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PostPosted: Wed May 11, 2011 1:55 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 |
+--------------------------+--------------------------+--------------------------+
| MO#4 1.00 1.05 | A114 Twin 0.75 -.-- | A115V0.9 1.00 0.90 |
| A114 1.00 0.85 | A114V3 -.-- 4.15 | A115V2 1.50 1.75 |
| A114V2 E1.50 1.30 | A115 1.50 1.55 | MO#5 H0.75 1.00 |
+--------------------------+--------------------------+--------------------------+

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

Abbreviations used in Rating Table on this page:
Est = Estimated rating by puzzle maker
E = Easy
H = Hard
Az = azpaull
G = gary w
M = Mike (mhparker)
N = Nasenbaer
P = Para
U = udosuk
Score = SudokuSolver v3.3.0 score, rounded to nearest 0.05
! indicates that the Score has changed at least 0.10 from the SS v3.2.1 score
R indicates lowest score out of 4 rotations of puzzle, for some harder puzzles
+------------------------+-----------+-------+-------+-------+-----------------------+-------+
| Puzzle | Made By | Est | Afmob | Andrew| Other Raters | Score |
+------------------------+-----------+-------+-------+-------+-----------------------+-------+
| Messy One #4 | Para | 1.00 | H1.00 | 1.00 |(G)1.0(Ed)H1.0 | 0.95 |
| Assassin 114 | Nasenbaer | E1.00 | 1.00 | H1.00 | | !1.15 |
| Assassin 114V2 | Nasenbaer | E1.25 | E1.50 | 1.50 |(M)1.50 | 1.30 |
| Assassin 114 Twin | Nasenbaer | H1.25 | | 0.75 |(U)0.50(P)H0.50(N)0.75 | |
| Assassin 114V3 | Nasenbaer | | | | No walkthroughs posted|!R5.30 |
| Assassin 115"Beijing" | udosuk | H1.25 | H1.50 | 1.50 |(Az)H1.25 | !1.85 |
| Assassin 115V0.9 | udosuk | H0.75 | | E1.25 |(M)1.0(U)0.75 | !1.00 |
| Assassin 115V2 | udosuk | E1.75 | 1.50 | 1.50 | | 1.80 |
| Messy One #5 | Nasenbaer | 0.75 | H0.75 | H0.75 |(Az)1.0 | 0.95 |
+------------------------+-----------+-------+-------+-------+-----------------------+-------+


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 1:16 am, edited 4 times in total.

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PostPosted: Tue Jun 14, 2011 8:43 pm 
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Grand Master
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Messy One #4 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:4096:4096:4096:6915:6915:6915:6915:2567:2567:5385:1546:3339:3339:4621:4621:6915:3344:2567:5385:1546:1546:6165:4621:4375:4375:3344:3344:5385:3612:3612:6165:4375:4375:5409:7458:7458:2596:2596:5926:6165:6165:6165:5409:7458:7458:8749:8749:5926:5168:5168:5409:5409:5409:3381:8749:8749:5926:5926:5168:5168:3381:3381:3381:8749:1344:1344:3650:3650:2628:2628:5190:5190:8749:8749:3402:3402:3650:3661:3661:3661:5190:
Solution:
+-------+-------+-------+
| 7 5 4 | 2 3 9 | 8 1 6 |
| 9 1 6 | 7 8 4 | 5 2 3 |
| 8 2 3 | 5 6 1 | 9 7 4 |
+-------+-------+-------+
| 4 6 8 | 3 2 5 | 1 9 7 |
| 3 7 2 | 1 9 6 | 4 8 5 |
| 1 9 5 | 8 4 7 | 3 6 2 |
+-------+-------+-------+
| 2 8 7 | 9 5 3 | 6 4 1 |
| 5 4 1 | 6 7 8 | 2 3 9 |
| 6 3 9 | 4 1 2 | 7 5 8 |
+-------+-------+-------+
Quote:
Para: Puzzle number 3 from me within a week. But will be a while before the next. Starting work again next week.
So enjoy this one! Just a regular killer, not too hard.
SS-score: 1.04. Estimated rating: 1.0. Have fun!

Afmob: That was a fun puzzle which had interesting ... to solve it. Thanks Para!
Rating: (Hard?) 1.0.

gary w: A very enjoyable puzzle...took me about an hour to solve.1.0 seems a pretty reasonable rating.

Andrew: Thanks Para for a fun puzzle. I hope the creators of the first three Messy Ones won't mind me saying that I found this one the most enjoyable so far. Maybe it fitted my solving style better?
I'll rate Messy One #4 as 1.0.

Ed: I missed that nice ... even though I found ... that started it. Nice one guys!

Walkthrough by Afmob:
That was a fun puzzle which had interesting Outies you could use to solve it. Thanks Para!

I'm not sure about the difficulty of my breakthrough move (step 4a), so my wt might be a bit too hard for a normal 1.0 Killer. After going through SudokuSolvers log I noticed that I could have made an easier cracking move by using the Killer pair (36) of 20(3) + 13(4) @ N9.

Messy One #4 Walkthrough:

1. R123
a) 6(3) = {123} locked for N1
b) 16(3) = {457} locked for R1+N1
c) 21(3) must have 4 xor 7 -> R4C1 = (47)
d) Outies N1 = 11(2) = {47} -> CPE: R4C4 <> 4,7
e) 13(2) <> 8
f) 21(3) = 8{49/67} -> 8 locked for C1
g) Innies N3 = 22(3) = 9{58/56} -> 9 locked for C7+N3
h) Outies R1 = 8(2) = [53/62/71]
i) 10(3) = {136} locked for N3, 6 also locked for R1
j) 13(3) = {247} locked for N3
k) Outies R1 = 8(2) = [53] -> R2C7 = 5, R2C9 = 3

2. R123
a) 27(5) = {23589} -> 2,3 locked for N2
b) 18(3) <> 1 because {189} blocked by R1C456 = (2389)
c) Hidden Single: R2C2 = 1 @ R2, R2C8 = 2 @ R2
d) 13(3) = {247} -> 4,7 locked for R3
e) Naked pair (89) locked in R13C7 for C7

3. N6
a) 29(4) = {5789} locked for N6
b) 21(5) must have one of (789) -> R6C6 = (789)
c) Innies+Outies N6: 5 = R6C6 - R6C9 -> R6C6 = (79), R6C9 = (24)

4. R6789 !
a) ! Outies = 7(2+1) <> 5,6,7,8,9; R5C3 <> 1 because R45C7 = {24} blocked by R6C9 = (24)
b) 34(7) must have 1 -> 1 locked for C1
c) 23(4) <> 1 because R5C3 = (234)
d) Hidden Single: R8C3 = 1 @ C3 -> R8C2 = 4, R6C1 = 1 @ N4
e) 13(4) = 1{237/246/345} -> 1 locked for R7+N9
f) 7 locked in R789C7 @ C7 for N9
g) 20(3) = 9{38/56} -> 9 locked for N9
h) 34(7) = {1235689} -> 8 locked for C2

5. N4
a) 10(2) <> 2,9; R5C1 <> 6
b) Killer pair (47) locked in R4C1 + 10(2) for N4

6. R6789
a) 21(5) = 136{29/47} -> 1 locked for C7 and 6 locked for R6
b) Outies R6789 = 7(2+1): R45C7 <> 2 because 1 locked there @ 21(5)
c) 2 locked in R6C79 @ N6 for R6
d) Hidden Single: R5C3 = 2 @ N4
e) Outies R6789 = 7(2+1) = {14}+2 -> 4 locked for C7+N6
f) 21(5) = {13467} because R45C7 = (14) -> R6C6 = 7
g) 10(2) = [37/82]
h) R6C9 = 2
i) Outies N9 = 10(2) = [82] -> R9C6 = 2, R8C6 = 8 -> R8C7 = 2
j) 14(3) @ N9 = {257} -> R9C7 = 7, R9C8 = 5

7. N58
a) Hidden Single: R9C5 = 1 @ R9
b) 14(3) = {167} -> 6,7 locked for R8+N8
c) 13(2) @ N8 = {49} -> R9C3 = 9, R9C4 = 4
d) 20(4) = {3458} -> R6C5 = 4, R6C4 = 8; {35} locked for R7
e) 24(5) = {13569}

8. Rest is singles.

Rating: (Hard?) 1.0.
Solving Outline by gary w:
A very enjoyable puzzle...took me about an hour to solve.1.0 seems a pretty reasonable rating.

easy to show r2c79=53 so r6c9=2/4 so r45c7<>6 so r5c3<>1 ..because r45c7+r5c3=7
so 1 in n4 not in c3
in n2 r1 has an 8 or a 9 so no 1 in 18(3) cage so 1 in r3 n2 so r2c2=1 so n4 1 in c1
in n7 r89c1+r9c2=14 this together with 5(2) cage -> 19(5) set which must contain a 1 which can only be at r8c3 so r8c2=4 so no 4(or 7) in 34(7) cage..this essentially cracks the killer.
Walkthrough by Andrew:
Thanks Para for a fun puzzle. I hope the creators of the first three Messy Ones won't mind me saying that I found this one the most enjoyable so far. Maybe it fitted my solving style better?

I also found Afmob's breakthrough step 4a, which I didn't consider to be a particularly difficult move since Ruud's combination calculator gives the 7(2+1) combinations. However like Afmob I didn't spot SS's killer pair 3,6 in N9; it's not an easy one to find unless one has had some reason to look again at the combinations for the 13(4) cage at R6C9. I could have used it after my step 8.

I'll rate Messy One #4 as 1.0.

Here is my walkthrough (3 typos corrected)

Prelims

a) R2C34 = {49/58/67}, no 1,2,3
b) R4C34 = {59/68}
c) R5C12 = {19/28/37/46}, no 5
d) R8C23 = {14/23}
e) R8C67 = {19/28/37/46}, no 5
f) R9C34 = {49/58/67}, no 1,2,3
g) 10(3) cage at R1C8 = {127/136/145/235}, no 8,9
h) 21(3) cage at R2C1 = {489/579/678}, no 1,2,3
i) 6(3) cage in N1 = {123}, locked for N1
j) 20(3) cage in N9 = {389/479/569/578}, no 1,2
k) 29(4) cage in N6 = {5789}, locked for N6
l) 13(4) cage at R6C9 = {1237/1246/1345}, no 8,9, must contain 1
m) 34(7) cage at R6C1 must contain 1

1. R1C123 = {457} (only remaining combination), locked for R1 and N1, clean-up: no 6,8,9 in R2C4

2. 21(3) cage at R2C1 = {489/678} (cannot be {579} because 5,7 only in R4C1), no 5, 8 locked for C1, clean-up: no 2 in R5C2
2a. 4 of {489} must be in R4C1 -> no 9 in R4C1

3. 45 rule on N1 1 innie R2C3 = 1 outie R4C1 + 2, no 8 in R4C1
3a. 8 in C1 locked in R23C1, locked for N1, clean-up: no 5 in R2C4, no 6 in R4C1 (step 3)
3b. 45 rule on N1 2 outies R2C4 + R4C1 = 11 = {47}, CPE no 4,7 in R4C4

4. 45 rule on N6 1 outie R6C6 = 1 innie R6C9 + 5, R6C6 = {6789}, no 6 in R6C9
4a. 6 in N6 locked in R456C7 + R6C8 for 21(5) cage -> no 6 in R6C6, clean-up: no 1 in R6C9 (step 4)
4b. 1 in 13(4) cage at R6C9 locked in R7C789, locked for R7 and N9, clean-up: no 9 in R8C6

5. 45 rule on R1 2 outies R2C79 = 8 = {17/26/35}, no 4,8,9

6. 45 rule on C123 3 outies R279C4 = 20 = {479/578} (cannot be {389/569} because R2C4 only contains 4,7), no 2,3,6, 7 locked for C4, clean-up: no 7 in R9C3

7. 45 rule on N3 3 innies R123C7 = 22 = {589/679}, 9 locked for C7 and N3, clean-up: no 5,6,7 in R2C9 (step 5), no 1 in R8C6
7a. 10(3) cage in N3 = {136} (only remaining combination), locked for N3, 6 locked in R1C89, locked for R1, clean-up: no 7 in R23C7 (step 7)
7b. R2C7 = 5, R2C9 = 3 (step 5), R13C7 = {89}, locked for C7 and N3, clean-up: no 8 in R6C6 (step 4), no 2 in R8C6
7c. Naked pair {16} in R1C89, locked for R1
7d. 2,3 in R1 locked in R1C456, locked for N2

8. 7 in C7 locked in R789C7, locked for N9
8a. 20(3) cage in N9 = {389/569}, no 4, 9 locked for N9

9. 45 rule on N9 3 outies R6C9 + R89C6 = 12, min R6C9 + R8C6 = 5 -> max R9C6 = 7

10. 18(3) cage in N2 = {459/468/567} (cannot be {189} which clashes with R1C4567), no 1
10a. 5 of {459/567} must be in R3C5 -> no 7,9 in R3C5
10b. {468} must be [486] (R2C56 cannot be {46} which clashes with R2C34, R2C56 cannot be {68} which clashes with R2C13) -> no 4,8 in R3C5

11. Killer pair 4,7 in R2C4 and 18(3) cage, locked for R2 and N2 -> R2C8 = 2, R2C2 = 1, clean-up: no 9 in R5C1, no 4 in R8C3

12. 17(4) cage at R3C7, R3C7 = {89} -> no 7,8,9 in R3C6 + R4C56

13. 45 rule on N69 3 outies R689C6 = 17 = {179/269/278/359/467} (cannot be{368/458} because R6C6 only contains 7,9)
13a. 1,2 of {179/278} must be in R9C6, 7 of {467} must be in R6C6 -> no 7 in R9C6
13b. 3 of {359} must be in R8C6 -> no 3 in R9C6

14. R9C678 = {167/248/257/347/356} (cannot be {158} because 1,5,8 only in R9C68)
14a. 8 of {248} must be in R9C8, 4 of {347} must be in R9C6 -> no 4 in R9C8

15. 45 rule on R6789 3 (2+1) outies R45C7 + R5C3 = 7 = {14}2/{23}2/{13}3/{12}4 (cannot be {24}1 which clashes with R6C9) -> no 6 in R45C7, R5C3 = {234}
15a. 6 in N6 locked in R6C78, locked for R6

16. Max R5C3 = 4 -> min R6C3 + R7C34 = 19, no 1
16a. R8C3 = 1 (hidden single in C3), R8C2 = 4, clean-up: no 6 in R5C1, no 6 in R8C67, no 9 in R9C4

17. R6C1 = 1 (hidden single in 34(7) cage at R6C1), clean-up: no 9 in R5C2
17a. 34(7) cage = {1235689}, no 7
17b. 8 in 34(7) cage locked in R679C2, locked for C2, clean-up: no 6 in R4C3, no 2 in R5C1
17c. 2 in C1 locked in R789C1, locked for N7 and 34(7) cage, no 2 in R6C2
17d. R7C3 = 7 (hidden single in N7)
17e. 1 in N6 locked in R45C7, locked for C7
17f. 7 in R6 locked in R6C56, locked for N5

18. R3C2 = 2 (hidden single in C2), R3C3 = 3

19. 2 in C3 locked in R56C3 -> 23(4) cage at R5C3 = {2579} (only remaining combination) -> R5C3 = 2, R6C3 + R7C4 = {59}, CPE no 5,9 in R6C4
19a. R1C3 = 4 (hidden single in C3)

20. R279C4 (step 6) = {479/578}
20a. 5 of {578} must be in R7C4 -> no 5 in R9C4, clean-up: no 8 in R9C3

21. R4C3 = 8 (hidden single in C3), R4C2 = 6, clean-up: no 4 in R5C1
21a. Naked pair {37} in R5C12, locked for R5 and N4 -> R4C1 = 4
21b. Naked pair {59} in R6C23, locked for R6 -> R6C6 = 7, R6C9 = 2 (step 4), clean-up: no 3 in R8C7

22. R5C3 = 2 -> R45C7 = [14] (step 15)
22a. Naked pair {36} in R67C7, locked for C7
22b. Naked pair {36} in R6C78, locked for R6
22c. Naked pair {48} in R6C45, locked for N5 and 20(4) cage at R6C4
22d. R6C45 = 12 -> R7C56 = 8 = {26/35}, no 9
22e. Killer pair 3,6 in R7C56 and R7C7, locked for R7

23. 13(4) cage at R6C9 = {1246} (only remaining combination), no 3,5 -> R7C7 = 6, R6C78 = [36], R1C89 = [16], R7C89 = [41], R3C89 = [74], clean-up: no 5 in 20(3) cage in N9 (step 8a)
23a. Naked pair {89} in R89C9, locked for C9 and N9 -> R45C9 = [75], R45C8 = [98], R89C8 = [35], R8C6 = 8, R89C9 = [98], R8C7 = 2, R9C7 = 7, R9C6 = 2 (step 14), R9C4 = 4, R9C3 = 9, R2C34 = [67], R6C23 = [95], R6C45 = [84]

24. Naked pair {35} in R7C56, locked for R7 and N8 -> R7C12 = [28], R7C4 = 9

25. 24(5) cage at R3C4 = {13569} -> R34C4 = [53]

and the rest is naked singles
Key Breakthrough Steps by Ed:
Andrew wrote:
I also found Afmob's breakthrough step 4a, which I didn't consider to be a particularly difficult move.
I missed that nice blocking move even though I found the outies that started it. Nice one guys!

