 3
→ 3 locked for n1/r1 at r1c13
b) Cells of 29(6) and 11(2) at n3 see each other : 29(6)+11(2) = 40(7) = {12346789} : no 5
→ 5 locked for n3/r1 at r1c79
c) Cells of 30(6) and 10(2) at n8 see each other : 30(6)+10(2) = 40(7) = {12346789} : no 5
→ 5 locked for n8/r9 at r9c46
d) Digit at r4c8 is locked for n3 at r1c79 → r4c8<>3
e) IO for n69 : r9c6=3+r4c8 → r9c6<>1,2,3 and r4c8<>7,8,9 → r4c8=(1246), r9c6=(4579)

2)a) IO for c9 : r7c8+r8c8 = 8 + r1c9+r2c9+r3c9
b) r7c8+r8c8 >= 14 : no 1,2,3,4
c) digits at r78c8 can't be at r123c9. Like this : if one of r78c8 was locked at r123c9, the other one would be equal to (step a) 8 + two of r123c9 >= 11.
d) We deduce from c) that r7c8 is locked at 14(2) at c9 since r7c8 sees r6789c9: r7c8=(5689)
e) From step a), r1c9+r2c9+r3c9=r7c8+r8c8 - 8 <= 17-8=9 : no 7,8,9 in r123c9
f) r67c9 <> 7. Like this : if one of r67c9 was 7, the other one and r7c8 would total 14 with the same combination as 14(2) at n9 since r7c8 is locked at 14(2), and there would be a clash between 14(2) and 21(3)
→ 7 locked for n9 at r89c9, clean up : r6c7<>1
g) At n6 : 21(3) <> 7 : {489}, 4 locked at r67c9 for c9
h) Innies for n6 : r4c8+r6c7+r6c9=14=h14(3) : permutations [239/428] ([158] is not possible since {58} blocks 14(2) at n6)
→ r4c8=(24), r6c7=(23) and r6c9=(89)
i) Last cell for 4 at 21(3) : r7c9=4

3)a) Outies for r89 : r6c5+r7c4+r7c5+r7c6 = h15(4)
→ r6c5 <> 7 since all combinations of h15(4) would be blocked by r7c9=4 and digit 5 locked at r9c46
b) Digit 7 of 40(7) (step 1)b)) is locked at n8 → r9c6<>7 → (step 1)e)) r4c8 <> 4
(that was proven in a different way in Andrew'WT)
c) From step 2)h), h14(3) at n6 is [239] → r7c7=5, r7c8=8
d) 18(3) at n9 (remind 7 is locked in) : last combination {279} → r8c8=9, r89c9= {27} locked

4)a) At n6 : 14(2)= {68} locked . At n3 : r1c9=5, r1c1=3, r23c9={13} locked and 11(2)={47} locked
b) Hidden pair : r45c7= {47} locked, r56c8={15} locked → r1c8=6 and r9c8=3. Hidden single : r1c7=2
c) r4c2 is locked for n1 at r1c3 : → r4c2 = (1478)
d) Outies for r123 : r4c24568 total 17 : {12356} since {12347} is blocked by r4c7=(47)
→ only valid candidate : r4c2 = 1
→ only valid candidate : r4c5=5
→ For the same reason : r4c6=3, and r4c4=6

Everything easy from here (singles, last cage combination)


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PostPosted: Fri Nov 04, 2011 9:56 pm 
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Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 154 by Ronnie G (May 2009) here
Puzzle Diagram: unfortunately, the original has been lost:
Image
Note: This is NOT a Killer-X
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:5888:4609:4609:4611:5124:7685:4358:4358:3592:4609:5888:4611:5124:5124:5124:7685:3592:4358:4609:4611:5888:4885:4885:4885:3592:7685:4358:4611:6172:6172:5888:4885:3592:3873:3873:7685:6172:6172:1830:4647:7464:4137:2858:3873:3873:4653:1830:1071:4647:7464:4137:2099:2858:4149:4406:4653:1071:4647:7464:4137:2099:4149:4670:4406:4406:4653:2882:7464:2882:4149:4670:4670:4936:4936:4936:4653:2882:4149:3918:3918:3918:
Solution:
+-------+-------+-------+
| 7 5 9 | 3 2 6 | 8 1 4 |
| 1 8 6 | 5 4 9 | 7 3 2 |
| 3 4 2 | 7 1 8 | 5 9 6 |
+-------+-------+-------+
| 5 9 7 | 6 3 2 | 1 4 8 |
| 2 6 4 | 1 8 5 | 9 7 3 |
| 8 3 1 | 9 7 4 | 6 2 5 |
+-------+-------+-------+
| 4 1 3 | 8 5 7 | 2 6 9 |
| 6 7 5 | 2 9 3 | 4 8 1 |
| 9 2 8 | 4 6 1 | 3 5 7 |
+-------+-------+-------+
Quote:
Ronnie G: I had hoped to post this on Friday, but it took me forever to solve this puzzle. Since there's been no A154 posted, I'll make this the official one!
I recently asked Afmob how he made the "criss-cross" cage pattern he used in A148. (I think other puzzle makers have used similar patterns.) I've always used Sumocue to draw out potential puzzle diagrams, and it's very cumbersome to include such a pattern. Afmob told me to try using JSudoku. Eureka!!!
You may be sorry he shared this with me, once you take a good look at this cage pattern. I went a little nuts with the criss-crosses. :shock:
I've used Richard's nifty "colouring" tool in the hopes it will provide some visual clarity. [Moderator edit: unfortunately, that image has been lost]
Enjoy!
SS score: 1.37.
udosuk: Been away for a long while. Thanks for a nicely easy puzzle as a welcome-back. :alien:

Afmob: I pretty much solved this Assassin the same way as udosuk, so no walkthrough from me.
I would rate this path of solving 1.0 - Hard 1.0
(Full post below, in the walkthroughs)

Andrew: Thanks Ronnie for a challenging Assassin.
Many of my steps were the same as those used in udosuk's walkthrough but I took longer to find the early key moves and didn't spot the important part of his final breakthrough...
I'll rate my walkthrough at least Easy 1.25 ...

Ed:
Ronnie G wrote:
it took me forever to solve this puzzle
Me too! I missed Andrew's step 12 (udosuk's line 11) :oops: :oops: . Always struggle to find innies. Have to go out and look for them.
Ronnie G wrote:
"criss-cross" cage pattern ...Sumocue's very cumbersome to include such a pattern.
Do you use Shift with click to make the cages? Perhaps JSudoku is even easier. Never tried.
Thanks Ronnie.

Walkthrough by udosuk:
Been away for a long while. Thanks for a nicely easy puzzle as a welcome-back. :alien:

Here is my WT for Assassin 154

30/4 @ r1c6={6789}
29/4 @ r5c5={5789}
Outies @ c789: r149c6=9
=> r1c6=6, r49c6={12} (6 not elsewhere @ 30/4)
Innie-outies @ n2: r1c4=r4c5 must be from {1234}
r1c4+r123c5={1234}
Hidden single @ c5: r9c5=6
=> 11/3 @ r8c4: r8c46=11-6=5=[14|23]
Innies @ c1234: r238c4=14=[{57}2|{58}1] (5 locked)
Innies @ r9: r9c46=5=[32|41]
Innies @ n58: r4c456=16-5=11=[632|641|731|8{12}]
(r4c4 must be from {678})
=> 9 @ c4 locked @ 18/3 @ r5c4={189|279|369} (no 4)
Outies @ r123: r4c19=24-11=13=[49|58|67]
Outies @ n1: r1c4+r4c14=14
=> max r1c4+r4c1=14-6=8, can't be [45]=9
=> r1c4 can't have 4
Hidden single @ c4: r9c4=4 (not elsewhere @ 18/4)

All very easy from here.
:sun:
Afmob's full post:
I pretty much solved this Assassin the same way as udosuk, so no walkthrough from me.

I would rate this path of solving 1.0 - Hard 1.0 since to get to the important hidden cages (Innies N58, Outies R123) you had to use other hidden cages (Innies R9, Innies N58), so they aren't immediately there.
Walkthrough by Andrew:
Thanks Ronnie for a challenging Assassin.

Many of my steps were the same as those used in udosuk's walkthrough but I took longer to find the early key moves and didn't spot the important part of his final breakthrough Outies @ n1: r1c4+r4c14=14; I'd looked at those outies earlier but maybe not after restricting r4c4 to {678}.

I'll rate my walkthrough at least Easy 1.25 because of step 12, which I found difficult to spot and because it's 5(3+2) innies, step 21 and I'd also rate the move I missed at this level.

Here is my walkthrough.

Prelims

a) 7(2) cage in N4 = {16/25/34}, no 7,8,9
b) 11(2) cage in N6 = {29/38/47/56}, no 1
c) R67C3 = {13}, locked for C3, clean-up: no 4,6 in R6C2
d) R67C7 = {17/26/35}, no 4,8,9
e) 19(3) cage in N7 = {289/379/469/478/568}, no 1
f) 11(3) cage in N8 = {128/137/146/236/245}, no 9
g) 30(4) cage at R1C6 = {6789}, CPE no 6,7,8 in R1C9
h) 14(4) cage at R1C9 = {1238/1247/1256/1346/2345}, no 9
i) R5678C5 = {5789}, locked for C5

1. 45 rule on N7 3 innies R7C23 + R8C3 = 9 = {126/135/234}, no 7,8,9
1a. 5 of {135} must be in R8C3 -> no 5 in R7C2

2. 45 rule on R9 3 innies R9C456 = 11 = {128/137/146/236/245}, no 9

3. 45 rule on N69 1 innie R4C9 = 1 outie R9C6 + 7 -> R4C9 = {89}, R9C6 = {12}
3a. 11(3) cage in N8 = {137/146/236/245} (cannot be {128} which clashes with R9C6), no 8
3b. Killer pair 1,2 in 11(3) cage and R9C6, locked for N8

4. 45 rule on C6789 3 innies R238C6 = 20 = {389/479/569/578}, no 1,2
4a. 8,9 of {389} must be in R23C6 -> no 3 in R23C6

