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PostPosted: Tue May 10, 2011 12:19 am 
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Grand Master
Grand Master

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

+--------------------------+--------------------------+--------------------------+
| Puzzle Rating Score | Puzzle Rating Score | Puzzle Rating Score |
+--------------------------+--------------------------+--------------------------+
| SOK1 E1.00 0.80 | A110XV2 H1.25 1.30 | A112V2 1.25 2.50 |
| SOK1 var 0.75 -.-- | A111 1.25 1.25 | A113 1.50 1.55 |
| A109 1.00 1.15 | A112 0.75 1.35 | A113-Lite 1.00 1.05 |
| A110X 1.00 0.80 | | |
+--------------------------+--------------------------+--------------------------+

Puzzle rating table, with links to archive entries; each of these has a link to the puzzle thread.
Assassin 113 and Assassin 113-Lite are in the same archive entry.

Abbreviations used in Rating Table on this page:
Est = Estimated rating by puzzle maker
E = Easy
H = Hard
F = Frank
G = gary w
Go = goooders
M = Mike (mhparker)
SOK = Special Occasion Killer
Score = SudokuSolver v3.3.0 score, rounded to nearest 0.05
! indicates that the Score has changed at least 0.10 from the SS v3.2.1 score
** in the Afmob column indicates that these puzzles were made by him,
for these ones the estimate is his rating.
+------------------------+-----------+-------+-------+-------+-----------------------+-------+
| Puzzle | Made By | Est | Afmob | Andrew| Other Raters | Score |
+------------------------+-----------+-------+-------+-------+-----------------------+-------+
| SOK1 | Nasenbaer | | E1.00 | E1.00 | | 0.80 |
| SOK1 var | Nasenbaer | | | 0.75 | | 0.75 |
| Assassin 109 | Afmob | E1.25 | 1.25 | H1.25 |(M)1.0 | 1.10 |
| Assassin 110X | Ed | | 1.00 | 1.00 |(M)1.0 | !1.00 |
| Assassin 110XV2 | Ed | 1.50 | H1.25 | H1.25 | | 1.35 |
| Assassin 111 | Nasenbaer | E1.50 | 1.25 | H1.50 |(G)1.25 | 1.30 |
| Assassin 112 | Afmob | H0.75 | ** | 0.75 |(F)H0.75(G)H0.5 | !0.80 |
| Assassin 112V2 | Afmob | 1.25 | ** | 1.50 |(M)1.50 | 2.50 |
| Assassin 113 | Para | H1.25 | 1.50 | 1.50 |(Go)1.50(Ed)2.0 | 1.55 |
| Assassin 113-Lite | Para | 1.00 | | 1.00 | | 1.10 |
+------------------------+-----------+-------+-------+-------+-----------------------+-------+


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


Last edited by Andrew on Thu Aug 04, 2011 1:15 am, edited 4 times in total.

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PostPosted: Tue May 10, 2011 2:58 am 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
SOK (Special Occasion Killer) 1 by Nasenbaer (June 2008) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:4864:4864:8194:8194:8194:8194:8194:3847:3847:4864:4864:2315:2315:3085:3085:4623:4623:3847:3346:11539:11539:2315:2326:11543:11543:4623:5146:3346:4124:4124:11539:2326:1568:1568:11543:5146:3620:11539:11539:11539:3624:3624:11543:11543:5146:3620:3620:3620:11539:3121:11543:3891:3891:5146:4406:11539:11539:3121:3121:11543:11543:11543:5146:4406:2624:2113:2113:3121:2116:2116:4166:4166:4406:2624:8266:8266:8266:8266:8266:4166:4166:
Solution:
+-------+-------+-------+
| 2 4 9 | 8 7 5 | 3 1 6 |
| 6 7 1 | 2 3 9 | 4 5 8 |
| 5 3 8 | 6 4 1 | 7 9 2 |
+-------+-------+-------+
| 8 9 7 | 1 5 4 | 2 6 3 |
| 3 5 2 | 7 8 6 | 9 4 1 |
| 4 1 6 | 9 2 3 | 8 7 5 |
+-------+-------+-------+
| 7 6 4 | 3 1 2 | 5 8 9 |
| 9 8 3 | 5 6 7 | 1 2 4 |
| 1 2 5 | 4 9 8 | 6 3 7 |
+-------+-------+-------+
Quote:
Nasenbaer: Sorry to do that (actually, no, I'm not ;) ), but I have a SOK - a Special Occasion Killer. I know, nobody has yet volunteered for A110 so there should be no other killer - but, just as the title says, it's a special occasion. :cheesey:
Yesterday evening there was the first semi-final in the European Soccer Championship, Germany vs. Turkey. :cheers: I wonder if you can deduce the final score from this killer. :cheesey:
BTW - how often can you find the score?
Rating by SSolver(3.2.0): 0.81 - sorry, not hard enough for a Messy One.
Have fun!

Afmob: Even if we play bad, we still win (apart from the game vs. Croatia). :cheesey:
To celebrate our victory, I'll post my shortest wt so far (again and again ...). By the way, it was ok to post a non-Assassin Killer or a variant since it's not Friday. If it had been Friday and A110 hadn't been claimed at that point, then you wouldn't have been allowed to post a variant.
BTW, I counted the score 9 times. I counted the cage sums and cells that were next to each other (also diagonally).
Rating: (Easy) 1.0.

Børge: Congratulations to all of you for Germany winning the semi-final yesterday.
Let's hope Germany will play a killer match on Sunday, so that we will not be Image but have to Image

Andrew: It was a good final. I was supporting Germany but on the day Spain were the better team.
Now I've done 3 Nasenbaer puzzles in a row! :)
A fun puzzle and an easy one. I'll also rate it Easy 1.0.
I think my walkthrough is sufficiently different from Afmob's to post. It's always good to have at least two posted walkthroughs unless they would be too similar.

Walkthrough by Afmob:
Even if we play bad, we still win (apart from the game vs. Croatia). :cheesey:

To celebrate our victory, I'll post my shortest wt so far (again and again ...). By the way, it was ok to post a non-Assassin Killer or a variant since it's not Friday. If it had been Friday and A110 hadn't been claimed at that point, then you wouldn't have been allowed to post a variant.

BTW, I counted the score 9 times. I counted the cage sums and cells that were next to each other (also diagonally).

SOK 1 Walkthrough:

1. R123
a) Outies R12 = 15(2) = [69] -> R3C4 = 6, R3C8 = 9
b) 9(3) = {126} -> 1,2 locked for R2
c) 18(3) = 9{36/45}
d) 15(3) = {168/258/267} <> 3,4 because other combos blocked by Killer pairs (34,35,56) of 18(3)
e) 32(5) = 89{267/357/456} -> 8,9 locked for R1

2. C789
a) Outies C9 = 6(3) = {123} locked for C8, 3 locked for N9
b) 3 locked in 16(4) @ N9 = 3{148/157/247/256} <> 9
c) 9 locked in 20(5) @ C9 = {12359} locked for C9
d) 15(3) = 6{18/27} -> 6 locked for C9+N1
e) 18(3) = {459} -> 4,5 locked for R2+N3

3. C123
a) 12(2) = {39} locked for R2+N2
b) Hidden Single: R1C3 = 9 @ N1
c) 32(5) = {35789} -> R1C7 = 3; {578} locked for R1+N2
d) R4C3 = 7, R4C2 = 9

4. N456
a) 9 locked in 45(9) @ N1 for C4+N5
b) 14(2) = {68} locked for R5+N5
c) 9(2) = {45} -> R3C5 = 4, R4C5 = 5
d) 6 locked in 45(9) @ N1 for R7+N7
e) Hidden Single: R4C8 = 6 @ 45(9) @ N2
f) 13(2) = {58} -> R4C1 = 8, R3C1 = 5
g) 15(2) = {78} locked for R6+N6

5. N78
a) 17(3) = {179} locked for C1+N7
b) 10(2) = {28} locked for C2+N7
c) 12(4) = {1236} because 4,5 only possible @ R7C4 -> R8C5 = 6

6. N14
a) 19(4) = {2467} -> R2C1 = 6, R2C2 = 7, R1C2 = 4, R1C1 = 2
b) R2C3 = 1, R3C2 = 3
c) 3,6 locked in 14(4) @ N4 = {1346} -> R6C2 = 1, R6C3 = 6; 4 locked for N4

7. Rest is singles.

Rating: (Easy) 1.0. I used Killer pairs once to eliminate some combos and thereby candidates.
Walkthrough by Andrew:
It was a good final. I was supporting Germany but on the day Spain were the better team.

Now I've done 3 Nasenbaer puzzles in a row! :)

A fun puzzle and an easy one. I'll also rate it Easy 1.0.

I think my walkthrough is sufficiently different from Afmob's to post. It's always good to have at least two posted walkthroughs unless they would be too similar.

Here is my walkthrough for SOK 1.

Prelims

a) R2C56 = {39/48/57}, no 1,2,6
b) R34C1 = {49/58/67}, no 1,2,3
c) R34C5 = {18/27/36/45}, no 9
d) R4C23 = {79}, locked for R4 and N4, clean-up: no 4,6 in R3C1, no 2 in R3C5
e) R4C67 = {15/24}
f) R5C56 = {59/68}
g) R6C78 = {69/78}
h) R89C2 = {19/28/37/46}, no 5
i) R8C34 = {17/26/35}, no 4,8,9
j) R8C67 = {17/26/35}, no 4,8,9
k) 9(3) cage at R2C3 = {126/135/234}, no 7,8,9
l) 12(4) cage at R6C5 = {1236/1245}, no 7,8,9, CPE no 1,2 in R9C5
m) R1C34567 = {26789/35789/45689}, no 1, 8,9 locked for R1
n) R9C34567 = {26789/35789/45689}, no 1, 8,9 locked for R9, clean-up: no 1,2 in R8C2

1. 45 rule on R1 3 outies R2C129 = 21 = {489/579/678}, no 1,2,3
1a. Killer triple 7,8,9 in R2C129 and R2C56, locked for R2

2. 45 rule on R2 2 outies R3C48 = 15 = [69], clean-up: no 4 in R4C1, no 3 in R4C5, no 6 in R6C7, no 2 in R8C3
2a. R3C4 = 6 -> R2C34 = 3 = {12}, locked for R2
2b. R4C8 = 9 -> R2C78 = 9 = {36/45}

3. 45 rule on R89 1 outie R7C1 = 1 innie R8C5 + 1, no 1,8,9 in R7C1

4. 45 rule on C9 3 outies R189C8 = 6 = {123}, locked for C8, clean-up: no 6 in R2C7 (step 2b)
4a. Killer triple 1,2,3 in R8C34, R8C67 and R8C8, locked for R8, clean-up: no 7 in R9C2

5. 15(3) cage at R1C8 = {168/258/267/348/357} (cannot be {456} which clash with R2C8)
[Oops! I missed that {348/357} also clash with R2C78. This would have simplified the next few steps. However I’ve left the walkthrough as I did it because I got there fairly quickly with step 7a.]
5a. R1C8 = {123} -> no 1,2,3 in R1C9
5b. 8 of {348} must be in R2C9 -> no 4 in R2C9

6. 16(4) cage in N9 = {1249/1258/1267/1348/1357/2347/2356} (cannot be {1456} because R89C8 must contain two of 1,2,3)
6a. R89C8 = {123} -> no 1,2,3 in R9C9
6b. 45 rule on C9 4 innies R1289C9 = 25 = {4579/4678}, 4,7 locked for C9

7. 45 rule on N3 3 innies R1C7 + R3C79 = 12 = {138/147/237/246} (cannot be {156/345} which clash with R2C78), no 5
7a. Hidden killer pair 1,2 in R1C7 + R3C79 and R1C8 for N3 -> R1C8 = {12}
7b. 15(3) cage at R1C8 (step 5) = {168/258/267}, no 4
7c. R1C7 + R3C79 (step 7) = {138/147/237} (cannot be {246} which clashes with 15(3) cage), no 6

8. 3 in C8 locked in R89C8, locked for N9, clean-up: no 5 in R8C6
8a. 16(4) cage in N9 (step 6) = {1348/1357/2347/2356}, no 9

9. 1,2,3,9 in C9 locked in R34567C9 = {12359} (only remaining combination), locked for C9

10. 15(3) cage at R1C8 (step 7b) = {168/267}, 6 locked for C9 and N3, clean-up: no 3 in R2C7 (step 2b)
10a. 16(4) cage in N9 (step 8a) = {1348/2347}, 4 locked for N9

11. Naked pair {45} in R2C78, locked for R2 and N3, clean-up: no 7,8 in R2C56
11a. Naked pair {39} in R2C56, locked for R2 and N2, clean-up: no 6 in R4C5
11b. R1C3 = 9 (hidden single in R1), R4C23 = [97], clean-up: no 1 in R8C4, no 1 in R9C2
11c. R8C1 = 9 (hidden single in C1), R79C1 = 8 = {26/35}/[71], no 4, no 7 in R9C1

12. 9 in 45(9) cage at R3C2 locked in R56C4, locked for C4 and N5, clean-up: no 5 in R5C56
12a. Naked pair {68} in R5C56, locked for R5 and N5, clean-up: no 1 in R3C5
12b. Naked quint {12345} in R4C45679, locked for R4, clean-up: no 8 in R3C1

13. 45 rule on R1 4 innies R1C1289 = 13 = {1237/1246} (cannot be {1345} because R1C9 only contains 6,7), no 5
13a. R1C9 = {67} -> no 6,7 in R1C12
13b. R1C9 = 6 (hidden single in R1)
13c. 5 in R1 locked in R1C456, locked for N2, clean-up: no 4 in R4C5

14. 45 rule on R9 4 innies R9C1289 = 13 = {1237/1246/1345}
14a. R9C9 = {47} -> no 4 in R9C2, clean-up: no 6 in R8C2

15. 45 rule on N4 3 innies R4C1 + R5C23 = 15 = {258/348/456} (cannot be {168} because 6,8 only in R4C1), no 1 in R5C23

16. 6 in 45(9) cage at R3C2 locked in R7C23, locked for R7 and N7, clean-up: no 2 in R79C1 (step 11c) , no 4 in R8C2, no 2 in R8C4

17. R4C8 = 6 (hidden single in 45(9) cage at R3C6), R4C1 = 8, R3C1 = 5, clean-up: no 9 in R6C7, no 3 in R79C1 (step 11c)
17a. R79C1 = [71], R2C1 = 6, R8C2 = 8, R9C2 = 2, R2C2 = 7, R2C9 = 8, R1C8 = 1 (step 10), R89C8 = [23], R9C9 = 7 (step 14), R8C9 = 4, clean-up: no 7 in R8C4, no 1,6 in R8C6, no 6 in R8C7

18. Naked pair {35} in R8C34, locked for R8 -> R8C567 = [671], R5C56 = [86], clean-up: no 1 in R4C5, no 5 in R4C6

19. R2C12 = [67] = 13 -> R1C12 = 6 = [24]

and the rest is naked singles.

It looks like a lot of the early steps can also be used in the variant which splits R8C67 and adds those cells to adjacent larger cages. If I do solve it, I'll tidy up step 5 and simplify the steps immediately following.


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PostPosted: Tue May 10, 2011 3:23 am 
Offline
Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
SOK (Special Occasion Killer) 1 variant by Nasenbaer (June 2008) here
Puzzle Diagram:
Image
The only difference to SOK 1 is in r8c67 which was originally a 8(2)-cage.
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:4864:4864:8194:8194:8194:8194:8194:3847:3847:4864:4864:2315:2315:3085:3085:4623:4623:3847:3346:11539:11539:2315:2326:11543:11543:4623:5146:3346:4124:4124:11539:2326:1568:1568:11543:5146:3620:11539:11539:11539:3624:3624:11543:11543:5146:3620:3620:3620:11539:4913:11543:3891:3891:5146:4406:11539:11539:4913:4913:11543:11543:11543:5146:4406:2624:2113:2113:4913:4913:4421:4421:4421:4406:2624:8266:8266:8266:8266:8266:4421:4421:
Solution:
+-------+-------+-------+
| 2 4 9 | 8 7 5 | 3 1 6 |
| 6 7 1 | 2 3 9 | 4 5 8 |
| 5 3 8 | 6 4 1 | 7 9 2 |
+-------+-------+-------+
| 8 9 7 | 1 5 4 | 2 6 3 |
| 3 5 2 | 7 8 6 | 9 4 1 |
| 4 1 6 | 9 2 3 | 8 7 5 |
+-------+-------+-------+
| 7 6 4 | 3 1 2 | 5 8 9 |
| 9 8 3 | 5 6 7 | 1 2 4 |
| 1 2 5 | 4 9 8 | 6 3 7 |
+-------+-------+-------+
Quote:
Nasenbaer: While working on SOK 1 (I was trying to make it harder but IMHO this one is easier) I created this version.
I think it's easier, but DON'T try to rate it with SSolver(3.2.0)! (Possible bug, see)
I tried to score the puzzle but after 40 minutes SSolver hadn't come up with anything. So I solved it by hand to verify that it has a unique solution (which it has).
After that I walked it through SSolver step by step to find out what the problem is.
(SudokuSolver output in hidden window below; Nasenbaer's comments are included there, since they might be "spoilers")

Richard(rcbroughton): Thanks for this, I've investigated and isolated the problem and now have a working fix for it. This will be included in the next release.
Richard's full reply here

Andrew: Many thanks for the variant! It was fun and I found it a lot easier that SOK1, which was also fairly easy. I've made some comments about the SS solution at the end.
This puzzle was very unusual because one would normally expect that splitting a 2-cell cage and adding the cells to larger cages would make it harder. In this case it made it easier!
I'll rate SOK1 Variant as 0.75. Maybe it ought to be even lower but I don't know whether step 3 would normally appear in a newspaper "deadly" so I think 0.5 would be too low; anyway I'm not sure if it would be so easy without elimination solving, which is how newspaper puzzles are normally solved.

SudokuSolver output:
SudokuSolver V3.2.0 by Richard Broughton
Successfully Imported Puzzle definition

Preliminaries
Cage 16(2) n4 - cells ={79}
Cage 6(2) n56 - cells only uses 1245
Cage 14(2) n5 - cells only uses 5689
Cage 15(2) n6 - cells only uses 6789
Cage 8(2) n78 - cells do not use 489
Cage 12(2) n2 - cells do not use 126
Cage 13(2) n14 - cells do not use 123
Cage 9(2) n25 - cells do not use 9
Cage 10(2) n7 - cells do not use 5
Cage 9(3) n12 - cells do not use 789
Cage 14(4) n4 - cells do not use 9
Cage 17(5) n9 - cells do not use 89
Cage 32(5) n123 - cells do not use 1
Cage 32(5) n789 - cells do not use 1

1. Naked pair {79} found at r4c23
1a. Cage sum in cage 13(2) n14 - removed 46 from r3c1
1b. Cage sum in cage 9(2) n25 - removed 2 from r3c5
1c. Combinations {1247} no longer valid in cage 14(4) n4

2. 45 Rule on r12 - outies r3c48 total 15
2a. Set candidate at r3c4 to 6
2b. Set candidate at r3c8 to 9
2c. Cage sum in cage 9(3) n12 - removed 345 from r2c3 - removed 345 from r2c4
2d. Cage sum in cage 13(2) n14 - removed 4 from r4c1
2e. Cage sum in cage 9(2) n25 - removed 3 from r4c5
2f. Cage sum in cage 15(2) n6 - removed 6 from r6c7
2g. Cage sum in cage 8(2) n78 - removed 2 from r8c3
2h. Combinations {159} {249} no longer valid in cage 15(3) n3
2i. Only combination {126} valid in cage 9(3) n12
2j. Combinations {378} {468} {567} no longer valid in cage 18(3) n3
2k. Combinations {49} no longer valid in cage 13(2) n14

3. Naked pair {12} found at r2c34
3a. Cage sum in cage 18(3) n3 - removed 78 from r2c7 - removed 78 from r2c8
3b. Combinations {189} {279} no longer valid in cage 18(3) n3

4. Candidate 1 locked in cage 17(5) n9 for n9 nowhere else in n9

5. Candidate 2 locked in cage 17(5) n9 for n9 nowhere else in n9

6. Candidate 3 locked in cage 17(5) n9 for n9 nowhere else in n9

7. Candidate 8 locked in cage 32(5) n123 for r1 nowhere else in r1

8. Candidate 9 locked in cage 32(5) n123 for r1 nowhere else in r1

9. Candidate 8 locked in cage 32(5) n789 for r9 nowhere else in r9
9a. Cage sum in cage 10(2) n7 - removed 2 from r8c2

10. Candidate 9 locked in cage 32(5) n789 for r9 nowhere else in r9
10a. Cage sum in cage 10(2) n7 - removed 1 from r8c2

11. Candidate 9 in c9 must be in cage 20(5) n369
11a. Removed combinations {12368} {12458} {12467} {13457} {23456} - cells ={12359}

12. Hidden single 8 found at r2c9 in c9
12a. Cage sum in cage 15(3) n3 - removed 7 from r1c8 - removed 7 from r1c9
12b. Cage sum in cage 12(2) n2 - removed 4 from r2c5 - removed 4 from r2c6
12c. Combinations {1378} {1468} {2368} {2458} no longer valid in cage 19(4) n1
12d. Combinations {267} {357} {456} no longer valid in cage 15(3) n3
12e. Combinations {48} no longer valid in cage 12(2) n2

13. Candidate 7 locked in n3 for c7 nowhere else in c7
13a. Cage sum in cage 15(2) n6 - removed 8 from r6c8

14. Candidate 7 locked in n6 for c8 nowhere else in c8

15. 45 Rule on r1 split cage 19(4) n1
15a. Included cells set as split cage 6(2) n1
15b. Excluded cells set as split cage 13(2) n1
15c. Combinations {1369} {2359} {3457} no longer valid in cage 19(4) n1

16. Candidate 1 locked in cage 20(5) n369 for c9 nowhere else in c9
16a. Cage sum in cage 15(3) n3 - removed 6 from r1c8

17. Candidate 2 locked in cage 20(5) n369 for c9 nowhere else in c9
17a. Cage sum in cage 15(3) n3 - removed 5 from r1c8

18. Candidate 3 locked in cage 20(5) n369 for c9 nowhere else in c9
18a. Cage sum in cage 15(3) n3 - removed 4 from r1c8

19. Candidate 5 locked in cage 20(5) n369 for c9 nowhere else in c9
19a. Cage sum in cage 15(3) n3 - removed 2 from r1c8
19b. Combinations {258} no longer valid in cage 15(3) n3

20. Candidate 8 locked in c8 for cage 45(9) n235689 nowhere else in cage 45(9) n235689

21. Killer LOL on n369
21a. No 8 in outies r367c6 - removed from innies at r9c7


22. Hidden single 8 found at r6c7 in c7
22a. Last cell in cage 15(2) n6 at r6c8 set to 7
22b. Combinations {69} no longer valid in cage 15(2) n6

23. Hidden single 8 found at r7c8 in cage 45(9) n235689

24. Hidden single 7 found at r5c4 in r5
24a. Cage sum in cage 8(2) n78 - removed 1 from r8c3

25. Candidate 6 locked in n6 for cage 45(9) n235689 nowhere else in cage 45(9) n235689

26. Killer LOL on n369
26a. No 6 in outies r367c6 - removed from innies at r19c7


27. Naked triple {459} found at r7c79 r9c7
27a. Only combinations valid in cage 17(5) n9

28. Naked pair {67} found at r89c9
28a. Cage sum in cage 15(3) n3 - removed 1 from r1c8
28b. Combinations {168} no longer valid in cage 15(3) n3

** Solver error: Candidate:6 not found in n3 trying
29. Naked Single 3 found at r1c8
29a. Cage sum in cage 18(3) n3 - removed 6 from r2c7 - removed 6 from r2c8
29b. Combinations {35789} no longer valid in cage 32(5) n123
29c. Combinations {369} no longer valid in cage 18(3) n3

Look at the Killer LOLs in steps 21 and 26 (highlighted in red). Step 21 works, but I don't know if the logic behind it is correct. It certainly fails in step 26 which removes 6 from r9c7.

I work with the standard solver options. BTW - is there a way to export the options so that you can post them here?
Walkthrough by Andrew:
Many thanks for the variant! It was fun and I found it a lot easier that SOK1, which was also fairly easy. I've made some comments about the SS solution at the end.

This puzzle was very unusual because one would normally expect that splitting a 2-cell cage and adding the cells to larger cages would make it harder. In this case it made it easier!

I'll rate SOK1 Variant as 0.75. Maybe it ought to be even lower but I don't know whether step 3 would normally appear in a newspaper "deadly" so I think 0.5 would be too low; anyway I'm not sure if it would be so easy without elimination solving, which is how newspaper puzzles are normally solved.

Here is my walkthrough, probably the shortest I've posted since I started listing Preliminaries first and using clean-up routinely. I've started with the same first two steps as for SOK1 but then step 3, the key breakthrough, is more powerful than for SOK1 which is why the Variant is a lot easier. Step 1 may not be necessary but it does simplify R2.

Prelims

a) R2C56 = {39/48/57}, no 1,2,6
b) R34C1 = {49/58/67}, no 1,2,3
c) R34C5 = {18/27/36/45}, no 9
d) R4C23 = {79}, locked for R4 and N4, clean-up: no 4,6 in R3C1, no 2 in R3C5
e) R4C67 = {15/24}
f) R5C56 = {59/68}
g) R6C78 = {69/78}
h) R89C2 = {19/28/37/46}, no 5
i) R8C34 = {17/26/35}, no 4,8,9
j) 9(3) cage at R2C3 = {126/135/234}, no 7,8,9
k) 19(4) cage at R6C5 = {12349/12358/12368/12457/13456}, CPE no 1 in R9C5
l) R1C34567 = {26789/35789/45689}, no 1, 8,9 locked for R1
m) 17(5) cage in N9 = {12347/12356}, no 8,9, 1,2,3 locked for N9
n) R9C34567 = {26789/35789/45689}, no 1, 8,9 locked for R9, clean-up: no 1,2 in R8C2

1. 45 rule on R1 3 outies R2C129 = 21 = {489/579/678}, no 1,2,3
1a. Killer triple 7,8,9 in R2C129 and R2C56, locked for R2

2. 45 rule on R2 2 outies R3C48 = 15 = [69], clean-up: no 4 in R4C1, no 3 in R4C5, no 6 in R6C7, no 2 in R8C3
2a. R3C4 = 6 -> R2C34 = 3 = {12}, locked for R2
2b. R4C8 = 9 -> R2C78 = 9 = {36/45}

3. 45 rule on C9 4 (3+1) outies R189C8 + R8C7 = 7 = {123} + 1 -> R8C7 = 1, R1C8 = 1, R89C8 = {23}, locked for C8 and N9, clean-up: no 6 in R2C7 (step 2b), no 5 in R4C6, no 7 in R8C34
3a. Killer pair 2,3 in R8C34 and R8C8, locked for R8, clean-up: no 7 in R9C2

4. 15(3) cage at R1C8 = {168} (only remaining combination) -> R12C9 = [68], clean-up: no 4 in R2C56, no 3 in R2C7 (step 2b)

5. R8C78 + R9C8 = 6 -> R89C9 = 11 = {47}, only remaining combination, locked for C9 and N9

6. Naked pair {45} in R2C78, locked for R2 and N3, clean-up: no 7 in R2C56
6a. Naked pair {39} in R2C56, locked for R2 and N2, clean-up: no 6 in R4C5
6b. Naked pair {67} in R2C12, locked for N1, clean-up: no 6 in R4C1
6c. Naked pair {58} in R34C1, locked for C1
6d. 7 in N3 locked in R13C7, locked for C7, clean-up: no 8 in R6C8

7. R1C3 = 9 (hidden single in R1), R4C23 = [97], clean-up: no 1 in R9C2
7a. 9 in C1 locked in R78C1
7b. R789C1 = {179/269}, no 3,4

8. 9 in 45(9) cage at R3C2 locked in R56C4, locked for C4 and N5, clean-up: no 5 in R5C56
8a. Naked pair {68} in R5C56, locked for R5 and N5, clean-up: no 1 in R3C5
8b. Naked quint {12345} in R4C45679, locked for R4 -> R34C1 = [58], R4C8 = 6, R6C8 = 7, R6C7 = 8, clean-up: no 4 in R4C5

9. R2C12 = {67} = 13 -> R1C12 = 6 = {24}, locked for R1 and N1 -> R2C34 = [12], clean-up: no 6 in R8C3
9a. Naked pair {35} in R8C34, locked for R8 -> R89C8 = [23], clean-up: no 7 in R8C2

10. R1C7 = 3 (hidden single in R1), R3C79 = [72], clean-up: no 2 in R4C5
10a. R7C8 = 8 (hidden single in C8)

11. Naked pair {38} in R3C23, locked for R3 and 45(9) cage -> R3C56 = [41], R4C5 = 5
11a. Naked pair {24} in R4C67, locked for R4 -> R4C4 = 1, R4C9 = 3
11b. Naked pair {59} in R7C79, locked for R7 and N9 -> R9C7 = 6, clean-up: no 4 in R8C2

12. 1 in N7 locked in R79C1, locked for C1
12a. R789C1 (step 7b) = {179} (only remaining combination) -> R8C1 = 9, R79C1 = {17}, locked for C1 and N7 -> R2C12 = [67]
12b. R5C1 = 3, R5C4 = 7, R5C9 = 1 (hidden singles in R5), then R5C7 = 9 (hidden single in R5), R67C9 = [59], R7C7 = 5, R2C78 = [45], R4C67 = [42], R5C8 = 4, R6C4 = 9

13. Naked pair {25} in R5C23, locked for N4 and 45(9) cage -> R6C1 = 4

and the rest is naked singles


After solving this puzzle I had a look at the SS solution that Nasenbaer posted. There are two main things that I'll comment on.