However, that didn't seem to make to much difference as it still came out quite easily. I used a hidden pair in n9 so will rate it at Hard 1.0. I missed the killer pair that SS uses because I was concentrating on the (58) in the 29(4)n6 & 20(3)n9 hoping for a record 3 generalized X-wing in the same 2 houses! Going for this record made this a memorable puzzle for me. Thanks Para!

Here is how to get to my key move in n9.

Messy ONe #4 Part WT

Afmob's step 1 to here
Code:
.-------------------------------.-------------------------------.-------------------------------.
| 457       457       457       | 123689    123689    123689    | 89        16        16        |
| 689       12        69        | 47        1246789   1246789   | 5         247       3         |
| 689       123       123       | 123456789 123456789 123456789 | 89        247       247       |
:-------------------------------+-------------------------------+-------------------------------:
| 47        5689      5689      | 1235689   123456789 123456789 | 1234678   5789      5789      |
| 1234679   12346789  123456789 | 123456789 123456789 123456789 | 1234678   5789      5789      |
| 12345679  123456789 123456789 | 123456789 123456789 123456789 | 1234678   123456789 124567    |
:-------------------------------+-------------------------------+-------------------------------:
| 12345679  123456789 123456789 | 123456789 123456789 123456789 | 123467    1234567   124567    |
| 12345679  1234      1234      | 123456789 123456789 12346789  | 1234678   3456789   456789    |
| 12345679  123456789 456789    | 456789    123456789 123456789 | 1234678   123456789 456789    |
'-------------------------------.-------------------------------.-------------------------------'

2. naked pair {89} in r13c7: both locked for c7

3. 29(4)n7 = {5789}: all locked for n6

4. deleted

5. 21(5)n6 must have at least one of 7-9 -> r6c6 = (789)

6. "45" n6: r6c6 - 5 = r6c9
6a. r6c6 = (79), r6c9 = (24)

7. generalized x-wing on 7's in 13(3)n3 & 29(4)n6: locked for c89

8. 20(3)n9 = {389/569}(no 4) = [3/5..]
8a. 9 locked for n9
8b. 3 in {389} must be in r8c8 -> no 8 in r8c8

9. 13(4)n6 = {1237/1246}(no 5) ({1345} = [4]{135} blocked by 20(3)n9 step 8)

10. hidden killer pair (58) in n9: 20(3)n9 has one 1 of 5/8 (step 8) -> r9c8 = (58)
(note: I missed the Killer Pair (36) in n9 in the 13(4)n6 (step 9) and 13(4)n9. Don't think it's as productive as the hidden killer pair though. Lucky!)

This unlocks the puzzle.


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PostPosted: Tue Jun 14, 2011 9:27 pm 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 114 by Nasenbaer (July 2008) here
Puzzle Diagram:
Image

Attention: This is a Killer X (1-9 cannot repeat on the diagonals)
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3:d:k:2304:2304:2304:3075:5380:3077:5894:5894:5894:4361:4361:4361:3075:5380:3077:1551:1551:1551:4882:4882:4882:3075:5380:3077:4120:4120:4120:3355:5404:2845:3102:3102:3102:4129:5410:2083:3355:5404:2845:3623:3623:3623:4129:5410:2083:3355:5404:2845:4912:4912:4912:4129:5410:2083:4150:4150:4150:2873:4410:4411:3900:3900:3900:3135:3135:3135:2873:4410:4411:3397:3397:3397:4424:4424:4424:2873:4410:4411:4430:4430:4430:
Solution:
+-------+-------+-------+
| 1 3 5 | 4 7 2 | 6 9 8 |
| 7 4 6 | 5 8 9 | 3 1 2 |
| 8 2 9 | 3 6 1 | 4 5 7 |
+-------+-------+-------+
| 4 9 8 | 6 1 5 | 2 7 3 |
| 6 7 1 | 9 2 3 | 5 8 4 |
| 3 5 2 | 7 4 8 | 9 6 1 |
+-------+-------+-------+
| 5 8 3 | 1 9 4 | 7 2 6 |
| 2 6 4 | 8 5 7 | 1 3 9 |
| 9 1 7 | 2 3 6 | 8 4 5 |
+-------+-------+-------+
Quote:
Nasenbaer: There was a lot of sweat here last week. Half of it came from the heat, the other half from Assassin 113 V1 and V2. And then there was Messy One 4 which almost made me miss the deadline for this post. :oops: So I thought this week should be a little easier and not so messy. Almost like a regular sudoku. But only almost.
The simple design gives you an easy start and should get you hooked, but be warned, there are still a few surprises. :twisted:
Rating SSolver(3.2.1): 0.84. My personal rating: (easy) 1.0.
V2 is already waiting to be posted, as is an Extra Special Version! :twisted: Have fun!

Afmob: Thanks Nasenbaer for a fun and most of the time relaxing Killer! :cheesey:
This puzzle even allowed me to be creative, so I spotted
Rating: 1.0.

Nasenbaer: Thanks for the thanks. :cheesey:
Very clever! :applause: I didn't see that one! As always you found a much smoother way through the puzzle than me. :applause:

Andrew: Thanks for a fun puzzle. I like the simple design. It was so easy to colour the cages. :)
Afmob's steps ... were really neat!
I'll rate A114 as Hard 1.0.

Walkthrough by Afmob:
Thanks Nasenbaer for a fun and most of the time relaxing Killer! :cheesey:
This puzzle even allowed me to be creative, so I spotted a nice Killer pair (step 2l and 4b) which doesn't seem to be a Killer pair at first sight.

A114 Walkthrough:

1. R123
a) 6(3) = {123} locked for R2+N3
b) 16(3) = {457} locked for R3
c) 19(3) = {289} locked for R3+N1
d) 17(3) = {467} locked for R2+N1
e) R3C5 = 6 -> R12C5 = 15(2) = {78} -> R2C5 = 8, R1C5 = 7
f) 9(3) = {135} locked for R1
g) Innies R1 = 13(3) = {247}

2. C456
a) 17(3) @ R7C5 = {359} locked for C5+N8
b) Innies C4 = 22(3) = 9{58/67} -> 9 locked for C4+N5
c) R2C4 = 5 -> R1C4 = 4 and R3C4 = 3; R1C6 = 2, R2C6 = 9, R3C6 = 1
d) 11(3) = {128} locked for C4+N8
e) Naked triple (467) locked in R789C6 for C6
f) 19(3): R6C6 <> 3,5 because R6C5 <= 4
g) R6C6 = 8 -> R6C45 = 11(2) = [74/92]
h) 14(3) <> 1 because R4C6 = (35)
i) Hidden Single: R4C5 = 1 @ N5
j) 12(3) @ R4C4 = {156} -> R4C4 = 6, R4C6 = 5
k) R5C6 = 3
l) R6C45 = 11(2) -> R5C5+R6C4 cannot be [29/47]

3. R789
a) Innies R8 = 20(3) = [857] -> R8C5 = 5, R8C6 = 7
b) Innies R9 = 11(3) = [236] -> R9C4 = 2, R9C5 = 3, R9C6 = 6
c) R7C4 = 1, R7C5 = 9, R7C6 = 4
d) 12(3) = 2{19/46} -> 2 locked for R8+N7
e) 13(3) = 3{19/46} -> 3 locked for N9

4. C123 + D/ !
a) Killer pair (47) locked in R2C2 + 21(3) for C2
b) ! Killer pair (47) locked in R3C7 + (R5C5+R6C4) for D/ (step 2l)
c) 17(3) @ N7 must have one of (47) -> R9C3 = (47)
d) ! R7C3 <> 8 because it sees all 8 of C9
e) 8 locked in 11(3) @ C3 = {128} locked for C3+N4
f) 21(3) = {579} locked for C2+N4
h) 17(3) @ N7 = {179} -> R9C3 = 7, R9C2 = 1, R9C1 = 9
i) R2C2 = 4, R2C3 = 6, R7C3 = 3

5. C789
a) R9C9 = 5
b) 8(3) = {134} locked for C9+N6
c) R3C9 = 7, R3C7 = 4, R3C8 = 5
d) 21(3) = {678} locked for C8+N6

6. Rest is singles without considering diagonals.

Rating: 1.0. I used Killer pairs and one of them wasn't so obvious.
Walkthrough by Andrew:
Nasenbaer wrote:
The simple design gives you an easy start and should get you hooked, but be warned, there are still a few surprises. :twisted:

This seemingly simple design makes it actually very hard to find a unique solution. After about 30 tries I gave up on the regular Killer version and tried it with Killer-X.

Thanks for a fun puzzle. I like the simple design. It was so easy to colour the cages. :)

I would think it ought to be possible to make a regular Killer with this cage pattern since there are half a dozen immediately available 45s making N2 and N8 into 3 cell cages both horizontally and vertically. However such a Killer might have too low a score to be appropriate as an Assassin.

I nearly finished this on Thursday evening but found that I'd reached an impossible position because I'd been careless with my manual eliminations. Then it was too late to rework it before bedtime.

Afmob's steps 2l and 4b were really neat! He could have done step 4b immediately after step 2l but I guess it didn't come under the heading C456. ;)

Afmob got to only remaining combinations in his step 1 more quickly than I did. I wasn’t searching for them, or checking with Ruud’s combination calculator, when there were plenty of naked triples that gave the same result.

I'll rate A114 as Hard 1.0. There was a lot of work to be done on the diagonals including, in my case, finding a hidden pair on one of them and several CPEs using the diagonals.

Here is my walkthrough for A114. It's probably more like the way Nasenbaer was expecting it to be solved.

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

Prelims

a) R1C123 = {126/135/234}, no 7,8,9
b) R123C5 = {489/579/678}, no 1,2,3
c) R1C789 = {689}, locked for R1 and N3
d) R2C789 = {123}, locked for R2 and N3
e) R3C123 = {289/379/469/478/568}, no 1
f) R456C2 = {489/579/678}, no 1,2,3
g) R456C3 = {128/137/146/236/245}, no 9
h) R456C8 = {489/579/678}, no 1,2,3
i) R456C9 = {125/134}, 1 locked for C9 and N6
j) R6C456 = {289/379/469/478/568}, no 1
k) R789C4 = {128/137/146/236/245}, no 9

1. Naked triple {457} in R3C789, locked for R3

2. Killer pair 2,3 in R2C9 and 8(3) cage at R4C9, locked for C9

3. 9(3) cage at R1C1 = {135/234}, 3 locked for R1 and N1
3a. 19(3) cage at R3C1 = {289} (only remaining combination), locked for R3 and N1 -> R3C5 = 6, clean-up: no 4 R1C123 (step 3)
3b. Naked triple {135} in R1C123, locked for R1 and N1
3c. Naked triple {467} in R2C123, locked for R2

4. R123C5 = {678} -> R12C5 = [78]

5. 45 rule on C5 3 innies R456C5 = 7 = {124}, locked for C5 and N5
5a. Naked triple {359} in R789C5, locked for N8

6. R789C4 = {128/146}, no 7, 1 locked for C4 and N8 -> R3C4 = 3, R3C6 = 1
6a. R3C4 = 3 -> R12C4 = 9 = [45], R12C6 = [29], clean-up: no 6 in R789C4 (step 6)
6b. Naked triple {128} in R789C4, locked for C4 and N8
6c. Naked triple {467} in R789C6, locked for C6

7. 45 rule on R9 3 innies R9C456 = 11 = {137/236/245} (cannot be {128} because 1,2,8 only in R9C4, cannot be {146} because 4,6 only in R9C6), no 8,9

8. 45 rule on R8 3 innies R8C456 = 20 = {578} (only remaining combination , cannot be {389} because 3,9 only in R8C5, cannot be {479/569} because R8C4 only contains 1,2,8) -> R8C456 = [857], R79C5 = [93]
8a. R9C456 (step 7) = {236} (only remaining combination) -> R9C46 = [26], R7C46 = [14]

9. R6C456 = {289/478} (cannot be {379/568} because R6C5 only contains 2,4, cannot be {469} because 6,9 only in R6C4) -> R6C6 = 8, locked for D\, R6C45 = [74/92], no 6

10. R4C456 = {156/237} (cannot be {129/147/246} because R4C6 only contains 3,5, cannot be {345} because 3,5 only in R4C6), no 4,9

11. R5C456 = {239/257/347} (cannot be {149/167} because R5C6 only contains 3,5, cannot be {356} because 3,5 only in R5C6), no 1,6

12. R4C45 = [61] (hidden singles in N5), 6 locked for D\, R4C6 = 5 (step 10), locked for D/, R5C6 = 3

13. R456C9 = {125/134}
13a. 2 of {125} must be in R4C9 -> no 2 in R56C9

14. R8C123 = {129/246}, no 3, 2 locked for R8 and N7
14a. R8C789 = {139/346}, 3 locked for N9

15. Killer pair 4,7 in R2C2 and R456C2, locked for C2

16. R1C1 + R8C8 = {13} (hidden pair on D\), CPE no 1 in R8C1

17. 5 on D\ locked in R7C7 + R9C9, locked for N9
17a. 15(3) cage at R7C7 = {258/267}
17b. 8 of {258} must be in R7C9 -> no 8 in R7C8

18. 8 in C9 locked in R17C9, CPE no 8 in R7C3
18a. 9 on D\ locked in R3C3 + R9C9, CPE no 9 in R9C3

19. R8C789 (step 14a) = {139/346}
19a. 9 of {139} must be in R8C9 -> no 9 in R8C7

20. R456C1 = {148/157/238/256/346} (cannot be {139} which clashes with R1C1, cannot be {247} which clashes with R456C2), no 9
20a. 9 in N4 locked in R456C2, locked for C2
20b. R456C2 = {489/579}, no 6

21. R9C123 = {179/458}
21a. 1 of {179} must be in R9C2 -> no 1 in R9C13
21b. 9 of {179} must be in R9C1 -> no 7 in R9C1

22. 1 on D/ locked in R2C8 + R8C2, CPE no 1 in R8C8 -> R8C8 = 3, locked for D\, R1C1 = 1
22a. R7C3 = 3 (hidden single on D/), R1C23 = [35]
22b. R7C3 = 3 -> R7C12 = 13 = [58/76/85], no 6 in R7C1

23. 3 in C1 locked in R456C1 (step 20) = {238/346}, no 5,7
23a. R7C1 = 5 (hidden single in C1), R7C2 = 8 (step 22b), R3C123 = [829], 9 locked for D\, R9C2 = 1, R8C2 = 6, locked for D/, R1C9 = 8, R2C8 = 1 (hidden singles on D/), R9C9 = 5 (hidden single on D\), clean-up: no 2 in R456C1 (step 23), no 4 in R456C2 (step 20b), no 2 in R4C9 (prelim i), no 9 in R8C1 (step 14), no 4 in R8C79 (step 14a), no 4 in R9C13 (step 21)
23b. R8C79 = [19], R9C13 = [97], 9 locked for D/, R6C4 = 7, locked for D/, R3C7 = 4, locked for D/, R5C5 = 2, locked for D\, R7C7 = 7, locked for D\, R9C78 = [84]
[I’ll leave the remaining naked singles for now.]

24. Naked triple {346} in R456C1, locked for C1 and N4

25. Naked triple {134} in R456C9, locked for C9 and N6

26. R456C7 = {259} (only remaining combination), locked for C7 and N6

and the rest is naked singles.


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PostPosted: Tue Jun 14, 2011 10:08 pm 
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Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 114 V2 by Nasenbaer (August 2008) here
Puzzle Diagram:
Image

Attention: This is a Killer X (1-9 cannot repeat on the diagonals)
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3:d:k:2304:2304:2304:2307:5636:3589:4870:4870:4870:4105:4105:4105:2307:5636:3589:3855:3855:3855:5138:5138:5138:2307:5636:3589:2840:2840:2840:4123:3100:4381:3102:3102:3102:3105:5154:3363:4123:3100:4381:5415:5415:5415:3105:5154:3363:4123:3100:4381:3120:3120:3120:3105:5154:3363:6198:6198:6198:5945:2618:3131:2108:2108:2108:2367:2367:2367:5945:2618:3131:5445:5445:5445:3144:3144:3144:5945:2618:3131:4174:4174:4174:
Solution:
+-------+-------+-------+
| 1 5 3 | 2 7 8 | 4 6 9 |
| 6 2 8 | 4 9 1 | 7 5 3 |
| 9 7 4 | 3 6 5 | 2 1 8 |
+-------+-------+-------+
| 7 1 9 | 5 3 4 | 6 8 2 |
| 5 3 2 | 7 8 6 | 1 9 4 |
| 4 8 6 | 1 2 9 | 5 3 7 |
+-------+-------+-------+
| 8 9 7 | 6 5 2 | 3 4 1 |
| 2 6 1 | 8 4 3 | 9 7 5 |
| 3 4 5 | 9 1 7 | 8 2 6 |
+-------+-------+-------+
Quote:
Nasenbaer: Time for V2 (actually more like V1.5). Usually that would be a slight variation of the design. But not this time. I like the simplicity of the design. So today we have a variation of the solution.
Rating SSolver(3.2.1): 1.29. My personal rating: (easy) 1.25.
I hope Afmob doesn't find another of those cracking moves to shorten the puzzle. ;) Have fun!