5. 45 rule on N6 3 innies R4C9 + R6C79 = 19 = {289/379/469/478/568}, no 1, clean-up: no 7 in R7C7
5a. 8,9 of {289} must be in R46C9 -> no 2 in R6C9

6. 45 rule on N47 1 innie R4C1 = 1 outie R9C4 + 1, no 1,2,3 in R4C1

7. 45 rule on C789 3 outies R149C6 = 9 -> R1C6 = 6, R49C6 = {12}, locked for C6
[This was there straight after the prelims. If not seen then I ought to have spotted it after step 4 from 45 rule on C6.]
7a. 7 of 30(4) cage at R1C6 locked in R2C7 + R3C8, locked for N3
7b. R2C7 + R3C8 + R4C9 = {789}, CPE no 8,9 in R23C9

8. 45 rule on N2 1 remaining innie R1C4 = 1 outie R4C5, no 5,7,8,9 in R1C4, no 6 in R4C5
8a. Naked quad {1234} in R1C45 + R23C5, locked for N2

9. R9C5 = 6 (hidden single in C5), clean-up: no 7 in R4C1 (step 6)
9a. 11(3) cage in N8 = {146/236}, no 5,7
9b. R9C456 (step 2) = {146/236} -> R9C46 = 5 = {14/23}, no 5,7,8, clean-up: no 6,8,9 in R4C1 (step 6)
9c. Naked pair {34} in R8C6 and R9C4, locked for N8, CPE no 4 in R8C3
9d. Naked pair {12} in R8C4 + R9C6, CPE no 1,2 in R8C7

10. R238C6 (step 4) = {389/479} (cannot be {569/578} because R8C6 only contains 3,4), no 5, 9 locked for C6 and N2
10a. 5 in N2 locked in R23C4, locked for C4

11. 45 rule on N4 3 innies R4C1 + R6C13 = 14, max R4C1 + R6C3 = 8 -> min R6C1 = 6
11a. R4C1 + R6C13 = {149/158/347} (cannot be {167} because 6,7 only in R6C1, cannot be {356} which clashes with 7(2) cage), no 6

12. 45 rule on N58 5(3+2) innies R4C456 + R9C46 = 16
12a. R9C46 = 5 (step 9b) -> R4C456 = 11 = {128/137/146/236}, no 9
12b. 6,7,8 only in R4C4 -> R4C4 = {678}

13. 9 in C4 locked in R567C4 = {189/279/369}, no 4

14. R9C789 = {159/258/357} (cannot be {249} which clashes with R9C46, cannot be {348} which clashes with R9C4), no 4, 5 locked for R9 and N9, clean-up: no 3 in R6C7
14a. 1 in R9 locked in R9C6789, CPE no 1 in R7C8

15. R9C123 = {289/379/478}
15a. 17(3) cage in N7 = {269/359/458/467} (cannot be {179/278/368} which clash with R9C123), no 1

16. 1 in N7 locked in R7C23, locked for R7, clean-up: no 7 in R6C7
16a. 1 locked in R7C23 + R8C3 (step 1) = {126/135}, no 4

17. 45 rule on N9 3 innies R7C78 + R8C7 = 12 = {237/246}, no 8,9, 2 locked in R7C78, locked for R7 and N9, clean-up: no 8 in R9C789 (step 14)

18. 8 in R9 locked in R9C123, locked for N7
18a. R9C123 (step 15) = {289/478}, no 3

19. R4C9 + R6C79 (step 5) = {289/469/568} (cannot be {379/478} because R6C7 only contains 2,5,6), no 3,7

20. 15(4) cage in N6 = {1239/1257/1347/1356} (cannot be {1248} which clashes with R4C9 + R6C79), no 8

21. 45 rule on R123 5 outies R4C14569 = 24
21a. 45 rule on N3 2 remaining outies R4C69 = 10 = [19/28]
21b. -> R4C145 = 14 = {248/257/347/356} (cannot be {158} which clashes with R4C69, cannot be {167/239} because R4C1 only contains 4,5), no 1, clean-up: no 1 in R1C4 (step 8)
21c. 4 of {347} must be in R4C1 -> no 4 in R4C5; clean-up: no 4 in R1C4 (step 8)

22. R9C4 = 4 (hidden single in C4), R4C1 = 5 (step 6), R9C6 = 1 (step 2), R4C9 = 8 (step 3), R8C46 = [23], R1C4 = 3, R4C56 = [32], R4C4 = 6 (step 21b), clean-up: no 2 in 7(2) cage in N4, no 3 in 11(2) cage in N6, no 7 in R567C4 (step 13), no 7 in R9C123 (step 18a), no 9 in R9C789 (step 14)

23. Naked pair {13} in R6C23, locked for R6 and N4

24. Naked pair {89} in R67C4, locked for C4 -> R5C4 = 1
24a. Naked pair {57} in R23C4, locked for N2
24b. Naked pair {89} in R23C6, locked for C6

25. R4C1 + R6C13 (step 11a) = {158} (last remaining combination) -> R6C13 = [81], R6C2 = 3, R5C3 = 4, R67C4 = [98], R7C3 = 3, clean-up: no 2 in R5C7, no 5 in R6C7, no 7 in R6C8
25a. R7C2 = 1 (hidden single in R7), R8C3 = 5 (step 16a)

26. Naked pair {57} in R5C6 + R6C5, locked for N5 -> R5C5 = 8, R6C6 = 4, clean-up: no 7 in R5C7
26a. Naked triple {256} in R6C789, locked for R6 and N6 -> R6C5 = 7, R57C6 = [57], R78C5 = [59], R5C7 = 9, R6C8 = 2, R67C7 = [62], R6C9 = 5, R2C7 = 7, R3C8 = 9, R23C4 = [57], R8C7 = 4, R4C7 = 1, R7C89 = [69], R7C1 = 4
26b. Naked pair {37} in R5C89, locked for R5 and N6 -> R4C8 = 4
26c. Naked pair {67} in R8C12 -> R8C89 = [81], R1C9 = 4
26d. Naked pair {37} in R59C9, locked for C9

27. R23C9 = {26} = 8 -> R1C78 = 9 = [81], R1C5 = 2, R2C8 = 3, R3C7 = 5

28. R23C1 = [13] (hidden pair in C1) = 4 -> R1C23 = 14 = [59]

and the rest is naked singles.


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PostPosted: Fri Nov 04, 2011 10:00 pm 
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Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 154 V1.5 by Ronnie G (May 2009) here
Puzzle Diagram:
Image[/url]
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:5376:4609:4609:4099:5636:7173:4358:4358:3592:4609:5376:4099:5636:5636:5636:7173:3592:4358:4609:4099:5376:4885:4885:4885:3592:7173:4358:4099:5660:5660:5376:4885:3592:4385:4385:7173:5660:5660:1830:5159:5160:4649:2858:4385:4385:5677:1830:1071:5159:5160:4649:1587:2858:4661:4406:5677:1071:5159:5160:4649:1587:4661:5182:4406:4406:5677:4162:4162:4162:4661:5182:5182:4936:4936:4936:5677:4162:4661:3406:3406:3406:
Solution:
+-------+-------+-------+
| 7 5 9 | 1 2 4 | 8 3 6 |
| 3 8 4 | 5 6 9 | 7 1 2 |
| 1 6 2 | 7 3 8 | 5 9 4 |
+-------+-------+-------+
| 5 9 7 | 4 1 2 | 3 6 8 |
| 2 4 6 | 3 8 5 | 9 7 1 |
| 8 1 3 | 9 7 6 | 4 2 5 |
+-------+-------+-------+
| 6 3 1 | 8 5 7 | 2 4 9 |
| 4 7 5 | 2 9 1 | 6 8 3 |
| 9 2 8 | 6 4 3 | 1 5 7 |
+-------+-------+-------+
Quote:
Ronnie G: Everything seems quiet around here, so I thought I'd try to liven it up a bit! Keep everyone occupied until manu's 156 and of course Ed's Anniversary 157. :thumbs:
Here's a slightly harder version of A154, let's call it V1.5 It has a slightly altered cage pattern and different solution from the V1.
SS score: 1.59.
Enjoy!

Afmob: Thanks for providing us with a V2, Ronnie! I used lots of combo clashes to solve this Killer, so maybe someone can find a neater way to crack it.
Rating: 1.5.

manu: Thanks Ronnie, for this puzzle that was resisting a lot !
It has got me thinking a lot because there are a lot of moves you need before cracking it.
I did not find an easier way than Afmob 's solution, but it is a different approach, so I post my wt

Andrew (in 2010): Having caught up with my backlog of unfinished V1s (A161 and A165) last month, I had a look to see if there were any other of Ronnie's puzzles that I hadn't yet done. I found that I'd got stuck on A154 V1.5 and hadn't yet started A150 V2 so I had a go at both of them in the last few days.
I'd got as far as step 21 and had a note of the innies in step 23 (I'd also used 45 rule on N58 for A154) but until this week I hadn't worked out how I could use them. Then after steps 24, 25 and 28 the puzzle was cracked.
Afmob wrote:
I used lots of combo clashes to solve this Killer, so maybe someone can find a neater way to crack it.
Maybe manu's step 2d is that way? It's a neat way, combining a "clone" and a short forcing chain. Many of my steps are similar to those used by Afmob but my breakthrough is different.
I'll also rate A154 V1.5 at 1.5.

Walkthrough by Afmob:
Thanks for providing us with a V2, Ronnie! I used lots of combo clashes to solve this Killer, so maybe someone can find a neater way to crack it.