First, SS didn't find my step 3 which I find rather surprising. It comes immediately from using mental arithmetic or Ruud's combination calculator for 7(3+1).

Second Nasenbaer comments on steps 21 and 26 of the SS solution. Richard has already said that he'll fix those steps in the next version of SS.

Superficially those steps appear correct if one takes LOL at face value. However on checking the 45 rule for these innies/outies the innies total 5 more than the outies so LOL cannot be used. The only time I've successfully used LOL in a killer was for one of Ruud's Assassins where there was a 45(9) cage overlapping with N5 and then it was clear that the innies had to equal the outies and to contain the same numbers. LOL is usually fairly easy to use for normal Jigsaws except when the same number can appear twice in the innies and outies. However one clearly has to be careful when using LOL for Killers and Jigsaw Killers.


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PostPosted: Tue May 10, 2011 3:48 am 
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Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 109 by Afmob (June 2008) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:3072:2305:2305:2305:3332:5637:5637:5637:3080:3072:5130:5131:5131:3332:2830:2830:2320:3080:5130:5130:2068:5131:3332:2830:3096:2320:2320:4379:6684:2068:6684:8991:5664:3096:5664:4387:4379:6684:6684:6684:8991:5664:5664:5664:4387:4379:3630:3630:8991:8991:8991:2099:2099:4387:3126:3126:3630:7481:8991:7481:2099:3389:3389:3126:4160:4160:7481:7481:7481:6213:6213:3389:4160:4160:6218:6218:6218:6218:6218:6213:6213:
Solution:
+-------+-------+-------+
| 4 2 1 | 6 3 5 | 9 8 7 |
| 8 9 7 | 4 2 1 | 3 6 5 |
| 6 5 3 | 9 8 7 | 4 2 1 |
+-------+-------+-------+
| 1 6 5 | 3 7 2 | 8 4 9 |
| 9 8 4 | 5 1 6 | 7 3 2 |
| 7 3 2 | 8 4 9 | 1 5 6 |
+-------+-------+-------+
| 5 4 9 | 7 6 3 | 2 1 8 |
| 3 1 6 | 2 9 8 | 5 7 4 |
| 2 7 8 | 1 5 4 | 6 9 3 |
+-------+-------+-------+
Quote:
Afmob: I had this Assassin ready after a couple of minutes but I decided to play around with the larger cages and I came up with those three U-cages and the middle unholy cross. :pallid:
Don't let it disctract you from solving!
SS Score (v3.2): 1.17. Estimated rating: Easy 1.25.

Mike(mhparker): Thanks for an enjoyable little puzzle! :-D Used a trick near the start (step 4a) that may not appeal to all (sorry, folks, just couldn't resist it... :wink:).
Estimated rating: 1.0 (with trick)

Andrew: Thanks Afmob for letting me have the time to solve this puzzle and post my WT first. I hope that you will now post your WT since it's clearly different from mine.
This was my first Assassin since returning from our holiday. Maybe that's why I found it a bit harder than the estimated rating. I'll rate it as Hard 1.25 the way I solved it.
(Comments about Mike's "trick" included in my walkthrough)

Afmob: Even though ...
Rating: 1.25. (for Afmob's posted walkthrough)

Walkthrough by Mike:
Thanks for an enjoyable little puzzle! :-D Used a trick near the start (step 4a) that may not appeal to all (sorry, folks, just couldn't resist it... :wink:).

Estimated rating: 1.0 (with trick)

Edit: Thanks to Andrew for a couple of corrections!

Assassin 109 Walkthrough (24 steps)
Prelims

a) 12(2) at R1C1, R1C9 and R3C7 = {39/48/57} (no 1,2,6)
b) 9(3) at R1C2 and R2C8 = {126/135/234} (no 7,8,9)
c) 22(3) at R1C6 = {589/679} (no 1..4); 9 locked for R1; cleanup: no 3 in R2C19
d) 20(3) at R2C2 = {389/569/578} (no 1,2,4) (Note: {479} blocked by R12C1 (Prelim a))
e) 20(3) at R2C3 = {389/479/569/578} (no 1,2)
f) 11(3) at R2C6 = {128/137/146/236/245} (no 9)
g) 8(2) at R3C3 = {17/26/35} (no 4,8,9)
h) 8(3) at R6C7 = {125/134} (no 6..9); 1 locked -> no 1 in R5C7 (CPE)

1. Innies R123: R3C37 = 7(2) = [25/34]
1a. -> R4C37 = [58/67]

2. Innies N7: R79C3 = 17(2) = {89}, locked for C3 and N7

3. Innies N9: R79C7 = 8(2) = [17/26/35/53]
3a. -> no 4 in R7C7; R9C7 = {3567}

4. Innies R1: R1C159 = 14(3)
4a. Uniqueness: R1C19 cannot sum to 12, because this would force R12C19 to a deadly rectangle
4b. -> no 2 in R1C5

5. 9(3) at R1C2 contains 1 of N1 and 2 of R1
5a. -> 9(3) at R1C2 = {126} (no 3..5), locked for R1
5b. 1 (of N1) locked in R1C23
5c. -> no 1 in R1C4

6. 22(3) at R1C6 (Prelim c) = {589} (no 7) (last combo), locked for R1
6a. cleanup: no 4,7 in R2C19

7. Naked triple (NT) at R1C78+R2C9 = {589}, locked for N3

8. Naked single (NS) at R3C7 = 4
8a. -> R3C3 = 3 (step 1)
8b. -> R4C37 = [58] (cage sums)
8c. cleanup: no 9 in R2C1, no 8 in R2C9

9. Hidden single (HS) in N3 at R1C8 = 8

10. 9(3) at R2C8 (Prelim b) = {126} (no 3) (last combo), locked for N3

11. 11(3) at R2C6 cannot contain both of {26} within R23C6, due to R1C4
11a. -> 11(3) at R2C6 = {137} (no 2,4..6,8)
11b. 1 locked in R23C6 for N2 and C6; 3 locked in R2C67 for R2

12. Innie/Outie (I/O) diff. N1: R2C3 = R1C4 + 1
12a. -> R2C3 = 7, R1C4 = 6

13. 11(3) at R2C6 (step 11) = [137] (last permutation)

14. R1C159 = [437]
14a. -> R2C19 = [85] (cage sums)

15. R1C67 = [59]

16. Split 13(2) at R23C4 = [49] (last permutation)
16a. -> R23C5 = [28]
16b. -> R2C28 = [96]

17. 17(3) at R4C9 = {269} (no 1,3,4) (last combo), locked for C9 and N6
17a. -> R3C89 = [21]

18. 8(3) at R6C7 = {125} (no 3,4) (last combo)
18a. -> R7C7 = 2; R6C78 = {15}, locked for R6 and N6
18b. R9C7 = 6 (step 3)

19. NS at R5C7 = 7
19a. -> R45C8 = {34}, locked for C8 and 22(5)
19a. -> R45C6 = 8(2) (cage split) = {26} (last combo), locked for C6 and N5

20. 14(3) at R6C2: R7C3 = {89}
20a. -> R6C23 must sum to 5 or 6 = [24/42/32] (no 6..8)
20b. -> 2 locked in R6C23 for R6 and N4

21. 17(3) at R4C1 = {179} (no 3,6) (last combo), locked for C1 and N4

22. HS in C2 at R5C2 = 8
22a. -> split 18(4) at R4C24+R5C34 = {1467/3456} (only possible combos)
22b. -> R4C2+R5C3 = {46}, locked for N4
22c. -> R45C4 = 8(2) (cage split) = [35/71]

23. R6C23 = [32]
23a. -> R7C3 = 9 (cage sum)

24. 13(3) at R7C8 = {148} (no 3,5,7) (last combo)
24a. -> R7C8 = 1; R78C9 = {48}, locked for N9

Rest is just singles and simple cage sums.
Walkthrough by Andrew:
In the Assassin Schedule thread, Afmob wrote:
By the way, if no one else posts a wt for A109 I'll post my solving path which is more standard (using Killer triples/Cage blockers and IOD) than Mike's approach.
Thanks Afmob for letting me have the time to solve this puzzle and post my WT first. I hope that you will now post your WT since it's clearly different from mine.

Mike wrote:
Used a trick near the start (step 4a) that may not appeal to all (sorry, folks, just couldn't resist it... :wink:).
Although I don't use URs, I've got no problem with other people using them except, possibly, in a "tag" solution where I'm also taking part. It was a very neat trick, probably not too hard to spot for those who use URs, because the R12C1 and R12C9 cages both have the same total, and made the solution a lot quicker. Mike's rating is clearly correct, assuming that the use of UR can be considered part of a 1.0 rating; I'm not sure about that.

This was my first Assassin since returning from our holiday. Maybe that's why I found it a bit harder than the estimated rating. I was stuck for a while after step 20 before spotting my breakthrough moves in steps 21 and 23. I'll rate it as Hard 1.25 the way I solved it. I'm not sure whether Ed will accept that rating since in previous off-forum discussions he's suggested my ratings when I've used combined cages were possibly a bit too low.

Here is my walkthrough. Afmob has commented that my steps 10b,c and 15a,b,c were complicated. Originally I just used steps 10 and 15 to reduce R2C3 to {47} and R2C7 to {37}, then while stuck at step 20 I noticed the clashes in C3 and C7 and felt that I ought to rework those steps. Steps 10b,c probably don't contribute anything to the solution but steps 15a,b,c make step 16 simpler. The clashes in steps 10 and 15 can be ignored if one replaces step 16 by innies-outies for N3 which also deletes 7,8 from R1C6; that was how I originally did that step. I would still rate A109 at Hard 1.25 on this alternative way.

Prelims

a) R12C1 = {39/48/57}, no 1,2,6
b) R12C9 = {39/48/57}, no 1,2,6
c) R34C3 = {17/26/35}, no 4,8,9
d) R34C7 = {39/48/57}, no 1,2,6
e) R1C234 = {126/135/234}, no 7,8,9
f) R1C678 = {589/679}, 9 locked for R1, clean-up: no 3 in R2C1, no 3 in R2C9
g) 20(3) cage at R2C2 = {389/479/569/578}, no 1,2
h) 20(3) cage at R2C3 = {389/479/569/578}, no 1,2
i) 11(3) cage at R2C6 = {128/137/146/236/245}, no 9
j) 9(3) cage in N3 = {126/135/234}, no 7,8,9
k) 8(3) cage at R6C7 = {125/134}, CPE no 1 in R5C7

1. 45 rule on R123 2 outies R4C37 = 13 = [58/67], clean-up: R3C3 = {23}, R3C7 = {45}
1a. 1 in N1 locked in R1C23, locked for R1
1b. R1C234 = {126/135}, no 4
1c. Killer pair 5,6 in R1C234 and R1C678, locked for R1, clean-up: no 7 in R2C1, no 7 in R2C9

2. 45 rule on N7 2 innies R79C3 = 17 = {89}, locked for C3 and N7
2a. 14(3) cage at R6C2 = {149/158/239/248} (cannot be {167/257347/356} because R7C3 only contains 8,9), no 6,7
2b. R7C3 = {89} -> no 8,9 in R6C2

3. 45 rule on N9 2 innies R79C7 = 8 = [17/26/35/53]

4. 45 rule on N3 2 outies R1C6 + R4C7 = 1 innie R2C7 + 10
4a. Max R1C6 + R4C7 = 17 -> max R2C7 = 7
4b. Min R1C6 + R4C7 = 12 -> min R2C7 = 2

5. 1 in N3 locked in 9(3) cage = {126/135}, no 4

6. R9C34567 = {12579/12678/13479/13569/13578/14568/23469/23478/23568} (cannot be {12489} because R9C7 only contains 3,5,6,7, cannot be {24567} because R9C3 only contains 8,9)
6a. R9C3 = {89} -> no 8,9 in R9C456

7. 45 rule on R789 3 innies R7C357 = 17 = {179/269/278/359/368/458} (cannot be {467} because R7C3 only contains 8,9)
7a. R7C3 = {89} -> no 8,9 in R7C5
7b. 1,2 of {179/269/278} must be in R7C7 -> no 1,2 in R7C5

8. 35(6) cage at R4C5 = {146789/236789/245789/345689}, 8,9 locked for N5

9. 4 in R1 locked in R1C159
9a. 45 rule on R1 3 innies R1C159 = 14 = {248/347}
9b. 2 of {248} must be in R1C5 -> no 8 in R1C5

10. 45 rule on N1 4 innies R1C23 + R23C3 = 13 = {1237/1246/1345}
10a. 4,7 only in R2C3 -> R2C3 = {47}
10b. {1246} must be [6142] (cannot be [1642] because R1C3 = 6 clashes with R34C3 = [26]), no 6 in R1C3
10c. {1345} must be [5143] (cannot be [1543] because R1C3 = 5 clashes with R34C3 = [35]), no 5 in R1C3

11. 20(3) cage at R2C3 = {479/578} (cannot be {389/569} because R2C3 only contains 4,7), no 3,6, CPE no 7 in R2C56

12. 20(3) cage at R2C2 = {389/569/578}, cannot be {479} which clashes with R2C3), no 4
[I had missed that {479} clashes with R12C1 after the Prelims]

13. 14(3) cage at R6C2 = {149/158/239/238} (step 2a)
13a. 8(3) cage at R6C7 = {125/134}
13b. 5 of {125} must be in R6C78 (R6C78 cannot be {12} which clashes with R6C23), no 5 in R7C7, clean-up: no 3 in R9C7 (step 3)

14. 45 rule on R789 3 innies R7C357 (step 7) = {179/269/278/359/368} (cannot be {458} because R7C7 only contains 1,2,3), no 4
14a. 5,6,7 only in R7C5 -> R7C5 = {567}

15. 45 rule on N3 4 innies R1C78 + R23C7 = 24 = {3489/3579/4578} (cannot be {2589/4569} which clashes with R12C9, cannot be {2679/3678} because R3C7 only contains 4,5), no 2,6
15a. {3489} must be [9834] (cannot be [8934] because R1C7 = 8 clashes with R34C7 = [48])
15b. {3579} must be [9735] (cannot be [7935] because R1C7 = 7 clashes with R34C7 = [57])
15c. {4578} must be [5874] (cannot be {78}{45} because R1C78 cannot be {78} which clashes with the combinations for R1C678, cannot be [8574] because R1C7 = 8 clashes with R34C7 = [48])
15d. -> R1C7 = {59}, R1C8 = {78}, R2C7 = {37}

16. R1C678 = {589/679}
16a. R1C8 = {78} -> no 7,8 in R1C6

17. 9(3) cage in N3 = {126} (cannot be {135} which clashes with R1C78 + R23C7), no 3,5
[Alternatively 2,6 in N3 locked in 9(3) cage = {126}]

18. 11(3) cage at R2C6 = {137/236} (cannot be {128/146/245} because R2C7 only contains 3,7), no 4,5,8, CPE no 3 in R2C5

19. 45 rule on R789 4 outies R6C2378 = 1 innie R7C5 + 5
19a. R7C5 = {567} -> R6C2378 = 10,11,12 = {1234/1235/1245}, 1,2 locked for R7

20. Hidden killer pair 1,2 in R6C2378 for R6, R6C23 contains one of 1,2 (step 2a) -> R6C67 must contain one of 1,2
20a. R6C78 cannot be {13/23} because cannot form 8(3) cage (no 4 in R7C7) -> no 3 in R6C78

21. Combined cage R1C1 + R23C12 = 32(5) = {35789/45689}, 5 locked for N1

22. R1C234 = {126/135} (step 1b)
22a. 5 of {135} must be in R1C4 -> no 3 in R1C4

23. 45 rule on N2 2 innies R1C46 = 2 outies R2C37 + 1
23a. R1C46 cannot total 12 -> R2C37 cannot total 11 -> no 4 in R2C3
23b. R2C3 = 7, R2C7 = 3, clean-up: no 5 in R1C2, no 9 in R2C9, no 1 in R3C6 (step 18)
23c. R1C46 = 11 (step 23) = [29/56/65]
23d. 3 in N2 locked in R13C5, locked for C5
23e. R123C5 = 3{19/28/46}, no 5,7

24. R3C6 = 7 (hidden single in R3), R2C6 = 1 (step 18), clean-up: no 9 in R23C5 (step 23e)

25. R1C7 = 9 (hidden single in N3)

26. 9 in N2 locked in R23C4, locked for C4
26a. 20(3) cage at R2C3 (step 11) = {479} (only remaining combination), no 5,8, 4 locked for C4 and N2, clean-up: no 6 in R23C5 (step 23e)

27. Naked triple {238} in R123C5, locked for C5 and N2
27a. Naked pair {56} in R1C46, locked for R1
27b. Naked triple {123} in R1C23 + R3C3, locked for N1, clean-up: no 9 in R2C9
27c. Naked pair {48} in R12C1, locked for C1 and N1

28. R456C1 = {179/269/359}, 9 locked for C1 and N4

29. R3C5 = 8 (hidden single in R3), R12C5 = [32], R3C3 = 3 (hidden single in R3), R4C3 = 5, R4C7 = 8 (step 1), R3C7 = 4, R23C4 = [49], R2C1 = 8, R1C1 = 4, R2C89 = [65], R2C2 = 9, R1C9 = 7, R1C8 = 8, R1C6 = 5 (step 16), R1C4 = 6, clean-up: no 3 in R456C1 (step 28)

30. R5C2 = 8 (hidden single in C2)

31. R456C9 = {269} (only remaining combination), locked for C9 and N6 -> R3C89 = [21]
31a. Naked triple {348} in R789C9, locked for N9

32. 8(3) cage at R6C7 = {125} (only remaining combination), no 4
32a. R7C7 = 2, R9C7 = 6 (step 3)
32b. Naked pair {15} in R6C78, locked for R6 and N6 -> R5C7 = 7
32c. Naked pair {34} in R45C8, locked for 22(5) cage
32d. Naked pair {26} in R45C6, locked for C6 and N5

33. 14(3) cage at R6C2 (step 2a) = {239/248}, 2 locked for N4, clean-up: no 6 in R456C1 (step 28)
33a. Naked triple {179}in R456C1, locked for C1 and N4
33b. 2 in C1 locked in R89C1, locked for N7

34. 13(3) cage in N9 = {148} (only remaining combination, cannot be {139/157} because 1,7,9 only in R7C8) -> R7C8 = 1, R78C9 = {48}, locked for C9 -> R8C7 = 5, R9C9 = 3, R9C6 = 4, R6C78 = [15]

35. R9C3 = 8 (hidden single in R9), R9C45 = {15} (step 6), locked for R9 and N8 -> R9C12 = [27], R89C8 = [79], R7C3 = 9

36. 12(3) cage in N7 = {345} (only remaining combination) -> R8C1 = 3

and the rest is naked singles and a cage sum

Now to catch up on other puzzles that have been posted since we went on holiday. Fortunately they have all had lower ratings than this one except for Ed's recently posted A110X V2. I probably won't post any walkthroughs for those puzzles, unless I come up with anything significantly different, but I hope to solve them all.
Walkthrough by Afmob:
Even though I'm not a fan of uniqueness moves, I have to admit that Mike's step 4a was quite neat. As you can see in my walkthrough, you only need to focus on one area to crack this killer and Andrew and Mike also had their main steps in R123.

A109 Walkthrough:

1. R123
a) Innies = 7(2) = [25/34]
b) 1 locked in 9(3) @ N1 = 1{26/35}; R1C4 <> 1 and 1 locked for R1
c) 4 locked in Innies R1 = 4{28/37};
d) Both 12(2) @ R1: R2C19 <> 3,7
e) 2 locked in Innies N1 = 13(4) = 12{37/46} must have one of (47) and it's only possible @ R2C3
-> R2C3 = (47)
f) 20(3) @ N2 = 7{49/58} because R2C3 = (47);
g) 8(2) = [26/35]
h) 12(2) @ R3 = [48/57]
i) 9(3): R1C4 <> 3 because R1C23 <> 5

2. R123 !
a) ! 9(3) can only have one (345) because two of (345) @ N3 must be @ R3C7 and 12(2) @ C9
-> 9(3) = {126} locked for N3
b) 22(3): R1C6 <> 7 because 6 only possible there
c) ! Innies+Outies N2: -1 = R2C37 - R1C46; R2C3 = (47):
- R2C7 <> 4,7 because R1C46 cannot be 12(2)
d) 7 locked in R1C789 @ N3 for R1
e) 12(2) @ N1 <> 5
f) 5 locked in 20(3) @ R2C2 @ N1 = 5{69/78}
g) Innies R1 = 14(3): R1C9 <> 3 because 7 only possible there
h) 12(2) @ C9 <> 9
i) Hidden Single: R2C7 = 3 @ N3
j) Killer pair (45) locked in 12(2) @ C9 + R3C7 for N3

3. R123
a) 5 locked in R1C46 @ R1 for N2
b) 20(3) @ N2 = {479} -> 9 locked for C4+N2
c) 22(3) must have one of (56) and it's only possible @ R1C6 -> R1C6 <> 8
d) 11(3) = 3{17/26}
e) 8 locked in 13(3) @ N2 = 8{14/23} -> 8 locked for C5
f) Killer pair (12) locked in 13(3) @ 11(3) for N2
g) 9(3): R1C23 <> 6 because R1C4 >= 5
h) Innies N1 = 13(4) = {1237} -> R2C3 = 7; 3 locked for N1
i) Hidden Single: R3C6 = 7 @ N2 -> R2C6 = 1
j) 13(3) = {238} locked for C5

4. R123
a) 12(2) @ R1C1 = {48} locked for C1+N1
b) Hidden Single: R3C5 = 8 @ R3, R3C3 = 3 @ R3 -> R4C3 = 5
c) 9(3) @ N1 = {126} -> R1C4 = 6
d) 22(3) = {589} -> R1C6 = 5; 8 locked for R1+N3
e) 12(2) @ C9 = {57} -> R1C9 = 7, R2C9 = 5
f) R3C7 = 4 -> R4C7 = 8

5. C789
a) 17(3) = {269} locked for C9+N6
b) 8(3) = {125} because R67C7 <> 3,4 -> R7C7 = 2, {15} locked for R6+N6
c) Innies N9 = R9C7 = 6
d) R3C9 = 1
e) 13(3) = {148} because R78C9 = (348) -> R7C8 = 1, {48} locked for N9
f) 22(5) = {23467} -> R5C7 = 7, 3 locked for C8, R45C6 = {26} locked for C6+N5

6. N47
a) Innies N7 = 17(2) = {89} locked for C3+N7
b) 14(3) = 2{39/48} because R7C3 = (89) -> 2 locked for R6+N4; R6C2 <> 8,9
c) 17(3) = {179} locked for C1+N4

7. N579
a) 35(6) = 4689{17/35} -> R7C5 = 6; {48} locked for N5
b) 26(5) = 468{17/35} -> R5C2 = 8; 4 locked for N4
c) R6C3 = 2, R6C2 = 3 -> R7C3 = 9
d) R9C3 = 8, R9C9 = 3
e) 24(5) = 168{27/45} -> 1 locked for R9+N8
f) 35(6) = {146789} -> 1,7 locked for N5 and 1 also locked for C5

8. Rest is singles.

Rating: 1.25. I used a Killer triple which can also be seen as a cage blocker.


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PostPosted: Tue May 10, 2011 4:12 am 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 110X by Ed (July 2008) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3:d:k:4353:3330:3330:3331:3331:2564:2564:2053:2053:4353:2310:2310:3331:3847:2312:2312:2825:2825:4353:5386:5386:3847:3847:8971:8971:4620:5901:4353:5386:5386:4622:4622:8971:4620:4620:5901:2575:2575:2575:4622:8971:4880:5901:5901:5901:2833:3602:3602:8971:4880:4880:4115:4115:5652:2833:3602:8971:8971:4629:4374:4115:4115:5652:1815:1815:4629:4629:4629:4374:3352:3352:5652:4121:4121:3610:3610:4374:4374:2075:2075:5652:
Solution:
+-------+-------+-------+
| 6 4 9 | 7 1 8 | 2 5 3 |
| 2 8 1 | 5 9 3 | 6 7 4 |
| 5 3 7 | 4 2 6 | 8 1 9 |
+-------+-------+-------+
| 4 5 6 | 3 7 2 | 9 8 1 |
| 7 1 2 | 8 5 9 | 3 4 6 |
| 8 9 3 | 1 6 4 | 7 2 5 |
+-------+-------+-------+
| 3 2 4 | 9 8 5 | 1 6 7 |
| 1 6 5 | 2 3 7 | 4 9 8 |
| 9 7 8 | 6 4 1 | 5 3 2 |
+-------+-------+-------+
Quote:
Ed:
In A106 thread, goooders wrote:
I like killersudokuonlines weekly offering because ... The cage sizes are much bigger
My original puzzle was designed around this idea. But.....couldn't solve it. So, had to keep chopping it down and kept getting lower and lower SSscores until finally found one I could solve! Sooo frustrating. Hope the result is not too easy.
I'll post an "old school" v2 of A110 (same solution: slightly different cage pattern) when I've had another crack myself first. So, a very big thankyou to Nasenbaer for volunteering for A111!!
SS(v3.2.1) score = 0.79

Afmob: Thanks Ed for an Assassin with an interesting cage structure! Especially the large 35(7) is nice. I tried the same (providing larger cages) with A109.
I think my step 3a is too advanced for a 0.75 Killer and since SS also uses this move SudokuSolver's rating is a bit off, in my opinion.
Rating: 1.0.