Afmob: I didn't find a shortcut this time and had to use ... After going through SudokuSolver's log I saw that this Killer allowed some really cool moves ... which I didn't find, so my moves are probably too difficult for this puzzle.
Rating: (Hard) 1.25 - (Easy) 1.5.

udosuk: Here is my walkthrough for v2: The first 5 steps are probably the standard opening moves until it gets interesting.
My steps 7,8,9 are probably not the most elegant tricks but at least they keep the solving path relatively short. :geek:

Mike(mhparker): Many thanks for this great little puzzle, Nasenbaer! You certainly seem to be becoming master of the Killer-X! :salute:
I'll certainly go along with the 1.5 rating.

Andrew: The walkthroughs by Afmob and udosuk, plus the partial one by Mike, were all interesting. I particularly liked ...
I'll rate A114V2 at 1.5 the way I solved it.

Walkthrough by Afmob:
I didn't find a shortcut this time and had to use some short chains to crack it. After going through SudokuSolver's log I saw that this Killer allowed some really cool moves along diagonals which I didn't find, so my moves are probably too difficult for this puzzle.

A114 V2 Walkthrough:

1. R789
a) 24(3) = {789} locked for R7+N7
b) 23(3) = {689} -> R7C4 = 6, {89} locked for C4+N8
c) 8(3) = 1{25/34} -> 1 locked for R7+N9
d) Innies R8 = 15(3) <> 7 because R8C4 = (89)
e) 7 locked in Innies R9 @ N8 for R9 = 17(3) = 7{19/28} because R9C4 = (89)
f) 7 locked in 21(3) @ R8 = 7{59/68}
g) 16(3) <> 5 because 5{29/38} blocked by Killer pair (35) of 8(3) and (25) of 12(3) @ N7
h) 5 locked in 12(3) @ R9 for N7 = 5{16/34}
i) 2 locked in R8C123 @ N7 for R8

2. C456
a) 9(3) = 3{15/24} -> 3 locked for C4+N2
b) 7 locked in R456C4 @ C4 for N5
c) 22(3) = 9{58/67} -> 9 locked for C5+N2
d) 9 locked in Innies C6 @ N5 = 19(3) = 9{28/46}
e) 3 locked in Innies C5 @ N5 for C5 = 13(3) = 3{28/46}
f) 1 locked in Innies C4 @ N5 for C4 = 13(3) = {157} locked for C4
g) 9(3) = {234} locked for N2
h) 14(3) = 1{58/67} -> 1 locked for C6
i) 12(3) @ N8 = {237} because R9C6 = (27) -> R8C6 = 3, R7C6 = 2, R9C6 = 7
j) Innies R9 = 17(3) = {179} -> R9C4 = 9, R9C5 = 1
k) R8C4 = 8

3. C456
a) Innie R7 = R7C5 = 5
b) R8C5 = 4
c) 22(3) = {679} locked for C5+N2
d) R5C5 = 8
e) 21(3) = {678} -> R5C4 = 7, R5C6 = 6
f) Innies R2 = 14(3) <> 8 because R2C5 >= 6
g) Innies R3 = 14(3) <> 8 because R3C5 >= 6
h) Hidden Single: R1C6 = 8 @ N2

4. R123
a) 19(3) = 9{37/46} -> 9 locked for R1+N2
b) 5 locked in 9(3) @ R1C1 @ R1 = {135} locked for R1+N1
c) 20(3) = {479} loocked for R3+N1
d) R3C5 = 6, R1C5 = 7, R2C5 = 9
e) 19(3) = {469} locked for R1+N3
f) R1C4 = 2, R3C4 = 3, R2C4 = 4
g) 11(3) = {128} locked for R3+N3
h) R3C6 = 5, R2C6 = 1

5. C123 !
a) 12(3) @ N7 = {345} locked for R9
b) 9(3) @ N7 = {126} locked for R8
c) 12(3) @ N4 <> 6,9 because {129/156/246} blocked by R28C2 = (126)
d) 12(3) @ N4 <> 5 since {345} blocked by R9C2 = (345)
e) 17(3) <> 1,7 because 7{19/46} blocked by R37C3 = (479) and {278}
blocked by Killer pair (78) of 12(3) @ N4
f) Hidden Killer pair (56) in 16(3) @ N4 for N4 since 17(3) can only have one of (56)
-> 16(3) <> {178/349}
g) 16(3) @ N4 <> {268} since R2C1 = (268)
h) 16(3) @ N4 <> {367} since (37) is a Killer pair of 12(3) @ N4
i) Hidden Killer triple (789) in R3C2 for C2 since 12(3) can only have one of (789)
j) ! R9C1 <> 4 because it leaves no 4 in N1:
- R9C1 = 4 -> R3C1 <> 4
- R9C1 = 4 -> R6C6 = 4 (HS @ N5) -> R3C3 <> 4

6. C123
a) 16(3) @ N4 <> 3,8 because {358} blocked by R9C1 = (35)
b) Killer triple (126) locked in R28C2 + 12(3) for C2
c) Killer triple (135) locked in R19C1 + 16(3) @ N4 for C1

7. C789 !
a) ! R3C7 <> 1 because it leaves no 1 in N1:
- R3C7 = 1 -> R4C4 = 1 (HS @ N5) -> R1C1 <> 1
- R3C7 = 1 -> R8C3 = 1 (HS @ N7) -> R1C3 <> 1
b) R3C7 = 2
c) Hidden Single: R9C8 = 2 @ C8
d) R9C9 = 6

8. C123
a) Hidden Single: R8C2 = 6 @ C2
b) R8C1 = 2, R8C3 = 1, R2C2 = 2
c) 2 locked in 17(3) @ C3 = {269} locked for C3+N4
d) Hidden Single: R1C1 = 1 @ N1

9. N4569
a) R4C4 = 5 -> R4C56 = 7(2) = [34] -> R4C5 = 3, R4C6 = 4
b) R6C6 = 9, R8C8 = 7
c) 20(3) = 9{38/56} -> 9 locked for C8+N6
d) 16(3) = {457} -> R4C1 = 7, {45} locked for C1+N4
e) R9C1 = 3, R2C8 = 5
f) 20(3) = {389} locked for N6+C8

10. Rest is singles without considering diagonals.

Rating: (Hard) 1.25 - (Easy) 1.5. I used some chains and Killer triples. Definitely easier than A113.
Walkthrough by udosuk:
Here is my walkthrough for v2:

0. Prelims

24/3 @ r7c1={789} (NP @ r7,n7)
=> 23/3 @ r7c4: r7c4=6, r89c4={89} (NP @ c4,n8)
=> 9/3 @ r1c4={135|234} (3 @ c4,n2 locked)
22/3 @ r1c5={589|679} (9 @ c5,n2 locked)
21/3 @ r5c4,r8c7 from {456789}
20/3 @ r3c1,r4c8 from {3456789}
19/3 @ r1c7 from {23456789}


1. n5

a. 7 @ c4,n5 locked @ r456c4
=> Innies @ c4: r456c4=13={157|247} must have 1|4

b. Innies @ c5: r456c5=13 from {1234568} can't be {148} (1a)
=> r456c5 can't have 1, must be {238|256|346}


2. n8

a. 1 @ c5,n8 locked @ 10/3 @ r7c5={127|145}

b. Innies @ r8: r8c456=15 with r8c4 from {89}
=> r8c56 can't have 7

c. 7 @ r9,n8 locked @ r9c56
=> Innies @ r9: r9c456=17=[827|872|917]

d. 12/3 @ r7c6 from {23457} with r9c6 from {27}
=> 12/3 @ r7c6 can't be {345}, must be {237}
=> r9c6=7, r78c6={23} (NP @ c6,n8)

e. r9c5=1, r78c5={45} (NP @ c5)
=> r89c4=[89] (2b)


3. n789

a. 21/3 @ r8c7 from {45679} must be {579} (NT @ r8,n9)
=> r78c5=[54]

b. 8/3 @ r7c7 from {1234} must be {134} (NT @ r7,n9)
=> r78c6=[23]

c. 16/3 @ r9c7={268} (NT @ r9)


4. n25

a. 22/3 @ r1c5 from {6789} must be {679} (NT @ c5,n2)
=> r5c5=8 (NS @ d\/), r46c5={23} (NP @ n5)

b. r456c4={157} (NT @ c4,n5) (1a)

c. 21/3 @ r5c4: r5c46=21-8=13=[76]

d. r46c6={49} (NP @ c6)


5. n123

a. Innies @ r2,r3: r2c456,r3c456=14 with r23c5 from {679}
=> r23c6 can't have 8

b. HS @ n2: r1c6=8
=> Innies @ r1: r1c45=9=[27|36]

c. 1 @ r1,n1 locked @ 9/3 @ r1c1={126|135}
=> 9/3 @ r1c1 & r1c4 form KNP {23} @ r1

d. 19/3 @ r1c7 from {45679} must be {469} (NT @ r1,n3)
=> r1c45=[27] (5b)
=> 9/3 @ r1c1={135} (NT @ n1)

e. 20/3 @ r3c1 from {46789} must be {479} (NT @ r3,n1)
=> r23c4=[43], r23c5=[96]

f. 11/3 @ r3c7 from {1258} must be {128} (NT @ r3,n3)
=> r23c6=[15]


6. c2 (!)

a. r9c2 from {345}, r28c2 from {126} must have 1|2,1|6,2|6
=> 12/3 @ r4c2 can't be {129|156|246|345}
=> 12/3 @ r4c2={138|147|237} must have 1|2,3|4,7|8

b. r28c2 & 12/3 @ r4c2 form KNT {126} @ c2
=> r1c2 from {35}

c. r19c2 & 12/3 @ r4c2 form KNT {345} @ c2
=> r3c2 from {79}


7. n15,d\/ (!!)

Turbot fish:
4 @ r3,n1 locked @ r3c13
4 @ c6,n5 locked @ r46c6
r3c3+r6c6 can't both be 4 (d\)
=> at least one of r3c1+r4c6 must be 4
=> r49c1, seeing r3c1+r4c6, can't be 4
=> r9c1 from {35}


8. d\/ (!!!)

d\: r1c1 from {135}, r4c4 from {15}
d/: r9c1 from {35}, r6c4 from {15}
=> r1c1+r4c4 can't be [31] (can't have two 5s @ d/)
=> 5 @ d\ locked @ r1c1+r4c4
=> r8c8 from {79}


9. c38,d\/ (!!)

Generalised x-wing:
7 @ d\ locked @ r3c3+r8c8
7 @ d/ locked @ r2c8+r7c3
=> 7 @ c3 locked @ r37c3, 7 @ c8 locked @ r28c8
=> 20/3 @ r4c8 from {345689} must be {389|569}
(9 @ c8,n6 locked)
=> r8c8=7


10. Mop up

r2c8+r9c1={35} (NP @ d/)
=> r46c4=[51]
=> d/: r3c7+r8c2=[26]
=> d\: r2c2+r9c9=[26]
=> 12/3 @ r6c4: r6c56=12-1=11=[29]
=> d\: r3c3=4
=> HS @ n7: r9c2=4
=> 12/3 @ r4c2 from {1378} must be {138} (NT @ c2,n4)
=> 17/3 @ r4c3 from {2569} must be {269} (NT @ c3,n4)
=> 16/3 @ r4c1={457} (NT @ c1,n4)
=> d/: r2c8+r9c1=[53]
=> 20/3 @ r4c8 from {3689} must be {389} (NT @ c8,n6)
=> d\: r1c1+r7c7=[13]
=> d/: r1c9+r4c6=[94]

All vanilla naked singles from here.

The first 5 steps are probably the standard opening moves until it gets interesting.

My steps 7,8,9 are probably not the most elegant tricks but at least they keep the solving path relatively short. :geek:
Partial Walkthrough by Mike:
Nasenbaer wrote:
Time for V2 (actually more like V1.5)...

Many thanks for this great little puzzle, Nasenbaer! You certainly seem to be becoming master of the Killer-X! :salute:

I'll certainly go along with the 1.5 rating. Indeed, I'm surprised that Afmob didn't quite want to put it in this category, after using a Turbot fish (step 5j), and a generalized Swordfish (step 7a). For me, anything requiring the use of chains is a 1.5 rating at least...

Unfortunately, I haven't got time for a full WT, but (unsurprisingly) managed to get to the following grid state pretty easily:

Code:
.-----------------------------------.-----------.-----------.-----------.-----------------------------------.
| 135         135         135       | 2         | 7         | 8         | 469         469         469       |
:-----------------------------------:           |           |           :-----------------------------------:
| 268         26          268       | 4         | 9         | 1         | 357         357         357       |
:-----------------------------------:           |           |           :-----------------------------------:
| 479         479         479       | 3         | 6         | 5         | 12          128         128       |
:-----------.-----------.-----------+-----------'-----------'-----------+-----------.-----------.-----------:
| 123456789 | 123456789 | 123456789 | 15          23          49        | 123456789 | 3456789   | 123456789 |
|           |           |           :-----------------------------------:           |           |           |
| 123459    | 123459    | 123459    | 7           8           6         | 123459    | 3459      | 123459    |
|           |           |           :-----------------------------------:           |           |           |
| 123456789 | 123456789 | 123456789 | 15          23          49        | 123456789 | 3456789   | 123456789 |
:-----------'-----------'-----------+-----------.-----------.-----------+-----------'-----------'-----------:
| 789         789         79        | 6         | 5         | 2         | 134         134         134       |
:-----------------------------------:           |           |           :-----------------------------------:
| 126         126         126       | 8         | 4         | 3         | 579         579         579       |
:-----------------------------------:           |           |           :-----------------------------------:
| 345         345         345       | 9         | 1         | 7         | 268         268         26        |
'-----------------------------------'-----------'-----------'-----------'-----------------------------------'

From here, I took the following route:

Next 5 steps

35. Skewed X-Wing: 7 in D/ and D\ locked in R37C3+R28C8
35a. -> no 7 elsewhere in C38

36. 20(3) at R4C8 (prelim d) = {389/569} (no 4)
36a. 9 locked for C8 and N6

37. Skewed X-Wing(2): 6 in D/ and D\ locked in R28C2+R19C9
37a. -> no 6 elsewhere in C29

38. Uniqueness: 13(3) at R4C9 cannot contain both of {57}, because it would force R28C78 to deadly rectangle on {57}
38a. -> {157} combo blocked
38b. -> 13(3) at R4C9 = {148/238/247} (no 5)

39. From step 38b, 13(3) either contains a 2, or...
39a. ...both of {18} -> R3C9 = 2
39b. -> 2 locked in R3456C9 for C9

This knocks the 2 off R9C9, which makes a big difference to the puzzle.

P.S. Haven't tried the twin killer version yet. Maybe a 0.5-rating for udosuk and a 1.0+ for everyone else? :)
Walkthrough by Andrew:
I effectively finished this before I started on A115 but only tidied up my walkthrough this evening with the re-work of steps 20 and 24.

The walkthroughs by Afmob and udosuk, plus the partial one by Mike, were all interesting. I particularly liked the grouped X-Wings that used the two diagonals. I've never looked for a grouped X-Wing on diagonals; something to remember for future Killer-Xs.

I also used a couple of short chains, the first one being my key breakthrough, so I'll rate A114V2 at 1.5 the way I solved it.