A154 V1.5 Walkthrough:

1. N7
a) 4(2) = {13} locked for C3
b) Innies N7 = 9(3) <> 7,8,9
c) 22(4): R6C1+R9C4 <> 1,2,3,4 because R7C2+R8C3 @ Innies N7 <= 8

2. C789
a) Outies C789 = 9(3) <> 7,8,9; R49C6 = (123) since R1C6 = (456)
b) Innies+Outies N69: -5 = R9C6 - R4C9 -> R4C9 = (678)
c) Outies N3 = 14(3): R1C6 <> 6 since R4C9 = (678) and 28(4) cannot have 6 and 7
d) 28(4) = 89{47/56} -> 9 locked for N3; CPE: R123C9 <> 8; R2C7+R3C8 <> 4,5 since R1C6 = (45)
e) Outies N9 = 12(2+1): R6C9 <> 1,2,3 since R6C7+R9C6 <= 8
f) Outies C789 = 9(3) = 3{15/24} -> 3 locked for C6

3. C123 !
a) Outies C123 = 11(3) <> 9; R14C4 = (1234) since R9C4 >= 5
b) Outies N1 = 10(3) <> 8,9; R4C1 <> 1,2,3 since R14C4 @ Outies C123 <= 6
c) 22(4) @ N7: R6C1 <> 5 because R6C9 <= 8 and R7C2+R8C3 @ Innies N7 <= 8
d) Innies N7 = 9(3) must have one of (25) but cannot have both of them which must be in R7C2 or R8C3
e) 22(4) = {1579/2389/2479/3568/4567} since other combos clash with step 3d
f) 22(4) <> {4567} since R7C2+R8C3 @ Innies N7 would be >= 9
g) Innies N7 = 9(3) must have one of (456) which can only be placed in R7C2 or R8C3
h) 22(4) <> {2389} since it contains no (456)
i) 22(4) = {1579/2479/3568} -> R7C2+R8C3 = {15/24/35}, R9C4 <> 5
j) Innies N7 = 3{15/24} (from step 3i) -> 3 locked for R7+N7

4. R678
a) 17(3) <> 5 since {458} blocked by Killer pair (45) of Innies N7
b) Innies R9 = 13(3): R9C5 <> 6,7,8,9 since R9C4 >= 6; R9C5 <> 1 since R9C46 <= 11
c) Hidden Killer pair (13) in 13(3) for R9 since Innies R9 = 13(3) cannot be {139}
-> 13(3) <> 2{47/56}
d) 13(3) = {139/157/238/346} because {148} blocked by Killer pair (48) of 19(3)
e) Innies N9 = 12(3) <> 3,8 because 3{18/27/45} blocked by Killer pairs (13,35,37) of 13(3)
f) R7C8+R8C7 <> 1 since they see all 1 of R9
g) Innies N9 = 12(3): R7C7 <> 5 because 1 only possible there
h) 6(2): R6C7 <> 1

5. N479 !
a) Outies R1234 = 14(4) <> 9
b) R7C8+R8C7 <> 9 since it sees all 9 of C9
c) Hidden Killer pair (12) in 13(3) for N9 since Innies N9 cannot be {129}
-> 13(3) = {139/157/238} <> 4,6
d) Innies N9 = 12(3) = {147/156/246} with R7C8+R8C7 = {26/46/47/56} since R7C7 = (124)
e) 18(4) = {1269/1467/2367/2457/3456} <> 8 since other combos clash with step 5d
f) ! Outies N9 = 12(2+1): R9C6 <> 2 because R6C79 can only be [46] and this leaves
no combo for 18(4) since 1,3 only possible @ R9C6

6. R789
a) 13(3) = {157/238} since {139} blocked by R9C6 = (13)
b) 9 locked in 19(3) @ R9 = 9{28/46} for N7
c) 5 locked in Innies N7 = 9(3) = {135} -> R8C3 = 5; 1 locked for R7+N7
d) 6(2) = {24} locked for C7
e) 1 locked in 13(3) @ N9 = {157} locked for R9+N9
f) R9C6 = 3, R8C7 = 6
g) 16(4) = {1249} locked for N8; 9 also locked for R8
h) 20(3) = {389} -> R7C9 = 9; 8 locked for R8
i) 22(4) = {3568} since R9C4 = (68) -> R7C2 = 3

7. R456
a) R7C3 = 1, R6C3 = 3
b) Innies N4 = 13(2): R4C1 = (57)
c) 7(2) = [25/61]
d) Killer pair (56) locked in 7(2) + Innies N4 for N4
e) 22(4) @ R4C2 = 49{18/27} -> 9 locked for R4
f) 9 locked in 11(2) @ N6 = {29} -> R5C7 = 9, R6C8 = 2
g) R7C8 = 4 -> R6C9 = 5, R6C2 = 1 -> R5C3 = 6, R6C7 = 4, R6C1 = 8, R9C4 = 6
h) Innie N6 = R4C9 = 8
i) Innie N4 = R4C1 = 5

8. R123+N4
a) 28(4) = {4789} since R2C7 = 7 -> R1C6 = 4, R2C7 = 7, R3C8 = 9
b) Outie N3 = R4C6 = 2
c) 18(3) = {567} locked for C6 since {189} blocked by R8C6 = (19); R6C6 = 6
d) Naked pair (79) locked in R6C45 for N5
e) R5C6 = 5, R7C6 = 7
f) Both 20(3) = 8{39/57} since R7C45 = (58) -> 8 locked for C45
g) Outies C123 = 5(2) = [14/23]
h) 16(4) = 15{28/46} -> R1C4 = 1
i) Outie N1 = R4C4 = 4

9. C123
a) 17(3) = {467} -> R7C1 = 6; 4 locked for R8+N7
b) Hidden Single: R2C3 = 4 @ C3 -> R3C2 = 6
c) 21(4) = {2478} since R3C3 = (278) -> 2,7,8 locked for N1

10. N2
a) R3C6 = 8
b) 19(4) = {1378} -> R4C5 = 1; 3,7 locked for R3+N2

11. Rest is singles.

Rating: 1.5. I used small combo analysis.
Walkthrough by manu:
Thanks Ronnie, for this puzzle that was resisting a lot !
It has got me thinking a lot because there are a lot of moves you need before cracking it.
I did not find an easier way than Afmob 's solution, but it is a different approach, so I post my wt

The hardest move is 2)d) : a short forcing chain that enables to get some time. I would have liked to avoid it, but this wt would be much longer without it.
Steps 2)d, 3)f) and 3)g) use the interesting cage pattern at n7 and n9 (cells r9c4 and r9c6 see many other ones)

Walkthrough A.154 V1.5

1)a) Outies for n7 : r6c1+r6c3+r9c4=17.
b) Max r6c3=3 → min r6c1+r9c4=14 : r6c1=(56789), r9c4=(56789)
c) IO for n47 : r9c4=1+r4c1 → r4c1 = (45678)
d) Outies for n1 : r4c1+r4c4+r1c4=10 = h10(3) (all these cells see each over)
→ r4c1<>8 → (step c)) r9c4<>9
e) r4c1 <> 4 since it would force r14c4 to be {15} (step d) ) and r9c4 to be 5 (step c)) which is not possible. Deduce (step c)) that r9c4 <> 5.

2)a) Outies for c789 : r1c6+r4c6+r9c6=9 with r1c6 >= 4 (cage 28(4)) → r1c6=(456), r4c6=(123) and r9c6=(123)
b) IO for n69 : r4c9=5+ r9c6 → r4c9=(678)
c) Outies for n9 : r6c7+r6c9+r9c6=12
d) Let us focus on cell r9c6 :
- If r9c6=1 or 2, this digit is locked for n9 at r7c7 (not at (20(3) and see all other cells), so
r9c6+r6c7=6 →(step c)) r6c9=6
- If r9c6 = 3, then r4c9=8
The conclusion is : r4c9=8 or r6c9=6 : we deduce r4c9<>6 →(step b)) r9c6 <> 1

3)a) Outies for r123 : r4c14569 total 20.
b) Min r4c4+r4c5+r4c6=6 → max r4c1 + r4c9 =14 : r4c1<>7 since r4c19 can't be [78]
→ (step 1)c) r9c4 <> 8
c) Innies for r9 : r9c456 = h13(3) : [652/643/742].
d) We deduce from c) that r9c5<>1 → 1 locked for r9 at n9 : 13(3)={139/148/157} : no 2,6.
e) Innies for n7 : r7c2+r7c3+r8c3 = h9(3).
f) Hidden killer pair {26} locked for r9 at r9c12346 → Cells r9c1234 contain at least one of {26} → CPE : r7c2+r8c3 cannot be {26}. We deduce that h9(3) at n7 cannot be {126}.
h9(3) = 3{15/24} : 3 locked for n7 at h9(3)
g) 3 is locked for r9 at r9c6789 → (CPE) r7c8, r8c7 <> 3
h) Innies of n9 : r7c7+r7c8+r8c7 = h9(3). No 1,3 for these cells → {246} locked for n9.
i) 6(2) at 6 = {24} locked for c7 → r8c7=6. R6c9<>6 → r9c6<>2 (step 2)d) : r9c6=3

4)a) r4c9=8 (step 2)b)
b) h13(3) at n9 = [643] (step 3)c)
c) Step 1)a) → r6c1+r6c3=11 : r6c1=8, r6c3=3 → r7c3=1
d) Innie for n4 : r4c1=5
e) Last combos : 22(4) at n478 = [8356] ; 7(2) at n4 = [61]
f) Hidden single for r7 : r7c1=6
g) Step 3)a) → r4c456 total 7 ={124} : r4c4=4, r4c56={12} locked for n5/r4
h) Step 1)d) → r1c4=1
i) Hidden singles : r4c8=6 then r4c7=3 → r4c23={79} locked for n4 → r5c12={24} locked for r5
j) Last combos at n6 : 17(4)={1367} with r5c89={45} locked for n6/r5, and 11(2)={29} with r5c7=9, r6c8=2. 6(2)=[42], and r7c8=4. R6c9=5
k) No 1,2,3,4 at r567c6 : 18(3)=[567]
l) r1c6=4 → r4c6=2 from step 2)a). r4c5=1
m) Last combo for 28(4) at n3 : {4789} : r2c7=7, r3c8=8
n) Hidden pair {24} locked for n3/c9 at r23c9. Hidden single for c9 : r1c9=6
o) Last combination for 17(4) at n3 : {2348} : r1c7=8, r1c8=3
p) Hidden pair {13} locked at r23c1 for c1/n1
q) Last combination for cage 18(4) at n1 : {1359} : r1c2=5, r1c3=9
r) 4 is locked for n1 at cells r2c3+r3c2 : 16(4)={1456} : r3c2=6, r2c3=4
s) Combination of cage 20(3) at n9 : {389} since 3 is locked at cage 20(3) for n9. r8c8=8, r8c9=3, r7c9=9.