Ed:
Afmob wrote:
I think my step 3a is too advanced for a 0.75 Killer and since SS also uses this move SudokuSolver's rating is a bit off, in my opinion...Rating: 1.0.
Great. :D It would have been too notorious to have the first 0.75 rating Assassin since...... :?:
Glad you liked that cage. I'm hoping it will yield a v2 with some nice triple cross-over moves (udosuk's name: but there is probably a more technically correct term, something something fish)

Mike(mhparker): I've just finished this one after more than 3 hours of chipping away at the candidates in 2 sessions :shock:, wondering how the puzzle could be so easy. (See hidden window for Mike's full post)

Andrew: Thanks Ed. A110X was a fun puzzle. I enjoyed the 35(7) cage along and "hugging" the diagonal and it was also very helpful for the solution.
I reached the result of Afmob's step 3a fairly early but using a slightly different route.
I'll also rate A110X at 1.0.

Walkthrough by Afmob:
Thanks Ed for an Assassin with an interesting cage structure! Especially the large 35(7) is nice. I tried the same (providing larger cages) with A109.

I think my step 3a is too advanced for a 0.75 Killer and since SS also uses this move SudokuSolver's rating is a bit off, in my opinion.

A110X Walkthrough:

1. R789
a) 16(2) = {79} locked for R9+N7
b) 14(2) = {68} locked for R9
c) 8(2) = {35} locked for R9+N9
d) 13(2) <> 8

2. N1478
a) Outies N14 = 5(2) = [23/32/41]
b) Innies N1478 = 13(2) <> 1,2,3; R7C4 <> 4,6
c) 7(2) <> 3,4 because (34) is a Killer pair of Outies N14
d) Innies N147 = 17(3) = 8{36/45} because 8 locked there @ N7 -> 8 locked for C3
and R8C3 <> 6,8 because 3 only possible there

3. N5 !
a) ! Using Innies N1478: Innies+Outies R6789: 8 = R5C6 - R6C4 -> R5C6 = 9, R6C4 = 1
b) Innies = 8(3) <> 6,7,8
c) Using Innies N1478: Outies N5 = 14(2) = [59/68/86]

4. C1 !
a) 7(2): R8C1 <> 6
b) ! Innies = 17(3) = 9{17/25/36} because R8C1 = (125) -> R9C1 = 9; R5C1 = (367)
c) R9C2 = 7

5. N5
a) Outies = 14(2) = {68} locked for R3+35(7)
b) Innies N1478 = 13(2) = {49} -> R7C3 = 4, R7C4 = 9
c) Innies = 8(3) = {125} locked for N5+D/
d) R8C2 = 6 -> R8C1 = 1

6. N478+C1
a) 11(2) = {38} -> R7C1 = 3, R6C1 = 8
b) R7C2 = 2, R8C3 = 5, R9C3 = 8, R9C4 = 6
c) Innies C1 = R5C1 = 7
d) 10(3) = {127} -> R5C2 = 1, R5C3 = 2
e) 14(3) = {239} -> 3,9 locked for R6+N4
f) R4C3 = 6

7. N13
a) 13(2) = {49} -> R1C2 = 4, R1C3 = 9
b) R3C7 = 8
c) 11(2) = {47} -> R2C8 = 7, R2C9 = 4
d) R1C9 = 3 -> R1C8 = 5
e) 10(2) = {28} -> R1C7 = 2, R1C6 = 8
f) 18(3) = 9{18/27/45} <> 3 because R3C8 = (19)

8. Rest is singles without considering the diagonals.

Rating: 1.0.
Mike's full post:
Hi folks,

I've just finished this one after more than 3 hours of chipping away at the candidates in 2 sessions :shock:, wondering how the puzzle could be so easy. After what was probably about 30 steps, including an implied cage blocker and a simple chain (!) I finally stumbled across Afmob's step 3a, which really killed the puzzle off. The rest then took less than 5 minutes. So Afmob's step 3a really seems to be the big chink in the armor for this puzzle: how long it takes to do the puzzle depends on how long it takes to see this move. Incidentally, this is one reason why I'm not so keen on rating puzzles depending on how long it takes to do them, since one person could see such a key move in a couple of minutes, and another after several hours.

"More hidden comments" BTW, the reason this move was difficult to see is that it's based on a split innie/outie (I/O) difference cage. I agree with Afmob that such a move deserves a rating of 1.0 rather than 0.75. It's IMO crucially important that any rating scheme considers such subtleties when evaluating a puzzle.

Last but not least, many thanks for the puzzle, Ed! :D
Walkthrough by Andrew:
Thanks Ed. A110X was a fun puzzle. I enjoyed the 35(7) cage along and "hugging" the diagonal and it was also very helpful for the solution.

I reached the result of Afmob's step 3a fairly early but using a slightly different route.

After going through Afmob's one, I now realise that my walkthrough could have been quite a lot shorter if I'd obtained the result of step 8 before I did step 3; unfortunately I never thought of revisiting step 3 after doing step 8.

I'll also rate A110X at 1.0. I don't think I can rate it lower with the chain of 45s that I used in step 5.

Here is my walkthrough. I've given eliminations on the diagonals; it's so easy for those of us doing manual eliminations to overlook them.

Prelims

a) R1C23 = {49/58/67}, no 1,2,3
b) R1C67 = {19/28/37/46}, no 5
c) R1C89 = {17/26/35}, no 4,8,9
d) R2C23 = {18/27/36/45}, no 9
e) R2C67 = {18/27/36/45}, no 9
f) R2C89 = {29/38/47/56}, no 1
g) R67C1 = {29/38/47/56}, no 1
h) R8C12 = {16/25/34}, no 7,8,9
i) R8C78 = {49/58/67}, no 1,2,3
j) R9C34 = {59/68}
k) R9C78 = {17/26/35}, no 4,8,9
l) R9C12 = {79}, locked for R9 and N7, clean-up: no 2,4 in R6C1, no 5 in R9C34, no 1 in R9C78
m) R5C123 = {127/136/145/235}, no 8,9
n) 19(3) cage in N5 = {289/379/469/478/568}, no 1

1. Naked pair {68} in R9C34, locked for R9, clean-up: no 2 in R9C78
1a. Naked pair {35} in R9C78, locked for R9 and N9, clean-up: no 8 in R8C78

2. 45 rule on N5 3 innies R4C6 + R5C5 + R6C4 = 8 = {125/134}, 1 locked for N5, D/ and 35(7) cage, clean-up: no 7 in R1C8, no 6 in R8C1

3. 45 rule on C1 3 innies R589C1 = 17 = {179/269/359/467}
3a. 1,2 of {179/269} must be in R8C1 -> no 1,2 in R5C1
3b. 6 of {467} must be in R5C1 -> no 4 in R5C1

4. 45 rule on C123 3 innies R789C3 = 17 = {368/458}, no 1,2, 8 locked for C3 and N7, clean-up: no 5 in R1C2, no 1 in R2C2, no 3 in R6C1

5. 45 rule on N14 3 innies R6C123 = 20 = {389/479/569/578}, no 1,2
[I noticed while checking through this walkthrough that taken together with the 14(3) cage this gives R6C1 = R7C2 + 6, R6C1 = {789}, R7C2 = {123}. This is obtained from step 8 so I haven’t tried to rewrite my steps.]
5a. 45 rule on N9 3 outies R6C789 = 14
5b. 45 rule on R6 3 innies R6C456 = 11 = {128/137/146} (cannot be {236/245} because R6C56 must total at least 10 for the 19(3) cage), no 5,9 -> R6C4 = 1
[Alternatively can use R6C456 = 11 together with 19(3) to get R5C6 = R6C4 + 8 -> R5C6 = 9, R6C4 = 1.]
5c. R6C56 = {28/37/46} = 10 -> R5C6 = 9, clean-up: no 1 in R1C7
5d. 18(3) cage in N5 = {378/468/567}, no 2

6. 35(7) cage at R3C6 = {1235789/1245689/1345679}, CPE no 9 in R3C4

7. R5C123 = {127/136/145/235}
7a. 7 of {127} must be in R5C1 -> no 7 in R5C23

8. 45 rule on N14 2 outies R7C12 = 5 = [23/32/41], no 5,6, no 4 in R7C2, clean-up: no 5,6 in R6C1

9. Killer pair 3,4 in R7C12 and R78C3, locked for N7
[I could have got that elimination after step 4 from the clash with R78C3.]

10. 45 rule on R12 2 outies R34C1 = 1 innie R2C5
10a. Min R34C1 = 3 -> min R2C5 = 3
10b. Max R2C5 = 9 -> max R34C1 = 9, no 9

11. 45 rule on N78 4 innies R7C1234 = 18, R7C12 = 5 (step 8) -> R7C34 = 13 = [49/58/67/85], no 3 in R7C3, no 2,3,4,6 in R7C4
11a. R7C34 = 13, R4C6 + R5C5 + R6C4 = 8 (step 2) -> R3C67 = 14 (cage sum) = [59/68/86], R3C6 = {568}, R3C7 = {689}

12. 35(7) cage at R3C6 = {1245689/1345679} (cannot be {1235789} which clashes with R4C6 + R5C5), 4 locked in R4C6 + R5C5 + R7C3, locked for D/, clean-up: no 7 in R2C9
12a. {1245689} must be {68}{25}1[49], {1345679} must be [59]{34}1[67] -> R7C34 = [49/67], no 5,8
12b. CPE no 6 in R3C3

13. R789C3 (step 4) = {368/458}
13a. 3 of {368} must be in R8C3 -> no 8 in R8C3
13b. R9C3 = 8 (hidden single in C3), R9C4 = 6
13c. R7C3 = {46} -> no 4,6 in R8C3
13d. 4 in N7 locked in R7C13, locked for R7

14. 45 rule on N69 2 outies R3C89 = 10 = {19/28/37/46}, no 5

15. 17(4) cage in N8 = {1457/2348} (cannot be {1358/2357} because R9C56 must contain two of 1,2,4), 4 locked in R9C56 (because they only contain 1,2,4), locked for R9 and N8
15a. R9C56 = {124} -> no 1,2 in R78C6
15b. R78C6 = {38/57}

16. Min R7C78 = {16} (cannot be {12} which clashes with R9C9) = 7 -> max R6C78 = 9, no 8,9 in R6C78
16a. Max R6C78 = 9 -> min R6C9 = 5 (step 5a)

17. R6789C9 = {1489/1579/1678/2479/2569/2578} (cannot be {4567} which doesn’t contain 1,2)
17a. R9C9 = {12} -> no 1,2 in R78C9

18. Hidden killer pair 1,2 in R7C78 and R9C9 for N9 -> R7C78 must contain 1 or 2
18a. Killer pair 1,2 in R7C12 and R7C78, locked for R7

19. R6C789 = 14 (step 5a)
19a. R7C78 must contain 1 or 2 (step 18) -> R6C78 cannot be {23} because cannot form 16(4) cage including all of 1,2,3 -> no 9 in R6C9 (cage sum)

20. 9 in R6 locked in R6C123, locked for N4

21. 5 in N7 locked in R8C123, locked for R8, clean-up: no 7 in R7C6 (step 15b)
21a. 18(4) cage at R8C3 = {1359/2358} (cannot be {1278} because R8C3 only contains 3,5), no 7
21b. 1 of {1359} must be in R8C5 -> no 9 in R8C5

22. 45 rule on N2 3 innies R123C6 = 17 = {278/368/458/467}, no 1, clean-up: no 9 in R1C7, no 8 in R2C7

23. R9C6 = 1 (hidden single in C6), R9C9 = 2, locked for D\, R9C5 = 4, clean-up: no 6 in R1C8, no 7 in R2C3, no 9 in R2C8, no 3,5 in R4C6 (step 2), no 3,8 in R78C6 (step 15b)
23a. R78C6 = [57], R7C4 = 9, R7C3 = 4 (step 12a), R4C6 = 2, R5C5 = 5 (step 2), 2,5 locked for D/, 5 locked for D\, R8C2 = 6, R8C1 = 1, 6 locked for D/, R3C7 = 8, R3C6 = 6, clean-up: no 9 in R1C2, no 7 in R1C3, no 3,4 in R1C7, no 2,3 in R1C8, no 3 in R2C3, no 2,3,4,7 in R2C7, no 3 in R2C8, no 3,5,6,9 in R2C9, no 7 in R6C1, no 3,8 in R6C5, no 8 in R6C6 (both step 5b)
23b. R2C89 = [74], 7 locked for D/, R1C9 = 3, R1C8 = 5, R9C1 = 9, R9C2 = 7, R9C78 = [53] , R6C1 = 8, R7C1 = 3, R7C2 = 2, R7C5 = 8, R8C3 = 5, clean-up: no 8 in R1C2, no 6 in R1C3, no 2 in R2C3
23c. R1C23 = [49], R1C6 = 8, R1C7 = 2, R2C6 = 3, R2C7 = 6, R2C23 = [81], 8 locked for D\, R2C5 = 9, R1C45 = [71], R1C1 = 6, locked for D\, R5C1 = 7, R6C6 = 4, R4C4 = 3 , 3,4 locked for D\, R3C3 = 7, locked for D\

and the rest is naked singles without using diagonals

I see that Ed's V2 also removes my early breakthrough by removing 3 innies for N14 (step 5).


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PostPosted: Tue May 10, 2011 4:33 am 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 110X V2 by Ed (July 2008) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3:d:k:4353:3330:3330:3331:3331:2564:2564:2053:2053:4353:2310:2310:3331:3847:2312:2312:2825:2825:4353:5386:5386:3847:3847:8971:8971:4620:5901:4353:5386:5386:4622:4622:8971:4620:4620:5901:5391:5391:5391:4622:8971:4880:5901:5901:5901:5391:3601:3601:8971:4880:4880:4114:4114:5651:5391:3601:8971:8971:4628:4373:4114:4114:5651:1814:1814:4628:4628:4628:4373:3351:3351:5651:4120:4120:3609:3609:4373:4373:2074:2074:5651:
Solution:
+-------+-------+-------+
| 6 4 9 | 7 1 8 | 2 5 3 |
| 2 8 1 | 5 9 3 | 6 7 4 |
| 5 3 7 | 4 2 6 | 8 1 9 |
+-------+-------+-------+
| 4 5 6 | 3 7 2 | 9 8 1 |
| 7 1 2 | 8 5 9 | 3 4 6 |
| 8 9 3 | 1 6 4 | 7 2 5 |
+-------+-------+-------+
| 3 2 4 | 9 8 5 | 1 6 7 |
| 1 6 5 | 2 3 7 | 4 9 8 |
| 9 7 8 | 6 4 1 | 5 3 2 |
+-------+-------+-------+
Quote:
Ed:
mhparker wrote:
After what was probably about 30 steps
Hope you kept your notes! This "old school" V2 has the same solution as the original, but takes out the "easy" opening (Afmob's step 3a). Finally worked out how to solve this next one nicely, so technically, it could be a 1.25 rating (no chains needed), but since it took me all week to find, I'll estimate it as 1.5.
Next time, I'll give myself 2 weeks to make the Assassin.
SS(v3.2.1) score = 1.31

Mike(mhparker): Both of these puzzles (A110X and A110X V2) were good, but the "V2" would IMO have been an even better "V1" than the original. Thanks once again, Ed, for providing us with the opportunity for making some interesting moves!
I found it difficult to write a WT for the V2, because, as is common for Killer-X puzzles, there were often several ways to go, and, when the puzzle starts to crack, things happen so quickly, that one tends to fall over one's toes (so to speak) trying to keep up.
Afmob commented on my A109 WT that I did too much in the preliminaries. He has a point, because for puzzles with a "racing start", like with my A107 cage pattern, it's even possible to get placements (!) within the preliminaries if one starts considering cage-external effects such as locked candidates and conflicting combinations (not to mention the resulting clean-ups). Therefore, starting with this WT, I am only doing combinational reductions at the preliminaries stage, like most automated solvers do. Any locked candidates, conflicting combinations, naked subsets, and so on, arising from the preliminaries stage are handled as regular WT steps.
This WT does not go "by the book", so is sprinkled with moves that are too complex for a V1.25. One move (step 27b) is even bordering on T&E :shock:, but was easy to spot after step 26 and should provide an interesting alternative to going down the "beaten track", as it were. So I decided to keep it in.
Hopefully, at least one other WT will be posted with a more conventional approach (Afmob?, Andrew?, ...), which will probably show that the SScore is spot on in this case.

Afmob: Ed was spot on with his description since the moves weren't too difficult (technical rating: 1.25, most difficult step: 4c) but it took me quite some time to find them. That's why I rate this Killer (hard) 1.25.
But I have to really wonder why it took me so long to find step 5c which was quite easy to see.

udosuk: Here is my 5-step walkthrough for V2. I haven't read the others' moves but I suspect the critical ones might be similar. :geek:
I like this puzzle because after the intense actions the mopping up is really short and sweet. :alien:

Andrew, in 2010: Continuing working through my backlog, the next variant I tried was A110X V2 which was another one which I found that I hadn't started.
Thanks Ed for another interesting and challenging variant. You have a knack for making good variants by just combining two cages, in this case the 10(3) cage at R5C1 and the 11(2) cage at R6C1.
Ed wrote:
mhparker wrote:
After what was probably about 30 steps
Hope you kept your notes! This "old school" V2 has the same solution as the original, but takes out the "easy" opening (Afmob's step 3a).
It might have helped if I'd looked to see how I'd solved the original but I didn't need to; I solved this as a new puzzle.
I'll rate my walkthrough for A110X V2 at Hard 1.25.

Walkthrough by Mike:
Hi folks,

Both of these puzzles (A110X and A110X V2) were good, but the "V2" would IMO have been an even better "V1" than the original. Thanks once again, Ed, for providing us with the opportunity for making some interesting moves!

I found it difficult to write a WT for the V2, because, as is common for Killer-X puzzles, there were often several ways to go, and, when the puzzle starts to crack, things happen so quickly, that one tends to fall over one's toes (so to speak) trying to keep up.

Afmob commented on my A109 WT that I did too much in the preliminaries. He has a point, because for puzzles with a "racing start", like with my A107 cage pattern, it's even possible to get placements (!) within the preliminaries if one starts considering cage-external effects such as locked candidates and conflicting combinations (not to mention the resulting clean-ups). Therefore, starting with this WT, I am only doing combinational reductions at the preliminaries stage, like most automated solvers do. Any locked candidates, conflicting combinations, naked subsets, and so on, arising from the preliminaries stage are handled as regular WT steps.

This WT does not go "by the book", so is sprinkled with moves that are too complex for a V1.25. One move (step 27b) is even bordering on T&E :shock:, but was easy to spot after step 26 and should provide an interesting alternative to going down the "beaten track", as it were. So I decided to keep it in.

Hopefully, at least one other WT will be posted with a more conventional approach (Afmob?, Andrew?, ...), which will probably show that the SScore is spot on in this case.

Edited to incorporate late feedback from Andrew. Thanks!

Assassin 110X V2 Walkthrough (35 steps)

Prelims

a) 13(2) at R1C2 and R8C7 = {49/58/67} (no 1..3)
b) 10(2) at R1C6 = {19/28/37/46} (no 5)
c) 8(2) at R1C8 and R9C7 = {17/26/35} (no 4,8,9)
d) 9(2) at R2C2 and R2C6 = {18/27/36/45} (no 9)
e) 11(2) at R2C8 = {29/38/47/56} (no 1)
f) 19(3) at R5C6 = {289/379/469/478/568} (no 1)
g) 7(2) at R8C1 = {16/25/34} (no 7..9)
h) 16(2) at R9C1 = {79} (no 1..6,8)
i) 14(2) at R9C3 = {59/68} (no 1..4,7)

1. Naked pair (NP) at R9C12 = {79}, locked for R9 and N7
1a. cleanup: no 5 in R9C34, no 1 in R9C78

2. NP at R9C34 = {68}, locked for R9
2a. cleanup: no 2 in R9C78

3. NP at R9C78 = {35}, locked for R9 and N9
3a. cleanup: no 8 in R8C78

4. Innies N5: R4C6+R5C5+R6C4 = 8(3) = {125/134} (no 6..9)
4a. 1 locked for N5 and D/
4b. cleanup: no 7 in R1C8, no 6 in R8C1
(Edit: Afmob pointed out that 1 is also locked for 35(7), although it becomes redundant after steps 8 & 9)

5. Outies N14: R7C12 = {14/23} (no 5,6,8) = {(3/4)..}

6. R7C12 (step 5) blocks {34} combo for 7(2) at R8C1
6a. -> no 3,4 in R8C12

7. Outies N69: R3C89 = 10(2) = {19/28/37/46} (no 5)

8. Innies N2369: R3C67 = 14(2) = {59/68} (no 1..4,7)

9. Innies N1478 or 35(7) cage split (steps 4 and 8): R7C34 = 13(2) = [49/67] (no 1..3,5,8; no 4,6 in R7C4)
(Note: {58} blocked by R3C67 (step 8))

10. Innie/Outie difference (IOD) N8: R79C4 = R8C3 + 10
10a. -> R79C34 = [4986/6786]
([6768] impossible (2 6s in C3); [4968] blocked because R79C4 cannot sum to 17, due to no 7 in R8C3)
10b. -> R9C34 = [86]
10c. -> R79C4 sum to 13 or 15
10d. -> R8C3 = {35} (no 1,2,4,6)
10e. cleanup: no 5 in R1C2, no 1 in R2C2

11. 6 in N7 locked in R7C3+R8C2 for D/
11a. cleanup: no 2 in R1C8, no 5 in R2C9, no 8 in R3C6

12. 8 in D/ locked in R2C8+R3C7 for N3
12a. cleanup: no 2 in R1C6, no 1 in R2C6, no 3 in R2C8, no 2 in R3C89 (step 7)

13. 5 in R7 locked in R7C56 for N8

14. 4 in N7 locked in R7C123 for R7

15. 18(4) at R7C5 = {1359/2358} (no 4,7) = {(1/2)..}
(Note: {1278} unplaceable, because none of these digits in R8C3, {1458/2349/2457} blocked by R9C56)
15a. -> no 3 in R8C6 (CPE)

16. 18(4) at R7C5, R9C5 and R9C6 form killer triple (KT) on {124} within N8
16a. -> no 1,2,4 in R78C6

17. 4 in N8 locked in R9C56 for R9

18. 22(4) at R6C9 cannot have both of {12}, otherwise cage sum unreachable
18a. -> no 1,2 in R678C9

19. 17(4) at R7C6 cannot have 2 of {789} due to cage sum
19a. -> no 7..9 in R7C6
(Alternatively, same result reachable via KT on {789} formed by 18(4), R7C4 and R8C6)

20. IOD R8: R8C69 = R7C5 + 7
20a. -> R7C5+R8C69 = [584/878] (Note: [979/896] blocked by R8C78)
20b. -> R7C5 = {58}, R8C6 = {78}, R8C9 = {48}
20c. 8 locked in R7C5+R8C6 for N8

21. 17(4) at R7C6: R9C56 (step 17) = {14/24} = 5 or 6
21a. -> R78C6 must sum to 11 or 12
21b. -> R78C6 = [38/57] = {(5/8)..} (no eliminations yet)

22. 35(7) at R3C6 = {1245689/1345679}
(Note: {1235789} blocked because none of these digits in R7C3,
{2345678} blocked because 35(7) must contain a 1 (within h8(3), step 4))
22a. 4 locked in R4C6+R5C5+R6C4+R7C3 for D/
22b. cleanup: no 7 in R2C9
(Note: CPE eliminations on {69} available in R3C34 here, but step 24 is more powerful)

23. Combined 18(4) cage at R13C89 cannot be {2367}, due to R2C89 (prelim e)
23a. -> R3C89 (step 7) cannot be {37}
23b. -> no 3,7 in R3C89

24. R3C89 = {19/46} = {(6/9)..}
24a. -> R3C67 (step 8) and R3C89 form KP on {69}
24b. -> no 6,9 elsewhere in R3

25. Innies N2: R123C6 = 17(3) = {179/269/368/467} (no 5)
(Note: {278} unplaceable due to R3C6, {359} blocked by R7C6, {458} blocked by R78C6 (step 21b))
25a. cleanup: no 4 in R2C7, no 9 in R3C7

26. 9 in D/ locked in R2C8+R9C1
26a. -> no 9 in R2C1 (CPE/crossover)

27. no 9 in 17(4) at R1C1. Here's how:
27a. {1259} blocked directly by R8C1
27b. {1349} forces R78C1 to [25]: impossible, since R78C1 can't sum to 7 (due to 7(2) at R8C12)
27c. -> no 9 in R14C1

(Note: could have seen step 27 much earlier, but it was only the crossover elimination in
step 26a that started me thinking as to whether I could squeeze the 9 in N1 even further)

28. 9 in N1 locked in 13(2) at R1C2 = {49}, locked for R1 and N1
28a. cleanup: no 1,6 in R1C67, no 5 in R2C23

29. Consider combined cages 10(2) at R1C6 and 8(2) at R1C8:
29a. Either R1C67 = [82], or R1C67 = {37} -> R1C89 = [62]
29a. -> 2 locked in R1C79 for R1 and N3
29b. cleanup: no 7 in R2C6, no 9 in R2C89

30. Hidden single (HS) in D/ at R9C1 = 9
30a. -> R9C2 = 7

31. 9 in N3 locked in R3C89 = {19}, locked for R3 and N3
31a. cleanup: no 7 in R1C9, no 8 in R2C6

32. Naked single (NS) at R3C6 = 6
32a. -> R3C7 = 8 (step 8), R7C3 = 4, R7C4 = 9 (step 9)
32b. cleanup: no 3 in R2C7, no 3 in R2C9, no 1 in R7C12 (step 5)

33. HS in N2 at R2C5 = 9
33a. -> split 6(2) at R3C45 = {24}, locked for R3 and N2

Note: next 2 moves intended to reach "all singles" stage quickly:

34. 8(3) at R4C6+R5C5+R6C4 = {125} (last combo), locked for N5 and D/
34a. cleanup: no 3,6 in R1C8, no 6 in R2C9, no 2,5 in R8C1

35. 9 in C6 locked 19(3) at R5C6 = {379/469} (no 8)
35a. 6 of {469} must go in R6C5
35b. -> no 4 in R6C5

All singles and simple cage sums from now on.
Walkthrough by Afmob:
Ed was spot on with his description since the moves weren't too difficult (technical rating: 1.25, most difficult step: 4c) but it took me quite some time to find them. That's why I rate this Killer (hard) 1.25.
But I have to really wonder why it took me so long to find step 5c which was quite easy to see.