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) R1C123 = {126/135/234}, no 7,8,9
b) R123C4 = {126/135/234}, no 7,8,9
c) R123C5 = {589/679}, 9 locked for C5 and N2
d) R1C789 = {289/379/469/478/568}, no 1
e) R3C123 = {389/479/569/578}, no 1,2
f) R3C789 = {128/137/146/236/245}, no 9
g) R456C8 = {389/479/569/578}, no 1,2
h) R5C456 = {489/579/678}, no 1,2,3
i) R7C123 = {789}, locked for R7 and N7
j) R789C4 = {689} -> R7C4 = 6, R89C4 = {89}, locked for C4 and N8
k) R7C789 = {125/134}, 1 locked for R7 and N9
m) R8C789 = {489/579/678}, no 1,2,3

1. R123C4 = {135/234}, 3 locked for C4 and N2
1a. 7 in C4 locked in R456C4, locked for N5

2. 9 in C6 locked in R456C9
2a. 45 rule on C6 3 innies R456C6 = 19 = {289/469}, no 1,3,5
2b. 3 in N5 locked in R46C5, locked for C5

3. R456C4 = {157} (only remaining combination, cannot be {247} which clashes with R456C6), locked for C4 and N5
3a. Naked triple {234} in R123C4, locked for N2
3b. 1 in N2 locked in R123C6, locked for C6

4. 45 rule on R8 3 innies R8C456 = 15 = {159/249/258/348} (cannot be {357} because R8C4 only contains 8,9), no 7

5. 7 in N8 locked in R9C56, locked for R9
5a. 45 rule on R9 3 innies R9C456 = 17 = {179/278} (cannot be {467} because R9C4 only contains 8,9), no 3,4,5

6. R8C789 must contain 7 = {579/678}, no 4

7. R789C6 must contain 3 = {237} (only remaining combination, cannot be {345} because R9C6 only contains 2,7) -> R9C6 = 7, R78C6 = {23}, locked for C6 and N8 -> R9C5 = 1, R9C4 = 9 (step 5a), R8C4 = 8, R78C5 = {45}, locked for C5, clean-up: no 8 in R123C5 (prelim c), no 8 in R456C6 (step 2a), no 6 in R8C789 (step 6)
7a. Naked triple {679} in R123C5, locked for C5 and N2 -> R5C5 = 8, locked for both diagonals
7b. R5C5 = 8 -> R5C46 = 13 = [76]
7c. R4C456 = {129/345}
7d. R6C456 = {129/345}

8. R1C123 = {126/135} (cannot be {234} which clashes with R1C4), no 4, 1 locked for R1 and N1

9. Naked triple {579} in R8C789, locked for R8 and N9 -> R8C5 = 4, R7C5 = 5, clean-up: no 2 in R7C789 (prelim k)
9a. Naked triple {134} in R7C789, locked for R7 and N9 -> R7C6 = 2, R8C6 = 3
9b. Naked triple {126} in R8C123, locked for N7

10. 45 rule on R1 3 innies R1C456 = 17 = {278/359/368} (cannot be {269/467} because R1C6 only contains 5,8, cannot be {458} because 5,8 only in R1C6), no 4
10a. 45 rule on R2 3 innies R2C456 = 14 = {149/257/356} (cannot be {158} because 1,5,8 only in R2C6, cannot be {167} because 6,7 only in R2C5, cannot be {239/347} which clash with R1C456, cannot be {248} because 2,4 only in R2C4), no 8
10b. Similarly 45 rule on R3 3 innies R3C456 = 14 = {149/257/356}, no 8
10c. R1C6 = 8 (hidden single in C6), R1C45 (step 10) = [27/36], no 9

11. 4,9 in R1 locked in R1C789 = {469}, locked for R1 and N3 -> R1C5 = 7, R1C4 = 2 (step 10)
11a Naked triple {135} in R1C123, locked for N1

12. R3C123 = {479} (only remaining combination), locked for R3 and N1 -> R3C45 = [36], R3C6 = 5 (step 10b), R2C456 = [491]
12a. Naked triple {268} in R2C123, locked for R2

13. R456C7 = {138/147/156/237/246/345} (cannot be {129} which clashes with R3C7), no 9
13a. R456C9 = {139/148/157/238/247} (cannot be {256} which clashes with R9C9, cannot be {346} which clashes with R456C7), no 6
13b. R456C7 (step 13) = {138/147/156/237/246} (cannot be {345} which clashes with R456C9)
13c. Killer pair 1,2 in R3C7 and R456C7, locked for C7

14. R456C2 = {138/147/237} (cannot be {129/156} which clash with R28C2, cannot be {246} which clashes with R2C2, cannot be {345} which clashes with R9C2), no 5,6,9

15. R456C3 = {269/359/368/458} (cannot be {179} which clashes with R7C3, cannot be {278} which clashes with R456C2, cannot be {467} which clashes with R37C3), no 1,7

16. R456C1 = {169/259/358/457} (cannot be {268} which clashes with R2C1, cannot be {178/349/367} which clash with R456C2)
16a. R456C3 (step 15) = {269/368/458} (cannot be {359} because R456C1 cannot be {268} and because R456C2 cannot be {246})
16b. 3 of {368} must be in R5C3 -> no 3 in R46C3

17. 1 on D\ locked in R1C1 + R4C4, CPE no 1 in R4C1

18. R456C3 = {269/368/458} (cannot be {359} which clashes with R379C3)

19. Killer pair 1,2 in R28C2 and R456C2, locked for C2

20. R4C456 (step 7c) = {129/345}
20a. R6C456 (step 7d) = {129/345}
20b. -> R4C4 + R6C6 = [14/59], R4C6 + R6C4 = [41/95] (other permutations clash with R4C456 and R6C456)
20c. No 4 in R7C7 because it forces a clash with step 20b
R7C7 = 4 => R6C6 = 9 => R3C3 = 7 => R8C8 = 5 => R4C4 = 1 => R4C4 + R6C6 = [19] clashes with step 20b -> R7C7 = 3, locked for D\
20d. Naked pair {15} in R1C1 + R4C4, locked for D\
[I originally did step 20c as
20c. R4C4 + R6C6 = [41] => R7C7 = 3 => R1C1 = 5
20d. R4C4 + R6C6 = [59] => R8C8 = 7 => R3C3 = 4 => R7C7 = 3
20e. -> R7C7 = 3, locked for D\ and no 5 in R8C8
but I reworked it because I know that Mike prefers chains to start with “no …” which in this case also gave a simpler step.]

21. R456C7 (step 13b) = {147/156/246}, no 8
21a. R9C7 = 8 (hidden single in C7)
21b. R456C9 (step 13a) = {139/157/238/247} (cannot be {148} which clashes with R7C9)

22. R456C8 = {389/569/578} (cannot be {479} which clashes with R8C8), no 4
22a. Killer pair 7,9 in R456C8 and R8C8, locked for C8
22b. R7C3 = 7 (hidden single on D/), R8C8 = 7 (hidden single on D\)

23. R456C8 (step 22) = {389/569}, 9 locked for N6

24. R456C9 (step 21b) = {157/238/247}
24a. R1C9 cannot be 4 because of clashes in C9
R1C9 = 4 => R7C9 = 1 => R3C9 = {28} -> no valid combinations for R456C9
[Another step that I reworked with “no …” and found that it was simpler]
24b. 4 on D/ locked in R4C6 + R9C1, CPE no 4 in R4C1

25. 8 in C9 locked in R3C9 and R46C9
25a. If R456C9 = {238} => R3C9 = 1
25b. -> no 2 in R3C9

26. Hidden killer pair 3,7 in R2C9 and R456C9 for C9 -> R2C9 = {37}

27. R456C1 (step 16) = {358/457} (cannot be {169} which clashes with R278C1, cannot be {259} which clashes with R1278C1), no 1,2,6,9, 5 locked for C1 and N4 -> R1C1 = 1, locked for D\ -> R4C4 = 5, R6C6 = 9 (step 20b), locked for D\, R3C3 = 4, R4C6 = 4, R6C4 = 1, both locked for D/, R9C1 = 3, locked for D/, R2C8 = 5, R3C7 = 2 , locked for D/, R8C2 = 6, locked for D/, R2C2 = 2, locked for D\, R2C79 = [73], R8C13 = [21], R9C23 = [45], R9C89 = [26], R4C5 = 3 (step 7c), R6C5 = 2, R1C23 = [53], R1C9 = 9, R8C79 = [95], clean-up: no 8 in R46C1 (step 16), no 7 in R46C2 (step 14), no 8 in R46C3 (step 15), no 4 in R56C7 (step 21), no 6 in R46C8 (step 23), no 1,8 in R45C9 (step 24)

and the rest is naked singles

Nasenbaer wrote:
Time for V2 (actually more like V1.5).
My personal rating: (easy) 1.25
I wonder what that rating was based on. Did the rest of us miss something? The grouped X-Wings using diagonals deserve at least a solid 1.25 while, as Mike has pointed out, any chains rate at least 1.5.

Afmob wrote:
After going through SudokuSolver's log I saw that this Killer allowed some really cool moves along diagonals which I didn't find, so my moves are probably too difficult for this puzzle.
Were they the grouped X-Wings or did SS find some other good moves along the diagonals?
Discussion by Mike about an interesting move found by SudokuSolver:
Andrew wrote:
Afmob wrote:
After going through SudokuSolver's log I saw that this Killer allowed some really cool moves along diagonals which I didn't find, so my moves are probably too difficult for this puzzle.
Were they the grouped X-Wings or did SS find some other good moves along the diagonals?

I suspect Afmob also meant moves like the following generalized Swordfish involving the diagonals:

Image

SudokuSolver wrote:
85. Swordfish on candidate 2 at r3c8=r9c8 - r9c9=r2c2 - r8c2=r3c7
85a. Removed candidate 2 from r3c9
85b. Removed candidate 2 from r3c9 r9c7
85c. Removed candidate 2 from r456c2 r3c9 r9c7
85d. Combinations {129} {237} no longer valid in cage 12(3) n4

Neat, isn't it? I seem to remember that JSudoku also found this move (or a similar one).


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PostPosted: Tue Jun 14, 2011 10:47 pm 
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Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 114 TwinKiller by Nasenbaer (August 2008) here
Puzzle Diagrams:
Image

Attention: Both killers are NOT unique when solved on their own! But together they have a unique solution.
Images with "udosuk style Killer Cages" by Børge:
Image

Image
Code: Select, Copy & Paste into solver:
Left Killer:
3x3::k:3072:3072:3072:4611:4356:2565:4102:4102:4102:4105:4105:4105:4611:4356:2565:4367:4367:4367:4370:4370:4370:4611:4356:2565:3096:3096:3096:2587:5404:3613:2846:2846:2846:5921:1826:3875:2587:5404:3613:3879:3879:3879:5921:1826:3875:2587:5404:3613:4912:4912:4912:5921:1826:3875:4406:4406:4406:3385:3898:4411:4412:4412:4412:3903:3903:3903:3385:3898:4411:2373:2373:2373:3400:3400:3400:3385:3898:4411:4942:4942:4942:
Right Killer:
3x3::k:4608:3841:3074:4355:4355:4355:2310:5383:3848:4608:3841:3074:3084:3084:3084:2310:5383:3848:4608:3841:3074:4117:4117:4117:2310:5383:3848:4379:4379:4379:3614:3359:4640:4385:4385:4385:5156:5156:5156:3614:3359:4640:2602:2602:2602:2093:2093:2093:3614:3359:4640:4659:4659:4659:4406:2359:4920:2873:2873:2873:3388:4413:3902:4406:2359:4920:5442:5442:5442:3388:4413:3902:4406:2359:4920:3403:3403:3403:3388:4413:3902:
Solution:
+-------+-------+-------+
| 1 7 4 | 6 9 2 | 3 5 8 |
| 9 2 5 | 8 1 3 | 4 7 6 |
| 8 6 3 | 4 7 5 | 2 9 1 |
+-------+-------+-------+
| 2 9 6 | 7 3 1 | 8 4 5 |
| 5 8 7 | 2 4 9 | 6 1 3 |
| 3 4 1 | 5 6 8 | 9 2 7 |
+-------+-------+-------+
| 7 1 9 | 3 2 6 | 5 8 4 |
| 4 3 8 | 9 5 7 | 1 6 2 |
| 6 5 2 | 1 8 4 | 7 3 9 |
+-------+-------+-------+
Quote:
Nasenbaer: While trying to create a decent Assassin with the now familiar design I stumbled upon this one. If you add the diagonals constrain to the left Killer you actually have a Killer-X with a unique solution. But SSolver(3.2.1) rating is 4.13, with extended T&E (several Bowman's Bingo). So I wouldn't advice you to try it, unless you are a complete masochist. ;)
Then I started thinking. What if I would rotate the design? I tried it, but now uniqueness was gone. Then I remembered those wonderful TwinKillers on DJApe's site. That was it! And it was solvable, even without the diagonals constrain! So here it is, the first Assassin TwinKiller! [creativity mode on] (Or should I say TwinAssassin? AssassinTwin? TwAssi? Or maybe ...) [creativity mode off] Ups, sorry about that. ;)

So here is something to do for you on this wonderful Sunday. ;) (Here it's cloudy with random rain showers)
SSolver(3.2.1) can't rate it. I tried it by adding all cages from the second killer as hidden cages to the first killer and then hit F7 for rating. But that didn't work. (BTW, that's what my question was about in this post, Richard.)
My personal rating: (hard) 1.25. Have fun!

udosuk: And it's an easy one too, probably a 0.5. I wouldn't make it an assassin. :ugeek:

Para: I agree with udosuk on the twin killer sudoku. I have done these frame killers before too and have also done loads of kakuros(which i'm not too bad at creating either). It's not that hard, prob 0.5 (hard). If you have done enough kakuros it's really a breeze and as kakuro has a lot of similarities to killer sudoku, it isn't really that hard. You just have to learn to think in a little bit of a different way.

Nasenbaer: I did the Twin Killer again, and I have to say it really is much too easy. :oops: Sorry about that. I would rate it now a 0.75 (because you still have to take a lot of steps). Somehow I was fooled by the orginal X-version of the left one which was rated very high.

Andrew: TwinKiller was a fun puzzle. Thanks Nasenbaer!
I've never tried to solve a Kakuro or a twin killer before; I just applied my normal methodical approach to solving it.
I'll rate TwinKiller at 0.75; it might even be considered to be Easy 1.0. The steps weren't difficult but it was long and at times it was hard to spot which 3-cell cage should be analysed next.
udosuk and Para suggest the rating should only be 0.5 even though udosuk then gave a very nice walkthrough using elimination solving. Mike's definition of 0.5 is a killer that can be solved with very few "pencil marks". If it can be solved (almost) as easily that way then I'll accept 0.5 but I feel that the combination analysis is much easier using elimination solving.
Another reason I'll give for rating it 0.75 is that 0.5 is defined as being equivalent to the hardest newspaper puzzles, which I think are equivalent to the hardest killers on http://www.sudoku.org. TwinKiller took me significantly longer to solve than any of those killers, even though it only required combination analysis with a few killer pairs.

Walkthrough by Andrew:
TwinKiller was a fun puzzle. Thanks Nasenbaer!

I've never tried to solve a Kakuro or a twin killer before; I just applied my normal methodical approach to solving it.

I'll rate TwinKiller at 0.75; it might even be considered to be Easy 1.0. The steps weren't difficult but it was long and at times it was hard to spot which 3-cell cage should be analysed next.

udosuk and Para suggest the rating should only be 0.5 even though udosuk then gave a very nice walkthrough using elimination solving. Mike's definition of 0.5 is a killer that can be solved with very few "pencil marks". If it can be solved (almost) as easily that way then I'll accept 0.5 but I feel that the combination analysis is much easier using elimination solving.

Another reason I'll give for rating it 0.75 is that 0.5 is defined as being equivalent to the hardest newspaper puzzles, which I think are equivalent to the hardest killers on http://www.sudoku.org. TwinKiller took me significantly longer to solve than any of those killers, even though it only required combination analysis with a few killer pairs.

Edit. I've added my walkthrough for completeness; don't know why I didn't post it at the time.

Here is my walkthrough for A114 Twin Killer

This is a TwinKiller. There are Vertical and Horizontal cages for each Nonet.