The rest is naked and hidden singles
Walkthrough by Andrew (in 2010):
Having caught up with my backlog of unfinished V1s (A161 and A165) last month, I had a look to see if there were any other of Ronnie's puzzles that I hadn't yet done. I found that I'd got stuck on A154 V1.5 and hadn't yet started A150 V2 so I had a go at both of them in the last few days.

I'd got as far as step 21 and had a note of the innies in step 23 (I'd also used 45 rule on N58 for A154) but until this week I hadn't worked out how I could use them. Then after steps 24, 25 and 28 the puzzle was cracked.

Afmob wrote:
I used lots of combo clashes to solve this Killer, so maybe someone can find a neater way to crack it.
Maybe manu's step 2d is that way? It's a neat way, combining a "clone" and a short forcing chain. Many of my steps are similar to those used by Afmob but my breakthrough is different.

Rating Comment. I'll also rate A154 V1.5 at 1.5 based on step 21c and step 23.

Here is my walkthrough for A154 V1.5.

Prelims

a) 7(2) cage in N4 = {16/25/34}, no 7,8,9
b) 11(2) cage in N6 = {29/38/47/56}, no 1
c) R67C3 = {13}
d) R67C7 = {15/24}
e) 20(3) cage at R5C4 = {389/479/569/578}, no 1,2
f) 20(3) cage at R5C5 = {389/479/569/578}, no 1,2
g) 19(3) cage in N7 = {289/379/469/478/568}, no 1
h) 20(3) cage in N9 = {389/479/569/578}, no 1,2
i) 28(4) cage at R1C6 = {4789/5689}, no 1,2,3
j) 14(4) cage at R1C9 = {1238/1247/1256/1346/2345}, no 9

1. Immediate follow-up steps
1a. Naked pair {13} in R67C3, locked for C3, clean-up: no 4,6 in R6C2
1b. 28(4) cage at R1C6 = {4789/5689}, CPE no 8 in R1C9

2. 45 rule on C123 3 outies R149C4 = 11 = {128/137/146/236/245}, no 9

3. 45 rule on C789 3 outies R149C6 = 9 = {126/135/234}, no 7,8,9
3a. R1C6 = {456} -> no 4,5,6 in R49C6
3b. 8,9 of 28(4) cage at R1C6 only in R2C7 + R3C8 + R4C9, CPE no 8,9 in R23C9

4. 45 rule on N7 3 innies R7C23 + R8C3 = 9 = {126/135/234}, no 7,8,9
4a. 45 rule on N7 3(2+1) outies R6C13 + R9C4 = 17
4b. Max R6C13 = 12 -> min R9C4 = 5
4c. Max R9C4 = 8 -> min R6C13 = 9, min R6C1 = 6

5. R149C4 (step 2) = {128/137/146/236/245}
5a. R9C4 = {5678} -> no 5,6,7,8 in R14C4

6. 45 rule on N9 3(2+1) outies R6C79 + R9C6 = 12
6a. Max R6C7 + R9C6 = 8 -> min R6C9 = 4

7. 45 rule on R1234 4 outies R5C1289 = 14 = {1238/1247/1256/1346/2345}, no 9

8. 45 rule on R9 3 innies R9C456 = 13
8a. Max R9C46 = 11 -> min R9C5 = 2
8b. Min R9C46 = 6 -> max R9C5 = 7
8c. 1 in R9 only in R9C6789, CPE no 1 in R7C8 + R8C7

9. 45 rule on N47 1 outie R9C4 = 1 innie R4C1 + 1 -> R4C1 = {4567}

10. 45 rule on N69 1 innie R4C9 = 1 outie R9C6 + 5 -> R4C9 = {678}
10a. 9 of 28(4) cage at R1C6 only in R2C7 + R3C8, locked for N3

11. 45 rule on N1 3(2+1) outies R1C4 + R4C14 = 10 = {127/136/145/235} (normal combinations apply because all three cells are common peers)
11a. 5 of {145} must be in R4C1 -> no 4 in R4C1, clean-up: no 5 in R9C4 (step 9)

12. 45 rule on N3 2 innies R2C7 + R3C8 = 1 outie R4C6 + 14
12a. Min R2C7 + R3C8 = 15, no 4,5
12b. 28(4) cage at R1C6 = {4789/5689}
12c. 4,5 only in R1C6 -> R1C6 = {45}

13. R149C6 (step 3) = {135/234}, 3 locked for C6

14. 45 rule on N2 2 innies R1C46 = 1 outie R4C5 + 4
14a. Max R1C46 = 9 -> max R4C5 = 5

15. 45 rule on N5 3 outies R7C456 = 3 innies R4C456 + 13
15a. Min R4C456 = 6 -> min R7C456 = 19, no 1

16. 45 rule on N7 2 outies R6C1 + R9C4 = 1 innie R7C3 + 13
16a. R7C3 = {13} -> R6C1 + R9C4 = 14,16 = [68/86/97], no 7 in R6C1
16b. R6C1 + R9C4 = 14,16 -> R7C2 + R8C3 = 6,8
16c. 22(4) cage at R6C1 = {68}[35]/[97][15]/[97]{24} -> no 5,6 in R7C2, no 6 in R8C3
16d. R7C23 + R8C3 (step 4) = {135/234}, 3 only in R7C23, locked for R7 and N7

17. 45 rule on N1 2 innies R2C3 + R3C2 = 1 outie R4C4 + 6
17a. Max R4C4 = 4 -> max R2C3 + R3C2 = 10, no 9 in R3C2

18. 45 rule on R4C14569 = 20 = {12368/12458/12467/13457/23456}
18a. 5 of {12458/13457/23456} must be in R4C1 -> no 5 in R4C5
18b. R1C46 (step 14) = R4C5 + 4
18c. Max R4C5 = 4 -> max R1C46 = 8, no 4 in R1C4

19. R9C456 = 13 (step 8) = {148/157/238/247/256/346}
19a. R9C4 = {678} -> no 6,7 in R9C5

20. 19(3) cage in N7 = {289/469/478/568}
20a. 17(3) cage in N7 = {179/269/278/467} (cannot be {458} which clashes with 19(3) cage), no 5
20b. R9C456 = (step 19) = {157/238/247/256/346} (cannot be {148} which clashes with 19(3) cage)

21. 13(3) cage at R9C7 = {139/157/238/346} (cannot be {148} which clashes with 19(3) cage at R9C1, cannot be {247/256} which clash with R9C456)
21a. 45 rule on N9 3 innies R7C78 + R8C7 = 12 = {129/147/156/246} (cannot be {138/237/345} which clash with 13(3) cage), no 3,8
21b. 1 of {156} must be in R7C7 -> no 5 in R7C7, clean-up: no 1 in R6C7
21c. 18(4) cage at R6C9 = {1269/1359/1467/2349/2367/2457/3456} cannot be {1368} because 1,3 only in R9C6, cannot be {1278/1458/2358} because R7C8 + R8C7 = {25/27/45} aren’t consistent with any combinations for R7C78 + R8C7 -> no 8 in R6C9
[This is how far I’d got when this puzzle first appeared. I had a note for the innies in step 23, which I’d used for A154, but at that time hadn’t worked out how to use them for this puzzle.]

22. 45 rule on N9 3 innies R7C78 + R8C7 (step 21a) = {129/147/156/246}
22a. 6 of {156/246} must be in R8C7 (R78C7 cannot be {15/24} because of CCC with R67C6), no 6 in R7C8
22b. 5 of {156} must be in R7C8 -> no 5 in R8C7

23. 45 rule on N58 5(3+2) innies R4C456 + R9C46 = 16 must contain at least one 1 in R4C456 + R9C6 (if R4C456 = {234} = 9 -> R9C46 = 7 can only be [61]), CPE no 1 in R56C6
23a. 1 in N5 only in R4C456, locked for R4
23b. 1 in R6 only in R6C23, locked for N4

24. Hidden killer triple 1,2,3 in 22(4) cage, 7(2) cage and R6C6 for N4, 7(2) cage contains one of 1,2,3, R6C6 = {13} -> 22(4) cage must contain one of 2,3
24a. 22(4) cage in N4 = {2479/2569/2578/3469/3478} (cannot be {3568/4567} which clash with 7(2) cage, cannot be {2389} which contains two of 2,3)

25. 45 rule on N4 3 innies R4C1 + R6C13 = 16 = {169/178/358} (cannot be {367} which clashes with 22(4) cage)
25a. 8,9 only in R6C1 -> R6C1 = {89}
25b. R6C1 + R9C4 (step 16a) = [86/97] -> R9C4 = {67}, clean-up: no 7 in R4C1 (step 9)
[Alternatively hidden killer pair 8,9 in 22(4) cage and R6C1 for N4, 22(4) cage contains one of 8,9 -> R6C1 must contain one of 8,9 -> R6C1 = {89}]
25c. R4C1 + R6C13 = {169/358}
25d. R4C1 + R9C4 = [56/67], CPE no 6 in R9C1

26. 7 in N4 only in 22(4) cage (step 24a) = {2479/2578/3478}, no 6
26a. 6 in N4 only in R4C1 + R5C3, CPE no 6 in R2C3

27. R9C456 (step 19) = {157/247/256/346}
27a. 4,5 only in R9C5 -> R9C5 = {45}

28. 1 in R4 only in R4C456
28a. R4C14569 (step 18) = {12368/12458/12467/13457}
28b. 7,8 only in R4C9 -> R4C9 = {78}, clean-up: no 1 in R9C6 (step 10)

29. 1 in R9 only in R9C789, locked for N9, clean-up: no 5 in R6C7

30. R7C78 + R8C7 (step 21a) = {246} (only remaining combination) -> R8C7 = 6, R7C78 = {24} (locked for R7 and N9), clean-up: no 5 in R6C8