A110X V2 Walkthrough:

1. R789
a) 16(2) = {79} locked for R9+N7
b) 14(2) = {68} locked for R9
c) 8(2) = {35} locked for R9+N9
d) 13(2) <> 8
e) Outies N14 = 5(2) = {14/23}
f) 7(2) <> 3,4 because (34) is a Killer pair of Outies N14
g) Innies N147 = 17(3) = 8{36/45} -> 8 locked for C3

2. N5
a) Innies N5 = 8(3) = 1{25/34} -> 1 locked for N5+D/+35(7)
b) Innies N2369 = 14(2) = {59/68}
c) Using Innies N2369: Outies N5 = 13(2) = [49/67] because
(58) is a Killer pair of Innies N2369

3. R789+D/
a) Innies N147 = 17(3): R8C3 = (35) because it's only possible there
b) Hidden Single: R9C3 = 8 @ C3, R9C4 = 6
c) 18(4) = 35{19/28} because R8C3 = (35) and R9C56 = (124) blocks 4{158/239/257}; CPE: R8C6 <> 3,5
d) 17(4) = 4{139/157/238} because 4 locked there @ N8; R78C6 <> 1,2,4 because R9C56 = (124)
and R7C6 = (35) because it's only possible there
e) 4 locked in R9C56 @ N8 for R9
f) 4 locked in R7C123 @ N7 for R7
g) 5 locked in R8C123 @ N7 for R8
h) 7(2): R8C1 <> 6
i) 6 locked in R7C3+R8C2 @ N7 for D/
j) 22(4): R678C9 <> 1,2 because R9C9 = (12)

4. R789 !
a) 3 locked in R8C345 @ R8 @ 18(4) -> R7C5 <> 3
b) Hidden Killer pair (12) in 18(4) for R8 since 7(2) can only have one of (12)
-> R7C5 <> 1,2
c) ! Innies+Outies R8: -7 = R7C5 - R8C69
- R7C5 <> 9 because (79) is a Killer pair of 13(2)
- R8C9 <> 7 (IOU @ N8)
- R8C9 <> 9 because R8C6 <> 3,6
- R8C6 <> 9, R8C9 <> 6 because (69) is a Killer pair of 13(2)
d) Innies+Outies R8: -7 = R7C5 - R8C69 -> R8C69 = 12/15(2) = 8{7/4} -> 8 locked for R8

5. C456+D/ !
a) Innies N2 = 17(3) <> 5 because 5{39/48} blocked by R7C6 = (35) and (58) is a Killer pair of 17(4)
b) 35(7) = 14569{28/37} because of Innies N5 and R7C3 = (46) -> 4,5 locked for D/
c) ! Killer pair (24) locked in R7C3+R8C2 + Innies N5 for D/
d) 7(2): R8C1 <> 2
e) 8(2) = [17/53]
f) 8 locked in R2C8+R3C7 @ D/ for N3
g) 11(2) = [74/83/92]

6. R123
a) Innies N2356 = 14(2) = [68/95]
b) Outies N69 = 10(2) <> 2,3,5,7 because R1C9 = (37)
c) Using Outies N69: Innies N3 = 16(3) <> 3 because R3C7 = (58) and R9C7 = (35) blocks {358}
and R2C7 <> 5 because R3C7 = (58)
-> Innies N3 = {178/259/268/457}
d) 10(2) <> 3,7 because R1C9 = (37); R1C6 <> 2
e) 9(2) @ N3 <> 4 and R2C6 <> 1,6
f) Killer pair (69) locked in R3C6 + Outies N69 for R3
g) Innies N3 = 16(3) <> 4 because (57) is a Killer pair of 8(2)
and R2C7 <> 1 because R1C7 <> 7,8
h) 9(2) <> 8
i) 10(2): R1C6 <> 6

7. R123+D/
a) 9 locked in R2C8+R9C1 @ D/ -> CPE: R2C1 <> 9
b) 9 locked in R1C123 @ N1 for R1
c) 10(2) <> 1
d) Innies N3 = 16(3) = {268} -> R3C7 = 8, {26} locked for C7+N3
e) 11(2) = {47} -> R2C8 = 7, R2C9 = 4
f) 9(2) @ N3 = {36} -> R2C6 = 3, R2C7 = 6
g) 10(2) = {28} -> R1C6 = 8, R1C7 = 2
h) R1C9 = 3 -> R1C8 = 5
i) 9(2) @ N1 = {18} -> R2C3 = 1, R2C2 = 8
j) 1 locked in R1C45 @ R1 @ 13(3) -> 13(3) = {157} -> R2C4 = 5, {17} locked for R1+N2

8. N17
a) 13(2) = {49} locked for R1+N1
b) R1C1 = 6, R2C1 = 2 -> R34C1 = 9(2) = [54] -> R3C1 = 5, R4C1 = 4
c) 35(7) = {124569} -> R7C4 = 9, R3C6 = 6, R7C3 = 4, {125} locked for N5+D/
d) R8C1 = 1, R7C1 = 3, R7C2 = 2
e) 21(5) = 123{69/78} -> R5C3 = 2, R5C2 = 1
f) 21(4) = {3567} -> 5,6 locked for R4+N4

9. Rest is singles

Rating: (Hard) 1.25. I used Killer pairs and Innies+Outies analysis.
Walkthrough by udosuk:
Here is my 5-step walkthrough for V2. I haven't read the others' moves but I suspect the critical ones might be similar. :geek:

Assassin 110X V2 walkthrough:


0. Preliminaries

10/2 @ r1c6={19|28|37|46}
8/2 @ r1c8={17|26|35}
9/2 @ r2c6={18|27|36|45}
16/2 @ r9c1={79} (NP @ r9,n7)
=> 14/2 @ r9c3={68} (NP @ r9)
=> 8/2 @ r9c7={35} (NP @ r9,n9)
=> 13/2 @ r8c7={49|67}, r9c569={124}


1. Simple innie & outies

Innies @ n5: /456=8={125|134} (1 @ n5,d/ locked)
Outies @ n14: r7c12=5={14|23} has 3|4
=> 7/2 @ r8c1 can't be {34}, must be [16|{25}]
Outies @ n69: r3c89=10={19|28|37|46}
Innies @ n23: r3c67=14={59|68} has 5|8
=> Innies @ n78: r7c34=13 can't be {58}, must be [49|67]
Innies @ n147: r789c3=17=[458|638] => r9c34=[86]
8 @ n3,d/ locked @ /23, 6 @ n7,d/ locked @ /78
=> r3c67=[{59}|68], r3c89=10={19|37|46}


2. n8 (!)

17/4 @ r7c6 can't be {35}+{12|14|24}
=> one of {35} @ n8 must be @ 18/4 @ r7c5
=> 18/4 @ r7c5={35(19|28)}
=> r8c6, seeing all 4 cells, can't have any of {35}
17/4 @ r7c6 can't have all of {124}
=> r78c6 can't have any of {124}
=> 4 @ r9,n8 locked @ r9c56 => r9c9 from {12}
=> 17/4 @ r7c6=[38{24}|39{14}|57{14}]


3. r789 (!)

22/4 @ r6c9 can't have both of {12}
=> r678c9 can't have any of {12}
=> {12} @ r8 locked @ r8c1245, one each @ r8c12 & r8c45
=> {12} @ n8 locked @ r8c45+r9c56 => r7c5 from {3589}
Innie-outies @ r8: r8c69=r7c5+7
13/2 @ r8c7={49|67} has 6|9 & 7|9
=> r8c69 can't be [97|96], with a min value of 7+4=11
=> r7c5+r8c69=[584|878]


4. c67 (!!)

Now r78c6=[38|57] has 3|5 & 5|8
=> Innies @ n2: r123c6=17 can't be [{39}5|{48}5]
=> r3c6 can't be 5
Innies @ n369: r123c7=16
8/2 @ r1c8=[17|62|{35}] has 1|2|5
=> r1c67+r23c7 can't be [1925]
r3c89=10={19|37|46} has 4|7|9
=> r12c7+r3c67 can't be [{47}95]
Also no 8 @ r12c7 => r12c7 can't be {38}
=> r123c7=16 can't be [{29}5|{38}5|{47}5]
=> r3c7 can't be 5 => r3c67=14=[68]


5. Mop up

35/7 @ r3c6: r7c34=[49] => /456={125} (NT @ n5,d/)
=> r8c123=[165] => r7c5=8 => 17/4 @ r7c6=[57{14}]
=> r89c9=[82] => r67c9=22-2-8=12=[57]
=> 8/2 @ r1c8=[53], 11/2 @ r2c8=[74]
=> 16/2 @ r9c1=[97], 8/2 @ r9c7=[53]
=> 13/2 @ r1c2=[49] => 10/2 @ r1c6=[82]
=> 9/2 @ r2c6=[36] => 9/2 @ r2c2=[81]
r345c9={169} => r5c78=23-1-6-9=7=[34]

All naked singles from here.

I like this puzzle because after the intense actions the mopping up is really short and sweet. :alien:
Walkthrough by Andrew, in 2010:
Continuing working through my backlog, the next variant I tried was A110X V2 which was another one which I found that I hadn't started.

Thanks Ed for another interesting and challenging variant. You have a knack for making good variants by just combining two cages, in this case the 10(3) cage at R5C1 and the 11(2) cage at R6C1.

Ed wrote:
mhparker wrote:
After what was probably about 30 steps
Hope you kept your notes! This "old school" V2 has the same solution as the original, but takes out the "easy" opening (Afmob's step 3a).
It might have helped if I'd looked to see how I'd solved the original but I didn't need to; I solved this as a new puzzle.

udosuk wrote:
Here is my 5-step walkthrough for V2. I haven't read the others' moves but I suspect the critical ones might be similar.
When I went through the walkthroughs posted by Afmob, Mike and udosuk I found that they had all used critical moves based on the innie-outie difference for R8. I therefore found that, even though there were already 3 posted walkthroughs, mine was significantly different because I'd used a different innie-outie difference and related combination analysis. Edit. Then when I had to re-work my later steps, see below, I found that I needed to use the innie-outie difference for R8 so it appears to be the critical step for this puzzle.

Rating Comment (amended). I'll rate my walkthrough for A110X V2 at Hard 1.25.
Here is my walkthrough for A110X V2. Thanks Afmob for pointing out that my original step 16 was flawed. I've re-worked from after step 15 and also made a few minor changes to earlier steps. Then thanks again to Afmob for finding that my new step 21 was flawed. I've found an alternative way to do that step.

Prelims

a) R1C23 = {49/58/67}, no 1,2,3
b) R1C67 = {19/28/37/46}, no 5
c) R1C89 = {17/26/35}, no 4,8,9
d) R2C23 = {18/27/36/45}, no 9
e) R2C67 = {18/27/36/45}, no 9
f) R2C89 = {29/38/47/56}, no 1
g) R8C12 = {16/25/34}, no 7,8,9
h) R8C78 = {49/58/67}, no 1,2,3
i) R9C12 = {79}
j)R9C34 = {59/68}
k) R9C78 = {17/26/35}, no 4,8,9
l) 19(3) cage in N5 = {289/379/469/478/568}, no 1

Steps resulting from Prelims
1a. Naked pair {79} in R9C12, locked for R9 and N7, clean-up: no 5 in R9C34, no 1 in R9C78
1b. Naked pair {68} in R9C34, locked for R9, clean-up: no 2 in R9C78
1c. Naked pair {35} in R9C78, locked for R9 and N9, clean-up: no 8 in R8C78

2. 45 rule on N14 2 outies R7C12 = 5 = {14/23}
2a. R8C12 = {16/25} (cannot be {34} which clashes with R7C12), no 3,4
2b. Killer pair 1,2 in R7C12 and R8C12, locked for N7
2c. 8 in N7 only in R789C3, locked for C3, clean-up: no 5 in R1C2, no 1 in R2C2

3. 45 rule on N69 2 outies R3C89 = 10 = {19/28/37/46}, no 5

4. 45 rule on N5 3 innies R4C6 + R5C5 + R6C4 = 8 = {125/134}, 1 locked for N5, D/ and 35(7) cage at R3C6, clean-up: no 7 in R1C8, no 6 in R8C1
4a. 45 rule on N2369 2 innies R3C67 = 14 = {59/68}
4b. 45 rule on N1478 2 innies R7C34 = 13 = [49/67] (cannot be {58} which clashes with R3C67), no 1,2,3,5,8, no 4,6 in R7C4
4c. 35(7) cage at R3C6 = {125}{68}[49]/{134}{59}[67](combining combinations in steps 4, 4a and 4b)
4d. 4 only in R4C6 + R5C5 + R6C4 + R7C3, locked for D/, clean-up: no 7 in R2C9
4e. 9 only in R3C67 + R7C4, CPE no 9 in R3C4
4f. 5 in R7 only in R7C56, locked for N8
4g. 8 on D/ only in R2C8 + R3C7, locked for N3, clean-up: no 2 in R1C6, no 1 in R2C6, no 3 in R2C8, no 2 in R3C89 (step 3)

5. 45 rule on C123 3 innies R789C3 = 17 = {368/458}
5a. 3,5 only in R8C3 -> R8C3 = {35}
5b. R9C3 = 8 (hidden single in C3), R9C4 = 6
5c. 4 in N7 only in R7C123, locked for R7
5d. 6 in N7 only in R7C3 + R8C2, locked for D/, clean-up: no 2 in R1C8, no 5 in R2C9, no 8 in R3C6 (step 4a)
5e. 6 in 35(7) cage at R3C6 only in R3C6 + R7C3, CPE no 6 in R3C3

6. 45 rule on N9 2 innies R7C78 = 1 outie R6C9 + 2
6a. Min R7C78 = {16} = 7 (cannot be {12} which clashes with R7C12) -> min R6C9 = 5
6b. Max R7C78 = 11 -> R7C78 must contain one of 1,2
6c. Killer pair 1,2 in R7C12 and R7C78, locked for R7

7. 45 rule on C1 2 innies R89C1 = 2 outies R5C23 + 7
7a. Max R89C1 = 14 -> max R5C23 = 7, no 7,8,9

8. 45 rule on C789 3 innies R123C7 = 16 = {169/178/259/268/457} (cannot be {349} which clashes with R3C89, cannot be {358} which clashes with R9C7, cannot be {367} because no 3,6,7 in R3C7), no 3, clean-up: no 7 in R1C6, no 6 in R2C6

9. 17(4) cage in N8 = {1259/1349/1457/2348} (cannot be {1358/2357} because R9C56 must contain two of 1,2,4)
9a. 17(4) cage only contains two of 1,2,4 -> no 1,2,4 in R8C6
9b. 5 of {1457} must be in R7C6 -> no 7 in R7C6

10. Hidden killer pair 3,5 in R78C6 and R7C5 + R8C45 for N8, R78C6 contains one of 3,5 -> R7C5 + R8C45 must contain one of 3,5 -> 18(4) cage at R7C5 must contain both of 3,5 = {1359/2358}, no 4,7, CPE no 3 in R8C6
10a. 3 in R8 only in R8C345, locked for 18(4) cage, no 3 in R7C5
10b. Hidden killer pair 3,5 in R7C5 + R8C45 and R7C6 for N8, R7C5 + R8C45 contains one of 3,5 -> R7C6 = {35}

11. 4 in N8 only in R9C56, locked for R9
11a. Killer pair 1,2 in R7C78 and R9C9, locked for N9

12. 45 rule on N8 4 remaining innies R78C45 = 22 = {1579/2389/2578}
12a. 5 of {1579/2578} must be in R7C5, 9 of {2389} must be in R7C4 -> no 9 in R7C5

13. 45 rule on R89 2 outies R7C56 = 2 innies R89C9 + 3
13a. R7C56 = [53/85] = 8,13 -> R89C9 = 5,10 = [41/82/91], no 6,7 in R8C9

14. 5,8 in R7 only in R7C56789
14a. 45 rule on R7 5 remaining innies R7C56789 = 27 = {15678/23589}, R7C56 = [53/85] -> R7C789 = {167/289} = 14,19
14b. R7C78 = {16/17/28/29} = 7,8,10,11 -> R6C9 (step 6) = {5689}, no 7

15. 22(4) cage at R6C9 = {1489/1678/2578} (cannot be {4567} which doesn’t contain one of 1,2, cannot be {1579/2479/2569} which clash with R8C78), 8 locked for C9
15a. 1,4 of {1489} must be in R89C9 -> no 9 in R8C9
15b. 7 of {1678} must be in R7C9 -> no 6 in R7C9

Re-worked from here
16. R7C789 = {167/289} (step 14a)
16a. 22(4) cage at R6C9 = {1489/2578} (cannot be {1678} which clashes with R7C789, CCC), no 6
16b. R6C9 = {589} -> R7C78 (step 6) = 7,10,11 = {16/28/29} (step 14b), no 7

[Looks like I need to use the same step as the other walkthroughs used although it’s simpler here after my earlier work.]
17. 45 rule on R8 2 innies R8C69 = 1 outie R7C5 + 7
17a. R7C5 = {58} -> R8C69 = 12,15 = [84/78], no 9, 8 locked for R8

18. 4 in N8 only in 17(4) cage (step 9) = {1457/2348} -> R78C6 = [38/57]
18a. 45 rule on N2 3 innies R123C6 = 17 = {179/269/368/467} (cannot be {278} because no 2,7,8 in R3C6, cannot be {359} which clashes with R7C6, cannot be {458} which clashes with R78C6), no 5, clean-up: no 4 in R2C7, no 9 in R3C7

19. 35(7) cage at R3C6 (step 4c) = {125}[68][49]/{134}[95][67], 5 locked for D/, clean-up: no 3 in R1C8, no 6 in R2C9, no 2 in R8C1
19a. Killer pair 2,6 in 35(7) cage at R3C6 and R8C2, locked for D/, clean-up: no 6 in R1C8, no 9 in R2C9
19b. R3C89 (step 3) = {19/46} (cannot be {37} which clashes with R1C9), no 3,7
19c. Killer pair 6,9 in R3C6 and R3C89, locked for R3
19d. 3 in N3 only in R12C9, locked for C9
19e. 5 in C9 only in R456C9, locked for N6

20. R1C67 = {19/46}/[82] (cannot be [37] which clashes with R1C9), no 3,7
20a. R123C7 (step 8) = {178/259/268} (cannot be {169} because R3C7 only contains 5,8, cannot be {457} which clashes with R1C89), no 4, clean-up: no 6 in R1C6
20b. R3C7 = {58} -> no 5 in R2C7, clean-up: no 4 in R2C6
20c. 7 of {178} must be in R2C7 -> no 1 in R2C7, clean-up: no 8 in R2C6

[Then I had to do further re-work, in this case only step 21, because my original step 21 was flawed.]
21. R2C67 = [27/36/72], R2C89 = [74/83/92] -> combined cage R2C6789 = [2783/3674/3692/7283]
21a. R123C7 (step 20a) = {259/268} (cannot be {178} = [178] which clashes with R2C6789 = [2783], CCC), no 1,7, 2 locked for C7 and N3, clean-up: no 9 in R1C6, no 2 in R2C6, no 9 in R2C8
[Having found that, I saw that Afmob used the simpler
9 on D/ only in R2C8 + R9C1, CPE no 9 in R2C1
9 in N1 only in R1C123, locked for R1 ...]


22. R9C1 = 9 (hidden single in D/), R9C2 = 7, clean-up: no 6 in R1C3, no 2 in R2C3

23. 9 in N1 only in R1C23 = {49}, locked for R1 and N1, clean-up: no 1 in R1C6, no 6 in R1C7, no 5 in R2C23
23a. R1C67 = [82], R2C7 = 6, R2C6 = 3, R2C9 = 4, R2C8 = 7, R1C9 = 3, placed for D/, R1C8 = 5, R3C7 = 8, R3C6 = 6 (step 4a), R78C6 = [57], R7C34 = [49], R1C23 = [49], R7C5 = 8, R7C9 = 7, R8C9 = 8, R7C7 = 1, R9C9 = 2, both placed for D\, R6C9 = 5 (step 16a), R7C8 = 6, R9C78 = [53], R2C23 = [81]

24. Naked pair {23} in R7C12, locked for N7 -> R8C3 = 5, R8C12 = [16]

25. R5C5 = 5, naked pair {12} in R4C6 + R6C4, locked for N5

26. Naked pair {17} in R1C45, locked for R1 and N2, R2C4 = 5 (cage sum)

27. Naked triple {169} in R345C9, locked for 23(5) cage at R3C9
27a. R345C9 = {169} = 16 -> R5C78 = 7 = [34], R56C6 = [94], R6C5 = 6 (cage sum)

and the rest is naked singles.


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PostPosted: Tue May 10, 2011 8:55 pm 
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Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 111 by Nasenbaer (July 2008) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:5120:5120:8706:4355:4355:1541:1541:2823:2823:5120:8706:8706:4108:4355:3854:3854:2823:6673:8706:4371:8706:4108:4108:8727:3854:6673:6673:4371:4371:8706:4108:8727:8727:5409:5409:6673:5668:4371:8706:8727:2856:8727:2858:5409:8748:5668:5668:4655:4655:2856:8727:2858:8748:8748:5668:4407:4655:4655:3898:8727:8748:3133:8748:3135:4407:4407:3898:3898:6212:6212:3133:8748:3135:1353:1353:1867:1867:6212:6212:3133:8748:
Solution:
+-------+-------+-------+
| 7 4 8 | 9 2 5 | 1 3 6 |
| 9 5 1 | 4 6 3 | 7 2 8 |
| 3 2 6 | 1 8 7 | 5 9 4 |
+-------+-------+-------+
| 1 8 4 | 3 9 2 | 6 7 5 |
| 2 6 7 | 5 4 1 | 9 8 3 |
| 5 9 3 | 8 7 6 | 2 4 1 |
+-------+-------+-------+
| 6 7 5 | 2 3 4 | 8 1 9 |
| 4 1 9 | 7 5 8 | 3 6 2 |
| 8 3 2 | 6 1 9 | 4 5 7 |
+-------+-------+-------+
Quote:
Nasenbaer: Hmmm. It's so weird. :scratch: Somehow the german word "Schnapszahl" pops around in my head. I'm confused, everything seems to be so messy. :scratch: I also can't remember which Assassin I'm supposed to post. There must be a clue somewhere, but I can't see it... :scratch: Maybe a little color could make me see through the cages...
Have fun!
PS: This is the first Assassin I intentionally made messy. The first version was a wonderful symmetrical killer (which will be posted in a couple of weeks as another Assassin), but with such a nice number my creativity suddenly went crazy... :twisted:
Rating SSolver(3.2.1): 1.27. (My personal rating: hard 1.25, maybe easy 1.5)

Afmob: Thanks Nasenbaer for this messy Killer. Nice cage pattern that fits the current Assassin. :cheesey:
It took me some time to find the cracking move ...
Rating: 1.25.

gary w: Yes,a nice puzzle.Like Afmob I'ld rate it about 1.25.
Many thanks for a very entertaining challenge.

Mike(mhparker): Congratulations to Afmob and Gary for completing this one. :applause:
I gave up, having missed Afmob's cracking combination. Maybe I took the word "schnapszahl" a bit too literally and am feeling too groggy from the night before? :)

Andrew: Thanks Nasenbaer. Nice cage pattern using the "continental" 1s!
I really struggled with A111. It took me several sessions over several days ... If I hadn't made a little progress in each of those sessions I'd have given up.
It's hard for me to give a rating for A111. Objectively from the steps in my walkthrough it's a 1.25. However I found several of the steps very difficult to find ... so subjectively it's a solid to hard 1.5. That's the rating that I would have given if I'd been the first to post a walkthrough.
A111 deserves more that one walkthrough so here is my one. Unlike the way that Afmob, Mike and some others try to optimise their walkthroughs, I've kept the key breakthrough where I found it since it wasn't an obvious one that I ought to have spotted earlier.
The key breakthrough steps are the same as those used by Afmob so I think this puzzle had a very narrow solving path.

Walkthrough by Afmob:
Thanks Nasenbaer for this messy Killer. Nice cage pattern that fits the current Assassin. :cheesey:

It took me some time to find the cracking move (step 2b) though it's not a particularly difficult move (at least for a 1.25 Killer), so my orginal walkthrough had some other moves in different regions to remove some possibilites but they led to nothing, so I took them off. It'll be interesting to see, if this Killer can be solved without it (in a not too difficult way).

A111 Walkthrough:

1. C6789
a) Outies C89 = 14(2) = {59/68}
b) Outies C6789 = 14(2) = {59/68}
c) 11(2) @ N6 <> 5,6 because (56) is a Killer pair of Outies C89
d) Killer triple (789) locked in 21(3) + 11(2) for N6
e) Innies N6 = 13(4) = 14{26/35} -> 4 locked for N6 and 1 locked for 34(7); CPE: R789C9 <> 4
f) 21(3) = 7{59/68} -> 7 locked for C8+N6

2. C6789 !
a) Innies+Outies N69: 12 = R89C6 - R4C9 -> R89C6 >= 14(2) <> 1,2,3,4 and R4C9 <> 6
b) ! Hidden Killer pair (14) in R89C7 @ N9 since 12(3) must have exacly one of (14)
c) ! 24(4): R89C7 <> 7 because 24(4) can't have both of (17)
and cannot be {47} since R89C6 must be >= 14 (step 2a)
d) 7 locked in R23C3 @ C3 @ 15(3) for N3 -> 15(3) = 7{26/35} and R2C6 <> 7
e) 26(4) = 89{36/45} -> 8 locked for N3
f) 7 locked in 34(7) @ N9 = 147{2389/2569/3568} -> 4 locked for N6
g) Innies N6 = 13(4) = {1345} because R4C9 = (35) -> 3,5 locked for N6 + 34(7) @ C9
h) 11(2) = {29} locked for C7+N6
i) Outies C89 = 14(2) = {68} locked for C7
j) 15(3) = {357} because R23C7 <> 2,6

3. C789
a) 2 locked in 11(3) @ N3 = 2{36/45}
b) Hidden Single: R1C7 = 1 @ N3 -> R1C6 = 5
c) R2C6 = 3 -> R23C7 = 12(2) = {57} locked for C7+N3
d) 24(4) = {3489} because R89C7 = {34} -> 3,4 locked for N9 and 8,9 locked for C6+N8
e) Outie N3 = R4C9 = 5
f) 34(7) = {1234789} -> R7C7 = 8; 2,9 locked for C9+N9
g) 11(3) = {236} locked for N3 and 3 also locked for R1

4. R789
a) 7(2) <> 3,4 because R9C7 = (34)
b) 15(3) = {357} because 6{27/45} blocked by Killer pairs (26,56) of 7(2)
-> 3,5,7 locked for N8
c) 7(2) = {16} locked for R9+N8
d) Naked pair (24) locked in R7C46 for R7
e) 5(2) = {23} locked for R9+N7
f) Innies N7 = 11(2) = {56} locked for R7+N7
g) 12(2) = {48} locked for C1+N7

5. N25
a) 34(7) = {1245679} -> R5C4 = 5, R4C5 = 9
b) R9C7 = 4, R8C7 = 3, R8C4 = 7, R7C5 = 3
c) 11(2) = {47} locked for C5+N5
d) 17(3) = {269} -> R1C4 = 9, {26} locked for C5+N2
e) 16(4) = {1348} -> R4C4 = 3, {14} locked for N2 and 4 also locked for C4

6. C123
a) Outies = 10(2) = {28} -> R7C4 = 2, R6C4 = 8
b) Innies = 8(2) = {35} -> R7C3 = 5, R6C3 = 3
c) R7C1 = 6, R1C1 = 7, R9C3 = 2, R9C2 = 3
d) Hidden Single: R7C2 = 7 @ N7
e) 22(4) = {2569} -> 2,5,9 locked for N4
f) R4C1 = 1
g) 17(4) = {1268} because R45C2 <> 5,9 -> R3C2 = 2; {68} locked for C2+N4
h) R1C2 = 4 -> R2C1 = 9

7. Rest is singles.

Rating: 1.25. I used a Killer triple, a Hidden Killer pair and IOD analysis.
gary w's Solving Outline:
Yes,a nice puzzle.Like Afmob I'ld rate it about 1.25.