Prelims, for the first diagram and then the second diagram

a) R123C6 = {127/136/145/235}, no 8,9
b) R456C1 = {127/136/145/235}, no 8,9
c) R456C2 = {489/579/678}, no 1,2,3
d) R4C456 = {128/137/146/236/245}, no 9
e) R456C7 = {689}, locked for C7 and N6
f) R456C8 = {124}, locked for C8 and N6
g) R6C456 = {289/379/469/478/568}, no 1
h) R8C789 = {126/135/234}, no 7,8,9
i) R9C789 = {289/379/469/478/568}, no 1
j) R123C7 = {135/234}, 3 locked for C7 and N3
k) R123C8 = {579/678}, 7 locked for C8 and N3
l) R5C123 = {389/479/569/578}, no 1,2
m) R6C123 = {125/134}, 1 locked for R6 and N4
n) R789C2 = {126/135/234}, no 7,8,9
o) R789C3 = {289/379/469/478/568}, no 1
p) R7C456 = {128/137/146/236/245}, no 9
q) R8C456 = {489/579/678}, no 1,2,3

1. R5C789 = {136} (only remaining combination because R5C7 only contains 6,8,9) = [613]
1a. Naked pair {57} in R46C9, locked for C9

2. R456C2 = {489/579} (cannot be {678} because R6C2 only contains 4,5), no 6, 9 locked for C2 and N4
2a. R6C2 = {45} -> no 4,5 in R45C2

3. R5C123 = {479/578}, 7 locked for R5 and N4
3a. 2 in R5 locked in R5C456, locked for N5

4. R4C789 = {278/458} (only remaining combinations) -> R4C7 = 8, R4C2 = 9, R6C7 = 9

5. R6C123 = {125/134}
5a. R6C23 = {45} -> no 4,5 in R6C13
5b. Killer pair 2,4 in R6C123 and R6C8, locked for R6

6. 6,8 in R6 locked in R6C456, locked for N5
6a. R6C456 = {568} (only remaining combination), locked for R6 and N5 -> R6C2 = 4, R6C8 = 2, R4C8 = 4, R6C9 = 7, R4C9 = 5, R5C2 = 8 (step 2)
6b. Naked pair {13} in R6C13, locked for N4

7. R789C2 = {126/135}, 1 locked for C2 and N7
7a. 7 in C2 locked in R123C2, locked for N1

8. R456C1 = {127/235} (only remaining combinations, cannot be {136} because 1,3 only in R6C1) -> R4C1 = 2, R4C3 = 6

9. R8C789 = {126/135/234}
9a. 6 of {126} must be in R8C8 -> no 6 in R8C9
9b. 3 of {135} must be in R8C8 -> no 5 in R8C8

10. R123C3 = {129/345} (cannot be {138} which clashes with R6C3), no 8
10a. R56C3 = [53/17]
10b. Killer pair 1,3 in R123C3 and R6C3, locked for C3
10c. Killer pair 1,5 in R123C3 and R56C3, locked for C3

11. 8 in N1 locked in R123C1, locked for C1
11a. R123C1 = {189/468}, no 3,5

12. R456C5 = {148/238/346} (cannot be {139/157} because 1,3,7 only in R4C5, cannot be {247} because 2,4 only in R5C5, cannot be {256} because 5,6 only in R6C5), no 5,7,9

13. R456C4 = {257} (only remaining combination, cannot be {149/239/248} because 2,4,9 only in R5C4, cannot be {158/356} because 5,6,8 only in R6C4, cannot be {167/347} because 1,3,7 only in R4C4) = [725], R5C56 = [49]

14. R123C6 = {127/145/235} (cannot be {136} which clashes with R4C6), no 6
14a. Killer pair 1,3 in R123C6 and R4C6, locked for C6

15. R789C6 = {278/458/467}
15a. R789C5 = {159/258/267} (cannot be {168} which clashes with R6C5, cannot be {357} which clashes with R789C6), no 3

16. 3 in N8 locked in R789C4, locked for C4
16a. R789C4 = {139/346}, no 8
16b. 8 in C4 locked in R123C4, locked for N2

17. R8C456 = {489/579/678}
17a. 6 of {678} must be in R8C4 -> no 6 in R8C56

18. R789C5 (step 15a) = {159/258/267}
18a. 7 of {267} must be in R8C5 -> no 7 in R79C5

19. R123C5 = {179/359} (cannot be {269} which clashes with R789C5), no 2,6, 9 locked for C5 and N2
19a. Killer pair 1,3 in R123C5 and R4C5, locked for C5

20. R123C4 = {468} (only remaining combination), locked for C4 and N2 -> R8C4 = 9
20a. 2 in N2 locked in R123C6, locked for C6

21. R8C456 (step 17) = {489/579}
21a. 4 of {489} must be in R8C6 -> no 8 in R8C6

22. R7C456 = {128/146/236} (cannot be {137} because 1,3 only in R7C4, cannot be {245} because R7C4 only contains 1,3), no 5,7
22a. 8 of {128} must be in R7C6 -> no 8 in R7C5

23. R9C456 = {148/157/238/346} (cannot be {247/256} because R9C4 only contains 1,3)
23a. 4,7 of {157/346} must be in R9C6 -> no 5,6 in R9C6

24. R9C123 = {148/157/238/247/256/346} (cannot be {139} which clashes with R9C4), no 9

25. 9 in N7 locked in R7C13, locked for R7
25a. R7C123 = {179/359} (cannot be {269} which clashes with R7C5), no 2,4,6,8
25b. Killer pair 1,3 in R7C123 and R7C4, locked for R7
25c. 1 in N9 locked in R8C79, locked for R8
25d. R8C789 = {126/135}, no 4

26. R789C3 = {289/478}
26a. R7C3 = {79} -> no 7 in R89C3

27. R789C1 = {359/467}
27a. 9 of {359} must be in R7C1 -> no 3,5 in R7C1
27b. 7 of {467} must be in R7C1 -> no 7 in R89C1

28. R7C123 (step 25a) = {179} (cannot be {359} because 3,5 only in R7C2) -> R7C2 = 1, R79C4 = [31], R7C13 = {79}, locked for R7
28a. R7C456 (step 22) = {236} -> R7C56 = [26], R6C56 = [68]
28b. R789C6 (step 15) = {467} (only remaining combination), R89C6 = {47}, locked for C6 and N8
28c. R123C6 (step 14) = {235} (only remaining combination), locked for C6 and N2 -> R4C56 = [31]

29. R9C7 = 7 (hidden single in C7), R89C6 = [74]
29a. 9 in R9 locked in R9C89 -> R9C789 = {379} -> R9C89 = [39], R8C8 = 6, clean-up: no 8 in R123C8 (prelim k), no 5 in R8C7 (step 25d)
29b. Naked triple {579} in R123C8, locked for C8 and N3 -> R7C8 = 8, R7C79 = [54], clean-up: no 1 in R123C7 (prelim j)
29c. Naked triple {234} in R123C7, locked for C7 and N3 -> R8C79 = [12]

30. R9C123 (step 24) = {256} (only remaining combination), R9C3 = 2, R9C12 = {56}, locked for R9 and N7 -> R8C123 = [438], R89C5 = [58], clean-up: no 6 in R123C1 (step 11a), no 1,9 in R123C3 (step 10)
30a. Naked triple {345} in R123C3, locked for C3 and N1 -> R56C3 = [71], R56C1 = [53], R7C13 = [79], R9C12 = [65]

31. R1C123 = {147/156} (cannot be {129/138} because 1,8,9 only in R1C1, cannot be {237/246/345} because R1C1 only contains 1,8,9) -> R1C1 = 1, R1C23 = [65/74]

32. R1C456 = {269/278} (cannot be {368/458} because R1C5 only contains 7,9, cannot be {359} because 3,5 only in R1C6, cannot be {467} because 4,6 only in R1C4) -> R1C6 = 2, no 4 in R1C4
32a. Naked pair {68} in R1C49, locked for R1 -> R1C2 = 7, R1C5 = 9, R1C8 = 5, R1C3 = 4, R1C7 = 3, R1C4 = 6 (step 32), R1C9 = 8

33. R3C789 = {129/147} (cannot be {246} because 2,4 only in R3C7) -> R3C9 = 1

and the rest is naked singles and a cage total.

The thread also included discussion about the similarity between Twin Killers and Kakuros starting here.

Nasenbaer also posted his Cookbook for creating Killers here.


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PostPosted: Tue Jun 14, 2011 11:18 pm 
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Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 114 V3 by Nasenbaer (August 2008) here

Thanks to Børge for providing the diagrams and code string for this variant.

Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3:d:k:3072:3072:3072:4611:4356:2565:4102:4102:4102:4105:4105:4105:4611:4356:2565:4367:4367:4367:4370:4370:4370:4611:4356:2565:3096:3096:3096:2587:5404:3613:2846:2846:2846:5921:1826:3875:2587:5404:3613:3879:3879:3879:5921:1826:3875:2587:5404:3613:4912:4912:4912:5921:1826:3875:4406:4406:4406:3385:3898:4411:4412:4412:4412:3903:3903:3903:3385:3898:4411:2373:2373:2373:3400:3400:3400:3385:3898:4411:4942:4942:4942:
Solution:
+-------+-------+-------+
| 1 7 4 | 6 9 2 | 3 5 8 |
| 9 2 5 | 8 1 3 | 4 7 6 |
| 8 6 3 | 4 7 5 | 2 9 1 |
+-------+-------+-------+
| 2 9 6 | 7 3 1 | 8 4 5 |
| 5 8 7 | 2 4 9 | 6 1 3 |
| 3 4 1 | 5 6 8 | 9 2 7 |
+-------+-------+-------+
| 7 1 9 | 3 2 6 | 5 8 4 |
| 4 3 8 | 9 5 7 | 1 6 2 |
| 6 5 2 | 1 8 4 | 7 3 9 |
+-------+-------+-------+
Quote:
Nasenbaer: While trying to create a decent Assassin with the now familiar design I stumbled upon this one. If you add the diagonals constrain to the left Killer you actually have a Killer-X with a unique solution. But SSolver(3.2.1) rating is 4.13, with extended T&E (several Bowman's Bingo). So I wouldn't advice you to try it, unless you are a complete masochist. ;)

Andrew (not posted): I once tried starting this variant and gave up after a few steps. I'm not that much of a masochist. ;)
SSv3.3.1 gives the score as 5.31. In Solver mode it took 301 steps to solve it!


(Archive Note) Also posted in the Unsolvables thread.


Last edited by Andrew on Fri May 04, 2012 11:58 pm, edited 1 time in total.

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PostPosted: Wed Jun 15, 2011 2:17 am 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 115 "Beijing" by udosuk (August 2008) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:5633:5633:5633:5633:5890:3587:3587:3587:3587:4100:1797:1797:5890:5890:5890:3078:3078:2823:4100:1797:2824:2824:10761:4106:4106:3078:2823:1803:1803:7948:2824:10761:4106:5133:3342:3342:7948:7948:7948:10761:10761:10761:5133:5133:5133:1807:1807:7948:4112:10761:4369:5133:3090:3090:2323:5652:4112:4112:10761:4369:4369:2325:2582:2323:5652:5652:3607:3607:3607:2325:2325:2582:4632:4632:4632:4632:3607:6681:6681:6681:6681:
Solution:
+-------+-------+-------+
| 5 8 3 | 6 9 1 | 2 7 4 |
| 9 4 1 | 7 2 5 | 3 8 6 |
| 7 2 6 | 4 8 3 | 9 1 5 |
+-------+-------+-------+
| 2 5 9 | 1 3 4 | 8 6 7 |
| 3 7 8 | 5 6 9 | 1 4 2 |
| 6 1 4 | 8 7 2 | 5 9 3 |
+-------+-------+-------+
| 1 6 5 | 3 4 8 | 7 2 9 |
| 8 9 7 | 2 5 6 | 4 3 1 |
| 4 3 2 | 9 1 7 | 6 5 8 |
+-------+-------+-------+
Quote:
udosuk: Here it is. Hope it won't distract you too much from the big event. :alien:
SSolver 3.2.1 Rating: 1.56. Personal Rating: 1.25 (Hard).
According to SS it is as hard as Para's Assassin 113. But there is a key move in the puzzle which if you can spot will make a huge breakthrough in the solving. And then I've found 2 different approaches to the "critical step", which I'll be glad if you guys can find a 3rd. :geek:
There is also a V2 available, but I suspect it won't be as hard as Para's Assassin 113 v2 which he is keeping to himself. (There is even a V3 but even I ain't gutsy enough to throw it out.) :ugeek:
Aussie Aussie Aussie! Oi Oi Oi! :mrgreen:

Afmob: Thanks for this challenging Killer!
It felt as difficult as Para's Assassin and I also didn't find a nice way to crack it.
Rating: (Hard?) 1.5.

Andrew: Assassin 115 deserves more than one walkthrough.
I struggled and had to resort to ...
I'll rate A115 as 1.5.

azpaull: I feel like the straggler coming in long after the medalists, but I finally solved the puzzle last night! Probably the toughest one I've completed to date - thanks, udosuk! (I don't think I've solved a 1.5 to date, so that rating sounds right to me.)
On a side note, it is great to see the image problem prompt a couple more of the invisible solvers to start posting! :applause: I hope that others join in the conversation every now and then.

udosuk: Thanks for the nice words, azpaull. Glad that you enjoyed my puzzle. I'm sure there will be many more for you in the future. :alien:

Walkthrough by Afmob:
Thanks for this challenging Killer!

It felt as difficult as Para's Assassin and I also didn't find a nice way to crack it.

A115 Walkthrough:

1. R123
a) Innie R1 = R1C5 = 9
b) Innies+Outies N3: -8 = R1C6 - R3C7 -> R1C6 = 1, R3C7 = 9
c) R3C1 = 7, R2C1 = 9
d) 7(3) = {124} locked for N1
e) Innies+Outies N1: R1C4 = R3C3 <> 2,4,7
f) 22(4) = {3568} locked for R1
g) 14(4) = {1247} -> 2,4,7 locked for N3
h) 7 locked in 23(4) @ R2 = 79{25/34}
i) Outies R12 = 8(3) = 1{25/34}; R3C2 = (24)
j) Hidden Single: R3C8 = 1 @ R3
k) 11(2): R2C9 <> 3,5

2. R789 !
a) Innie R9 = R9C5 = 1
b) Innies+Outies N7: 4 = R9C4 - R7C3 -> R9C4 <> 2,3,4 and R7C3 <> 6,7,8,9
c) Outies N7 = 20(3) <> 1,2
d) 42(7) must have 9 -> 9 locked for R5+N5
e) ! Innies N9 = 26(4) <> 1,3 because {3689} blocked by Killer pair (36) of 9(3)
f) Innies+Outies N9: R9C6 = R7C7 <> 3,9
g) 3 locked in 18(4) @ R9 for N7
h) 9(2) <> 2,6
i) 22(3) = {679} locked for N7 since {589} blocked by Killer pair (58) of 9(2)
j) Innies+Outies N7: 4 = R9C4 - R7C3 -> R9C4 <> 7

3. C456
a) Hidden Single: R4C4 = 1 @ N5
b) 11(3) = 1{28/46}; R3C4 = (24)
c) Innies+Outies N1: R1C4 = R3C3 = (68)
d) 16(3) @ N2 = 9{25/34}
e) Outies N9 = 17(3) <> 5 because 5{39/48} blocked by Killer pairs (35,45) of 16(3) @ N2
f) Innies C6789 = 20(3) <> 2; R58C6 <> 3 because R2C6 <> 8,9
g) Innies C6789 = 20(3): R5C6 <> 4,6 since 9 only possible there
h) Innies C1234 = 14(3) <> 8 because {248} blocked by R3C4 = (24)
i) Hidden pair (68) in R1C4+R3C5 @ N2 -> R3C5 = (68)
j) 2 locked in R46C6 @ N5 for C6
k) Innies+Outies N9: R9C6 = R7C7 <> 2,5

4. R3456
a) Naked pair (24) locked in R3C24 for R3
b) 16(3) @ N2: R4C6 <> 3,5
c) 7(2) @ R4C1 <> 6
d) Killer pair (24) locked in 7(2) + R4C6 for R4
e) 13(2) <> 9
f) Hidden Single: R4C3 = 9 @ R4
g) 9 locked in 12(2) @ N6 = {39} locked for R6+N6
h) 7(2) @ R6C1 <> 4
i) Innies R1234 = 19(2+1) = 6+[58/76/85] / 8+[38] since 8+{56}
blocked by Killer pair (56) of 13(2) -> R4C5 <> 6 and R4C7 <> 7
j) 6 locked in R4C789 @ R4 for N6

5. R789
a) 2 lockend in R8C45 @ N8 for R8 -> 14(4) = 12{38/47/56}
b) 9(3): R7C8 <> 4,6 because 2 only possible there and R8C7 <> 5,6 since R78C8 <> 1
c) 10(2): R7C9 <> 8
d) 14(4): R8C5 <> 8 because R8C6 <> 2,3

6. C456
a) 8 locked in 42(7) @ C5 -> R5C6 <> 8
b) 3 locked 42(7) @ N5 -> R7C5 <> 3
c) Innies C6789 = 20(3): R8C6 <> 5 because R25C6 <> 6,8
d) 14(4): R8C45 <> 6 because R8C6 <> 2,5
e) 6 locked in 42(7) @ C5 -> R5C4 <> 6
f) Hidden Killer pair (68) in Outies N7 for C4
-> Outies N7 = 20(3) = {389/569/578} <> 4
g) 16(3) @ N7 <> {268/349/367} because R7C3 <> 3,6,7; R6C4 <> 3,4,9 and
R1C4 = (68) blocks {268}

7. C789
a) Innies C9 = 24(5) = {12489/12579/13479/23478} <> 6 since R6C9 = (39)
and because of Killer pairs (35,36,68) of 11(2)
b) 7 locked in 26(4) @ R9 = 7{289/469/568}
c) Innies N9 = 26(4) = 7{289/469/568} since R9C4 = R7C7 (step 2f) -> 7 locked for N9
d) 10(2) <> 3
e) 3 locked in 9(3) @ N9 = 3{15/24}

8. R789 !
a) ! Innies R89 = 32(6): R8C9 <> 9 since R8C23 would be {67} -> Innies = {145679}
-> no combo for 9(3) since R7C9 = 1 @ 10(2)
b) Hidden Single: R8C2 = 9 @ R8
c) 10(2): R7C9 <> 1
d) 1 locked in R8C79 @ N9 for R8
e) 9(2): R7C1 <> 8
f) Hidden Killer pair (12) in Outies R89 for R7 since besides Outies R89 only R7C3 can have 1 or 2
g) Outies R89 = 18(4) = 2{169/457} <> 3 because R7C2 = (67), R7C1 = (145)
and R7C8 = (235); 2 locked for R7+N9
h) 3 locked in R8C78 @ N9 for R8
i) 14(4) = 12{47/56}

9. C456
a) 8 locked in Outies N9 @ C6 = 17(3) = 8{27/36}
b) Hidden Single: R5C6 = 9 @ C6
c) 9 locked in Outies N7 @ C4 = 20(3) = 9{38/56}
d) 16(3) @ N7 = {169/358} -> R7C4 = (39), R6C4 = (68)
e) Innies+Outies N7: 4 = R9C4 - R7C3 -> R9C4 <> 6,8
f) Innies+Outies N9: R9C6 = R7C7 <> 4
g) Innies C6789 = 20(3) = 9{47/56}

10. N9
a) Killer pair (45) locked in 9(3) + 26(4) for N9
b) 10(2) <> 6
c) Hidden Single: R2C9 = 6 @ C9 -> R3C9 = 5
d) R3C6 = 3 -> R4C6 = 4
e) Hidden Single: R7C4 = 3 @ R7 -> R6C4+R7C3 = 13(2) = [85] -> R6C4 = 8, R7C3 = 5
f) Hidden Single: R7C1 = 1 @ R7 -> R8C1 = 8

11. Rest is singles.

Rating: (Hard?) 1.5. I used a small contradiction chain.
Walkthrough by Andrew:
Assassin 115 deserves more than one walkthrough.