31. Naked pair {13} in R7C23, locked for N7, R8C3 = 5 (step 4), clean-up: no 2 in R6C2

32. Naked pair {24} in R67C7, locked for C7, clean-up: no 7,9 in R6C8

33. 18(4) cage at R6C9 (step 21c) = {2367/3456} -> R9C6 = 3, R4C9 = 8 (step 10), clean-up: no 3 in 11(2) cage in N6
33a. 5,7 of 18(4) cage only in R6C9 -> R6C9 = {57}

34. 3 in N9 only in 20(3) cage = {389} (only remaining combination) -> R7C9 = 9, R8C89 = [83]

35. Naked triple {157} in R9C789, locked for R9 -> R9C5 = 4, R9C4 = 6, R4C1 = 5 (step 9), clean-up: no 2 in R5C3

36. Naked triple {578} in R7C456, locked for R7 and N8 -> R7C1 = 6
36a. R4C14569 (step 28a) = {12458} (only remaining combination) -> R4C456 = {124} -> R4C4 = 4, R1C4 = 1 (step 2), R4C56 = {12}, locked for R4 and N5, CPE no 2 in R3C6

37. R4C8 = 6 (hidden single in R4), clean-up: no 5 in R5C7
37a. Naked pair {24} in R6C78, locked for N6
37b. Naked pair {24} in R67C8, locked for C8

38. Naked pair {79} in R25C7, locked for C7 -> R4C7 = 3

39. Naked pair {79} in R4C23, locked for N4 -> R6C1 = 8, R6C3 = 3 (step 25), R7C23 = [31], R6C2 = 1, R5C3 = 6

40. R4C78 = [36] = 9 -> R5C89 = 8 = {17}, locked for R5 and N6 -> R5C7 = 9, R6C8 = 2, R67C7 = [42], R7C8 = 4, R6C9 = 5, R2C7 = 7, R3C8 = 9, R1C6 = 4 (cage sum)

41. R4C6 = 2 (step 3), R1C9 = 6 -> R2C8 + R3C7 = 6 = {15}, locked for N3 -> R1C78 = [83], R4C5 = 1

42. Naked pair {29} in R8C45, locked for R8 -> R8C6 = 1

43. R23C1 = {13} (hidden pair in C1) = 4 -> R1C23 = 14 = [59], R4C23 = [97]

44. R2C3 = 4 (hidden single in C3), R3C2 = 6 (cage sum), R23C9 = [24], R2C2 = 8, R3C3 = 2, R1C1 = 7, R1C5 = 2, R8C45 = [29]

45. 22(4) cage in N2 = {2569} (only remaining combination) -> R2C456 = {569}, locked for R2 and N2

46. 45 rule on C6 2 remaining innies R23C6 = 17 = [98]

and the rest is naked singles.


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PostPosted: Fri Nov 04, 2011 10:47 pm 
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Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 155 by Afmob (May 2009) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:3329:3329:3331:3331:5378:4613:4613:5127:5127:3329:5380:5378:5378:4613:7942:4613:5127:3080:5380:5380:5378:7942:7942:7942:7942:3080:3080:5380:7694:2827:2827:7689:7689:7689:7689:3080:7694:7694:4877:4877:2827:3084:3084:7689:3850:4111:7694:7694:4882:4877:3084:7689:3084:3850:4111:4111:4882:4882:4882:4883:4883:4883:2836:2576:2576:4881:4882:4629:4629:2070:2836:2836:2576:4881:4881:4629:2070:2070:4887:4887:4887:
Solution:
+-------+-------+-------+
| 5 6 4 | 9 2 1 | 3 8 7 |
| 2 1 9 | 3 8 7 | 6 5 4 |
| 8 3 7 | 5 4 6 | 9 1 2 |
+-------+-------+-------+
| 9 4 1 | 7 6 2 | 8 3 5 |
| 6 7 2 | 8 3 5 | 1 4 9 |
| 3 8 5 | 1 9 4 | 7 2 6 |
+-------+-------+-------+
| 4 9 3 | 2 7 8 | 5 6 1 |
| 1 2 8 | 6 5 9 | 4 7 3 |
| 7 5 6 | 4 1 3 | 2 9 8 |
+-------+-------+-------+
Quote:
Afmob: I decided to make a messy Assassin since it's been a long time since Messy One #6. I hope you can see which cell belongs to which cage. :twisted:
SS Score: 1.08. Estimated rating: Hard 1.0.

Ronnie G: Thanks Afmob for another fun puzzle! :applause: Didn't get to look at it until today, and it took me a good 2 hours, maybe a little longer, to solve. I'm getting slow in my old age!
It seems I made tons of eliminations before finding the first placement, and even when I thought I had it cracked, many more moves were needed to finally get to "singles only".
I kept some rough & sloppy notes & will try to sort them out into a coherent WT, but I must take a break for now.
Thanks again! :thumbs:

manu: My first solving path was using some contradictions between combos which was not satisfying, until I found step 2a of my WT ( two cloned cages).
This is a hard to solve puzzle that resists a lot, until the end ! Thanks Afmob.

Ronnie G: Well I FINALLY finished my walk-through. :bouncy:
Maybe it took so long because I have so little experience in writing them for an audience other than myself.
I don't rate puzzles, but don't think I used anything harder than a hidden killer pair. However it took me MANY small steps to complete this puzzle. This could be the longest WT in history for a relatively easy puzzle! Haven't yet looked at manu's version, maybe he came up with something clever to shorten the path.

Andrew:
Afmob wrote:
I decided to make a messy Assassin since it's been a long time since Messy One #6.
The cage pattern was certainly a good Messy One but this Assassin was waaaay harder than any in the Messy One series, even though the SS score is fairly low.
Ronnie G wrote:
... and it took me a good 2 hours, maybe a little longer, to solve. I'm getting slow in my old age!
IMHO that's quick for solving an Assassin. This one took me a lot longer; I got stuck and came back to it later. I did manage to solve A156, including writing my walkthrough as I went along, in that sort of time but the only other Assassins that I can remember solving in that time were Ruud's very early ones when I was writing walkthroughs but not posting them.
I get the impression that Afmob is a much quicker solver than I am. I don't know how long others take.
manu wrote:
My first solving path was using some contradictions between combos which was not satisfying, until I found step 2a of my WT ( two cloned cages).
So did I; see step 14 of my first walkthrough.
manu wrote:
This is a hard to solve puzzle that resists a lot, until the end !
It certainly does!
manu's step 2a is really neat! The immediate result in step 2b was obtained more directly by Ronnie's step 7 and my step 11. However its importance became obvious with manu's steps 2d and 2e. These led more quickly to the first placement and to a shorter WT.
Congratulations Ronnie on your first posted WT! An excellent effort! :applause:
After mentioning to Ed that I'd used difficult combination analysis (step 14) and an ALS block (step 16) he told me that I must have missed something easy. I therefore went back and eventually found what I'd missed. As a result I've decided to post both my walkthroughs for A155. For clarity I've posted them as separate messages and given them separate ratings.
I'll rate my first walkthrough at Hard 1.25

Andrew's Second Walkthrough: As I said above, I went back to find what I'd missed first time through.
Ronnie G wrote:
This could be the longest WT in history for a relatively easy puzzle!
And this one is even longer! ;)
The solving path this time is fairly similar to the way Ronnie solved it but I've decided to post this anyway.
I'll rate my second walkthrough at (Very) Hard 1.0; that's also my rating for A155.
Here is my second walkthrough. The first 11 steps are the same as before; I've repeated them for clarity and in case anyone goes directly to my easier solving path.

Walkthrough by manu:
My first solving path was using some contradictions between combos which was not satisfying, until I found step 2a of my WT ( two cloned cages).
This is a hard to solve puzzle that resists a lot, until the end ! Thanks Afmob.

ASSASSIN 155 WALKTHROUGH

1)a) Outies for n89 : r6c4+r7c3=4 : {13} locked for thr rest of cage 19(5)
b) Innies-outies for n7 : r6c1 = r7c3 = (13) → r7c12 <> 1,2,3
c) Naked pair {13} locked at r6c14 for r6
d) Outies for r123 : r4c1+r4c9=14 : [86/95] since max r4c9=6 (12(4) cage)
e) Innies for n4 : r46c1+r45c3 total 15 : {1239/1248} since r4c1=(89) → 2 is locked at r45c3 for n4 and c3
f) Innies-outies for c12 : r9c2=r6c3 → r9c2<>2,3
g) 2 is locked at cage 10(3) for n7 : {127/235}.
h) Killer pair {13} locked at r7c3 and cage 10(3) for n7 → no 3 for cage 19(3) (n7) (important for the next step 2)b))

2)a) This is the main step : let us focus on cage 19(3) at n8. Digits that form this cage must be
locked for n7 at cages 10(3) and 19(3) ; but the sum of two of these is at least 10, so
cage 10(3) contains at most one of them → 19(3) (at n7) contains at least two
of them → 19(3) (at n7) contains all these digits since the cage total is also 19.
Conclusion : cages 19(3) at n7 and 19(3) at n8 have the same combinations.
b) We deduce from a) and 1)h) there is no 3 at cage 19(3) at n8 → hidden pair {13} locked
at r7c39 for r7.
c) Innies-outies for r89 : r8c4=5+r7c9 → r8c4=(68)
d) No 2 for 19(3) at n7 → no 2 for 19(3) at n8 (step a). 2 is locked for r7 at r7c45.
e) Using step d) , 19(5) (n8) = {12358/12367} → r7c45=(257)

3)a) Innies-outies for n9 : r7c6=4+r8c7.
→ r7c6 > 4 : no 4
b) Hidden killer triple {134} locked for n8 at 18(3) and r9c56.
18(3) at n8 cannot contain two of {134} → r9c56=(134). We deduce that combination
of cage 8(3) at n9 is {134}, locked for cage 19(3) at n9 that sees all cells of 8(3)
c) 19(3) at n9 : {289/568} 8 locked for n9/r9
d) r8c7=(134) →(step a) r7c6 = (578).
e) Combinations of 19(3) at n8 : No 2,3 → {469/478/568}
But {469} is not possible since r7c6=(578) → 19(3) = 8{47/56} with r7c6=8.
f) step a → r8c7=4

4)a) r7c78={56} locked r7/n9
b) r7c45={27} locked r7/n8 and r8c4=6
c) 19(3) at n9 : {289} locked r9/n9
d) r7c12={49} locked n7 → r6c1=3, r6c4=1, r7c3=3 and r7c9=1.
e) Step 1)e) : r46c1+r45c3 = {1239} : r4c1=9, r4c3=1 and r5c3=2
f) Step 1)d) : r4c9=5

5)a) Combination of cage 12(4) at n3 : {1245} with r3c8=1, r23c9={24} locked for n3/c9
b) Hidden single for n8 : r9c4=4
c) Last combination : 19(3) at n7 is {568} → r8c3=8, {56} locked for n7/r9 and r6c3=(56)
(step 1)f))
d) Naked pair {56} locked for c3 at r69c3 → 13(2) (at n1) is [49], and r23c3={79} locked
for n1/c3 and the rest of 21(4).
e) 31(5) at n2 must contain 9 → r3c7=9
f) At n3 : 20(3)={578} → r8c9=3 (hidden single for c9), r8c8=7)
g) Hidden single for n6 : r5c7=1 : 30(6) at n5 does not contain {15} : 30(6) = {234678}
h) At n2 : 21(4) = {2379} with {23} locked for n2
i) 3 locked for r3 at r3c12 → 21(4) = {1389} : r2c2=1, r3c1=8 r3c2=3
j) Hidden single : r3c9=2 → r2c9=4
k) At n6 : 15(2) = {69} locked for n6/c9 since {78} is blocked by r1c9=(78).