All my early moves centred on N6 the innies of N69 and the outies of the 34(7) cage N69.Not too difficult to show that r4567c7 has an 8 and a 9 and that r4c9=3/5 with 7 at r45c8.Combine this with what pair must be missing from the 34(7) cage -> r4c9=5 and this cracks the puzzle.So,unusually all the early "action" plus cracking moves took place in a small part of the grid.Very nice.

Many thanks for a very entertaining challenge.
Walkthrough by Andrew:
Thanks Nasenbaer. Nice cage pattern using the "continental" 1s!

Afmob wrote:
It took me some time to find the cracking move (step 2b) though it's not a particularly difficult move (at least for a 1.25 Killer), so my orginal walkthrough had some other moves in different regions to remove some possibilites but they led to nothing, so I took them off.

I really struggled with A111. It took me several sessions over several days to get from the position after step 16 to the key breakthrough in step 24, which I then realised had been available for a long time. If I hadn't made a little progress in each of those sessions I'd have given up.

Then I made a mistake in my original step 26 which I found while checking my walkthrough; it had reached the correct solution using a flawed move.

After that I forgot about re-working my walkthrough until today when I managed to find my new step 26; the remaining steps were fairly straightforward.

It's hard for me to give a rating for A111. Objectively from the steps in my walkthrough it's a 1.25. However I found several of the steps, including the important steps 24 and 26, very difficult to find so subjectively it's a solid to hard 1.5. That's the rating that I would have given if I'd been the first to post a walkthrough.

A111 deserves more that one walkthrough so here is my one. Unlike the way that Afmob, Mike and some others try to optimise their walkthroughs, I've kept the key breakthrough where I found it since it wasn't an obvious one that I ought to have spotted earlier.

The key breakthrough steps are the same as those used by Afmob so I think this puzzle had a very narrow solving path.

Prelims

a) R1C67 = {15/24}
b) R56C5 = {29/38/47/56}, no 1
c) R56C7 = {29/38/47/56}, no 1
d) R89C1 = {39/48/57}, no 1,2,6
e) R9C23 = {14/23}
f) R9C45 = {16/25} (cannot be {34} which clashes with R9C23)
g) 20(3) cage in N1 = {389/479/569/578}, no 1,2
h) 11(3) cage in N3 = {128/137/146/236/245}, no 9
i) 21(3) cage in N6 = {489/579/678}, no 1,2,3
j) 26(4) cage at R2C9 = {2789/3689/4589/4679/5678}, no 1
k) Each of the 34(7) cages must contain 1

1. Killer pair 1,2 in R9C23 and R9C45, locked for R9

2. 45 rule on C123 2 outies R67C4 = 10, no 5
2a. R67C3 = 8 = {17/26/35}, no 4,8,9

3. 45 rule on C6789 2 outies R4C5 + R5C4 = 14 = {59/68}
3a. 1 locked in R34567C6 for C6 because 34(7) cage must contain 1, clean-up: no 5 in R1C7
3b. R56C5 = {29/38/47} (cannot be {56} which clashes with R4C5 + R5C4), no 5,6

4. 45 rule on C89 2 outies R47C7 = 14 = {59/68}
4a. R56C7 = {29/38/47} (cannot be {56} which clashes with R47C7), no 5,6
4b. Killer triple 7,8,9 in 21(3) cage and R56C7, locked for N6
4c. 1 in N6 locked in R5C9 + R6C89, locked for 34(7) cage -> no 1 in R78C9

5. 45 rule on N7 2 innies R7C13 = 11 = [47/56/65/83/92], no 1, no 2,3,7 in R7C1, clean-up: no 7 in R6C3 (step 2a)

6. 45 rule on N69 2 outies R89C6 = 1 innie R4C9 + 12
6a. Max R89C6 = 17 -> max R4C9 = 5
6b. Min R4C9 = 2 -> min R89C6 = 14, no 2,3,4
6c. Min R89C6 = 14 -> max R89C7 = 10, no 8,9 in R8C7

7. 45 rule on N6 4 innies R45C9 + R6C89 = 13 = {1246/1345}, 4 locked for N6, CPE no 4 in R789C9, clean-up: no 7 in R56C7
7a. Killer pair 8,9 in R47C7 and R56C7, locked for C7
7b. 7 in N6 locked in R45C8, locked for C8

8. 45 rule on N1 2 outies R45C3 = 1 innie R3C2 + 9
8a. Max R45C3 = 17 -> max R3C2 = 8

9. 45 rule on N8 4 innies R7C46 + R89C6 = 23
9a. Min R89C6 = 14 (step 6b) -> max R7C46 = 9, no 9, clean-up: no 1 in R6C4 (step 2)

10. 45 rule on C789 3 outies R1289C6 = 25 = {2689/3589/4579/4678} (cannot be {3679} because R1C6 only contains 2,4,5)
10a. 2 of {2689} must be in R1C6 -> no 2 in R2C6

11. R7C46 + R89C6 = 23 (step 9) = {1679/2489/2579/3479/3578} (cannot be {1589/2678/3569/4568} which clash with R9C45)
11a. 8 of {2489/3578} must be in R89C6 (because min R89C6 = 14) -> no 8 in R7C46, clean-up: no 2 in R6C4 (step 2)

12. 9 in N3 locked in 26(4) cage at R2C9 = {2789/3689/4589/4679}
12a. 2,3 of {2789/3689} must be in R4C9 -> no 2,3 in R2C9 + R3C89

13. 45 rule on N7 4 innies R7C13 + R9C23 = 16 = {1249/1348/1456/2356} (cannot be {1258/1267/1357} which cannot be fitted into R7C13 and R9C23, cannot be {2347} which clashes with R89C1), no 7, clean-up: no 1 in R6C3 (step 2a), no 4 in R7C1 (step 5)

14. Combined cage R4567C7 = 25 = {2689/3589}
14a. 15(3) cage at R2C6 = {159/168/249/267/348/357/456} (cannot be {258} which clashes with R4567C7)
14b. 8,9 of {249/348} must be in R2C6
14c. 4 of {456} must be in R23C7 (R23C7 cannot be {56} which clashes with R47C7)
14d. -> no 4 in R2C6

15. 45 rule on C12 2 outies R89C3 = 2 innies R2C2 + R3C1 + 3
15a. Min R2C2 + R3C1 = 3 -> min R89C3 = 6, no 1 in R8C3

16. Hidden killer quad 6,7,8,9 in R7C13, R89C1 and 17(3) cage for N7 -> 17(3) cage must have two of 6,7,8,9 = {179/269/278/368} (cannot be {359/458} which only have one of 6,7,8,9, cannot be {467} because cannot then place 1 in N7), no 4,5

17. 34(7) cage at R1C3 must contain 1
17a. If R3C2 = 1 => R45C3 must contain 1 => cannot place 1 in C1
17b. -> no 1 in R3C2
17c. 1 in N1 locked in 34(7) -> no 1 in R45C3
[When I mentioned this step to Ed, at the time when I was struggling to make any progress, he said he’d used the more elegant CPE for 1,2 in C1 -> no 1,2 in R45C3 which then locks 1 in 34(7) for N1 -> no 1 in R3C2.]

18. 11(3) cage in N3 = {128/137/146/236/245}
18a. 5 of {245} must be in R2C8 (R1C89 cannot be {25/45} which clash with R1C67), no 5 in R1C89

19. 45 rule on N23 2 outies R4C49 = 1 innie R3C6 + 1
19a. Min R4C49 = 3 -> min R3C6 = 2
19b. Max R3C6 = 9 -> max R4C49 = 10, no 9 in R4C4

20. R45C9 + R6C89 (step 7) = {1246/1345}
20a. 34(7) cage at R5C9 = {1234789/1235689/1245679/1345678}
20b. {1234789} must have {134} in N6 with R4C9 = 5 (cannot have {124} in N6 because no 6 in R4C9)
20c. {1235689} must have {126/135} in N6 with R4C9 = 4
20d. {1245679} must have {145} in N6 with R4C9 = 3 (cannot have {124} in N6 because no 6 in R4C9, cannot have {126} in N6 because no 4 in R789C9)
20e. {1345678} must have {134/145} in N6 with R4C9 = {35} (cannot have {135} in N6 because no 4 in R789C9)
20f. -> R4C9 = {345}, no 2

21. 45 rule on N3 3 outies R12C6 + R4C9 = 13
21a. Min R1C6 + R4C9 = 5 -> max R2C6 = 8
21b. Min R4C9 = 3 -> max R12C6 = 10

22. R1289C6 (step 10) = {2689/3589/4579/4678}
22a. 5 of {3589} must be in R1C6
22b. 5,6 of {4579/4678} must be in R2C6 (R12C6 cannot be [47/48] because max R12C6 = 10)
22c. -> no 7 in R2C6, no 5 in R89C6
22d. R89C6 = {69/78/79/89}, min R89C6 = 15

23. 24(4) cage at R8C6 = {1689/2589/2679/3489/3579/3678/4569/4578}
23a. 2 of {2679} must be in R8C7
23b. 7 of {3579/3678/4578} must be in R89C6 (R89C6 cannot be {68} because min R89C6 = 15)
23c. -> no 7 in R8C7

24. R789C8 = {129/138/156/246/345}
24a. Hidden killer pair 1,4 in R789C8 and R89C7 for N9, R789C8 contains one of 1,4 -> R89C7 must contain one of 1,4
24b. 24(4) cage at R8C6 (step 23) = {1689/3489/4569/4578} (cannot be {2589/2679/3579/3678} which don’t contain 1 or 4), no 2
24c. 1 of {1689} must be in R8C7, 4,5 of {4569/4578} must be in R89C7 -> no 6 in R8C7, no 7 in R9C7
24d. R789C8 = {129/138/156/345} (cannot be {246} which clashes with R89C7 = [16] when R789C8 contains 4)
24e. R89C6 = {69/78/89} = 15 or 17 -> R89C7 = 7 or 9 = [16]/{34/45}

25. 7 in C7 locked in R23C7, locked for N3
25a. 15(3) cage at R2C6 (step 14a) = {267/357}, no 1,4,8
25b. 6 of {267} must be in R2C6 -> no 6 in R23C7

26. 7 in N9 locked in 34(7) cage at R5C9 = {1234789/1245679/1345678}, 4 locked in R5C9 + R6C78 for N6
26a. R45C9 + R6C89 (step 7) = {1345} (cannot be {1246} because R4C9 only contains 3,5), locked for N6, clean-up: no 8 in R56C7, no 9 in R7C7 (step 4)

27. Naked pair {29} in R56C7, locked for C7 and N6, clean-up: no 4 in R1C6, no 5 in R7C7 (step 4)
27a. Naked pair {68} in R47C7, locked for C7, clean-up: no 1 in R8C7 (step 24e)

28. R1C7 = 1 (hidden single in C7), R1C6 = 5

29. 15(3) cage at R2C6 (step 25a) = {357} (only remaining combination) -> R2C6 = 3, R23C7 = {57}, locked for C7 and N3

30. Naked pair {34} in R89C6, locked for N9
30a. R89C7 = {34} -> R89C6 = {89} (step 24b), locked for C6 and N8
30b. R89C6 = {89} -> R7C46 = {24} (step 11), locked for R7 and N8, clean-up: no 6 in R6C3 (step 2a), no 3,4,7,9 in R6C4 (step 2), no 9 in R7C1 (step 5), no 5 in R9C45
30c. Naked pair {16} in R9C45, locked for R9 and N8, clean-up: no 4 in R9C23
30d. Naked pair {23} in R9C23, locked for R9 and N7 -> R89C7 = [34], clean-up: no 5 in R6C3 (step 2a), no 8 in R7C1 (step 5), no 8,9 in R8C1, no 9 in R9C1
30e. Naked pair {56} in R7C13, locked for R7 and N7 -> R7C7 = 8, R4C7 = 6, clean-up: no 8 in R5C4 (step 3), no 7 in R89C1
30f. R89C1 = [48] -> R89C6 = [89], R9C89 = [57], R7C89 = [19], R7C2 = 7, R7C5 = 3, R8C23 = [19], R8C8 = 6 (step 24), R8C9 = 2, clean-up: no 8 in R56C5

31. 34(7) cage at R5C9 (step 26) = {1234789} (only remaining combination) -> R5C9 + R6C89 = {134}, locked for N6 -> R4C9 = 5, clean-up: no 9 in R5C4 (step 3)

32. 26(4) cage at R2C9 (step 12) = {4589} (only remaining combination) -> R3C8 = 9, R23C9 = {48}, locked for C9 and N3 -> R12C8 = [32], R1C9 = 6, R6C8 = 4, clean-up: no 7 in R5C5

33. R34567C6 = {12467}, locked for 34(7) cage at R3C6, clean-up: no 8 in R4C5 (step 3) -> R4C5 = 9, R5C4 = 5, R8C45 = [75], clean-up: no 2 in R56C5 -> R56C5 = [47]
33a. Naked triple {126} in R456C6, locked for C6 and N5 -> R37C6 = [74], R23C7 = [75], R46C4 = [38], R7C4 = 2, R6C3 = 3, R7C3 = 5 (step 2a), R7C1 = 6, R56C9 = [31], R9C23 = [32]

34. R12345C3 = {14678}, locked for 34(7) cage at R1C3
34a. R12345C3 = 26 -> R2C2 + R3C1 = 8 = [53], R12C1 = [79], R1C2 = 4 (prelim g), R1C345 = [892], R2C5 = 6 (cage sum)

and the rest is naked singles.


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PostPosted: Tue May 10, 2011 9:24 pm 
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Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 112 by Afmob (July 2008) here
Puzzle Diagram:
Image
Note the remote cage 22(4) at R19C19
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:5632:2561:5378:2563:2563:2563:2310:5383:5632:2561:2561:5378:5378:4621:2310:2310:5383:5383:5138:6163:6163:6163:4621:3351:3351:3351:4634:5138:5138:4893:3358:4621:3358:5409:4634:4634:5138:4893:4893:1831:3358:2345:5409:5409:4634:5165:5165:4893:1831:5169:2345:5409:3380:3380:5165:5165:2104:2104:5169:2875:2875:3380:3380:4927:4927:2625:3138:5169:3138:5445:4166:4166:5632:4927:2625:2625:3138:5445:5445:4166:5632:
Solution:
+-------+-------+-------+
| 5 6 4 | 7 1 2 | 3 9 8 |
| 1 3 9 | 8 6 4 | 2 5 7 |
| 2 8 7 | 9 5 3 | 6 4 1 |
+-------+-------+-------+
| 4 5 8 | 3 7 6 | 1 2 9 |
| 9 2 3 | 5 4 1 | 8 7 6 |
| 7 1 6 | 2 9 8 | 5 3 4 |
+-------+-------+-------+
| 3 9 2 | 6 8 7 | 4 1 5 |
| 8 4 5 | 1 3 9 | 7 6 2 |
| 6 7 1 | 4 2 5 | 9 8 3 |
+-------+-------+-------+
Quote:
Afmob: I messed aound with some cage patterns and noticed that this one messed up SS rating because of its remote cage. I'll post V2 (or more fitting V1.5) on Sunday which has an SS score of 2.0+.
SS Score: 1.33. Estimated rating: (Hard) 0.75.

Frank: Very enjoyable Afmob. I think (hard) 0.75 is about right.
Clearly there was nothing on TV here in Vancouver this evening :)
Many thanx.

Nasenbaer: As Frank said: very enjoyable. Thanks, Afmob!
Took me long enough to be the first for a walkthrough ;) but here it is:
There is so much going on in so many different places that I think it's hard to keep a clean walkthrough for this one. I hope this one wasn't too messy. ;)

gary w: Many thanks Afmob for a nice (gentle this time) puzzle.Took me about 25-30 minutes,perfect for a Friday evening.I'ld rate it about the same as The Times "deadly" killers...about 0.5+ ??

Andrew: I'm still struggling with those "continental" 1s in A111 so after doing SOK1 Variant yesterday I did A112 today.
I'll rate it at 0.75. I note Gary's slightly lower rating and agree that after the simple opening steps it's straightforward but there's still a reasonable amount of work which I feel is probably better done using elimination solving which is why I haven't rated this puzzle lower.
My solution path is fairly similar to Nasenbaer's but I'll post it anyway.

Walkthrough by Nasenbaer:
As Frank said: very enjoyable. Thanks, Afmob!

Took me long enough to be the first for a walkthrough ;) but here it is:

Walkthrough A112

Preliminaries
0. 10(3) @ r1c2, r1c4 and r8c3 = {127|136|145|235} -> no 8,9
0a. 24(3) @ r3c2 = {789} -> 7,8,9 locked for r3
0b. 21(3) @ r1c3, r1c8 and r8c7 = {489|579|678} -> no 1,2,3
0c. 9(3) @ r1c7 = {126|135|234} -> no 7,8,9
0d. 7(2) @ r5c4 = {16|25|34} -> no 7,8,9
0e. 9(2) @ r5c6 = {18|27|36|45} -> no 9
0f. 8(2) @ r7c3 = {17|26|35} -> no 4,8,9
0g. 11(2) @ r7c6 = {29|38|47|56} -> no 1
0h. 19(3) @ r8c1 = {289|379|469|478|568} -> no 1
0i. 13(4) @ r6c8 = 1{237|246|345} -> no 8,9
0j. 20(3) @ r6c5 = {389|479|569|578} -> no 1,2

1. 45 on n5: r46c5 = h16(2) = {79} -> 7,9 locked for n5 and c5

2. 45 on c5: r159c5 = h7(3) = {124} -> 1,2,4 locked for c5
Cleanup:
2a. 18(3) @ r2c5 = {369|378|567}
2b. 20(3) @ r6c5 = {389|569|578}
2c. 9(2) @ r5c6 = {18|36|45} -> no 2
2d. 13(3) @ r4c4 = {148|238|256|346}
2e. 13(3) @ r3c6 = 6{25|34} -> 6 locked for r3
2f. 18(3) @ r2c5 : no 3,5 in r2c5

3. 45 on r3: r3c159 = h8(3) = 1{25|34} -> 1 locked for r3
3a. no 3,5 in r3c19

4. 45 on r1289: r28c5 = h9(2) = [63]
4a. -> r3467c5 = [5798]
4b. 13(3) @ r3c6 = {346} -> 3,4,6 locked for r3, 6 locked in r3c78 for n3
4c. 10(3) @ r1c4 = {127} -> 1,2,7 locked for r1 and n2
4d. 3,4 locked in r23c6 for c6 and n2
4e. 8,9 locked in r23c4 for c4 and n2
4f. 7 locked in r3c23 for n1
4g. -> 21(3) @ r1c3 = {489} -> 4 locked in r12c3 for c3 and n1
4h. naked quad {4789} in n1 -> no 8,9 in r1c1
4i. 10(3) @ r1c2 = 3{16|25} -> 3 locked for n1
4j. 9(3) @ r1c7 = 3{15|24} -> no 3,4,5 in r2c7
4k. 21(3) @ r1c8 = 9{48|57} -> 9 locked for n3
4l. 9(2) @ r5c6 = {18} -> 1,8 locked for n5 and c6
4m. 13(3) @ r4c4 = [526|625|346] -> no 2,4 in r4c4, no 2 in r4c6, 6 locked in r4c46 for r4 and n5
4n. 7(2) @ r5c4 = {25|34}
4o. 8(2) @ r7c3 : no 5 in r7c3
4p. 11(2) @ r7c6 : no 3,7 in r7c7
4q. 12(3) @ r8c4 : combinations {138|237|345} removed

5. 45 on r12: r1c19 = h13(2) = [58]
5a. 10(3) @ r1c2 = {136} -> r1c2 = 6, 1,3 locked in r2c12 for r2 and n1
5b. r3c19 = [21], 9(3) r2c6 = [423], r3c6 = 3, r1c3 = 4 (hidden single), r1c8 = 9
5c. r9c19 = h9(2) = [72]|{36}
5d. 11(2) @ r7c6 : no 9 in r7c6

6. 1 locked in r456c7 for c7, n6 and 21(4) -> 21(4) = 1{389|479|569|578} -> no 2
6a. r7c8 = 1 (locked for 13(4))

The rest is easy cleanup.

7. 45 on n6: r7c9 = 5
7a. r2c89 = [57]
7b. 13(4) @ r6c8 = {1345} -> r6c89 = {34} -> 3,4 locked for r6 and n6
7c. 8(2) @ r7c3 = {26} -> 2,6 locked for r7
7d. 11(2) @ r7c67 = [74], r3c78 = [64], r6c89 = [34], r1c456 = [712]
7e. 7(2) @ r5c4 = {25} -> 2,5 locked for c4 and n5
7f. 13(3) @ r4c4 = [346], r7c34 = [26], 12(3) @ r8c4 = [129], r9c46 = [45]
7g. r9c9 = 3 (hidden single), r9c1 = 6, r9c3 = 1 (hidden single), r8c3 = 5
7h. r56c3 = [36] (hidden singles), r3c3 = 7 (hidden single), r9c7 = 9 (hidden single), r8c7 = 7, r9c8 = 8, r4c89 =[29], r5c89 = [76], r8c89 = [62], r9c2 = 7, r24c3 = [98], r23c4 = [89], r3c2 = 8, r8c12 = [84]
7i. r5c2 = 2, r56c4 = [52],r5c1 = 9, r7c12 = [39], r2c12 = [13], r4c12 = [45], r6c12 = [71], r56c6 = [18], r456c7 = [185]

puzzle completed!

EDIT: correction made in step 7h. Thanks, Andrew.

There is so much going on in so many different places that I think it's hard to keep a clean walkthrough for this one. I hope this one wasn't too messy. ;)
gary w's Solving Outline:
Many thanks Afmob for a nice (gentle this time) puzzle.Took me about 25-30 minutes,perfect for a Friday evening.I'ld rate it about the same as The Times "deadly" killers...about 0.5+ ??

In c5 r159={124}
Innies of r1289 -> r28c5=9={36}
r3c159=8 -> r3c5=5 > r2c5=6 -> r4c5=7
And it comes out pretty quickly now.
Walkthrough by Andrew:
I'm still struggling with those "continental" 1s in A111 so after doing SOK1 Variant yesterday I did A112 today.

I'll rate it at 0.75. I note Gary's slightly lower rating and agree that after the simple opening steps it's straightforward but there's still a reasonable amount of work which I feel is probably better done using elimination solving which is why I haven't rated this puzzle lower.

Here is my walkthrough. My solution path is fairly similar to Nasenbaer's but I'll post it anyway.

Prelims

a) R56C4 = {16/25/34}, no 7,8,9
b) R56C6 = {18/27/36/45}, no 9
c) R7C34 = {17/26/35}, no 4,8,9
d) R7C67 = {29/38/47/56}, no 1
e) 10(3) cage in N1 = {127/136/145/235}, no 8,9
f) 21(3) cage at R1C3 = {489/579/678}, no 1,2,3
g) R1C456 = {127/136/145/235}, no 8,9
h) 9(3) cage at R1C7 = {126/135/234}, no 7,8,9
i) 21(3) cage in N3 = {489/579/678}, no 1,2,3
j) R3C234 = {789}, locked for R3
k) R678C5 = {389/479/569/578}, no 1,2
l) 19(3) cage in N7 = {289/379/469/478/568}, no 1
m) 10(3) cage at R8C3 = {127/136/145/235}, no 8,9
n) 21(3) cage at R8C7 = {489/579/678}, no 1,2,3
o) 13(4) cage at R6C8 = {1237/1246/1345}, no 8,9

1. 45 rule on C5 3 innies R159C5 = 7 = {124}, locked for C5

2. R3C678 = {256/346}, no 1, 6 locked for R3
2a. Killer pair 3,5 in R3C5 and R3C678, locked for R3

3. 45 rule on N5 2 innies R46C5 = 16 = {79}, locked for C5 and N5, clean-up: no 2 in R56C6

4. R234C5 = {369/378/567}
4a. 6,8 only in R2C5 -> R2C5 = {68}

5. 45 rule on R1289 2 innies R28C5 = 9 = [63], R3C5 = 5, R4C5 = 7 (step 4), R67C5 = [98], clean-up: no 2 in R3C678 (step 2), no 5 in R7C3, no 3 in R7C7
5a. Naked triple {346} in R3C678, locked for R3

6. R1C456 = {127} (only remaining combination), locked for R1 and N2
6a. Naked pair {34} in R23C6, locked for C6 and N2, clean-up: no 5,6 in R56C6, no 7 in R7C7
6b. Naked pair {89} in R23C4, locked for C4
6c. Naked pair {18} in R56C6, locked for C6 and N5, clean-up: no 6 in R56C4
6d. 7 in R3 locked in R3C23, locked for N1
6e. 6 in N5 locked in R4C46, locked for R4

7. Killer pair 2,4 in R56C4 and R5C5, locked for N5

8. 21(3) cage at R1C3 = {489} (only remaining combination), 4 locked in R12C3, locked for C3 and N1
8a. 10(3) cage in N1 = {136/235}, 3 locked for N1
8b. Naked quad {4789} in R12C3 + R3C23, locked for N1

9. 45 rule on R12 2 remaining innies R1C19 = 13 = [58], no 5,8 in R9C19, clean-up: no 2 in 10(3) cage in N1 (step 8a)
9a. R1C19 = 13 -> R9C19 = 9 = {27/36}, no 1,4,9
9b. 10(3) cage in N1 = {136} -> R1C2 = 6, R2C12 = {13}, locked for R2 and N1 -> R3C1 = 2, R3C9 = 1, R2C6 = 4, R1C7 = 3, R2C7 = 2 (cage sum), R3C6 = 3, clean-up: no 9 in R7C6, no 7 in R9C9 (step 9a)
9c. Naked pair {46} in R3C78, locked for N3 -> R1C8 = 9, R1C3 = 4
[I then spotted that R3C2 cannot be 7 because that would force UR in R23C34 for 8,9 but I don’t use UR]

10. 13(4) cage at R6C8 = {1237/1246/1345}
10a. 1 locked in R67C8, locked for C8
10b. R7C8 = 1 (hidden single in N9), clean-up: no 7 in R7C34

11. 45 rule on N6 1 remaining outie R7C9 = 5, R2C89 = [57], clean-up: no 3 in R7C3, no 6 in R7C67
11a. 13(4) cage at R6C8 = {1345} (only remaining combination) -> R6C89 = {34}, locked for R6 and N6, clean-up: no 3,4 in R5C4

12. Naked pair {25} in R56C4, locked for C4 and N5 -> R7C34 = [26], R4C46 = [36], R5C5 = 4, R7C6 = 7, R7C7 = 4, R1C456 = [712], R3C78 = [64] , R6C89 = [34], R9C5 = 2, clean-up: no 7 in R9C1 (step 9a)

13. Naked pair {39} in R7C12, locked for N7 -> R9C1 = 6, R9C9 = 3

14. 12(3) cage in N8 = {129} (only remaining combination) -> R8C46 = [19], R9C46 = [45]

15. 21(3) cage at R8C7 = {579} (only remaining combination) -> R89C7 = [79], R9C8 = 8, R4C89 = [29], R5C89 = [76], R8C89 = [62], R8C3 = 5, R9C23 = [71], R4C3 = 8, R23C3 = [97], R2C4 = 8, R3C24 = [89], R56C3 = [36], R8C12 = [84],

16. R7C12 = {39} = 12 -> R6C12 = 8 = [71]

and the rest is naked singles.