I struggled and had to resort to step 21 that's somewhere between a chain and a hypothetical. There's also some combination analysis but not too heavy; step 34 doesn't use hypotheticals, I've just written it that way to avoid a long string of permutations which would only lead to one candidate elimination.

I'll rate A115 as 1.5.

Here is my walkthrough. Thanks Afmob for the alternative (better) way to do step 35 and to Ed for an interesting (fun) alternative to step 22.

Prelims

a) R23C1 = {79}, locked for C1 and N1
b) R23C9 = {29/38/47/56}, no 1
c) R4C12 = {16/25/34}, no 7,8,9
d) R4C89 = {49/58/67}, no 1,2,3
e) R6C12 = {16/25/34}, no 7,8,9
f) R6C89 = {39/48/57}, no 1,2,6
g) R78C1 = {18/36/45}, no 2
h) R78C9 = {19/28/37/46}, no 5
i) 7(3) cage in N1 = {124}, locked for N1
j) 11(3) cage at R3C3 = {128/137/146/236/245}, no 9
k) 22(3) cage in N7 = {589/679}, 9 locked for N7
l) 9(3) cage in N9 = {126/135/234}, no 7,8,9
m) R1C6789 = {1238/1247/1256/1346/2345}, no 9
n) 14(4) cage in N8 = {1238/1247/1256/1346/2345}, no 9
o) R9C6789 = {2789/3689/4589/4679/5678}, no 1
p) 42(7) cage at R3C5 = {3456789}, no 1,2

1. 45 rule on R1 1 innie R1C5 = 9
1a. 9 in 42(7) cage locked in R5C46, locked for R5 and N5
1b. 9 in N4 locked in R46C3, locked for C3

2. 45 rule on R9 1 innie R9C5 = 1

3. 45 rule on N3 1 innie R3C7 = 1 outie R1C6 + 8 -> R1C6 = 1, R3C7 = 9, R23C1 = [97], clean-up: no 2,4 in R2C9, no 2 in R3C9

4. 45 rule on N1 1 outie R1C4 = 1 innie R3C3 -> R1C4 = {3568}
4a. Naked quad {3568} in R1C1234, locked for R1
4b. Naked triple {247} in R1C789, locked for N3

5. R3C7 = 9 -> R34C6 = 7 = {25/34}, no 6,7,8

6. 45 rule on R123 1 innie R3C5 = 2 outies R4C46 + 3
6a. Min R4C46 = 3 -> min R3C5 = 6
6b. Max R3C5 = 8 -> max R4C46 = 5, no 5,6,7,8, no 4 in R4C4, clean-up: no 2 in R3C6 (step 5)

7. 45 rule on R1234 4(3+1) innies R3C5 + R4C357 = 28 = 6{589}/6{679}/8{389}/8{479}/8{569}/8{578}, no 1,2 in R4C37

8. 9 in N6 must be in R46C89
8a .Combined cage R46C89 = 25 = {3589/3679/4579}
8b. 3,9 of {3589} must be in R6C89, 4,9 of {4579} must be in R4C89 -> no 4,8 in R6C89

9. 45 rule on N7 1 outie R9C4 = 1 innie R7C3 + 4, no 6,7,8 in R7C3, no 2,3,4 in R9C4

10. Max R7C3 = 5 -> min R67C4 = 11, no 1 in R6C4
10a. R4C4 = 1 (hidden single in C4), clean-up: no 6 in R4C12
10b. R4C4 = 1 -> R3C34 = 10 = [64/82], R1C4 = {68} (step 4)
10c. Naked pair {68} in R3C35, locked for R3, clean-up: no 3,5 in R2C9
10d. Naked pair {68} in R1C4 + R3C5, locked for N2, CPE no 6,8 in R5C4

11. 45 rule on R12 3 remaining outies R3C289 = 8 = {125/134}
11a. 2,4 only in R3C2 -> R3C2 = {24} -> R3C8 = 1
11b. Naked pair {24} in R3C24, locked for R3, clean-up: no 3 in R4C6 (step 5)
11c. Killer pair 2,4 in R4C12 and R4C6, locked for R4, clean-up: no 9 in R4C89

12. R4C3 = 9 (hidden single in R4)
12a. 9 in N6 locked in R6C89 = {39}, locked for R6 and N6, clean-up: no 4 in R6C12
12b. 3 in N5 locked in R4C5 + R5C456 for 42(7) cage, no 3 in R7C5

13. 45 rule on C6789 3 innies R258C6 = 20 = {389/479/569/578}, no 2
13a. 3 of {389} must be in R2C6 -> no 3 in R58C6
13b. 9 of {479/569} must be in R5C6 -> no 4,6 in R5C6
13c. 6 in 42(7) cage locked in R34567C5, locked for C5

14. 45 rule on C1234 3 innies R258C4 = 14 = {239/257/347/356} (cannot be {248} which clashes with R3C4), no 8

15. 45 rule on N7 3 outies R679C4 = 20 = {389/479/569/578}, no 2
15a. 2 in N5 locked in R46C6, locked for C6

16. 45 rule on N9 1 outie R9C6 = 1 innie R7C7, no 1,2 in R7C7, no 9 in R9C6
16a. 45 rule on N9 3 outies R679C6 = 17 = {269/278/368/467} (cannot be {359/458} which clash with R258C6), no 5, clean-up: no 5 in R7C7 (step 16)

17. R3C5 + R4C357 (step 7) = 6{589}/6{679}/8{389} (cannot be 8{569} which clashes with R4C89)
17a. 6{679} must be 6[976] -> no 6 in R4C5, no 7 in R4C7
17b. 6 in R4 locked in R4C789, locked for N6

18. 9(3) cage in N9 = {126/135/234}
18a. 1 of {126/135} must be in R8C7 -> no 5,6 in R8C7

19. 14(4) cage in N8 = {1238/1247/1256/1346}
19a. 8 of {1238} must be in R8C6 -> no 8 in R8C5
19b. 8 in C5 locked in R34567C5, locked for 42(7) cage, no 8 in R5C6

20. R258C6 (step 13) = {389/479/569/578}
20a. 6,8 of {569/578} must be in R8C6 -> no 5 in R8C6

21. R3C5 = R4C46 + 3 (step 6), R4C4 = 1 -> R3C5 = R4C6 + 4 -> R3C5 + R4C6 = [62/84]
21a. 45 rule on R789 2 outies R6C46 = 1 innie R7C5 + 6
21b. If R3C5 = 6 => no 6 in R567C6 => R6C46 must contain 6 => R6C46 = {46/67/68} (cannot be {56} which clashes with R6C12) => no 5,6 in R7C5
21c. If R3C5 = 8 => R4C6 = 4 => R7C5 = 4 (only remaining position in 42(7) cage)
21d. -> no 5,6 in R7C5
21e. 5 in 42(7) cage locked in R4C5 + R5C456 + R6C5, locked for N5

22. 45 rule on R6789 4(3+1) innies R6C357 + R7C5 = 20 = 4{178}/4{457}/7{148}/8{147} (cannot be 4{268}/7{157}/7{256}/8{156}/8{246} which clash with R6C12, cannot be 7{247} because R6C12 then cannot be {25} so 5 cannot be placed in R6), no 2,6
[Ed pointed out an interesting alternative.
R3C5 = 6 => 6 in N5 must be in R6C46
R3C5 = 8 => R4C6 = 4(step 21) => R6C6 = 2
-> R6C46 must have 2 or 6
Killer pair 2,6 in R6C12 and R6C46, locked for R6.]


23. 2 in N6 locked in R5C789, locked for R5
23a. Combined cage R46C12 must contain 2 = {1256/2345}, 5 locked for N4

24. 16(3) cage at R6C4 = {169/178/349/358/457} (cannot be {259} because no 2,5,9 in R6C4, cannot be {268/367} because they would make R679C4 {68}6/{67}7), no 2, clean-up: no 6 in R9C4 (step 9)
24a. 9 of {169} must be in R7C4 -> no 6 in R7C4

25. 2 in N7 locked in R9C123, locked for R9
25a. R9C1234 = {2349/2358/2457} (cannot be {2367} because R7C3 = 3 when R9C4 = 7), no 6

26. 2 in N8 locked in R8C45, locked for R8, clean-up: no 8 in R7C9
26a. 9(3) cage in N9 = {126/135/234}
26b. 6 of {126} must be in R8C8 -> no 6 in R7C8
26c. 2 of {234} must be in R7C8 -> no 4 in R7C8

27. 14(4) cage in N8 = {1238/1247/1256}
27a. 6 of {1256} must be in R8C6 -> no 6 in R8C4
27b. 6 in N8 locked in R789C6, locked for C6

28. R258C4 (step 14) = {239/257/347}
28a. R679C4 (step 15) = {389/569/578} (cannot be {479} which clashes with R258C4), no 4

29. 6 in R9 locked in R9C6789
29a. R9C6 = R7C7 (step 16) -> 6 locked in R7C7 + R9C789, locked for N9, clean-up: no 4 in R78C9
29b. 9(3) cage in N9 = {135/234}, 3 locked for N9, clean-up: no 7 in R78C9, no 3 in R9C6 (step 16)

30. 3 in R9 locked in R9C123, locked for N7, clean-up: no 6 in R78C1, no 7 in R9C4 (step 9)
30a. 6 in N7 locked in 22(3) cage = {679}, locked for N7

31. R6C46 = R7C5 + 6 (step 21a)
31a. R7C5 = {478} -> R6C46 = [64/82/67/68], no 7 in R6C4
31b. Naked pair {68} in R16C4, locked for C4, clean-up: no 4 in R7C3 (step 9)

32. Killer pair 1,5 in R78C1 and R7C3, locked for N7

33. 16(3) cage at R6C4 (step 24) = {169/178/358}
33a. 5 of {358} must be in R7C3 -> no 5 in R7C4

34. R46C12 (step 23a) = {1256/2345}
34a. R46C1 cannot be {25} which clashes with R4C12 and R6C12, R46C1 cannot be [35] because R46C2 = [42] clashes with R3C2, R46C1 cannot be [56] because R46C2 = [21] clashes with R23C3
34b. 45 rule on C1 5 innies R14569C1 = {12368/23456}
34c. 6 of {12368} must be in R1C1 (R56C1 cannot be {16} which clashes with R6C12) -> no 8 in R1C1
Consider key points for the permutations of {23456}
34da. If 2 is in R46C1 => no 5 in R4C1 (or R6C1)
34db. If 2 is in R9C1 and 3 in R1C1, R46C1 cannot be [56] => no 5 in R4C1
34dc. If 2 is in R9C1 and 6 in R1C1 => R6C1 = 5
34dd. If 5 is in R1C1 => no 5 in R4C1
34e. -> no 5 in R4C1, clean-up: no 2 in R4C2

35. 45 rule on C9 5 innies R14569C9 = 24 = {13479/13569/23478} (cannot be {12678/14568/24567} because R6C9 only contains 3,9, cannot be {12489/12579/13578/23469} which clash with R78C9, cannot be {23568} which clashes with R23C9), 3 locked for C9 -> R3C9 = 5, R2C9 = 6, R3C6 = 3, R4C6 = 4 (step 5), clean-up: no 3 in R4C12, no 7,8 in R4C8
[This step has been available for a long time but I only spotted it after step 34b and then only realised how powerful it was after I’d eliminated the non-valid combinations, some of which only worked after R78C9 had been reduced to two combinations in step 29.]
[Afmob pointed out that an alternative here is hidden killer triple 3,8,9 for C9. R6C9 = {39}, R78C9 must contain one of 8,9 -> R23C9 cannot be [83] -> R23C9 = [65]. This is simpler than my step but possibly harder to spot.]

36. R4C12 = [25] -> R4C789 = [867], R4C5 = 3, R2C78 = [38]
36a. Naked pair {16} in R6C12, locked for R6 and N4 -> R6C4 = 8, R1C4 = 6, R3C5 = 8, R3C3 = 6, R3C4 = 4 (step 10b), R3C2 = 2, R8C3 = 7, R6C3 = 4, R6C7 = 5, R6C56 = [72], R7C5 = 4, R2C23 = [41], R7C3 = 5, R7C4 = 3 (step 33), R9C4 = 9 (step 9), R5C456 = [569], R8C45 = [25], R2C456 = [725], R8C6 = 6 (step 27), R78C2 = [69], R6C12 = [61], R7C67 = [87], R9C6 = 7, R7C1 = 1, R7C89 = [29], R8C1 = 8

and the rest is naked singles

udosuk wrote:
According to SS it is as hard as Para's Assassin 113. But there is a key move in the puzzle which if you can spot will make a huge breakthrough in the solving. And then I've found 2 different approaches to the "critical step", which I'll be glad if you guys can find a 3rd. :geek:
I'll be interested to see what that is. I don't think I found it.

Now to try V0.9 and see if it is easier. Don't know whether I'll try V2, certainly not before I try A113 and/or A113-Lite which I haven't yet started.
Walkthrough by udusok:
0. Prelims

16/2 @ r2c1={79} (NP @ c1,n1)
7/3 @ r2c2={124} (NT @ n1)
11/2 @ r2c9={29|38|47|56}
42/7 @ r3c5={3456789}
7/2 @ r4c1 & r6c1={16|25|34}
13/2 @ r4c8={49|58|67}
12/2 @ r6c8={39|48|57}
9/2 @ r7c1={18|36|45} (not {27} for 16/2 @ r2c1)
22/3 @ r7c2={589|679} (9 @ n7 locked)
9/3 @ r7c8={126|135|234}
10/2 @ r7c9={19|28|37|46}


1. r12349,c4

a. Innies @ r1: r1c5=9

b. Innies @ r9: r9c5=1

c. Innie-outies @ n3: r3c7=r1c6+8
=> r1c6=1, r3c7=9
=> 16/2 @ r2c1=[97]
=> 16/3 @ r3c6: r34c6=16-9=7={25|34}

d. Outies @ n7: r679c4=20 from {3456789}
=> HS @ c4: r4c4=1
=> 11/3 @ r3c3: r3c34=11-1=10=[64|82]
=> 7/2 @ r4c1={25|34}

e. Outies @ n1: r134c4=11
=> r1c13=11-1=10=[64|82]
=> 22/4 @ r1c1={3568} (NQ @ r1)
=> r1c789={247} (NT @ n3)


2. n23

a. 7 @ r2,n2 locked @ r2c456
=> 23/4 @ r1c5: r2c456=23-9=14={257|347}
=> HP @ n2: r1c4+r3c5={68}
=> r3c35={68} (NP @ r3)

b. Innies @ n2: r3c46=7=[25|43]
=> r3c69={35} (NP @ r3)
=> r3c8=1
=> 11/2 @ r2c9=[65|83]


3. r456

a. 16/3 @ r3c6: r34c6=[34|52]
=> 7/2 @ r4c1 & r4c6 form KNP {24} @ r4
=> 13/2 @ r4c8={58|67}
=> HS @ r4: r4c3=9

b. 9 @ r5,n5,42/7 locked @ r5c46
=> 9 @ r6,n6 locked @ 12/2 @ r6c8={39} (NP @ r6,n6)
=> 7/2 @ r6c1={16|25}

c. 2 @ c6,n5 locked @ r46c6
=> 2 @ r8,n8 locked @ r8c45


4. n79,r9 (!)

a. Innie-outies @ n7: r9c4=r7c3+4
=> r7c3 from {12345}, r9c4 from {56789}

b. Innies @ n9: r7c7+r9c789=26
But 10/2 @ r7c9={19|28|37|46} must have 3|6|8|9
=> r7c7+r9c789 can't be {3689}, can't have 3

c. Innie-outies @ n9: r7c7=r9c6
=> r7c7 & r9c6 must be both from {45678}

d. 3 @ r9,n7 locked @ r9c123
=> 9/2 @ r7c1={18|45} must have 5|8
=> 22/3 @ r7c2 cant be {589}, must be {679} (NT @ n7)
=> r7c3 from {1245}
=> r9c4=r7c3+4 from {5689}

e. 7 @ r9 locked @ 26/4 @ r9c6
But r7c7=r9c6
=> 7 @ n9 locked @ r7c7+r9c789
=> 10/2 @ r7c9=[19|91|28|46|64]

f. 3 @ n9 locked @ 9/3 @ r7c8
=> 9/3 @ r7c8={135|234}=[234|243|315|513]
=> r7c8 from {235}, r8c7 from {134}, r8c8 from {345}


5. c89 (!!)