6)a) r23c3=[97] → r3c4=5, r3c56={46} locked for n2/r3 → r2c6=7 (cage combination)
b) cage combinations : r5c4=8, r6c5=9 and r4c4 + r5c5 ={37} locked for n5
c) r56c9=[96] → r6c3=5 → r5c6=5 (hidden single)
d) last combination for 12(4) : {1245} → r6c68={24} locked for r6

The rest is singles

Edit : Ed has pointed out some typos : thanks ;)
Walkthrough by Ronnie G:
Well I FINALLY finished my walk-through. :bouncy:

Maybe it took so long because I have so little experience in writing them for an audience other than myself. Please send your comments & corrections.

Thanks, Andrew, for providing a few corrections & clarifications, in blue:

I don't rate puzzles, but don't think I used anything harder than a hidden killer pair. However it took me MANY small steps to complete this puzzle. This could be the longest WT in history for a relatively easy puzzle! Haven't yet looked at manu's version, maybe he came up with something clever to shorten the path.

Walk-Through A155:

Beginning with Sumocue mark-ups.

1) 8(3) @ R8C7: 1 locked -> no 1 @ R8C456 (CPE)

2) R123 outies: R4C19 = 14(2) = [86/95]
2a) R4C9 = {56} -> no 5,6 elsewhere in 12(4) cage ->
1,2 locked in 12(4) cage for N3

3) 30(5) in N4 can’t have both (89), because R4C1 = {89} ->
Remaining combos for 30(5): 567{39/48} -> no 1,2 in 30(5) cage
3a) 567 locked in 30(5) for N4
3b) KP (89) in N4 @ R4C1 and 30(5) cage -> not elsewhere in N4
3c) R5C3 = {234}, so rest of 19(3) cage R5C4+R6C5 = {6789}

4) R789 outies: R6C14 = 4(2) = {13}, locked for R6
4a) 2 locked in N4 & C3 @ R45C3
4b) 1 locked in 12(4) cage @ R5C67, locked for R5. No 6 @R5C67
4c) 16(3) @ R6C1 must have one of (13), no combos have both ->
R7C12 = {456789}

5) IOD C12: R6C3 = R9C2, no 2,3 in R9C2
5a) 2 locked in 10(3) cage in N7 -> 10(3) = 2{17/35}, no 4,6

6) IOD for N7: R6C1 = R7C3 -> R7C3 = {13}
6a) 19(5) @ R6C4 has both 1&3 (@R6C4 & R7C3): 1,3 not elsewhere in cage 19(5)
6b) KP (13) for N7 @ R7C3 & 10(3) cage -> not elsewhere in N7
6c) 1 locked in N8 for R9, not elsewhere in R9

7) R89 outies: R7C459 = h10(3) cage (after removing 4(2) cage @ R6C4 + R7C3):
h10(3) = {127/145/235} (can’t be {136} since 1,3 are only in R7C9) ->
R7C45 = {2457}, R7C9 = {13}
7a) NP (13) @ R7C39, locked for R7
7b) 11(3) in N9 <> {245}, since it must have 1 or 3 -> no 5 in 11(3) cage

8) IOD R89: R8C4 = R7C9 + 5 -> R8C4 = {68}

9) Hidden KP (89) for N7 @ R7C12 & 19(3) cage -> 16(3) @ R6C1 <> {367}
9a) KP (89) for R7: since R7C12 must have one of (89), 19(3) @ R7C678 can’t have
both -> 19(3) <> {289}, no 2 in 19(3) cage
NOTE: Andrew points out the uncertainty of calling this a KP versus HKP. I often
confuse them myself, when I realize there are only two places for them to go.
My use of conventions is still lacking in many such regards!

9b) 2 locked @ R7C45 for 19(5) cage & N8: 19(5) = 123 {58/67}, no 4
9c) 8(3) @ R8C7: R8C7 <> 5, since R9C56 can’t be {12}

10) IOD N9: R8C7 + 4 = R7C6 -> R7C6 = {5678}
10a) 9 locked in 18(3) cage in N8: 18(3) = 9{18/36/45}, no 7 in 18(3)
10b) 7 locked in N8 for R7, not elsewhere in R7
At this point, I should have realized 6 locked in 19(3) for R7, not elsewhere in R7,
which would lead to a faster placement: R6C1=3. Oh well.


11) Outies N9: R7C6 + R9C56 = 12(3) = {138/147/345}
(combo {156} blocked by KP (56) in 19(5) cage)
11a) No 5 @ R9C56, since R7C6 can’t be 3 or 4. No 6 @ R7C6.

12) 8(3) @ R8C7 = {134}: no 2 @ R8C7, no 3,4 @ R9C789 & R8C56 (CPE’s)
12a) Remaining combos for 19(3) in N9: 8{29/56}, 8 locked for N9 & R9, no 7
12b) Hidden triple (134) @ R9C456, locked for R9

13) 6 locked in N8 for R8, not elsewhere in R8
13a) 11(3) in N9 = {137} (last combo), locked for N9 & 7 is locked for R8
13b) FINALLY A PLACEMENT!! R8C7 = 4 (NS), R9C4 = 4 (HS)

14) R8C56 = {59}, since {68} blocked by R8C4. (59) locked for N8 & R8
14a) R8C34 = [86]
14b) R9C23 = {56} locked for R9 & N7
14c) R9C1 = 7 (HS), R8C12 = {12}, locked for R8 & N7
14d) R7C39 = [31], R6C14 = [31]
14e) R7C6 = 8 (HS), R7C78 = {56}

15) 30(5) in N4 = {45678} (last combo), locked for N4
15a) R4C1 = 9, R45C3 = [12], R7C12 = [49]
15b) R5C4 + R6C5 = {89}, locked for N5
15c) R4C4 + R5C5 = {37} (can’t be {46} since R4C4 contains no 4 or 6),
(37) locked for N5

16) 30(6) @ R4C5 = 2347{59/68}
16a) 3 & 7 locked in 30(6) in N6, not elsewhere in N6
16b) 15(2) in N6 = {69}, locked for N6 & C9
16c) NS: R4C9 = 5
16d) HS: R3C8 = 1, R23C9 = {24}, locked for N3 & C9

17) NS: R9C9 = 8
17a) 20(3) @ N3 must have 3 or 7: 20(3) = 8{39/57}, no 6. 8 locked for N3 & C8
17b) 6 locked in N3 for C7 -> R7C78 = [56]
17c) HP (58) @ R12C8 -> R1C9 = 7

18) R8C89 = [73]

19) 12(4) @ R5C6 = {1245}, no 6
19a) R6C7 <> 2, 5 locked in 12(4) cage for N5 & C6
19b) R8C56 = [59]

20) (Should have put this after step 17)
13(2) @ R1C3 = [49], since {58} blocked by R1C8

21) 13(3) @ N1 = 2{38/56} -> 2 locked for N1, no 1 in 13(3) cage
(Step re-written for greater clarity)

21a) KP’s (35) & (36) locked in 13(3) & 21(4) for N1 ->
R23C3 = {79}, (79) locked for N1, C3, & 21(4) cage
21b) 21(4) @ R1C5 = {2379} -> R1C5 + R2C4 = {23}, locked for N2

22) 18(4) @ R1C6 = {1368} -> no 4,7,9
22a) (36) locked @ R12C7 for C7 & 18(4) cage -> R3C7 = 9

And at last!! All singles from here.
First Walkthrough by Andrew:
Afmob wrote:
I decided to make a messy Assassin since it's been a long time since Messy One #6.
The cage pattern was certainly a good Messy One but this Assassin was waaaay harder than any in the Messy One series, even though the SS score is fairly low.

Ronnie G wrote:
... and it took me a good 2 hours, maybe a little longer, to solve. I'm getting slow in my old age!
IMHO that's quick for solving an Assassin. This one took me a lot longer; I got stuck and came back to it later. I did manage to solve A156, including writing my walkthrough as I went along, in that sort of time but the only other Assassins that I can remember solving in that time were Ruud's very early ones when I was writing walkthroughs but not posting them.

I get the impression that Afmob is a much quicker solver than I am. I don't know how long others take.

manu wrote:
My first solving path was using some contradictions between combos which was not satisfying, until I found step 2a of my WT ( two cloned cages).
So did I; see step 14 of my first walkthrough.

manu wrote:
This is a hard to solve puzzle that resists a lot, until the end !
It certainly does!

manu's step 2a is really neat! The immediate result in step 2b was obtained more directly by Ronnie's step 7 and my step 11. However its importance became obvious with manu's steps 2d and 2e. These led more quickly to the first placement and to a shorter WT.