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PostPosted: Tue May 10, 2011 9:55 pm 
Offline
Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 112 V2 by Afmob (July 2008) here
Puzzle Diagram:
Image
Note the remote cage 22(4) at R19C19
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:5632:2561:5378:2563:2563:2563:2310:5383:5632:2561:2561:5378:5378:4621:2310:2310:5383:5383:5906:5906:6164:6164:4621:2583:2583:5145:5145:5906:4124:6164:3358:4621:3358:2583:5666:5145:5906:4124:4124:1831:3358:2345:5666:5666:5145:5165:5165:4124:1831:5169:2345:5666:3380:3380:5165:5165:2104:2104:5169:2875:2875:3380:3380:4927:4927:2625:3138:5169:3138:5445:4166:4166:5632:4927:2625:2625:3138:5445:5445:4166:5632:
Solution:
+-------+-------+-------+
| 5 6 4 | 7 1 2 | 3 9 8 |
| 1 3 9 | 8 6 4 | 2 5 7 |
| 2 8 7 | 9 5 3 | 6 4 1 |
+-------+-------+-------+
| 4 5 8 | 3 7 6 | 1 2 9 |
| 9 2 3 | 5 4 1 | 8 7 6 |
| 7 1 6 | 2 9 8 | 5 3 4 |
+-------+-------+-------+
| 3 9 2 | 6 8 7 | 4 1 5 |
| 8 4 5 | 1 3 9 | 7 6 2 |
| 6 7 1 | 4 2 5 | 9 8 3 |
+-------+-------+-------+
Quote:
Afmob: Here is V2 with a SS rating that's totally of the charts. :brickwall:
But don't worry! Take a closer look (and have a calculator nearby :cheesey:) to spot its weakness.
SS Score: 2.49. Estimated rating: (Easy) 1.25 - 1.25.

udosuk: I don't know how easy V2 is supposed to be? But using a very ingenius algebraic trick I'm able to crack it in 4 quick steps: :ugeek:

Mike(mhparker):
udosuk wrote:
But using a very ingenius algebraic trick...
Indeed it is! Nice way of looking at this move (your step 3), which an automated solver would probably pick up by analyzing all possible permutations.
Last but not least, thanks for the puzzle, Afmob! Hopefully, Richard is listening and will be able to fix the problem with SS not finding ... ;)
(Further comments included in Mike's walkthrough)

Afmob: Congratulations for finding the weakness of V2! Mike is right, this move helps to crack my killer. My initial estimated rating might be a bit low since I also used the same Killer triple as udosuk. The important move (in my case step 2c) can be applied in an easier way which justifies a 1.25 rating, I think:
(see Afmob's snippet below for further comments on the rating)

udosuk: Yep, the way Afmob put the cracking move is much more straight forward than mine.
Thanks for the nice puzzle! :alien:

Andrew, in 2010: Continuing working through my backlog, the next variant I tried was A112 V2 where I'd originally got stuck at step 13; when I resumed I edited that step before continuing from that position.
My solving path was very different from the three already posted.
I'll rate my walkthrough for A112 V2 at Easy 1.5 to 1.5; I'm inclined to go for the latter because I found some steps hard to find.
(further comments in walkthrough)
(there's also further discussion in a hidden window after Andrew's walkthrough)

Walkthrough by udosuk:
I don't know how easy V2 is supposed to be? But using a very ingenius algebraic trick I'm able to crack it in 4 quick steps: :ugeek:

0. Prelims

21/3 @ r1c3={489|579|678}
24/3 @ r3c3={789}


1. c5,n5

Innies @ c5: r159c5=7={124} (NT @ c5)
Innies @ n5: r46c5=16={79} (NP @ c5,n5)
Innies @ r1289: r28c5=9={36} (NP @ c5)
=> r37c5={58}
=> 18/3 @ r2c5=[387|657], 20/3 @ r6c5=[956|983]
=> r46c5=[79]


2. c3,n1

21/3 @ r1c3: r12c3 must have at least one of {789}
=> r1234c3 form Killer NT {789} @ c3
=> 7 @ c3,n1 locked @ r123c3


3. n1 (!!)

Innie-outies @ n1: r2c4+r45c1=r1c1+r3c3+9
=> r1c1=r45c1+r2c4-r3c3-9
=> 2*r1c1=r145c1+r2c4-r3c3-9
=> Max value of 2*r1c1 is 24+9-7-9=17
=> Max value of r1c1 is 8
=> r1c1 can't be 9


4. r12 (!)

Innies @ r12: r1c19+r2c5=19
But r1c1 can't be 7|9
=> r1c19 can't be {79}, can't sum to 16
=> r2c5=19-r1c19 can't be 19-16=3
=> r2c5=6


5. Huge mop-up

18/3 @ r2c5=[657], 20/3 @ r6c5=[983]
10/3 @ r1c4 from {1234789}={127} (NT @ r1,n2)
HS @ 24/3 @ r3c3: r3c3=7
21/3 @ r1c3 from {45689}={489}
=> r124c3={489} (NT @ c3)
=> 4 @ c3,n1,21/3 locked @ r12c3
=> r23c4={89} (NP @ c4,n2)
=> r23c6={34} (NP @ c6)

r19c1=19-r2c5=19-6=13
r1c1 from {3568}, r1c9 from {345689}
=> r19c1={58} (NP @ r1,22/4)
9/3 @ r1c7 has r1c7+r2c6 from {3469}
=> r1c7+r2c6={34}, r2c7=9-3-4=2

10/3 @ r1c2 from {135689} must be {136}
=> r1c2=6, r2c12={13} (NP @ r2,n1)
=> r23c6=[43]
=> r2c34={89} (NP @ r2,21/3)
=> r2c89={57} (NP @ n3)
=> r1c13789=[54398]

9/2 @ r5c6 from {12568} must be {18} (NP @ c6,n5)
7/2 @ r5c4 from {23456} must be {25|34}
=> 6 @ r4,n5 locked @ r4c46
10/3 @ r3c6 has r3c6=3
=> r34c7=10-r3c6=10-3=7=[61]

20/4 @ r3c8 has r3c89={14}
=> r45c9=20-1-4=15=[96]
=> r24c3=[98]
23/4 @ r3c1 has r3c12={28}
=> r45c1=23-2-8=13=[49]

Innies @ n6: r6c89=45-22-1-9-6=7={25|34} has 2|4
=> 13/4 @ r6c8: r7c89=13-7=6={15} (NP @ r7,n9,13/4)
=> r6c89={34} (NP @ r6,n6)
=> 7/2 @ r5c4={25} (NP @ c4,n5)
=> 8/2 @ r7c3=[26]
=> 11/2 @ r7c6=[74]
=> r7c12=[39]
=> r2c12=[13]

12/3 @ r8c4 from {12459}={129}
=> r8c46=[19], r9c5=2
22/4 @ r1c1: r9c19=22-5-8=9=[63]
20/4 @ r6c1: r6c12=20-3-9=8=[71]


All naked singles from here.
:alien:
Mike's walkthrough:
udosuk wrote:
But using a very ingenius algebraic trick...

Indeed it is! Nice way of looking at this move (your step 3), which an automated solver would probably pick up by analyzing all possible permutations.

It would be interesting to know whether this is the move Afmob intended when giving the puzzle a relatively low 1.25. I would personally rate your step 3 as a little harder than this (maybe around the 1.5 level), because it's a non-obvious move of its type. However, Afmob tends to consistently rate puzzles 1 category lower than I do, so I guess your move could be the "weakness" that Afmob was referring to.

I missed udosuk's ingenious step, so had to resort to somewhat more complex moves, otherwise my initial solving path was very similar to his (what else?). Therefore, I've chosen to speed things up and just post a modified copy of his initial steps, with my alternative highlighted in blue:

0. Prelims

21/3 @ r1c3={489|579|678}
24/3 @ r3c3={789}
10/3 @ r1c2={127|136|145|235}

1. c5,n5

Innies @ c5: r159c5=7={124} (NT @ c5)
Innies @ n5: r46c5=16={79} (NP @ c5,n5)
Innies @ r1289: r28c5=9={36} (NP @ c5)
=> r37c5={58}
=> 18/3 @ r2c5=[387|657], 20/3 @ r6c5=[956|983]
=> r46c5=[79]

2. c3,n1

21/3 @ r1c3: r12c3 must have at least one of {789}
=> r1234c3 form Killer NT {789} @ c3
=> 7 @ c3,n1 locked @ r123c3
r12c3, r3c3 and r4c3 form Killer HT {789} @ c3
=> r12c3 only contains 1 of {789}
=> 21/3 @ r1c3: r2c4={789}


3. Outies @ r89: r1c19+r7c5=21=[498|588|678|768|795|858|948|975]
=> r1c19<>1,2,3

4. 8s @ n1, 24/3:
4a. Either 8 @ r3c123, or
4b. 8 @ r12c3 -> r3c4=8, or
4c. 8 @ r1c1 -> r1c9+r7c5=[58] (step 3)
=> 8 locked in r3c1234+r7c5
=> r3c5<>8 (CPE)
=> split 11/2 @ r23c5=[65]
=> r78c5=[83]


...

Last but not least, thanks for the puzzle, Afmob! Hopefully, Richard is listening and will be able to fix the problem with SS not finding that important hidden cage... ;)
Afmob's snippet:
Congratulations for finding the weakness of V2! Mike is right, this move helps to crack my killer. My initial estimated rating might be a bit low since I also used the same Killer triple as udosuk. The important move (in my case step 2c) can be applied in an easier way which justifies a 1.25 rating, I think:

A112 V2 wt snippet:

1. C456
a) Innies C5 = 7(3) = {124} locked for C5
b) Innies R1289 = 9(2) = {36} locked for C5
c) 20(3) = 9{38/56} because R8C5 = (36) -> 9 locked for C5
d) Innies N5 = 16(2) = {79} -> R4C5 = 7, R6C5 = 9
e) 24(3) = {789} -> 7 locked for R3

2. C123 !
a) ! Killer triple (789) locked in 21(3) + R34C3 for C3
-> 21(3) can only have one of (789) @ R12C3 -> R2C4 <> 4,5,6
b) 7 locked in R123C3 @ C3 for N1
c) ! Innies+Outies N1: 33 = R23C4 + R45C1+R4C3 - R1C1: R1C1 <> 9 because
R23C4 @ N2 <= 17 and R45C1+R4C3 @ N4 <= 24
[alternative: IOD N14: 4 = R23C4 - (R1C1+R6C12) -> R1C1 <> 9 because
R6C12 = 1{2/3} blocked by Killer pairs (12,13) of 16(4)]

By the way, when you score A112 V1 and V2 with the default SS options you get the following results -
V1: 0.92 SudokuSolver spots the hidden cage.
V2: 1.37. SudokuSolver finds udosuk's cracking move.

Those scores are pretty close to our rating, so SudokuSolver's rating is right after all. :lol:
Walkthrough by Andrew, in 2010:
Continuing working through my backlog, the next variant I tried was A112 V2 where I'd originally got stuck at step 13; when I resumed I edited that step before continuing from that position.

I finished this puzzle about a week ago and glanced at the walkthroughs posted by udosuk, Mike and Afmob but I only went through their walkthrougs today.

I was puzzled by the discussion about a possible weakness in A112 V2 and its focus on the hidden 19(3) cage in R12 which I found very early or, in Mike's case, the three outies for R89. I don't see that as a weakness and I very much doubt that SudokuSolver missed that. I also doubt that any software solver would have found udosuk's step 3; while adding the same thing to both sides of an equation is a known mathematical technique, it's unusual and ingenious for a sudoku solving step. In my view the weakness, if there is one, is that the solving path becomes a lot easier after eliminating 9 from R1C1 or fixing R23C5. I took a very long time to reach that position. Udosuk, Mike and Afmob all post optimised walkthroughs so either they made that elimination a lot earlier than I did or they subsequently optimised their walkthroughs to focus on it. Out of the three walkthroughs posted already, Afmob used the most direct way to make that elimination but all three used difficult steps to achieve it so I don't really think that this puzzle can be considered to have a weakness.

My solving path was very different from the three already posted. I hope that you will find some of my steps interesting.

Rating Comment. I'll rate my walkthrough for A112 V2 at Easy 1.5 to 1.5; I'm inclined to go for the latter because I found some steps hard to find. I used some interesting interactions between cages, for example in steps 16 and 19, and then used a short contradiction move for my final breakthrough.

If I was rating the walkthroughs already posted, I would put Afmob's step 2c at Very Hard 1.25 with both udosuk's and Mike's breakthrough steps at least in the 1.5 range.
Here is my walkthrough for A112 V2.

Prelims

a) R56C4 = {16/25/34}, no 7,8,9
b) R56C6 = {18/27/36/45}, no 9
c) R7C34 = {17/26/35}, no 4,8,9
d) R7C67 = {29/38/47/56}, no 1
e) 10(3) cage in N1 = {127/136/145/235}, no 8,9
f) 21(3) cage at R1C3 = {489/579/678}, no 1,2,3
g) 10(3) cage at R1C4 = {127/136/145/235}, no 8,9
h) 9(3) cage at R1C7 = {126/135/234}, no 7,8,9
i) 21(3) cage in N3 = {489/579/678}, no 1,2,3
j) 24(3) cage at R3C3 = {789}
k) 10(3) cage at R3C6 = {127/136/145/235}, no 8,9
l) 20(3) cage at R6C5 = {389/479/569/578}, no 1,2
m) 19(3) cage in N7 = {289/379/469/478/568}, no 1
n) 10(3) cage at R8C3 = {127/136/145/235}, no 8,9
o) 21(3) cage at R8C7 = {489/579/678}, no 1,2,3
p) 13(4) cage at R6C8 = {1237/1246/1345}, no 8,9

1. 45 rule on C5 3 innies R159C5 = 7 = {124}, locked for C5

2. 45 rule on N5 2 innies R46C5 = 16 = {79}, locked for C5 and N5, clean-up: no 2 in R56C6
2a. 9 in N2 locked in R23C4, locked for C4

3. 45 rule on R1289 2 innies R28C5 = 9 = {36}, locked for C5

4. R678C5 = {389/569} (cannot be {578} because 5,8 only in R7C5) -> R6C5 = 9, R4C5 = 7
4a. 7 in 24(3) cage at R3C3 only in R3C34, locked for R3
4b. R23C5 = 11 = [38/65]

5. 45 rule on R12 3 innies R1C19 + R2C5 = 19
5a. R2C5 = {36} -> R1C19 = 13 or 16 = {49/58/67/79}, no 1,2,3
5b. R1C19 = 13 or 16 -> R9C19 = 6 or 9, no 9 in R9C19

6. 10(3) cage at R1C4 = {127/145} (cannot be {136} which clashes with R2C5, cannot be {235} which clashes with R23C5), no 3,6, 1 locked for R1 and N2

7. 9(3) cage at R1C7 = {126/135/234}
7a. 1 of {126/135} must be in R2C7 -> no 5,6 in R2C7

8. 21(3) cage at R1C3 = {489/579/678}
8a. 4,5,6 must be in R12C3 (R12C3 cannot be {78/79/89} which clash with R34C3, ALS block) -> no 4,5,6 in R2C4
8b. Killer triple 7,8,9 in R12C3 and R34C3, locked for C3, clean-up: no 1 in R7C4
8c. 21(3) cage at R1C3 and 24(3) cage at R3C3 must both contain 9 (because there’s no other 9 in C34) -> 21(3) cage = {489/579}, no 6
8d. 7 in C3 only in R123C3, locked for N1, clean-up: no 6 in R1C9 (step 5a)

9. 10(3) cage in N1 = {136/235} (cannot be {145} which clashes with 21(3) cage at R1C3), no 4, 3 locked for N1

10. 9 in N2 only in R23C4
10a. 45 rule on N2 4 innies R23C46 = 24
10b. R23C4 = {79/89} = 16,17 -> R23C6 = 7,8 = {26/34} (cannot be {25} which clashes with 10(3) cage at R1C4, cannot be {35} which clashes with R23C5), no 5
10c. R56C6 = {18/45} (cannot be {36} which clashes with R23C6), no 3,6

11. 13(3) cage in N5 = {238/256/346} (cannot be {148} which clashes with R56C6), no 1
11a. R5C5 = {24} -> no 2,4 in R4C46

12. 10(3) cage at R3C6 = {136/145/235}
12a. 4 of {145} must be in R3C6 -> no 4 in R34C7

13. R23C6 (step 10b) = {26/34} = 7 or 8 -> R1234C7 = 11 or 12 (from sum of cage totals) = {1235/1236/1245}, 1,2 locked for C7, clean-up: no 9 in R7C6
13a. {1235} can only be [21]{35} (because no 1,5 in R23C6 and cannot be {23}{15} because R23C6 cannot both be 4, which can be seen alternatively as R34C7 cannot total 1 more than R12C7)
{1236} can only be {23}{16} (because no 1,5 in R23C6)
{1245} can only be {24}{15} (because 4 only in R12C7)
13b. -> R12C7 = [21]/{23}/{24}, no 5,6, 2 locked for C7 and N3, CPE no 2 in R2C6, clean-up: no 6 in R3C6 (step 10b)
13c. 6 in N2 only in R2C56, locked for R2
13d. 9 in N8 only in R89C6, CPE no 9 in R8C7

14. Hidden killer pair 1,2 in R2C12 and R2C7 for R2, R2C12 contains one of 1,2 -> R2C7 = {12}
14a. 10(3) cage in N1 (step 9) = {136/235}
14b. R2C12 contains 1,2 -> no 2 in R1C2

15. 9(3) cage in R1C7 = {126/234}
15a. 4 of {234} must be in R2C6 (R2C67 cannot be [32] which clashes with R2C12) -> no 4 in R1C7, no 3 in R2C6, clean-up: no 4 in R3C6 (step 10b)
15b. R1234C7 (step 13) = {1235/1236}, 3 locked for C7, clean-up: no 8 in R7C6

16. 3 in R1 only in R1C27, 1 in R2 only in R2C127
16a. 9(3) cage in R1C7 must contain 3 in R1C7 or 1 in R2C7 -> 10(3) cage in N1 must contain 3 in R1C2 or 1 in R2C12 -> 10(3) cage = 3{25}/6{13} (cannot be 5{23}), no 5 in R1C2

17. 45 rule on R67 4 innies R6C3467 = 1 outie R8C5 + 18
17a. R8C5 = {36} -> R6C3467 = 21,24 = {1578/2478/2568/3468/3567/3678/4578}
17b. 7,8 of {1578} must be in R6C67 -> no 1 in R6C6, clean-up: no 8 in R5C6

18. Hidden grouped killer triple 7,8,9 in R1C19, 10(3) cage at R1C4 and 21(3) cages at R1C3 and R1C8 for R12, the two 21(3) cages must each have two of 7,8,9 -> R1C19 + 10(3) cage at R1C4 must contain two of 7,8,9
18a. R1C19 (step 5a) = {49/58}/[67/97] + 10(3) cage at R1C4 (step 6) = {127/145} must contain two of 7,8,9 -> combined cage R1C14569 = {49/58}{127}/[97]{145} (cannot be [67]{145} which only contains one of 7,8,9), no 6 in R1C1, 7 locked for R1

19. 4 in R3 only in R3C1289
19a. 21(3) cages at R1C3 and R1C8 must have different combinations (to avoid clash between R2C34 and R2C89, ALS block)
19b. Only one of the 21(3) cages can contain 7 (because no 7 in R1C38) -> one of the 21(3) cages must be {489}
19c. 4 locked in one of the 21(3) cages and R3C1289 for N13, no 4 in R1C19
19d. R1C19 (step 18a) = {58}/[97], no 9 in R1C9
19e. Killer pair 5,7 in R1C19 and 10(3) cage at R1C4, locked for R1

20. 21(3) cage at R1C3 = {489/579}
20a. 5 of {579} must be in R2C3 -> no 7 in R2C3
20b. R3C3 = 7 (hidden single in C3)
20c. R3C4 + R4C3 = {89}, CPE no 8 in R4C4
20d. 8 in N5 only in R46C6, locked for C6

21. 45 rule on N8 4 remaining innies R79C46 = 22 = {1579/2479/2569/4567} (cannot be {3469} which clashes with R8C5), no 3, clean-up: no 5 in R7C3, no 8 in R7C7
21a. 3 in N8 only in R8C456, locked for R8

22. 12(3) cage in N8 = {129/138/147/156/246} (cannot be {237/345} which clash with R79C46)
22a. 8 of {138} must be in R8C4 -> no 3 in R8C4
22b. 3 in C4 only in R456C4, locked for N5

23. 10(3) cage at R8C3 = {127/136/145/235}
23a. 3 of {136} must be in R9C3 -> no 6 in R9C3

24. 21(3) cage at R1C3 = {489/579}
24a. Cannot be {579}, here’s how
{579} = [957] => R1C19 = {58} clashes with 10(3) cage at R1C4 = {145}
24b. 21(3) cage at R1C3 = {489}, no 5,7

25. Naked pair = {89} in R23C4, locked for C4 and N2 -> R3C5 = 5, R2C5 = 6 (step 4b), R2C6 = 4, R3C6 = 3 (step 10b), R78C5 = [83], clean-up: no 5 in R56C6, no 7 in R7C7
25a. Naked pair {89} in R2C34, locked for R2 and 21(3) cage at R1C3 -> R1C3 = 4

26. Naked pair {57} in R2C89, locked for R2 and N3 -> R1C9 = 8, R1C1 = 5 (step 19d)

27. Naked triple {127} in 10(3) cage at R1C4, locked for R1 -> R1C7 = 3, R2C7 = 2 (step 15), R1C2 = 6, R1C8 = 9
27a. Naked pair {13} in R2C12, locked for N1

28. R56C6 = [18], clean-up: no 6 in R56C4
28a. 6 in N5 only in R4C46, locked for R4

29. R3C6 = 3 -> R34C7 = 7 = [61], clean-up: no 5 in R7C6

30. R3C89 = {14} = 5 -> R45C9 = 15 = [96], R4C3 = 8, R3C4 = 9, R2C34 = [98]

31. R3C12 = {28} = 10 -> R45C1 = 13 = [49]

32. R1C19 = [58] = 13 -> R9C19 = 9 = [27/63/72], no 1,4, no 3 in R9C1

33. 45 rule on N6 2 remaining innies R6C89 = 7 = {25/34}, no 7
33a. R6C89 = 7 -> R7C89 = 6 = {15} (cannot be {24} which clashes with R6C89), locked for R7, N9 and 13(4) cage at R6C8, clean-up: no 2 in R6C89, no 3 in R7C3, no 7 in R7C4, no 6 in R7C6

34. Naked pair {34} in R6C89, locked for R6 and N6, clean-up: no 3,4 in R5C4
34a. Naked pair {25} in R56C4, locked for C4 and N5 -> R4C6 = 6, R4C4 = 3, R5C5 = 4, R7C34 = [26], R7C6 = 7, R7C7 = 4, clean-up: no 7 in R9C9 (step 32)

35. 21(3) cage at R8C7 = {579} (only remaining combination) -> R8C7 = 7, R9C7 = 9, R9C6 = 5

and the rest is naked singles.
Further Discussion:
Mike:

Andrew wrote:
I was puzzled by the discussion about a possible weakness in A112 V2 and its focus on the hidden 19(3) cage in R12 which I found very early or, in Mike's case, the three outies for R89. I don't see that as a weakness and I very much doubt that SudokuSolver missed that.

Software can have bugs, you know! ;). Furthermore, the hidden cage in question is neither of the ones you listed, but the innies of R1289 at R28C5 = h9(2). The latest SS version still scores this puzzle at 2.48 :shock:, because it misses this hidden cage and hence fails to make the first two easy placements. One can see this quite clearly by stepping through the puzzle after scoring it. But the interesting thing is that, if the scoring options are not loaded, SS finds this hidden cage without difficulty!

Andrew wrote:
I also doubt that any software solver would have found udosuk's step 3;

They don't need to. They just permute the cage, taking into account any blocking constraints. Indeed, by doing so, and taking into account that R3C3+R45C1 cannot contain {89} due to R4C3, SS was able to show (without using udosuk's step 2) that R1C1 not only cannot be a 9, but cannot be a 7, either. Also, if one looks at the solver log (see below), which also lists two other constraints (23(4) cage and R3C5) that were not necessary to get the eliminations, then it's pretty clear that it's doing a full permutational analysis:

SudokuSolver v3.3.1 wrote:
34. 45 Rule on n1 - outies r45c1 r2c4 minus innies r1c1 r3c3 equals 9
34a. Cage 23(4) n14 no placement with r4c1 r5c1 = {38}
34b. Cell at r4c3 restricts combinations with cells r3c3 r4c1 r5c1 containing {389} {489} {589} {689} {789} {889} {899}
34c. Cell at r3c5 restricts combinations with cells r2c4 r3c3 containing [58]
34d. Removed candidates 79 from r1c1
34e. Removed candidate 1 from r4c1
34f. Removed candidate 1 from r5c1


Andrew wrote:
while adding the same thing to both sides of an equation is a known mathematical technique, it's unusual and ingenious for a sudoku solving step.

Yes, it was a clever way of deterministically formulating it. Most of us lesser mortals ;) would have (with the same IOD cage) probably have reached the same result via induction (i.e., "R1C1 cannot be 9. Here's how: if R1C1 = 9, then ...").

Andrew wrote:
I don't really think that this puzzle can be considered to have a weakness.

Nor do I. The fact that it's sometimes possible to find a quick way in doesn't IMO necessarily make a puzzle poorer. Indeed, if the "trick" is quite clever, it can be quite motivating for the solver to find it, and arguably be more enjoyable than endlessly shaving off candidates without encountering any really obvious "breakthrough".

Andrew:

Thanks Mike, and Afmob by PM, for clarifying which hidden cage was missed; it hadn't been clear from the earlier posts in the thread. It's interesting that SS missed it when using scoring options but found it without scoring options loaded.

I'll have to remember udosuk's algebraic method of adding the same thing to both sides of an equation next time I come across innie-outie differences where one side has multiple cells and the other side only one cell; it seems that it might be useful for that case.

Ed:

Mike wrote:
Furthermore, the hidden cage in question is neither of the ones you listed, but the innies of R1289 at R28C5 = h9(2). The latest SS version still scores this puzzle at 2.48 :shock:, because it misses this hidden cage ......
True but it misses it because of a different hidden cage.
SS with slight mod to scoring routine wrote:
23. 45 Rule on r7 - innies r7c12589 total 26
23a. Found a hidden cage cage h26(5) n789
Getting this the next step is
Quote:
24. 45 Rule on r3456 - outies r28c5 total 9
24a. After removing cage h26(5) n789
So, by using the Scoring settings and just increasing the "hidden cages" to 5 (from 4) it gives it 1.02! (down from 2.48). Of course, to make this one change to the Scoring routine means other puzzles get messed up (not this dramatically though!) Just did a batch score on the main 115 puzzles that Richard and I used for v3.3.0 and the overall correlation drops about 1.5% but this is not significant. It still means that about 20 out of 100 Assassins will be more than 0.20 score off.

Haven't thought about SS this deeply for ages!