Now we have:
11/2 @ r2c9=[65|83]
12/2 @ r6c8={39}
9/3 @ r7c8=[234|243|315|513]

Considering these 3 cages altogether:
r236c9+r678c8=[65????|839324]
=> r2c9+r8c8=[63|64|65|84] must have 6|4
=> 10/2 @ r7c9, seeing r2c9+r8c8, can't be {46}
=> 10/2 @ r7c9=[19|28|91]

(Another way of seeing this:
Either r6c8=3 or r6c9=3
If r6c8=3, 9/3 @ r7c8=[234] => 10/2 @ r7c9 can't be {46}
If r6c9=3, 11/2 @ r2c9=[65] => 10/2 @ r7c9 can't be {46}
Therefore 10/2 @ r7c9 can't be {46}, must be [19|28|91])


Alternative step 5. r78 (!!)

a. Outies @ r89: r7c1289=18
Also {12} @ r7 locked @ r7c1389
=> r7c1289 must have at least one of {12}
=> r7c1289 can't be [5634]
=> r7c29 can't be [64]

b. r7c1 from {1458}, r7c8 from {235}
=> r7c18 can't sum to 5
=> r7c29 can't sum to 18-5=13, can't be [76|94]

c. 9/3 @ r7c8 & 10/2 @ r7c9 can't both have 4
=> 9/3 @ r7c8=[234|243] & 10/2 @ r7c9={46} can't both happen
=> r7c89 can't be [24|26]
=> r7c1289 can't be [5724|1926]
=> r7c29 can't be [74|96]

d. Therefore r7c29 can't be [64|74|94|66|76|96]
=> r7c9 can't be 4|6
=> 10/2 @ r7c9 can't be [46]
=> 10/2 @ r7c9=[19|28|91]


6. c9 (!)

10/2 @ r7c9=[19|28|91] has 8|9
12/2 @ r6c8={39}: r6c9 has 3|9
=> r678c9 together has 3|8
=> 11/2 @ r2c9 can't be [83], must be [65]


7. Mop up

r3c46=[43], r4c6=4 (2b,3a)
=> r3c2=2
=> 7/2 @ r4c1=[25]
=> 13/2 @ r4c8=[67]
=> r4c7=8

r1c4=r3c3=6 (1d,1e)
=> r8c3=7
=> r78c2={69} (NP @ c2)
=> 7/2 @ r6c1=[61]
=> r2c23=[41]

HS @ 42/7 @ r3c5: r7c5=4
=> r7c3=5, r9c4=9 (4a)
HS @ n5: r6c6=2
=> 17/3 @ r6c6: r7c67=17-2=15=[87]
=> 9/2 @ r7c1=[18]


All naked singles from here.

Many thanks for Andrew who helped me correct a few typos. Note my "Prelims" are not complete in the sense that they don't cover all automatical eliminations made by a program based on cage sums at the start of solving. I just put them there to help make the solving path more fluent. :geek:

There is also discussion in this thread about the best way to post links to diagrams stored on image hosting sites.


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PostPosted: Wed Jun 15, 2011 2:58 am 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 115 V0.9 "Beijing" by udosuk (August 2008) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:5633:5633:5633:5633:5915:3587:3587:3587:3587:4100:1797:1797:5915:5915:5915:3078:3078:2823:4100:1797:2824:2824:2845:4106:4106:3078:2823:1803:1803:7948:2824:2845:4106:5133:3342:3342:7948:7948:7948:5146:5146:5146:5133:5133:5133:1807:1807:7948:4112:2846:4369:5133:3090:3090:2323:5652:4112:4112:2846:4369:4369:2325:2582:2323:5652:5652:3612:3612:3612:2325:2325:2582:4632:4632:4632:4632:3612:6681:6681:6681:6681:
Solution:
+-------+-------+-------+
| 5 8 3 | 6 9 1 | 2 7 4 |
| 9 4 1 | 7 2 5 | 3 8 6 |
| 7 2 6 | 4 8 3 | 9 1 5 |
+-------+-------+-------+
| 2 5 9 | 1 3 4 | 8 6 7 |
| 3 7 8 | 5 6 9 | 1 4 2 |
| 6 1 4 | 8 7 2 | 5 9 3 |
+-------+-------+-------+
| 1 6 5 | 3 4 8 | 7 2 9 |
| 8 9 7 | 2 5 6 | 4 3 1 |
| 4 3 2 | 9 1 7 | 6 5 8 |
+-------+-------+-------+
Quote:
udosuk: On the other hand, after reading mhparker's post :arrow: here, here is a v0.9 specially catered for him:
(Warning: potential spoiler to anyone who'd like to tackle the official version seriously)
SSolver 3.2.1 Rating: 0.92. Personal Rating: 0.75 (Hard).

Mike(mhparker): Nice that you created a 3rd (bronze?) version just for me! :-) Thanks!
I'm doing it right now, but haven't finished it yet. Indeed... as I speak... just realized that I've hit a contradiction! :doh: Will have to unwind a few steps. Fortunately, I save the grid after each move or two, even if I don't keep a WT. A bit like a climber hammering pitons into the rock face to limit his fall...

Mike(mhparker) in a later post: Thanks once again to udosuk, for creating a puzzle specially dedicated to me! Fame at last! :dance: (Although, I suppose, being only a "bronze medal" version, I could take it both ways - but I prefer to take it as a compliment...)
I don't know how many people have already tackled it, or intend to tackle it, but it's actually not actually that easy, being of typical Assassin V1 standard, from the time that Ruud used to make 'em.
Indeed, at first I thought that it would even be a candidate for a 1.25 rating, having several moves of this category in my original walkthrough, but I eventually found a way through that restricts itself to simpler moves. Even so, I'll rate it as no less than 1.0. For example, it took me much longer to do this puzzle than Messy Ones #1 and #3 (from Ed and Nasenbaer, respectively), both of which were also rated as 1.0.
Despite one or two of udosuk's comments earlier in this thread, which make this puzzle sound like an easy (even almost trivial :shock:) option, it's actually totally unsuitable for beginners and, on the contrary, requires a solid understanding of the principles of solving Killers, as can be seen from the WT below:

Børge
mhparker wrote:
P.S. No idea what the "spoiler" for the Original version was that udosuk was referring to...
Oh Mike, come on, I'm :shock:
Even I, who have never attempted to solve a Killer can spot the spoiler, I think.

Hidden Text:
udosuk wrote that V1 and v0.9 have identical solutions.

Look at the cross in the middle of V1 with 7 cells with sum = 42.

In v0.9 this cross is replaced with three cages. Two cages with two cells each and both cages have sum = 11, and one cage with three cells and sum = 20.
2 + 2 + 3 = 7
11 + 11 + 20 = 42

Image

udosuk replied to Mike's posts: Glad you didn't take it the "other" way... ;) I made that version for you because I fully understand your post, as I also enjoy puzzles which are fully solvable without any help from programs. (That said, I also enjoy the process of making a nice elegant walkthrough for a challenging puzzle with a little help from a program. :ugeek:)
I gave it a 0.75 (hard), although it is very close to a 1.0 (easy) from the SS rating of 0.92. I will study your walkthrough and see if there's any shortcut you've missed. :study:

Andrew: After A115 I've now done the V0.9. As Mike said it's still a good challenge.
Congratulations to Mike for re-working it with simpler steps to get the rating down to 1.0. I missed his step 30 which isn't available in A115 and isn't an easy one to spot.
Although V0.9 is definitely easier than A115, I found that several sub-steps ... were no longer available so I had to find other ways to make those eliminations.
I'll rate V0.9 as Easy 1.25 the way I solved it. If I'd found Mike's step 30 it would have been 1.0.

udosuk: Børge has explained correctly what the "spoiler" is. I know the 2 puzzles each has some constraints the other hasn't but if you combined these 2 together you can solve each puzzle much more easily. So in a way v1 is also a "potential spoiler" to v0.9. This is probably what annoys many experts here.

udosuk, when posting his walkthrough: Notice the shortness of the solving path. People can argue some moves are quite tricky but I still think it's a 0.75. :ugeek:

Walkthrough by Mike:
Thanks once again to udosuk, for creating a puzzle specially dedicated to me! Fame at last! :dance: (Although, I suppose, being only a "bronze medal" version, I could take it both ways - but I prefer to take it as a compliment...)

I don't know how many people have already tackled it, or intend to tackle it, but it's actually not actually that easy, being of typical Assassin V1 standard, from the time that Ruud used to make 'em.

Indeed, at first I thought that it would even be a candidate for a 1.25 rating, having several moves of this category in my original walkthrough, but I eventually found a way through that restricts itself to simpler moves. Even so, I'll rate it as no less than 1.0. For example, it took me much longer to do this puzzle than Messy Ones #1 and #3 (from Ed and Nasenbaer, respectively), both of which were also rated as 1.0.

Despite one or two of udosuk's comments earlier in this thread, which make this puzzle sound like an easy (even almost trivial :shock:) option, it's actually totally unsuitable for beginners and, on the contrary, requires a solid understanding of the principles of solving Killers, as can be seen from the WT below:

Edit: Error in original step 34 by-passed and subsequent steps adapted. Many thanks to udosuk for pointing this out.
Edit2: Incorporated even more corrections from eagle-eyed Andrew. Many thanks for these, Andrew!

Assassin 115 Beijing v0.9 "Bronze" Walkthrough (45 steps)

Prelims

a) 14(4) at R1C6 and R8C4 = {1238/1247/1256/1346/2345} (no 9)
b) 16(2) at R2C1 = {79} (no 1..6,8)
c) 7(3) at R2C2 = {124} (no 3,5..9)
d) 11(2) at R2C9, R3C5 and R6C5 = {29/38/47/56} (no 1)
e) 11(3) at R3C3 = {128/137/146/236/245} (no 9)
f) 7(2) at R4C1 and R6C1 = {16/25/34} (no 7..9)
g) 13(2) at R4C8 = {49/58/67} (no 1..3)
h) 20(3) at R5C4 = {389/479/569/578} (no 1,2)
i) 12(2) at R6C8 = {39/48/57} (no 1,2,6)
j) 9(2) at R7C1 = {18/27/36/45} (no 9)
k) 22(3) at R7C2 = {589/679} (no 1..4)
l) 9(3) at R7C8 = {126/135/234} (no 7..9)
m) 10(2) at R7C9 = {19/28/37/46} (no 5)
n) 26(4) at R9C6 = {2789/3689/4589/4679/5678} (no 1)

1. Naked pair (NP) at R23C1 = {79}, locked for C1 and N1
1a. cleanup: no 2 in R78C1

2. Naked triple (NT) at R2C23+R3C2 = {124}, locked for N1

3. 9 in 22(3) at R7C2 (prelim k) locked for N7

4. Innie R1: R1C5 = 9
4a. cleanup: no 2 in R3467C5

5. Innie R9: R9C5 = 1

6. Innie/Outie difference (IOD), N3: R3C7 = R1C6 + 8
6a. -> R1C6 = 1, R3C7 = 9
6b. cleanup: no 2 in R23C9

7. R23C1 = [97]
7a. cleanup: no 4 in R4C5 and R2C9

8. IOD, N1: R1C4 = R3C3
8a. -> no 2,4,7 in R1C4

9. Naked quad (NQ) at R1C1234 = {3568}, locked for R1

10. NT at R1C789 = {247}, locked for N3

11. Split 7(2) at R34C6 = {25/34} (no 6..8)

12. Innies R1234: R4C37 = 17(2) = [98]
12a. cleanup: no 3 in R3C5; no 4,5 in R4C89; no 4 in R6C89

13. NP at R4C89 = {67}, locked for R4 and N6
13a. cleanup: no 4,5 in R3C5; no 1 in R4C12; no 5 in R6C89

14. NP at R6C89 = {39}, locked for R6 and N6
14a. cleanup: no 4 in R6C12, no 8 in R7C5

15. Hidden single (HS) in R4 at R4C4 = 1

16. Split 10(2) at R3C34 = [64/82]
16a. -> no 3,5 in R3C3; no 3,5,6,8 in R3C4
16b. -> no 3,5 in R1C4 (step 8)

17. NP at R1C4+R3C5 = {68}, locked for N2

18. NP at R3C35 = {68}, locked for R3
18a. cleanup: no 3,5 in R2C9

19. Outies R123: R4C56 = 7(2) = [34/52]
19a. -> no 3,5 in R4C6
19b. cleanup: no 2,4 in R3C6 (step 11)

20. Hidden pair (HP) in R3 at R3C24 = {24} (no 1)

21. HS in R3 at R3C8 = 1
21a. -> split 11(2) at R2C78 = [38/56/65]
21b. -> no 3 in R2C8

22. Innies R6789: R6C37 = 9(2) = [45/54/72/81] (no 6)
22a. no 1,2 in R6C3

23. Outies R789: R6C456 = 17(3) = {278/458/467} (no eliminations) = {(4/7)..}

24. 9 of N5 locked in 20(3) at R5C4 = {389/569} (no 4,7) = {(3/5)..}
(Note: {479} blocked by R6C456 (step 23))

25. 20(3) at R5C4 (step 24) and R4C5 form killer pair (KP) on {35} within N5
25a. -> no 5 in R6C456
25b. cleanup: no 6 in R7C5

26. IOD, N7: R9C4 = R7C3 + 4
26a. -> no 6..8 in R7C3; no 2..4 in R9C4

27. IOD, N9: R7C7 = R9C6
27a. -> no 1 in R7C7; no 8,9 in R9C6

28. Outies N7: R679C4 = 20(3) = {389/479/569/578} (no 2)

29. 2 in N5 locked in R46C6 for C6
29. cleanup: no 2 in R7C7 (step 27)

30. 2 in C5 locked in C5 Innies: R258C5 = 13(3) = {238/256} (no 4,7)
(Note: {247} blocked by R5C5)

31. HP in C5 at R67C5 = {47} (no 3,5,6,8)

32. 2 in N8 locked in split 13(3) at R8C456 = {238/256} (no 4,7)
(Note: {247} blocked by R7C5)
32a. 2 locked in R8C45 for R8
32b. cleanup: no 8 in R7C9

33. Innies C6789: R258C6 = 20(3) = {389/569/578} (no 4)
(Note: {479} blocked by R8C6)
33a. 3 of {389} must go in R2C6
33b. -> no 3 in R58C6
33c. 9 of {569} must go in R5C6
33d. -> no 6 in R5C6

34. 4 in N2 locked in R23C4 for C4

35. R679C4 (step 28) = {389/569/578} = {(6/8)..}
35a. -> R679C4 and R1C4 form KP on {68} within C4
35b. -> no 6,8 in R58C4

36. Innies C1234: R258C4 = 14(3) = {239/257} (no 4)
(Note: {347} blocked by R58C4)
36a. 9 of {239} must go in R5C4
36b. -> no 3 in R5C4
36c. 5 of {257} must go in R5C4
36d. -> no 5 in R28C4

37. HS in N2 at R3C4 = 4
37a. -> R3C3 = 6 (cage sum)
37b. -> R1C4 = 6 (step 8)
37c. cleanup: no 2 in R7C3 (step 26)

38. R34C5 = [83]
38a. -> R4C6 = 4 (step 19)
38b. -> R3C6 = 3 (cage sum)
38c. cleanup: no 8 in R2C9; no 3,4 in R7C7 (step 27); no 7 in R7C5

39. R23C9 = [65]
39a. -> R2C78 = [38], R4C89 = [67]
39b. cleanup: no 3,4 in R78C9

40. Naked single (NS) at R3C2 = 2
40a. -> R4C12 = [25]
40b. cleanup: no 4 in R6C7 (step 22)

41. R67C5 = [74]
41a. -> R6C4 = 8
41b. -> R6C6 = 2 (step 23)
41c. cleanup: no 1 in R6C7 (step 22), no 5 in R8C1

42. R6C37 = [45]
42a. -> R2C23 = [41]
42b. cleanup: no 5 in R9C4 (step 26), no 5 in R9C6 (step 27)

43. Split 8(2) at R7C34 = {35} (no 7,9) (last combo), locked for R7
43a. cleanup: no 4,6 in R8C1

44. NS at R7C8 = 2
44a. -> split 7(2) at R8C78 = [43] (last permutation)
44b. -> R6C89 = [93]
44c. cleanup: no 6 in R7C1, no 8 in R78C9

45. NS at R8C4 = 2
45a. -> split 11(2) at R8C56 = {56} (no 8) (last combo), locked for R8 and N8
45b. cleanup: no 6 in R7C7 (step 27)

Nothing but naked singles remaining now.