Congratulations Ronnie on your first posted WT! An excellent effort! :applause:

After mentioning to Ed that I'd used difficult combination analysis (step 14) and an ALS block (step 16) he told me that I must have missed something easy. I therefore went back and eventually found what I'd missed. As a result I've decided to post both my walkthroughs for A155. For clarity I've posted them as separate messages and given them separate ratings.

I'll rate my first walkthrough at Hard 1.25 because of steps 14 and 16.

Here is my first walkthrough.

Prelims

a) R1C34 = {49/58/67}, no 1,2,3
b) R56C9 = {69/78}
c) 20(3) cage in N3 = {389/479/569/578}, no 1,2
d) 11(3) cage at R4C3 = {128/137/146/236/245}, no 9
e) 19(3) cage at R5C3 = {289/379/469/478/568}, no 1
f) R7C678 = {289/379/469/478/568}, no 1
g) 10(3) cage in N7 = {127/136/145/235}, no 8,9
h) 19(3) cage in N7 = {289/379/469/478/568}, no 1
i) R9C789 = {289/379/469/478/568}, no 1
j) 11(3) cage in N9 = {128/137/146/236/245}, no 9
k) 8(3) cage at R8C7 = {125/134}, CPE no 1 in R8C456 (also R9C789 if I hadn’t done that 19(3) cage first)
l) 12(4) cage at R2C9 = {1236/1245}, no 7,8,9
m) 12(4) cage at R5C6 = {1236/1245}, no 7,8,9

1. 45 rule on R123 2 outies R4C19 = 14 = [95/86]
1a. 12(4) cage at R2C9 = {1236/1245}
1b. 1,2 locked in R2C9 + R3C89, locked for N3
1c. R4C9 = {56} -> no 5,6 in R2C9 + R3C89

2. 45 rule on R789 2 outies R6C14 = 4 = {13}, locked for R6
2a. 12(4) cage at R5C6 = {1236/1245}
2b. 1 locked in R5C67, locked for R5
2c. 1,3 of {1236} must be in R5C67 -> no 6 in R5C67
2d. Max R6C1 = 3 -> min R7C12 = 13, no 1,2,3

3. 45 rule on N7 1 outie R6C1 = 1 innie R7C3 -> R7C3 = {13}
[Alternatively 45 rule on N89 2 outies R6C4 + R7C4 = 4 = {13}]
3a. Naked pair {13} in R6C4 + R7C3, locked for 19(5) cage
3b. 1 in N8 locked in R9C456, locked for R9

4. 45 rule on N9 1 outie R7C6 = 1 innie R8C7 + 4, no 2,3,4 in R7C6

5. 45 rule on R89 1 innie R8C4 = 1 outie R7C9 + 5, no 5,6,7,8 in R7C9, no 2,4,5 in R8C4

6. 45 rule on N89 3 innies R7C45 + R8C4 = 15 = {249/258/267/456}
6a. 8,9 of {249/258} must be in R8C4 -> no 8,9 in R7C45
[Alternatively R7C45 + R8C4 = 15 from 19(5) cage with R6C4 + R7C3 = {13} = 4]

7. 45 rule on N4 4 innies R46C1 + R45C3 = 15 = {1239/1248} (only valid combinations because R4C1 must contain one of 8,9), no 5,6,7, 1,2 locked for N4
7a. 2 locked in R45C3, locked for C3, CPE no 2 in R5C5
7b. R4C1 = {89} -> no 8,9 in R45C3
7c. Max R5C3 = 4 -> min R5C4 + R6C5 = 15, no 2,3,4,5

8. 11(3) cage at R4C3 = {128/137/146/236/245}
8a. 8 of {128} must be in R5C5 -> no 8 in R4C4

9. 45 rule on N1 3 innies R123C3 = 1 outie R4C1 + 11
9a. R4C1 = {89} -> R123C3 = 19,20, no 1

10. 45 rule on C12 1 outie R6C3 = 1 innie R9C2, no 2,3 in R9C2
10a. 2 in N7 locked in 10(3) cage = {127/235}, no 4,6
10b. Killer pair 1,3 in R7C3 and 10(3) cage, locked for N7

11. 45 rule on N7 3 innies R7C123 = 16
11a. 45 rule on R7 remaining innies R7C459 = 10 = {127/145/235} (cannot be {136} which clashes with R7C3), no 6
11b. 1,3 only in R7C9 -> R7C9 = {13}, clean-up: no 7,9 in R8C4 (step 5)
11c. Naked pair {13} in R7C39, locked for R7
[At this stage I ought to have listed the combinations for R7C123 or for the 19(3) cage in N7. See alternative walkthrough in my next message.]

12. 11(3) cage in N9 = {128/137/146/236} (cannot be {245} because R7C9 only contains 1,3), no 5

13. R7C45 + R8C4 (step 6) = {258/267/456}
13a. 45 rule on N9 3 outies R7C6 + R9C56 = 12 = {129/138/147/345} (cannot be {156/246} which clash with R7C45 + R8C4, cannot be {237} which clashes with combinations of 8(3) cage at R8C7), no 6, clean-up: no 2 in R8C7 (step 4)

14. 45 rule on N9 3 innies R7C78 + R8C7 = 15 = {249/258/456} (cannot be {159/357} because R7C678 cannot be 5{59}/7{57}, cannot be {168/267/348} which clash with 11(3) cage), no 1,3,7, clean-up: no 5,7 in R7C6 (step 4)

15. 8(3) cage at R8C7 = {125/134}
15a. 1 locked in R9C56, locked for R9
15b. R8C7 = {45} -> no 4,5 in R9C56

16. 18(3) cage in N8 = {279/378/459} (cannot be {369} which clashes with R7C6 + R8C4 (ALS block), cannot be {468/567} which clash with R7C45 + R8C4), no 6

17. R8C4 = 6 (hidden single in N8), R7C9 = 1 (step 5), R7C3 = 3, R6C4 = 1, R6C1 = 3, clean-up: no 7 in R1C3, no 5 in 10(3) cage in N7 (step 10a)
17a. R7C9 = 1 -> R8C89 (step 12) = {28/37}, no 4
17b. Killer pair 2,7 in R8C12 and R8C89, locked for R8
17c. Naked triple {127} in 10(3) cage, locked for N7, clean-up: no 7 in R6C3 (step 10)
17d. 18(3) cage in N8 (step 16) = {378/459} (cannot be {279} because 2,7 only in R9C4), no 2
17e. 7 of {378} must be in R9C4 -> no 3 in R9C4

18. R3C8 = 1 (hidden single in C8)
18a. R234C9 = {236/245}, 2 locked for C9, clean-up: no 8 in R8C8 (step 17a)

19. R6C1 = 3 -> R46C1 + R45C3 (step 7) = {1239} (only remaining combination) -> R4C1 = 9, R45C3 = [12], R4C9 = 5 (step 1), clean-up: no 3 in R23C9 (step 18a), no 9 in R9C2 (step 10)
19a. Naked pair {24} in R23C9, locked for C9 and N3

20. R5C3 = 2 -> R5C4 + R6C5 = 17 = {89}, locked for N5

21. R5C7 = 1 (hidden single in R5)
21a. 12(4) cage at R5C6 = {1236/1245}, 2 locked in R6C68, locked for R6

22. R4C3 = 1 -> R4C4 + R5C5 = 10 = [37/46/73], no 2 in R4C4, no 4,5 in R5C5

23. 30(6) cage at R4C5 = {234678} (only remaining combination), no 9, 8 locked in R4C78 + R5C8 + R6C7, locked for N6, clean-up: no 7 in R56C9

24. Naked pair {69} in R56C9, locked for C9 and N6

25. 6 of 30(6) cage at R4C5 (step 23) locked in R4C56, locked for R4 and N5, clean-up: no 4 in R4C4 (step 22)
25a. Naked pair {37} in R4C4 + R5C5, locked for N5
25b. 5 in N5 locked in R56C6, locked for C6

26. R6C1 = 3 -> R7C12 = 13 = [49/58/85], no 6 in R7C1, no 4,6 in R7C2
26a. 6 in N7 locked in R9C23, locked for R9

27. R7C678 = {469/568} (cannot be {289} which clashes with R7C12), no 2
27a. Killer pair 8,9 in R7C12 and R7C6, locked for R7

28. R7C45 = {27}(hidden pair in R7), locked for N8, clean-up: no 3,8 in 18(3) cage (step 17d)

29. R7C6 = 8 (hidden single in N8), R8C7 = 4 (step 4), R8C56 = [59], R9C4 = 4, R8C3 = 8, clean-up: no 9 in R1C3, no 5 in R1C4, no 4 in R6C3 (step 10), no 5 in R7C12 (step 26), no 2 in R8C8 (step 17a)
29a. R7C12 = [49]

30. Naked pair {37} in R8C89, locked for R8 and N9 -> R9C9 = 8, clean-up: no 5 in R9C78 (prelim i)
30a. R9C1 = 7 (hidden single in R9)

31. X-Wing for 7 in R4C4 + R5C5 and R7C45, no other 7 in C45, clean-up: no 6 in R1C3

32. R23C3 = {79} (hidden pair in C3), locked for N1 and 21(4) cage at R1C5
32a. R23C4 = {79} = 16 -> R1C5 + R2C4 = 5 = {23} (cannot be {14} because 1,4 only in R1C5), locked for N2
[Naked pair {56} in R69C3, locked for C3 -> R1C3 = 4, R1C4 = 9 …, as in step 28 of my second walkthrough, followed by naked pair {79} in R23C4 … is more direct.]