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PostPosted: Wed May 11, 2011 1:29 am 
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Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 113 by Para (July 2008) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:6400:2817:2817:2817:4612:4612:5126:4615:4615:6400:6400:2827:2827:2827:4612:5126:5126:4615:2322:6400:4628:4885:4885:4885:5126:3097:2586:2322:4380:4628:4628:4885:2080:2080:3097:2586:4644:4380:4380:3879:3879:3879:2858:2858:2586:4644:2094:1583:1583:4145:5170:5170:2858:3893:4644:2094:5688:4145:4145:4145:5170:3389:3893:3135:5688:5688:3138:5187:5187:5187:3389:3389:3135:3135:5688:3138:3138:5453:5453:5453:3389:
Solution:
+-------+-------+-------+
| 4 1 2 | 8 5 9 | 7 3 6 |
| 5 7 3 | 2 6 4 | 1 8 9 |
| 6 9 8 | 7 3 1 | 4 5 2 |
+-------+-------+-------+
| 3 4 9 | 1 8 6 | 2 7 5 |
| 7 8 5 | 9 4 2 | 6 1 3 |
| 2 6 1 | 5 7 3 | 9 4 8 |
+-------+-------+-------+
| 9 2 4 | 3 1 5 | 8 6 7 |
| 1 5 7 | 6 9 8 | 3 2 4 |
| 8 3 6 | 4 2 7 | 5 9 1 |
+-------+-------+-------+
Quote:
Para: Here's my assassin. One of those puzzles i like. A lot of playing with 45-tests. There's 2 more version available. An easier and a harder. Your choice if i post a second one and also which one that is then.
SS-Score: 1.56. Rating: 1.25 (hard).

Afmob: This was certainly a difficult Killer, so thanks Para!
My breakthrough step 5a is not the nicest move, so I hope someone comes up with something more elegant. Note that my walkthrough is fairly long since I couldn't find a way to shorten the end game. I would like a V2 but an easier version might be better since V1 is already quite tough.
Rating: 1.5.

Ed:
Afmob wrote:
an easier version might better since V1 is already quite tough.
I'll second that. Thought I was finally getting somewhere and oops - contradiction. No more time for now.
Love having a typical Para toughie. Can't seem to find any decent "45"s. Congratulations to Afmob for getting it so quickly ;clapclap; !
Wish me luck!

Mike(mhparker: And I'll third it!
Nevertheless, thanks for the challenge so far, Para! Also, thanks for posting the "Lite" version instead of (perish the thought!) an even harder "V2"! :sun:
(Full comments by Mike, and others, in a hidden window below)

Para: :whistle: I clearly said playing with 45-tests. I never said they seemed useful. You've all read some of my walk-throughs. :whistle:
You know how 1 candidate can make or break a puzzle. I can make one with 81 1 cell cages too. :lol: (Or maybe 80 to make it that more challenging.)

goooders: As a proponent of time based difficulty I'd agree the original was about 1.5 as it took me north of 3 hours to finish sitting on a sunny Greek island ... Top class puzzle.

Andrew: I've at last tried and solved A113. :)
My solving path had a lot in common with Afmob's walkthrough. There is clearly a narrow solving path for the key breakthrough ...
I thought about rating A113 as Hard 1.25 but on consideration I'll make it 1.5. It's hard to know how to rate ...

Para: That was how i did it too/intended it. So you could just join me in my hard 1.25 rating ;).

Ed: WARNING: 3-monthly grumpy post.
I disagree with Afmob and Andrew rating their walk-throughs for A113 as 1.5 ...
(Ed's full post and replies to it are in a hidden window below)

Walkthrough by Afmob:
This was certainly a difficult Killer, so thanks Para!

My breakthrough step 5a is not the nicest move, so I hope someone comes up with something more elegant. Note that my walkthrough is fairly long since I couldn't find a way to shorten the end game. I would like a V2 but an easier version might be better since V1 is already quite tough.

A113 Walkthrough:

1. R6789
a) Innies N7 = 11(2) <> 1; R7C1 <> 2,3,7
b) Outies R89 = 10(2) <> 5; R7C3 <> 1,2
c) 8(2): R6C2 <> 7
d) Outies N89 = 27(4) = 9{378/468/567} <> 1,2 -> 9 locked for R6
e) 1,2 locked in 13(4) @ N9 = 12{37/46}
f) Innies+Outies R6789: 3 = R5C1 - R6C8 -> R5C1 <> 1,2,3 and R6C8 <> 7,8
g) Innies+Outies N8 : -17 = R6C5 - (R8C56+R9C6) -> R6C5 <> 8,9

2. R1234
a) Innies N3 = 7(2) = [34/43/52]
b) Outies R12 = 13(2) <> 1,2,3
c) 12(2): R4C8 <> 3,4,5
d) 10(3): R45C9 <> 4 because R3C9 <> 1,5
e) Outies N3 = 15(3): R45C9 <> 7 because R4C8 >= 7
f) Innies+Outies R1234: -1 = R5C9 - R4C2 -> R4C2 <> 1,5,8,9
g) Outies N2 = 14(3+1) -> R4C5 <> 9

3. C789
a) Outies C9 = 11(3) <> 9
b) Innies+Outies C89: -11 = R5C7 - R29C8 -> R5C7 <> 7,8 and R2C8 <> 1,2
c) Innies N36 = 19(3) <> 1,5 because {568} blocked by Killer pair (56) of 10(3)
d) Innies N36 = 19(3): R6C79 <> 6 because R4C7 <> 4,9
e) 8(2): R4C6 <> 3,7
f) 15(2): R7C9 <> 9
g) 18(3): R1C8 <> 1,2 because 1{89} / 2{79} blocked by Killer pairs (79,89) of 15(2)
h) 18(3) <> {459} since (45) is a Killer pair of Innies N3
i) Hidden Killer pair (89) in 18(3) @ C9 since 15(2) can only have one of (89)
-> 18(3) <> {567} -> 18(3) <> 5
j) 5 locked in 10(3) @ C9 = 5{14/23} -> 5 locked for N6

4. R456
a) Innies+Outies R6789: 3 = R5C1 - R6C8 -> R5C1 <> 8
b) Innies+Outies R1234: -1 = R5C9 - R4C2 -> R4C2 <> 7

5. N3689 !
a) ! Innies N36 = 19(3) <> 4 because it forces Outies N89 = [6948] (because of 20(3) @ N6)
so that 20(3) = [947] and Innies N36 = [748] -> Two 7s in C7
b) 4 locked in 11(3) @ N6 = {146} locked for N6
c) 10(3) = {235} locked for C9
d) 8(2): R4C6 <> 2
e) Innies N36 = 19(3) = 9{28/37} -> 9 locked for R6+N6
f) 12(2) <> 3

6. C89+R789 !
a) 2 locked in Outies C9 @ C8 = 11(3) = 2{18/36}
b) ! 9 locked in Innies C89 @ C8 = 22(4) = 49{18/36} because R56C8 = (146)
-> 4 locked in R56C8 for C8+N5 and R29C8 <> 6
c) R3C8 = 5 -> R4C8 = 7
d) Innie N3 = R3C9 = 2
e) 10(3) = {235} -> 3 locked for N6
f) R4C7 = 2 -> R4C6 = 6

7. R456
a) Innies+Outies R1234: -1 = R5C9 - R4C2 -> R4C2 = 4, R5C9 = 3
b) R4C9 = 5
c) Outies N89 = 27(4) = {3789} because R6C79 = (89)
-> 3,7 locked for R6+N6 and 8 locked for R6
d) 17(3) = 4{58/67}
e) 2 locked in 15(3) @ R5 for N5 = 2{49/58}
f) Outies R6789 = 14(3) = 1{49/67} because R5C78 = (146) -> 1 locked for N6 and R5C1 = (79)
g) 9(2) = [18/63/81]
h) 15(2): R7C9 <> 8
i) 9 locked 18(3) @ R4 = 9{18/36}; R3C3 <> 9

8. R123
a) 18(3) @ N1: R3C3 <> 3 because 6 only possible there
b) 3 locked in 19(4) @ R3 = 13{69/78} -> 3 locked for N2
c) Hidden Single: R3C7 = 4 @ R3
d) Outie R12 = R3C2 = 9
e) 19(4) = {1378} -> 7 locked for N2; CPE: R12C5 <> 1,8
f) 6 locked in R3C13 @ R3 for N1
g) 25(4) = 59{38/47} because R3C13 = (168) blocks {1789} -> 5 locked for N1
h) 18(3) @ N2 = 4{59/68} because (189) blocked by R3C456 = (1378) -> 4 locked for N2

9. N9
a) 4 locked in 13(4) @ C9 = {1246} -> 6 locked for N9
b) R7C9 = 7, R6C9 = 8
c) 20(3) @ R6 = {389} -> R6C6 = 3, R6C7 = 9, R7C7 = 8
d) R9C8 = 9, R9C7 = 5 -> R9C6 = 7
e) 20(3) @ R8 = {389} -> R8C7 = 3, {89} locked for R8+N8

10. C123
a) 8(2) = [26/53/62]
b) Innies N7 = 11(2) = [56/92]
c) 18(3) @ C1 = 7{29/56} -> R5C1 = 7; R6C1 <> 5
d) Hidden Single: R7C1 = 9 @ C1, R4C3 = 9 @ C3, R4C1 = 3 @ N4 -> R3C1 = 6
e) 8(2) = {26} locked for C2
f) 12(3) = 3{18/45} -> R9C2 = 3
g) 25(4) = {4579} -> R2C2 = 7, {45} locked for C1+N1

11. R123
a) 11(3) @ R1 = {128} locked for R1 because R1C2 = (18)
b) 11(3) @ R2 = 2{18/36}
c) Killer pair (16) locked in 11(3) + R2C7 for R2

12. Rest is singles.

Rating: 1.5. I used combo analysis to crack it.
Full comments by Mike and others:
Mike:

Ed wrote:
Afmob wrote:
an easier version might better since V1 is already quite tough.
I'll second that.

And I'll third it! Spent a couple of hours on Saturday morning on a second session whilst sitting out in the sun, with a printed copy of the grid and killer combination table. Managed to eliminate a grand total of another (trumpets, please!) 10 candidates. So "only" 589 left now!! :-)

Ed wrote:
Can't seem to find any decent "45"s. Congratulations to Afmob for getting it so quickly ;clapclap; !

Ditto! Most "45"s seem to be specially crafted to only shave off 1 or 2 candidates. :brickwall:

Ed wrote:
Wish me luck!

Will do. I can certainly empathize with that request!

Nevertheless, thanks for the challenge so far, Para! Also, thanks for posting the "Lite" version instead of (perish the thought!) an even harder "V2"! :sun:

Para:

Ed wrote:
Can't seem to find any decent "45"s. Congratulations to Afmob for getting it so quickly ;clapclap; !

mhparker wrote:
Ditto! Most "45"s seem to be specially crafted to only shave off 1 or 2 candidates. :brickwall:

:whistle: I clearly said playing with 45-tests. I never said they seemed useful. You've all read some of my walk-throughs. :whistle:

mhparker wrote:
Managed to eliminate a grand total of another (trumpets, please!) 10 candidates. So "only" 589 left now!! :-)

You know how 1 candidate can make or break a puzzle. I can make one with 81 1 cell cages too. :lol: (Or maybe 80 to make it that more challenging.)

goooders:

As a proponent of time based difficulty I'd agree the original was about 1.5 as it took me north of 3 hours to finish sitting on a sunny Greek island.A lot of combination crunching on the right hand side got you a fair way then the middle needed the same(unlike Lite where once started the middle was tame by comparison).Top class puzzle.

Mike:

goooders wrote:
As a proponent of time based difficulty I'd agree the original was about 1.5 as it took me north of 3 hours to finish sitting on a sunny Greek island.

That sounds like the way to do 'em! Going green with envy here... :mrgreen:

goooders wrote:
A lot of combination crunching on the right hand side got you a fair way then the middle needed the same(unlike Lite where once started the middle was tame by comparison).Top class puzzle.

Actually, I preferred the Lite version myself. The original seemed to me to have a very narrow (although interesting) solving path. So I'll be surprised if anyone can post a WT for it that is substantially different from Afmob's. By contrast, the Lite version provided more opportunity for finding alternatives, if one missed the "optimal" way forward. There's certainly scope for more than one WT for the Lite. For this reason, I think it would have made a better V1, with the original being posted as a V2.

Like you, I used to prefer the really tough puzzles, where it was (in the "good old days") really satisfying to find a way (possibly the only way) through, and get a kick from telling everyone about it. The problem these days, however, is that the automated solvers have become so good that they have taken a lot of the mystery and adventure out of the game that used to be present. Anyone can find out in a few seconds how to do these beasts by simply taking JSudoku or SudokuSolver out for a spin on them.

The consequence of all this is that I find myself these days unwilling to spend an hour or more trying to find the breakthrough move that JS or SS can find for me anyway in a tiny fraction of the time. The frustration simply isn't worth it for me (especially because I've got lots of other stuff to be getting on with at the moment). Therefore, when I seem to be grinding to a halt, I give myself about 15 minutes to find a way of progressing the game significantly, after which (if unsuccessful) I give up.

In contrast, I still find easier puzzles fun to do, where it's irrelevant that an automated solver can quickly make mincemeat of them. Although I find my solving path a bit ragged at times, they often have enough redundancy built in to them that I can make it to the end without hitting a frustrating brick wall. Obviously there's a compromise to be found here. A puzzle where each move takes only a matter of a few seconds to spot is probably too trivial to be worthwhile.

Andrew:

I haven't yet tried either version so can't comment on the puzzles.

Mike wrote:
Actually, I preferred the Lite version myself. The original seemed to me to have a very narrow (although interesting) solving path. So I'll be surprised if anyone can post a WT for it that is substantially different from Afmob's. By contrast, the Lite version provided more opportunity for finding alternatives, if one missed the "optimal" way forward. There's certainly scope for more than one WT for the Lite. For this reason, I think it would have made a better V1, with the original being posted as a V2.

Like you, I used to prefer the really tough puzzles, where it was (in the "good old days") really satisfying to find a way (possibly the only way) through, and get a kick from telling everyone about it. The problem these days, however, is that the automated solvers have become so good that they have taken a lot of the mystery and adventure out of the game that used to be present. Anyone can find out in a few seconds how to do these beasts by simply taking JSudoku or SudokuSolver out for a spin on them. ...

In one way I can understand what Mike is saying but at the same time there's still the same challenge from the really tough puzzles. As is well known, I haven't downloaded any software solvers and have no plans to do so at the moment. If I had them, I wouldn't be looking to see how they solved a puzzle until I had either finished or got stuck and given up after trying on and off for several days. In that sense therefore I don't see how software solvers take the fun/challenge out of the puzzles. Are they really much different from reading a posted WT?
Mike's analysis of Afmob's breakthrough step, with Afmob's reply and Ed's comments:
Mike:

Yes, I know it's already history (in the meantime there have been MO4 and A114, with A114V2 imminent), but I'd like to pick up on Afmob's breakthrough move for the A113 Original.

Before jumping in at the deep end, I assume that most people reading this are aware of regular 3-cell IOU. If not, please refer to Ed's original post here.

Now back to Afmob's breakthrough:

Afmob wrote:
5. N3689 !
a) ! Innies N36 = 19(3) <> 4 because it forces Outies N89 = [6948] (because of 20(3) @ N6)
so that 20(3) = [947] and Innies N36 = [748] -> Two 7s in C7

This sounds like a fairly arbitrary contradiction move until one takes a look at it in more detail.

Here's the state of the grid at the time this move was made:

Image

Note that the two green cells plus the yellow cell make up the 20(3) cage at R6C6, whilst the two red cells plus

the yellow cell make up the hidden 19(3) cage representing the N36 innies (Afmob's step 3c).

Because both of these cages have the yellow cell in common, the sum of the two green cells (the "outies") must be 1 higher than the sum of the two red cells (the "innies").

Furthermore, note that one of the outies (R6C6) can "see" one of the innies (R6C9), whilst the other outie (R7C7) can see the other innie (R4C7).

Due to this arrangement of the 4 cells, neither outie can be 1 higher than the innie it can see, otherwise the other innie and outie (which can see each other) would be forced to the same value.

This is an example of 4-cell IOU (IOU4). It explains why, in Afmob's step 5a, the [6948] permutation for R6C5679 (N89 outies) is blocked, since R6C6 cannot be 1 higher than R6C9 for the reason stated above.

Interestingly, although Afmob's step 5a only listed the [6948] permutation, it's also interesting to consider the effects of the hidden 16(3) N3689 outie cage at R4C6+R6C56 (= "h16(3)n5")on the full set of permutations for R6C5679:

The candidate combinations for h27(4) at R6C5679 are: {3789/4689/5679}. Let's look at all possible permutations for these combinations (hope I didn't miss any!), based on the grid image shown above:

[3789] - ok
[3798] - ok
[3879] - ok
[3897] - blocked by IOU4 (see above)
[3978] - blocked by h16(3)n5, IOU4
[3987] - blocked by h16(3)n5
[7389] - ok
[7398] - ok
[7839] - ok
[7938] - blocked by h16(3)n5, IOU4
[4689] - blocked by h16(3)n5, 20(3)n569
[4698] - blocked by h16(3)n5
[6489] - blocked by h16(3)n5, 20(3)n569
[6498] - blocked by h16(3)n5
[6849] - blocked by 20(3)n569
[6948] - blocked by IOU4
[5679] - blocked by h16(3)n5, 20(3)n569
[5697] - blocked by h16(3)n5
[6579] - blocked by h16(3)n5
[6597] - blocked by h16(3)n5

Thus, the only non-blocked combination for R6C5679 is {3789} (no 4..6), all locked for R6. Possible permuations are (from above): [3789/3798/3879/7389/7398/7839] -> no 9 in R6C6, no 7 in R6C9.

IOU4 is a complex move, as is enumeration of individual permutations. Killer Sudoku doesn't get much more difficult than this, without using chains or resorting to hypotheticals. Therefore, objectively, there would be an argument for rating this puzzle at 1.75, even though some people (not including me!) may have found it subjectively easier.

Comments and feedback welcome.

Afmob:

Wow, I haven't looked at my move this way.

I think if one had used this move to its full extend - namely eliminating all blocked permutations of the Outies of N89 then this move could be rated 1.75. But I only used a small part of it to remove 4 from R6C7, so in this case I wouldn't rate it higher than 1.5.

Ed:

mhparker wrote:
IOU4 is a complex move..Therefore, objectively, there would be an argument for rating this puzzle at 1.75
Thanks Mike for the really interesting explanation for this move. :applause: I used a contradiction chain that was similiar to Mike's explanation (but two 7's into r7c79) and further into the solution (r6c5 to (357) but the contradiction was because of r6c5679. I tried to find a way to make it less chain like but....

Subjectively, I have no hesitation in giving this a 1.75 rating. Took 2 weekends of solving time. Even after locking those cells, it was still a continual battle. Also, reached contradictions twice - so 3rd time lucky! Never been superstitious before but "13" has me worried now!!

Unfortunately, won't be backing up my rating with a WT. Once again, congrats to Afmob! Very thankful Nasenbaer has been merciful with this week's puzzle (relief emoticon).

Another never-to-be-forgotten puzzle from Para!
Walkthrough by Andrew:
I've at last tried and solved A113. :)

My solving path had a lot in common with Afmob's walkthrough. There is clearly a narrow solving path for the key breakthrough, the elimination of 4 from R6C7.

Mike's IOU4 for Afmob's step 5a was a very interesting concept.

I thought about rating A113 as Hard 1.25 but on consideration I'll make it 1.5. It's hard to know how to rate steps 17a and 20a. I look at them as combined cages formed from a complete cage and a partial cage, which subjectively makes these steps easy ones.

Here is my walkthrough. I used the interaction of Outies N3689 and R6C5679 for the key breakthrough. This Assassin was solved by combination analysis but none too difficult. Thanks Afmob for the corrections.

Prelims

a) R34C1 = {18/27/36/45}, no 9
b) R34C8 = {39/48/57}, no 1,2,6
c) R4C67 = {17/26/35}, no 4,8,9
d) R67C2 = {17/26/35}, no 4,8,9
e) R6C34 = {15/24}
f) R67C9 = {69/78}
g) R1C234 = {128/137/146/236/245}, no 9
h) R2C345 = {128/137/146/236/245}, no 9
i) R345C9 = {127/136/145/235}, no 8,9
j) 11(3) cage in N6 = {128/137/146/236/245}, no 9
k) 20(3) cage at R6C6 = {389/479/569/578}, no 1,2
l) R8C567 = {389/479/569/578}, no 1,2
m) R9C678 = {489/579/678}, no 1,2,3
n) 1,2 in N9 locked in 13(4) cage = {1237/1246}, no 5,8,9

1. 45 rule on R12 2 outies R3C27 = 13 = {49/58/67}, no 1,2,3

2. 45 rule on R89 2 outies R7C38 = 10 = {37/46}/[82/91], no 1,2,5 in R7C3

3. 45 rule on N3 2 innies R3C89 = 7 = [34/43/52], clean-up: no 3,4,5 in R4C8

4. 45 rule on N7 2 innies R7C12 = 11 = [47/56/65/83/92], no 1, no 2,3,7 in R7C1, clean-up: no 7 in R6C2

5. 45 rule on R1234 1 innie R4C2 = 1 outie R5C9 + 1, no 1,9 in R4C2

6. 45 rule on R6789 1 outie R5C1 = 1 innie R6C8 + 3, no 1,2,3 in R5C1, no 7,8 in R6C8

7. 45 rule on N89 4 outies R6C5679 = 27 = {3789/4689/5679}, no 1,2, 9 locked for R6

8. R345C9 = {127/136/145/235}
8a. 4 of {145} must be in R3C9 -> no 4 in R45C9, clean-up: no 5 in R4C2 (step 5)

9. 45 rule on C89 2 innies R29C8 = 1 outie R5C7 + 11
9a. Max R29C8 = 17 -> max R5C7 = 6
9b. Min R29C8 = 12, no 1,2 in R2C8

10. 45 rule on C12 2 innies R18C2 = 1 outie R5C3 + 1
10a. Min R18C2 = 3 -> min R5C3 = 2

11. 45 rule on C1 3 outies R239C2 = 19 = {289/379/469/478/568}, no 1

12. 45 rule on C9 3 outies R178C8 = 11 = {128/137/146/236/245}, no 9

13. 45 rule on C9 4 innies R1289C9 = 20 = {1469/1478/2369/2378/2459} (cannot be {1289/1379/1568/2468/2567/3467} which clash with R67C9, cannot be {3458} because no 5,8 in R89C9 and 18(3) cage in N3 cannot be {558})
13a. R345C9 (step 8) = {136/145/235} (cannot be {127} which clashes with R1289C9), no 7, clean-up: no 8 in R4C2 (step 5)

14. Hidden killer pair 8,9 in R12C9 and R67C9 for C9 -> R12C9 must contain one of 8,9
14a. 18(3) cage in N3 = {189/279/369/378/468} (cannot be {459} which clashes with R3C89, cannot be {567} because R12C9 must contain one of 8,9), no 5
14b. R12C9 cannot be {27} because no 8 in R89C9, R12C9 cannot be {29} because R89C9 cannot be {36} and no 5 in R89C9, R12C9 cannot be {79} which clashes with R67C9 -> 18(3) cannot be {279} = {189/369/378/468}, no 2
14c. 1 of {189} must be in R12C9 (R12C9 cannot be {89} which clashes with R67C9), no 1 in R1C8
14d. 4 of {468} must be in R12C9 (R12C9 cannot be {68} which clashes with R67C89, no 4 in R1C8

15. 5 in C9 locked in R45C9, locked for N6, clean-up: no 3 in R4C6, no 8 in R5C1 (step 6)
15a. R345C9 (step 13a) = {145/235}, no 6, clean-up: no 7 in R4C2 (step 5)
15b. Max R4C2 = 6 -> min R5C23 = 11, no 1 in R5C2

16. 45 rule on R6 3 remaining innies R6C128 = 12 = {138/156/237/246} (cannot be {147/345} which clash with R6C34)
16a. 7,8 of {138/237} must be in R6C1 -> no 3 in R6C1

17. 11(3) cage in N6 = {128/137/146/236}
17a. 1 locked in 11(3) cage and R45C9 for N6, clean-up: no 7 in R4C6

18. 45 rule on N3689 3 outies R4C6 + R6C56 = 16 = {169/178/259/268/358/367} (cannot be {349} because R4C6 only contains 1,2,5,6, cannot be {457} because R6C56 clashes with R6C5689), no 4
18a. R6C5679 (step 7) = {3789/4689/5679}
18b. R4C6 + R6C56 cannot be {169} because R6C5679 = [6948] would make R7C79 both 7 and because no 5 in R6C79
18c. R4C6 + R6C56 cannot be {268} because R6C5679 = [6849] would make 20(3) cage at R6C6 [848]
18d. -> R4C6 + R6C56 = {178/259/358/367}
18e. 6 of {367} must be in R4C6 -> no 6 in R6C56
18f. R6C56 = {37/38/59/78} -> R6C5679 = {3789/5679}, no 4, 7 locked for R6

19. 4 in N6 locked in 11(3) cage (step 17) = {146}, locked for N6, clean-up: no 4 in R3C9 (step 15a), no 3 in R3C8 (step 3), no 2 in R4C2 (step 5), no 2 in R4C6, no 9 in R4C8, no 5,6 in R5C1 (step 6), no 5 in R6C56 (step 18f), no 9 in R7C9
19a. Naked triple {235} in R345C9, locked for C9
19b. Naked quad {3789} in R6C5679, locked for R6, clean-up: no 5 in R7C2, no 6 in R7C1 (step 4)
19c. 9 in N6 locked in R6C79, locked for R6
19d. 18(3) cage in N3 (step 14b) = {189/369/378/468}
19e. 3 of {378} must be in R1C8 -> no 7 in R1C8

20. 2 in C8 locked in R78C8 -> R178C8 (step 12) = {128/236}, no 4,7
20a. Killer triple 1,4,6 in R178C8 and R56C8, locked for C8, clean-up: no 8 in R4C8
20b. 4 in C8 locked in R45C8, locked for N6
20c. R34C8 = [57], R3C9 = 2 (step 3), clean-up: no 8 in R3C27 (step 1), no 4 in R4C1, no 3 in R4C2 (step 5), no 1 in R4C6, no 8 in R7C9

21. Naked pair {35} in R45C9, locked for N6 -> R4C7 = 2, R4C6 = 6, R4C2 = 4, R5C9 = 3 (step 5), R4C9 = 5, clean-up: no 3,4,7 in R3C1, no 9 in R3C7 (step 1), no 2 in R6C4
21a. Naked pair {89} in R6C79, locked for R6
21b. Naked pair {37} in R6C56, locked for N5, CPE no 3,7 in R7C6
21c. R4C2 = 4 -> R5C23 = 13 = {58/67}, no 2,9

22. Killer pair 6,7 in R7C9 and 13(4) cage, locked for N9
22a. R9C678 = {489/579}, 9 locked for R9
22b. 7 of {579} must be in R9C6 -> no 5 in R9C6

23. 45 rule on N5 3 remaining innies R4C45 + R6C4 = 14 = {149/158}, 1 locked for N5
23a. 4,5 only in R6C4 -> R6C4 = {45}, clean-up: no 5 in R6C3
23b. 1 in N5 locked in R4C45, locked for R4, clean-up: no 8 in R3C1
23c. 1 in R6 locked in R6C123, locked for R6

24. 20(3) cage at R6C6 = {389/479/578}
24a. 3 of {389} must be in R6C6 -> no 3 in R7C7

25. 18(3) cage at R3C3 = {189/369} (cannot be {378} because R4C34 = [38] clashes with R4C1, cannot be {468} because 4,6 only in R3C3), no 4,7
25a. 6 of {369} must be in R3C3 -> no 3 in R3C3

26. 3 in R3 locked in R3C456, locked for N2
26a. 19(4) cage at R3C4 = {1369/1378}, no 4

27. R3C7 = 4 (hidden single in R3), R3C2 = 9 (step 1)
27a. 4 in C9 locked in R89C9 -> 13(4) cage at R8C8 = {1246}, locked for N9
27b. R7C9 = 7, R6C9 = 8, R6C7 = 9, clean-up: no 1 in R6C2, no 4 in R7C1 (step 4)
27c. Naked quad {1246} in R5678C8, locked for C8
27d. Naked pair {58} in R79C7, locked for C7 and N9 -> R8C7 = 3, R9C8 = 9
27e. R8C7 = 3 -> R8C56 = {89} (prelim l), locked for R8 and N8

28. 45 rule on N89 1 remaining innie R7C7 = 1 outie R6C5 + 1 -> R6C5 = 7, R7C7 = 8, R6C6 = 3, R9C7 = 5, R9C6 = 7 (step 22a), clean-up: no 3 in R7C2 (step 4), no 5 in R6C2
28a. Naked pair {26} in R67C2, locked for C2, clean-up: no 7 in R5C3 (step 21c)

29. R567C1 = {189/279/567}
29a. 6 of {567} must be in R6C1 -> no 5 in R6C1
[Afmob pointed out that cannot be {189} because I’ve eliminated 8 from the cage. I’ve left this step unchanged because the next step makes this elimination.]