33. R3C4 = 5 (hidden single in C4)
33a. 31(5) cage at R2C6 = {35689/45679}, no 1
33b. 9 locked in R3C57, locked for R3 -> R23C3 = [97]

34. 20(3) cage in N3 = {389/578} (cannot be {569} because R1C9 only contains 3,7), no 6, 8 locked in R12C8, locked for C8 and N3
34a. R1C9 = {37} -> no 3,7 in R12C8

35. R7C8 = 6 (hidden single in C8), R7C7 = 5

36. 5 in N3 locked in 20(3) cage (step 34) = {578} -> R1C9 = 7, R12C8 = {58}, R8C89 = [73]
36a. Naked triple {369} in R123C7, locked for C7 -> R9C78 = [29]

37. 45 rule on N23 1 remaining innie R1C4 = 9, R1C3 = 4, R5C4 = 8, R6C5 = 9, R56C9 = [96], R6C3 = 5, R5C1 = 6, R9C23 = [56]
37a. R2C6 = 7 (hidden single in R2)
37b. R3C7 = 9 (hidden single in R3), clean-up: no 8 in R3C5 (step 33a)
37c. R5C6 = 5 (hidden single in R5)

38. Naked pair {46} in R3C56, locked for R3 and N2

39. 5 in N1 locked in R12C1 -> 13(3) cage = {256} (only remaining combination) -> R1C2 = 6, R12C1 = {25}, locked for C1 and N1

and the rest is naked singles.
Second Walkthrough by Andrew:
As I said above, I went back to find what I'd missed first time through.

Ronnie G wrote:
This could be the longest WT in history for a relatively easy puzzle!
And this one is even longer! ;)

The solving path this time is fairly similar to the way Ronnie solved it but I've decided to post this anyway.

I'll rate my second walkthrough at (Very) Hard 1.0; that's also my rating for A155.

Here is my second walkthrough. The first 11 steps are the same as before; I've repeated them for clarity and in case anyone goes directly to my easier solving path.

Prelims

a) R1C34 = {49/58/67}, no 1,2,3
b) R56C9 = {69/78}
c) 20(3) cage in N3 = {389/479/569/578}, no 1,2
d) 11(3) cage at R4C3 = {128/137/146/236/245}, no 9
e) 19(3) cage at R5C3 = {289/379/469/478/568}, no 1
f) R7C678 = {289/379/469/478/568}, no 1
g) 10(3) cage in N7 = {127/136/145/235}, no 8,9
h) 19(3) cage in N7 = {289/379/469/478/568}, no 1
i) R9C789 = {289/379/469/478/568}, no 1
j) 11(3) cage in N9 = {128/137/146/236/245}, no 9
k) 8(3) cage at R8C7 = {125/134}, CPE no 1 in R8C456 (also R9C789 if I hadn’t done that 19(3) cage first)
l) 12(4) cage at R2C9 = {1236/1245}, no 7,8,9
m) 12(4) cage at R5C6 = {1236/1245}, no 7,8,9

1. 45 rule on R123 2 outies R4C19 = 14 = [95/86]
1a. 12(4) cage at R2C9 = {1236/1245}
1b. 1,2 locked in R2C9 + R3C89, locked for N3
1c. R4C9 = {56} -> no 5,6 in R2C9 + R3C89

2. 45 rule on R789 2 outies R6C14 = 4 = {13}, locked for R6
2a. 12(4) cage at R5C6 = {1236/1245}
2b. 1 locked in R5C67, locked for R5
2c. 1,3 of {1236} must be in R5C67 -> no 6 in R5C67
2d. Max R6C1 = 3 -> min R7C12 = 13, no 1,2,3

3. 45 rule on N7 1 outie R6C1 = 1 innie R7C3 -> R7C3 = {13}
[Alternatively 45 rule on N89 2 outies R6C4 + R7C4 = 4 = {13}]
3a. Naked pair {13} in R6C4 + R7C3, locked for 19(5) cage
3b. 1 in N8 locked in R9C456, locked for R9

4. 45 rule on N9 1 outie R7C6 = 1 innie R8C7 + 4, no 2,3,4 in R7C6

5. 45 rule on R89 1 innie R8C4 = 1 outie R7C9 + 5, no 5,6,7,8 in R7C9, no 2,4,5 in R8C4

6. 45 rule on N89 3 innies R7C45 + R8C4 = 15 = {249/258/267/456}
6a. 8,9 of {249/258} must be in R8C4 -> no 8,9 in R7C45
[Alternatively R7C45 + R8C4 = 15 from 19(5) cage with R6C4 + R7C3 = {13} = 4]

7. 45 rule on N4 4 innies R46C1 + R45C3 = 15 = {1239/1248} (only valid combinations because R4C1 must contain one of 8,9), no 5,6,7, 1,2 locked for N4
7a. 2 locked in R45C3, locked for C3, CPE no 2 in R5C5
7b. R4C1 = {89} -> no 8,9 in R45C3
7c. Max R5C3 = 4 -> min R5C4 + R6C5 = 15, no 2,3,4,5

8. 11(3) cage at R4C3 = {128/137/146/236/245}
8a. 8 of {128} must be in R5C5 -> no 8 in R4C4

9. 45 rule on N1 3 innies R123C3 = 1 outie R4C1 + 11
9a. R4C1 = {89} -> R123C3 = 19,20, no 1

10. 45 rule on C12 1 outie R6C3 = 1 innie R9C2, no 2,3 in R9C2
10a. 2 in N7 locked in 10(3) cage = {127/235}, no 4,6
10b. Killer pair 1,3 in R7C3 and 10(3) cage, locked for N7

11. 45 rule on N7 3 innies R7C123 = 16
11a. 45 rule on R7 remaining innies R7C459 = 10 = {127/145/235} (cannot be {136} which clashes with R7C3), no 6
11b. 1,3 only in R7C9 -> R7C9 = {13}, clean-up: no 7,9 in R8C4 (step 5)
11c. Naked pair {13} in R7C39, locked for R7

And now the step that I missed first time.

12. R7C123 = {169/178/349/358} (cannot be {367} which clashes with 19(3) cage)
[Alternatively hidden killer pair 8,9 in R7C12 and 19(3) cage for N7, 19(3) only contains one of 8,9 -> R7C12 must contain one of 8,9]
12a. R7C678 = {469/478/568} (cannot be {289} which clashes with R7C12), no 2
12b. 2 in R7 locked in R7C45, locked for N8
12c. R7C45 + R8C4 (step 6) = {258/267}, no 4

13. 8(3) cage at R8C7 = {125/134}
13a. 2 of {125} must be in R8C7 -> no 5 in R8C7, clean-up: no 9 in R7C6 (step 4)

14. 9 in N8 locked in 18(3) cage = {189/369/459}, no 7
14a. 1 of {189} must be in R9C4 -> no 8 in R9C4
14b. 7 in N8 locked in R7C456, locked for R7

15. 45 rule on N9 3 outies R7C6 + R9C56 = 12 = {138/147/345} (cannot be {156} which clashes with R7C45 + R8C4), no 6, clean-up: no 2 in R8C7 (step 4), no 5 in R9C56 (step 13)

16. Naked triple {134} in 8(3) cage at R1C7, CPE no 3,4 in R8C56 and R9C789
16a. R9C456 = {134} (hidden triple in N8), locked for R9, clean-up: no 4 in R6C3 (step 10)
[Alternatively 1,3,4 of 18(3) cage (step 14) only in R9C4 -> R9C4 = {134}
Naked triple {134} in R9C456, locked for R9, clean-up: no 4 in R6C3 (step 10)]

17. 45 rule on R9, 1 outie R8C3 = 1 remaining innie R9C1 + 1, no 4,5,7,9 in R8C3, no 2 in R9C1

18. 2 in R9 locked in R9C789, locked for N9
18a. R9C789 = {289}, locked for R9 and N9, clean-up: no 8,9 in R6C3 (step 10)

19. R7C678 = {478/568} -> R7C6 = 8, R8C4 = 6, R8C3 = 8, R8C7 = 4 (step 4), clean-up: no 7 in R1C3, no 5 in R1C4
19a. Naked pair {59} in R8C56, locked for R8 and N8, R9C4 = 4 (step 14), clean-up: no 9 in R1C3
19b. Naked triple {567} in R9C123, locked for N7

20. R7C12 = {49}, R7C3 = 3 (step 12), R6C14 = [31], R7C9 = 1
20a. R8C12 = {12} -> R9C1 = 7 (step 10a), clean-up: no 7 in R6C3 (step 10)

21. R46C1 + R45C3 (step 7) = {1239} (only remaining combination) -> R4C1 = 9, R45C3 = [12], R4C9 = 5 (step 1), R7C12 = [49]

22. R5C3 = 2 -> R5C4 + R6C5 = 17 = {89}, locked for N5

23. R4C3 = 1 -> R4C4 + R5C5 = 10 = {37} (only remaining combination), locked for N5

24. 5 in N5 locked in R56C6, locked for C6 -> R8C56 = [59]
24a. 5 locked in R56C6 -> 12(4) cage at R5C6 = {1245}, no 3,6, R5C7 = 1, 2 locked in R6C68, locked for R6

25. 30(6) cage at R4C5 = {234678} (only remaining combination), no 9, 8 locked in R4C78 + R5C8 + R6C7, locked for N6, clean-up: no 7 in R56C9

26. Naked pair {69} in R56C9, locked for C9 and N6
26a. 6 in N5 locked in R4C56, locked for R4

27. R4C1 = 5 -> 12(4) cage at R2C9 = {1245}, R2C8 = 1, R23C9 = {24}, locked for C9 and N3 -> R9C9 = 8

28. Naked pair {56} in R69C3, locked for C3 -> R1C3 = 4, R1C4 = 9, R5C4 = 8, R6C5 = 9, R56C9 = [96], R6C3 = 5, R5C1 = 6, R9C23 = [56]
28a. R5C6 = 5 (hidden single in R5)

29. Naked pair {79} in R23C9, locked for N1 and 21(4) cage at R1C5
29a. R23C9 = {79} = 16 -> R1C5 + R2C4 = 5 = {23}, locked for N2

30. R3C4 = 5 (hidden single in C4)
30a. 31(5) cage at R2C6 = {35689/45679}, no 1, R3C7 = 9, R23C3 = [97], R9C78 = [29]
30b. 31(5) cage at R2C6 = {45679} (only remaining combination) -> R2C6 = 7, R3C56 = {46}, locked for R3 and N2, R1C6 = 1, R2C5 = 8, R23C9 = [42], R9C56 = [13], R3C12 = [83], R2C2 = 1 (cage sum), R7C12 = [12]
30c. R1C2 = 6 (hidden single in N1)

31. R1C6 + R2C5 = [18] = 9 -> R12C7 = 9 = [36]

and the rest is naked singles.


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