30. R6C128 (step 16) = {246} (only remaining combination) -> R6C8 = 4, R6C12 = {26}, locked for R6 and N4 -> R6C34 = [15], clean-up: no 7 in R5C2 (step 21c)
30a. Naked pair {58} in R5C23, locked for R5 and N4 -> R4C13 = [39] , R5C1 = 7, R3C1 = 6, R3C3 = 8, R4C45 = [18], R3C456 = [731], R5C23 = [85], R67C1 = [29], R67C2 = [62], R8C56 = [98], R9C2 = 3

31. 12(3) cage in N8 = {246} (only remaining combination), locked for N8 -> R7C456 = [315], R7C8 = 6, R7C3 = 4, R5C78 = [61], R8C8 = 2
31a. 2 in N8 locked in R9C45, locked for R9 -> R9C3 = 6, R8C3 = 7, R8C2 = 5 (cage sum)

32. R12C7 = {17} = 8, R3C7 = 4 -> R2C8 = 8 (cage sum), R1C8 = 3, R12C3 = [23]
32a. R2C3 = 3 -> R2C45 = 8 = {26}, locked for R2 and N2

and the rest is naked singles

Now to try A113-Lite.
Ed's post on the ratings for A113 and various replies:
Ed:

WARNING: 3-monthly grumpy post.

I disagree with Afmob and Andrew rating their walk-throughs for A113 as 1.5 . I also now disagree with my rating of 1.75 but the 1.5s bother me the most. I now think they (we) should have given ratings as high 1.75 or even better, a 2.0 for the breakthrough moves (Afmob's 5a, Andrew's 18b, my chain move). I'm not suggesting we don't use those moves, but I think they should get a higher rating.

The reason is they use 3 or more elements to find a contradiction. To me, it's not about how difficult they are to find or follow, but about the satisfaction/niceness/elegance of the solution. I found similar contradictions (especially to Andrew's), but chose not to use them until much further into the solution.

I don't mean to pick on Afmob and Andrew here - it's just they have given us the WTs (thanks guys!) .
Afmob's Step 5a wrote:
Innies N36 = 19(3) <> 4 because it forces Outies N89 = [6948] (because of 20(3) @ N6)
so that 20(3) = [947] and Innies N36 = [748] -> Two 7s in C7
This uses cage-placement in a hidden cage in n6, cage-placement in a hidden cage r6 and cage-placement in the 20(3)r6c6. It uses 3 (or 4?) elements: 2 hidden cages & a regular cage (plus cage-placement?). If it had just done any 2 of these, it would be standard Assassin level stuff.

Andrew wrote:
18b. R4C6 + R6C56 cannot be {169} because R6C5679 = [6948] would make R7C79 both 7 and because no 5 in R6C79
This uses a hidden cage in n5, a hidden cage in r6 and a combination of 2 cages linked by n9 (& cage-placement?). This is 3 (or 4?) different elements.

By contrast (at least to me!), Mike's IOU4 way uses two (1 of which is complex) elements to assess some of the permutations in r6c5679 (note: some of the permutations he considers rely on a different single element). With IOU4, he first establishes one general relationship between two hidden cages (In same r or c: outie cells cannot be Innie cell +1), then considers cage-placement in one of the hidden cages.

If IOU4 was sufficient on its own to crack the puzzle, then I'd agree with the 1.75 rating Mike gave. Presumably it could have been sufficient if used further into the solution. But this is not really my point. No matter what rating, IOU4 is in my view, one powerful, elegant piece of logic which is far more satisfying.

Returning now to the problem with (especially) the 1.5 ratings given to this puzzle. First, IOU4 will have no chance of the prestige it deserves when 1.5 ratings are given for less elegant moves! To me, it is an anomaly to have the most satisfying solution getting the higher (and therefore less prestigious) rating!

Second, there is no way I'm going to put in the effort to make a walk-through for my solution which chooses not to use 3-elements analysis so early when I will have to give my way a higher rating.

Third, I've realized that I don't like reading a WT with this type of analysis being given a low rating. I feel a bit cheated after I put so much effort in trying to avoid using those sorts of moves.

These last couple of paragraphs are starting to drift into more general comments about how I've felt for quite a while about some of the ratings being suggested. Every now and then I need to let out my frustrations because it is affecting my morale.

A bit discouraged about Assassins above 1.25 rating.

Afmob:

I think what makes my breakthrough move easier than a standard contradiction chain is that those cells are directly connected with each other.

When we look at the Outies of N89 = R6C5679 = 27(4) we can see that two of those cells also belong to the Innies of N36 namely R6C79, so the placement of those Innies is directly linked to the placement of Outies N89. This relationship is easier to see than a normal 1.75 chain, were a placement in one cage does not directly influence the placement of another cage since you have to go over several cages to see the relation. So I think a rating of 1.5 is justified but my chain/combo analysis might be more of (hard) 1.5 rating since it needs a contradiction to eliminate a candidate (I usually prefer forcing chains, where two different placements lead to the same conclusion).

When I look at "classic" 1.75 Killers like Maverick 2 or Maverick 4, I notice that they don't fall after just one of those moves. They need several "chainy" moves to be cracked but A113 was cracked after our different moves.

Mike:

Ed wrote:
These last couple of paragraphs are starting to drift into more general comments about how I've felt for quite a while about some of the ratings being suggested. Every now and then I need to let out my frustrations because it is affecting my morale.

I know exactly how you feel, because I've been thinking exactly the same way for some time, too.

Ed wrote:
A bit discouraged about Assassins above 1.25 rating.

Unfortunately, the ones at 1.25 and below don't seem to be much better! In particular, I was very annoyed about the A115 "0.9" version that was "dedicated" to me, but I don't want to go into detail about that right now. Suffice to say, given a similar scenario again, I will ignore the puzzle and let somebody else do it instead.

Would really like to go into more detail here, and will try to post a more worthy reply to your very interesting post in a few days' time. I also look forward to trying your great-looking A117 Hammer Throw then, too.

In the meantime, many thanks for your post, Ed.

Ed:

About IOU4 mhparker wrote:
Because both of these cages have the yellow cell in common, the sum of the two green cells (the "outies") must be 1 higher than the sum of the two red cells (the "innies")
I think IOU4 is simpler than this and should work without the common yellow cell or the hidden cages. The important part it is that the two outies of n36 each sees one of the innies. (See examples below).

Here is an alternative (Hard 1.5 Rating?) way to get to Afmob and Andrew's key elimination of 4 from r6c7 for A113 using a couple of "simple" IOU4 moves, and another variation on IOU (IOOU) . These moves are all courtesy of SudokuSolver, but not with the default scoring order. BTW - the default scoring order never gets rid of that 4 but instead (eventually) finds a hidden single 5 in c8.

Starting with Andrew's WT step 16 to here: (select and paste marks into A113 in SudokuSolver or JSudoku)
Code:
.-------------------------------.-------------------------------.-------------------------------.
| 123456789 12345678  12345678  | 12345678  123456789 123456789 | 123456789 3678      1346789   |
| 123456789 23456789  12345678  | 12345678  12345678  123456789 | 123456789 3456789   1346789   |
| 12345678  456789    123456789 | 123456789 123456789 123456789 | 456789    345       234       |
:-------------------------------+-------------------------------+-------------------------------:
| 12345678  2346      123456789 | 123456789 123456789 12567     | 12367     789       1235      |
| 45679     23456789  23456789  | 123456789 123456789 123456789 | 12346     1234678   1235      |
| 1245678   12356     1245      | 1245      3456789   3456789   | 346789    12346     6789      |
:-------------------------------+-------------------------------+-------------------------------:
| 45689     23567     346789    | 123456789 123456789 123456789 | 3456789   123467    6789      |
| 123456789 123456789 123456789 | 123456789 3456789   3456789   | 3456789   123467    123467    |
| 123456789 23456789  123456789 | 123456789 123456789 456789    | 456789    456789    123467    |
'-------------------------------.-------------------------------.-------------------------------'


17. 45 Rule on n36 - innies r6c79 + r4c7 total 19 = {289/379/469/478/569}
17a. Removed candidate 1 from r4c7
17b. 6 in {469/568} must be in r4c7 -> Removed candidate 6 from r6c7 & r6c9
17c. Cage sum in cage 8(2) n56 - removed 7 from r4c6
17d. Cage sum in cage 15(2) n69 - removed 9 from r7c9

18. deleted

NOW IOU4
Image
19. "45" n36: 2 outies r6c6 + r7c7 - 1 = 2 innies r6c9 + r4c7
-> 1 outie cannot be the i/o difference (in this case 1) more than the one innie it sees as this would force the remaining innie & outie to be equal which is impossible since they can see each other. (IOU4)
19a. when 3 in r7c7, 2 is blocked from r4c7 by IOU4 -> min. innies n36 = [67] = 13 -> 2 outies = 14: but max. sum in 2 outies = 12 when r7c7 = 3
19b. -> no 3 in r7c7

20. 45 Rule on n8: innies r8c56 + r9c6 = outie r6c5 + 17
20a. max. 3 innies in n8 = 24 -> max. r6c5 = 7 (Removed candidates 89 from r6c5)

21. 45 Rule on n5: innies r6c56 + r4c6 = 16 (After removing cage h14(3) n5)
21a. = {169/178/259/268/358/367/457} ({349} not possible with r4c6)
21b. 8 & 9 in {169/259/268/358} must be in r6c6; 5 & 6 in {367/457} must be in r4c6
21c. -> no 5 or 6 in r6c6

22. Limited placement of candidates in cage 20(3) n569
22a. Removed combination {569} - no valid placement
22b. Removed 6 from r7c7

23. 45 Rule on n789 - outies r6c5679 total 27 (After removing cage h11(2)n7)
23a. = {3789/4689}(no 5) ({5679} blocked by (56) only in r6c5)
23b. 6 in {4689} must be in r6c5 -> no 4 in r6c5
23c. Candidate 8 locked in r6c679 for r6
23d. Combinations {5679} no longer valid in cage h27(4)r6
23e. Combinations {138} no longer valid in cage h12(3)r6c128
23f. Combinations {168} {258} {348} no longer valid in cage h15(3)n4

24. Combinations {1249} {1258} no longer valid in cage 16(4) n58
24a. Removed redundant candidates 9

25. Combinations {259} no longer valid in cage h16(3) n5 ((25) only in r4c7)

A variation on IOU: IOOU eliminates 7 from r6c9.
Image

26. 45 Rule on n89: outies r6c59 = innie r7c7 + 7
26a. IOOU: When the 2 Outies see each other AND when 1 of the two outies = the i/o difference (in this case 7) -> the innie and the other outie must share a candidate Unequal to the i/o difference. If they don't, then the original outie cannot = i/o difference. [edit: clarity. Thanks Mike and Andrew]
26b. Removed candidate 7 from r6c9 (NOTE: when r6c9 = 7 -> 1 innie = 1 outie = 7 (no other digits are common in r6c5 & r7c7): but this means 2 7's in r6)
26c. Cage sum in cage 15(2) n69 - removed 8 from r7c9

NOW back to IOU4
Image
27. "45" n36: 2 outies r6c6 + r7c7 - 1 = 2 innies r6c9 + r4c7
27a. no 9 in r6c6 as it forces r6c9 = 8 (which is not possible according to IOU4) [typo fixed: thanks Afmob]

28. deleted

29. Candidate 9 locked in r6 for n6 nowhere else in n6
29a. Cage sum in cage 12(2) n36 - removed 3 from r3c8
29b. Combinations {39} no longer valid in cage 12(2) n36
29c. Cage sum in innie/outie cells (r4c8)-(r3c9)=5 - removed 4 from r3c9
29d. Combinations {159} no longer valid in cage h15(3) n6

30. Combinations {145} no longer valid in cage 10(3) n36
30a. Removed redundant candidates 1
30b. Cage sum in innie/outie cells (r4c2)-(r5c9)=1 - removed 2 from r4c2

31. Naked triple {235} found at r345c9

32. Candidate 2 locked in n9 for c8 nowhere else in c8
32a. Cage sum in innie/outie cells (r5c1)-(r6c8)=3 - removed 5 from r5c1


Now the next nice move from SS. (NOTE: This one is more obvious than first appears since r7c9 = (67))
Image
33. "45" n89: 1 outie r6c5 + 8 = 2 innies r7c79
33a. -> no 7 in r7c7
NOTE: 7 in r7c7 -> r7c9 = 6 -> 2 innies = 13: but this is blocked by no 5 in r6c5
33b. no 7 in r7c7
33c. 20(3)n5 = {389/479/578} = [3/7..]

34. h12(3)r6c128 = {156/246}(no 3 or 7) ({237} blocked by [3/7..] in 20(3)n5 step 33c)
34a. no 5 in r7c2
34b. no 6 in r7c1 (h11(2))
34c. 6 locked for r6

36. 7 in r6 only in h27(4) = {3789}(no 4): all locked for r6

Now the key elimination of 4 from r6c7 is made, which makes things more straightforward from here.


Assassin 113-Lite by Para (July 2008) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:4608:4353:4353:4353:4356:4356:5382:3335:3335:4608:4608:2827:2827:2827:4356:5382:5382:3335:2578:4608:5396:4885:4885:4885:5382:1305:3354:2578:4892:5396:5396:4885:2848:2848:1305:3354:2340:4892:4892:4135:4135:4135:4906:4906:3354:2340:2862:2351:2351:3633:4658:4658:4906:2613:2340:2862:4920:3633:3633:3633:4658:5693:2613:4671:4920:4920:4162:4419:4419:4419:5693:5693:4671:4671:4920:4162:4162:3149:3149:3149:5693:
Solution:
+-------+-------+-------+
| 6 5 3 | 9 4 7 | 1 2 8 |
| 4 7 2 | 1 8 6 | 5 9 3 |
| 8 1 9 | 2 3 5 | 6 4 7 |
+-------+-------+-------+
| 2 6 7 | 5 9 8 | 3 1 4 |
| 1 9 4 | 6 7 3 | 8 5 2 |
| 3 8 5 | 4 1 2 | 7 6 9 |
+-------+-------+-------+
| 5 3 6 | 7 2 4 | 9 8 1 |
| 7 4 1 | 8 6 9 | 2 3 5 |
| 9 2 8 | 3 5 1 | 4 7 6 |
+-------+-------+-------+
Quote:
Para: Here's the Lite version of assassin 113.
SS-Score: 1.03. Rating: 1.0.

Mike(mhparker): Since no-one's posted one yet, here's my walkthrough for the Lite version. Despite its name, it still proved to be non-trivial, and proved to be good fun to do. So, thanks very much, Para!

Andrew: A113-Lite was a nice, straightforward puzzle once I'd got over my early mistake.
I'll rate A113-Lite at 1.0.

Walkthrough by Mike:
Since no-one's posted one yet, here's my walkthrough for the Lite version. Despite its name, it still proved to be non-trivial, and proved to be good fun to do. So, thanks very much, Para!

The puzzle fell pretty quickly after the first few placements, so I've decided not to take the WT to the usual singles stage, which also has the advantage that it should only take 10 minutes or so for someone to follow it through.

[collapse=Assassin 113 Lite Walkthrough (first 16 steps)]Prelims

a) 11(3) at R2C3 = {128/137/146/236/245} (no 9)
b) 10(2) at R3C1 and R6C9 = {19/28/37/46} (no 5)
c) 21(3) at R3C3 = {489/579/678} (no 1..3)
d) 5(2) at R3C8 = {14/23} (no 5..9)
e) 19(3) at R4C2 and R5C7 = {289/379/469/478/568} (no 1)
f) 11(2) at R4C6 and R6C2 = {29/38/47/56} (no 1)
g) 9(3) at R5C1 = {126/135/234} (no 7..9)
h) 9(2) at R6C3 = {18/27/36/45} (no 9)
i) 14(4) at R6C5 = {1238/1247/1256/1346/2345} (no 9)

1. Innies N3: R3C89 = 11(2) = [29/38/47] (no 1,5,6; no 2..4 in R3C9)
1a. cleanup: no 4 in R4C8

2. Outies N3: R4C89+R5C9 = 7(3) = {124} (no 3,5..9)
2a. {12} locked for N6; 4 locked in R45C9 for N6 and C9
2b. cleanup: no 2 in R3C8; no 9 in R3C9 (step 1); no 7,9 in R4C6; no 6 in R67C9; no 8,9 in R7C9

3. Outies R12: R3C27 = 7(2) = {16/25} (no 3,4,7..9)
(Note: {34} blocked by R3C8)

4. 13(3) at R3C9 = {247/148} = {(2/8)..}
4a. -> {28} combo blocked for 10(2) at R6C9
4b. -> no 2,8 in R67C9

5. 10(2) at R6C9 = {37}/[91] = {(1/7)..}
5a. -> 13(3) at R3C9 and 10(2) at R6C9 form killer pair on {17} within C9
5b. -> no 1,7 in R1289C9

6. Innies N6: R4C7+R6C79 = 19(3) = {379} (no 5,6,8) (last combo)
(Note: {568} blocked because none of these digits available in R6C9)
6a. {379} locked for N6
6b. cleanup: no 3,5,6 in R4C6

7. Innie/Outie diff. (IOD) R6789: R6C8 = R5C1 + 5
7a. -> R5C1+R6C8 = [16/38] (no 2,4,5; no 6 in R5C1)

8. 5 in N6 locked in R5C78 for R5

9. 16(3) at R5C4 = {169/178/349/367} (no 2) = {(1/3)..}
(Note: {268} blocked by R5C78)
9a. -> 16(3) at R5C4 and R5C1 form killer pair on {13} within R5
9b. -> no 1,3 in R5C239

10. 1 in N6 locked in R4C89 for R4
10a. cleanup: no 9 in R3C1

11. IOD R1234: R4C2 = R5C9 + 4
11a. -> R4C2 = {68} (no 2..5,7,9)

12. 19(3) at R4C2 = {289/469} (no 7) = {(2/4)..}
(Note: {478} blocked by R4C2+R5C9 (step 11))
12a. can have only 1 of {68}, which must go in R4C2
12b. -> no 6,8 in R5C23
12c. 9 locked in R5C23 for R5 and N4
12d. cleanup: no 1 in R3C1, no 2 in R7C2

13. Innies N7: R7C12 = 8(2) = [17/26/35/53] (no 4,8,9; no 6 in R7C1)
13a. cleanup: no 2,3,7 in R6C2

14. 10(2) at R3C1 = {28/37/46} = {(2/3/4)..}
14a. -> {234} blocked for 9(3) at R5C1
14b. furthermore, {126} combo blocked by R5C1+R6C8 (step 7)
14c. -> 9(3) at R5C1 = {135} (no 2,4,6), locked for C1
14d. cleanup: no 7 in R34C1; no 6 in R7C2 (step 13)
14e. -> no 5 in R6C2

15. Outies N7: R5C1+R6C12 = 12(3) = [138/354] (no 6, no 1 in R6C1)
(Note: [156/318] both blocked by R5C1+R6C8 (step 7))
15a. if [354], then R6C8 = 8 (step 7)
15b. -> 8 locked in R6C28 for R6
15c. cleanup: no 1 in R6C34, no 5 in R7C2
15d. -> no 3 in R7C1 (step 13)

16. Hidden single (HS) in N4 at R5C1 = 1
16a. -> R67C1 = [35]
16b. -> R7C2 = 3 (step 13)
16c. -> R6C2 = 8
16d. -> R4C2 = 6, R6C8 = 6
16e. cleanup: no 2,4 in R3C1; no 2 in R5C23 (step 12); no 7 in R67C9

Rest is pretty easy now.
Walkthrough by Andrew:
A113-Lite was a nice, straightforward puzzle once I'd got over my early mistake. Initially when I did step 2 I made the eliminations for it in R7C37 rather than in R7C38. :bigoops: I did in fact make quite a lot of progress after that but when I came to steps like Outies N3689, which I'd used for A113, I thought this isn't a 1.0 puzzle and checked my earlier steps.

I liked Mike's use of I/O differences for R1234 and R6789 which led to a slightly quicker solution but not necessarily the simplest one. I used fairly simple combination analysis although maybe part of step 9 is comparable in difficulty with Mike's steps.

I'll rate A113-Lite at 1.0.

Here is my walkthrough

Prelims

a) R34C1 = {19/28/37/46}, no 5
b) R34C8 = {14/23}
c) R4C67 = {29/38/47/56}, no 1
d) R67C2 = {29/38/47/56}, no 1
e) R6C34 = {18/27/36/45}, no 9
f) R67C9 = {19/28/37/46}, no 5
g) R2C345 = {128/137/146/236/245}, no 9
h) 21(3) cage at R3C3 = {489/579/678}, no 1,2,3
i) 19(3) cage in N4 = {289/379/469/478/568}, no 1
j) R567C1= {126/135/234}, no 7,8,9
k) 19(3) cage in N6 = {289/379/469/478/568}, no 1
l) 14(4) cage at R6C5 = {1238/1247/1256/1346/2345}, no 9

1. 45 rule on R12 2 outies R3C27 = 7 = {16/25/34}, no 7,8,9

2. 45 rule on R89 2 outies R7C38 = 14 = {59/68}

3. 45 rule on N3 2 innies R3C89 = 11 = [29/38/47], clean-up: no 4 in R4C8
3a. Min R3C9 = 7 -> max R45C9 = 6, no 6,7,8,9

4. 45 rule on N7 2 innies R7C12 = 8 = [17]/{26/35}, no 4,8,9, clean-up: no 2,3,7 in R6C2

5. 45 rule on N3 3 outies R4C8 + R45C9 = 7 = {124}, locked for N6, 4 locked in R45C9 for C9, clean-up: no 2 in R3C8, no 9 in R3C9 (step 3), no 7,9 in R4C6, no 6 in R6C9, no 6,8,9 in R7C9

6. 45 rule on R1234 1 innie R4C2 = 1 outie R5C9 + 4, R4C2 = {568}

7. 45 rule on R6789 1 innie R6C8 = 1 outie R5C1 + 5, no 5,6 in R5C1, no 3,5 in R6C8

8. 5,6 in C9 locked in R1289C9
8a. 45 rule on C9 4 innies R1289C9 = 22 = {2569/3568}, no 1,7

9. 13(3) cage in N3 = {238/256} (cannot be {139} which clashes with R1289C9, cannot be {148/157/247} because 1,4,7 only in R1C8, cannot be {346} which clashes with R3C8), no 1,4,7,9, 2 locked for N3, clean-up: no 5 in R3C2 (step 1)
9a. 2 of {238} must be in R1C8 (R12C9 cannot be {23/28} which clash with R1289C9) -> no 3,8 in R1C8

10. 19(3) cage in N4 = {289/469/478/568} (cannot be {379} because R4C2 only contains 5,6,8), no 3

11. 45 rule on C89 4 innies R2569C8 = 27 = {3789/4689/5679}, no 1,2, 9 locked for C8, clean-up: no 5 in R7C3 (step 2)

12. 45 rule on C9 3 outies R178C8 = 13 = {238/256} (cannot be {148} because R1C8 only contains 2,5,6, cannot be {157} because 1,7 only in R8C8, cannot be {247/346} which clash with R34C8), no 1,4,7, 2 locked for C8 -> R4C8 = 1, R3C8 = 4, R3C9 = 7 (step 3), clean-up: no 9 in R3C1, no 3 in R3C27 (step 1), no 3,6 in R4C1, no 5 in R4C2 (step 6), no 3 in R67C9
12a. 3 of {238} must be in R8C8 -> no 8 in R8C8
12b. Naked pair {24} in R45C9, locked for C9, clean-up: no 8 in R6C9
12c. R67C9 = [91], clean-up: no 2 in R4C6, no 4 in R5C1 (step 7), no 2 in R7C2, no 6 in R7C1, no 7 in R7C2 (both step 4), no 4 in R6C2

13. 22(4) cage in N9 = {3568} (only remaining combination), locked for N9
13a. R1C8 = 2 (hidden single in C8)

14. 19(3) cage in N6 = {568} (only remaining combination), locked for N6, 5 locked in R5C78 for R5, clean-up: no 3,5,6 in R4C6, no 2 in R5C1 (step 7)
14a. Naked pair {37} in R46C7, locked for C7
14b. Naked triple {249} in R789C7, locked for C7 and N9 -> R9C8 = 7
14c. R2C8 = 9, R8C8 = 3 (hidden singles in C8)
14d. R2569C8 (step 11) = {5679} (only remaining combination) -> R56C8 = [56], R5C7 = 8, R7C8 = 8, R5C1 = 1 (step 7), R7C3 = 6 (step 2), clean-up: no 9 in R4C1, no 5 in R67C2, no 3 in R6C3, no 3,8 in R6C4, no 2,3 in R7C1 (step 4)
14e. R46C2 = [68], R7C12 = [53], R6C1 = 3 (prelim j), R46C7 = [37], R4C6 = 8, clean-up: no 2 in R3C1, no 1 in R3C7 (step 1), no 7 in R4C1, no 2 in R6C3, no 1,2 in R6C4
14f. Naked pair {45} in R6C34, locked for R6
14g. Naked pair {56} in R89C9, locked for C9
14h. Naked triple {247} in R7C456, locked for R7, N8 and 14(4) cage -> R7C7 = 9, R6C5 = 1, R6C6 = 2
14i. Naked pair {24} in R4C19, locked for R4

15. 21(3) cage at R3C3 = {579} (only remaining combination)
15a. 5 in N4 locked in R46C3, locked for C3 -> R3C3 = 9
15b. Naked pair {57} in R4C34, locked for R4 -> R4C5 = 9
15c. R5C2 = 9 (hidden single in R5), R5C3 = 4 (cage sum), R45C9 = [42], R6C34 = [54], R4C34 = [75], R4C1 = 2, R3C1 = 8

16. 18(3) cage in N7 = {279} (only remaining combination) -> R89C1 = [79], R9C2 = 2, R3C2 = 1, R12C3 = [32], R89C9 = [83], R8C2 = 4, R89C7 = [24], R9C6 = 1 (cage sum), R89C3 = [18], R3C7 = 6 (step 1)

17. R8C7 = 2 -> R8C56 = 15 = [69]

and the rest is naked singles.


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