P.S. No idea what the "spoiler" for the Original version was that udosuk was referring to...

P.P.S. No problem with discussing slightly off-topic stuff here, Rick and Joe-Casey. Far from it, it's great to have you around. Fortunately, Afmob will be glad to see that, with this post, we're back to having way over 10% of the posts on this thread containing walkthroughs... :)
Walkthrough by Andrew:
After A115 I've now done the V0.9. As Mike said it's still a good challenge.

Congratulations to Mike for re-working it with simpler steps to get the rating down to 1.0. I missed his step 30 which isn't available in A115 and isn't an easy one to spot.

Although V0.9 is definitely easier than A115, I found that several sub-steps using the 42(7) cage were no longer available so I had to find other ways to make those eliminations.

My breakthrough was a hidden killer triple, the move that Afmob suggested as an alternative to step 35 in my A115 walkthrough, so I'll rate V0.9 as Easy 1.25 the way I solved it. If I'd found Mike's step 30 it would have been 1.0.

Here is my walkthrough. Edit. Step 19 deleted, step 22a edited, parts of step 22b deleted and errors in step numbering corrected.

Prelims

a) R23C1 = {79}, locked for C1 and N1
b) R23C9 = {29/38/47/56}, no 1
c) R34C5 = {29/38/47/56}, no 1
d) R4C12 = {16/25/34}, no 7,8,9
e) R4C89 = {49/58/67}, no 1,2,3
f) R6C12 = {16/25/34}, no 7,8,9
g) R67C5 = {29/38/47/56}, no 1
h) R6C89 = {39/48/57}, no 1,2,6
i) R78C1 = {18/36/45}, no 2
j) R78C9 = {19/28/37/46}, no 5
k) 7(3) cage in N1 = {124}, locked for N1
l) 11(3) cage at R3C3 = {128/137/146/236/245}, no 9
m) R5C456 = {389/479/569/578}, no 1,2
n) 22(3) cage in N7 = {589/679}, 9 locked for N7
o) 9(3) cage in N9 = {126/135/234}, no 7,8,9
p) R1C6789 = {1238/1247/1256/1346/2345}, no 9
q) 14(4) cage in N8 = {1238/1247/1256/1346/2345}, no 9
r) R9C6789 = {2789/3689/4589/4679/5678}, no 1

1. 45 rule on R1 1 innie R1C5 = 9, clean-up: no 2 in R34C5, no 2 in R67C5

2. 45 rule on R9 1 innie R9C5 = 1

3. 45 rule on N3 1 innie R3C7 = 1 outie R1C6 + 8 -> R1C6 = 1, R3C7 = 9, R23C1 = [97], clean-up: no 2,4 in R2C9, no 2 in R3C9, no 4 in R4C5

4. 45 rule on N1 1 outie R3C3 = 1 innie R1C4 -> R1C4 = {3568}
4a. Naked quad {3568} in R1C1234, locked for R1
4b. Naked triple {247} in R1C789, locked for N3

5. R3C7 = 9 -> R34C6 = 7 = {25/34}, no 6,7,8

6. 45 rule on R123 3 outies R4C456 = 8 = {125/134} -> R4C4 = 1, R4C56 = [34/52], clean-up: no 3,4,5 in R3C5, no 2,4 in R3C6 (step 5), no 6 in R4C12
6a. R4C4 = 1 -> R3C34 = 10 = [64/82], R1C4 = {68} (step 4)
6b. Naked pair {68} in R3C35, locked for R3, clean-up: no 3,5 in R2C9
6c. Naked pair {68} in R1C4 and R3C5, locked for N2
6d. Killer pair 2,4 in R4C12 and R4C6, locked for R4, clean-up: no 9 in R4C89
6e. R6C89 = {39/48} (cannot be {57} which clashes with R4C89), no 5,7

7. 45 rule on R12 3 remaining outies R3C289 = 8 = {125/134}
7a. 2,4 only in R3C2 -> R3C2 = {24} -> R3C8 = 1
7b. R3C8 = 1 -> R2C78 = {38/56}

8. R4C3 = 9 (hidden single in R4)
8a. 45 rule on R1234 1 remaining innie R4C7 = 8, clean-up: no 3 in R2C8 (step 7b), no 5 in R4C89, no 4 in R6C89
8b. Naked pair {67} in R4C89, locked for N6
8c. Naked pair {39} in R6C89, locked for R6 and N6, clean-up: no 4 in R6C12, no 8 in R7C5

9. 45 rule on R789 3 outies R6C456 = 17 = {278/458/467}
9a. Killer pair 2,4 in R4C6 and R6C456, locked for N5

10. 9 in N5 locked in R5C456 = {389/569}, no 7
10a. Killer pair 3,5 in R4C5 and R5C456, locked for N5, clean-up: no 6 in R7C5
10b. 7 in N5 locked in R6C456, locked for R6

11. 45 rule on R6789 2 innies R6C37 = 9 = [45/54/81], no 1,2,6 in R6C3, no 2 in R6C7
11a. 2 in N6 locked in R5C789, locked for R5
11b. 2 in N4 locked in combined cage R46C12 = {1256/2345}, 5 locked for N4, clean-up: no 4 in R6C7 (step 11)
11c. 4 in N6 locked in R5C789, locked for R5

12. 45 rule on N7 1 outie R9C4 = 1 innie R7C3 + 4, no 6,7,8 in R7C3, no 2,3,4 in R9C4
12a. 45 rule on N7 3 outies R679C4 = 20 = {389/479/569/578}, no 2
12b. 6 of {569} must be in R6C4 -> no 6 in R79C4, clean-up: no 2 in R7C3 (step 12)
12c. 2 in N5 locked in R46C6, locked for C6
12d. 2 in N8 locked in R8C45, locked for R8, clean-up: no 8 in R7C9

13. 45 rule on C6789 3 innies R258C6 = 20 = {389/479/569/578}
13a. 3 of {389} must be in R2C6 -> no 3 in R58C6
13b. 9 of {569} must be in R5C6 -> no 6 in R5C6

14. 45 rule on C1234 3 innies R258C4 = 14 = {239/257/347/356} (cannot be {248} which clashes with R3C4), no 8

15. 45 rule on N9 1 outie R9C6 = 1 innie R7C7, no 1,2 in R7C7, no 8,9 in R9C6
15a. 45 rule on N9 3 outies R679C6 = 17 = {269/278/368/467} (cannot be {359/458} which clash with R258C6), no 5, clean-up: no 5 in R7C7 (step 15)

16. 45 rule on N9 4 innies R7C7 + R9C789 = 26 = {2789/4589/4679/5678} (cannot be {3689} which clashes with 9(3) cage), no 3, clean-up: no 3 in R9C6 (step 15)

17. 9(3) cage in N9 = {126/135/234}
17a. 1 of {126/135} must be in R8C7 -> no 5,6 in R8C7
17b. 2 of {126} must be in R7C8 -> no 4,6 in R7C8

18. 14(4) cage in N8 = {1238/1247/1256/1346}
18a. 8 of {1238} must be in R8C6 -> no 8 in R8C5

19. Deleted

20. 2 in N7 locked in R9C123, locked for R9
20a. R9C1234 = {2349/2358/2457} (cannot be {2367} because R7C3 = 3 when R9C4 = 7), no 6

21. 6 in R9 locked in R9C6789
21a. R9C6 = R7C7 (step 15) -> 6 locked in R7C7 + R9C789, locked for N9, clean-up: no 4 in R78C9
21b. 9(3) cage in N9 = {135/234}, 3 locked for N9, clean-up: no 7 in R78C9

22. R23C9 = [65/83]
22a. Hidden killer triple 3,8,9 in R23C9, R6C9 and R78C9 for C9, R6C9 = {39}, R78C9 contains 8 or 9 -> R23C9 cannot contain more than one of 3,8 -> R23C9 = [65], R2C78 = [38], R4C89 = [67], R3C6 = 3, R4C6 = 4 (step 5), R4C5 = 3 (step 6), R3C5 = 8, R1C4 = 6, R3C3 = 6, R3C4 = 4 (step 6a), R3C2 = 2, R4C12 = [25], clean-up: no 6 in R6C56 (step 9), no 5,7 in R7C5
22b. R6C456 = [872], R6C3 = 4, R2C23 = [41], R7C5 = 4, clean-up: no 5 in R8C1, no 7 in R8C46 (step 18)
22c. R6C9 = 3 (hidden single in C9), R6C8 = 9, R9C3 = 2 (hidden single in R9)

23. 1 in C2 locked in R56C2, locked for N4 -> R6C12 = [61], R6C7 = 5, clean-up: no 3 in R78C1
23a. R5C5 = 6 (hidden single in R5)

24. R6C4 = 8 -> R7C34 = 8 = {35}, locked for R7 -> R7C8 = 2, R8C78 = [43] (step 17), R5C8 = 4, R1C8 = 7, R1C79 = [24], R5C79 = [12], R9C8 = 5, clean-up: no 8 in R8C9
24a. R7C4 = 3 (hidden single in C4), R7C3 = 5, R1C1 = 5 (hidden single in R1), R9C9 = 8 (hidden single in C9), clean-up: no 8 in 22(3) in N7 (prelim n)
24b. R8C3 = 7, R9C12 = [43], R1C23 = [83], R5C123 = [378]

25. 7 in R7 locked in R7C67
25a. R6C6 = 2 -> R7C67 = 15 = [87]

and the rest is naked singles

I'm not sure what the "spoiler" was in V0.9. The only thing I can think of is that Innies N9 and the 26(4) cage have the same total. If it's not that, maybe udosuk can tell us what the "spoiler" is.
Walkthrough by udusok:
This is my walkthrough for v0.9 (version for mhparker)

0. Prelims

16/2 @ r2c1={79} (NP @ c1,n1)
7/3 @ r2c2={124} (NT @ n1)
11/2 @ r2c9 & r3c5 & r6c5={29|38|47|56}
20/3 @ r5c4={389|479|569|578}
7/2 @ r4c1 & r6c1={16|25|34}
13/2 @ r4c8={49|58|67}
12/2 @ r6c8={39|48|57}
9/2 @ r7c1={18|36|45} (not {27} for 16/2 @ r2c1)
22/3 @ r7c2={589|679} (9 @ n7 locked)
9/3 @ r7c8={126|135|234}
10/2 @ r7c9={19|28|37|46}


1. r123469

a. Innies @ r1: r1c5=9

b. Innies @ r9: r9c5=1

c. Innie-outies @ n3: r3c7=r1c6+8
=> r1c6=1, r3c7=9
=> 16/2 @ r2c1=[97]

d. Innies @ r1234: r4c37=17=[98]
=> 13/2 @ r4c8={67} (NP @ r4,n6)
=> 12/2 @ r6c8={39} (NP @ r6,n6)
=> 7/2 @ r6c1={16|25}
=> 7/2 @ r4c1={25|34}
=> HS @ r4: r4c4=1
=> 11/3 @ r3c3: r3c34=11-1=10=[64|82]
=> 11/2 @ r3c5=[65|83]
=> r3c35={68} (NP @ r3)

e. Outies @ r123: r4c456=8
=> r4c56=8-1=7=[34|52]
=> 16/3 @ r3c6: r34c6=16-9=7=[34|52]

f. Outies @ n1: r134c4=11
=> r13c4=11-1=10=[64|82]
=> r1c4+r3c5={68} (NP @ n2)
=> 22/4 @ r1c1={3568} (NQ @ r1)
=> r1c789={247} (NT @ n3)
=> r3c69={35} (NP @ r3)
=> r3c8=1
=> 12/3 @ r2c7: r2c78=12-1=11=[38|56|65]
=> 11/2 @ r2c9=[65|83]


2. c456

a. Outies @ r789: r6c456 from {245678}=17 must have 2|4
=> r4c6 & r6c456 form KNP {24} @ n5

b. 20/3 @ r5c4 from {356789} must have 3|5
=> r4c5 & r5c456 form KNP {35} @ n5
=> r6c456 from {24678}=17={278|467} (7 @ r6,n5 locked)

c. Innies @ c5: r258c5=13 from {2..8} with r5c5 from {3568}
=> r258c5 can't be {157|247}, can't have 7
=> 7 @ c5 locked @ 11/2 @ r6c5={47} (NP @ c5)

d. Outies @ n7: r679c4=20 can't have 2
=> 2 @ c6,n5 locked @ r46c6
=> 2 @ r8,n8 locked @ r8c45

e. 14/4 @ r8c4: r8c456=14-1=13 must have 2
But r7c5 from {47}
=> r8c456 can't be {247}
=> r8c456={238|256}

f. Innies @ c6789: r258c6=20 with r8c6 from {3568}
=> r258c6 can't be {479}, can't have 4
=> r258c6=20 from {356789} must have 3|5
=> r3c6 & r258c6 form KNP {35} @ c6

g. Max r2c6=7 => Min r58c6=20-7=13
=> r58c6 can't have 3
=> 3 @ c6,n2 locked @ r23c6

h. 7 @ r2,n2 locked @ 23/4 @ r1c5: r2c456=23-9=14
But r2c5 from {25}
=> r2c456={257} (NT @ r2,n2)


3. Mop up

a. 1e: r34c6=[34], 1f: r13c4=[64]
=> 11/2 @ r2c9=[65], 11/2 @ r3c5=[83], 11/2 @ r6c5=[74]
=> r2c78=[38], r3c23=[26], r6c4=8
=> 7/2 @ r4c1=[25], 13/2 @ r4c8=[67]

b. Innies @ r6789: r6c37=9 from {1245}=[45]
=> r2c23=[41]
=> 16/3 @ r6c4: r7c34=16-8=8 from {23579}={35} (NP @ r7)
=> 9/3 @ r7c8: r7c8=2, r8c78=[43]
=> 2e: r8c456={256} (NT @ r8,n8)
=> 2d: r79c4=20-8=12=[39]


All naked singles from here.

Notice the shortness of the solving path. People can argue some moves are quite tricky but I still think it's a 0.75. :ugeek:


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PostPosted: Wed Jun 15, 2011 3:39 am 
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Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Assassin 115 V2 "Beijing" by udosuk (August 2008) here
Puzzle Diagram:
Image

The original diagram is no longer accessible. I've used Børge's cover version of udosuk's original diagram
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:5633:5633:5633:5633:7962:3587:3587:3587:3587:4100:1797:1797:7962:7962:7962:3078:3078:2823:4100:1797:2824:2824:7962:4106:4106:3078:2823:1803:1803:7948:2824:7708:4106:5133:3342:3342:7948:7948:7948:7708:7708:7708:5133:5133:5133:1807:1807:7948:4112:7708:4369:5133:3090:3090:2323:5652:4112:4112:4635:4369:4369:2325:2582:2323:5652:5652:4635:4635:4635:2325:2325:2582:4632:4632:4632:4632:4635:6681:6681:6681:6681:
Solution:
+-------+-------+-------+
| 5 8 3 | 6 9 1 | 2 7 4 |
| 9 4 1 | 7 2 5 | 3 8 6 |
| 7 2 6 | 4 8 3 | 9 1 5 |
+-------+-------+-------+
| 2 5 9 | 1 3 4 | 8 6 7 |
| 3 7 8 | 5 6 9 | 1 4 2 |
| 6 1 4 | 8 7 2 | 5 9 3 |
+-------+-------+-------+
| 1 6 5 | 3 4 8 | 7 2 9 |
| 8 9 7 | 2 5 6 | 4 3 1 |
| 4 3 2 | 9 1 7 | 6 5 8 |
+-------+-------+-------+
Quote:
udosuk: Anyway, this is the much anticipated :?: v2:
SSolver 3.2.1 rating: 1.73. Personal rating: 1.75 (easy).
You can find some common moves from v1, but probably need to find a different critical move. :ugeek:
I can guarantee it's easier than defeating the USA men's basketball team. :king:

Afmob: Thanks for this interesting puzzle! Strangely though, I found it a bit easier than V1.
Rating: 1.5. I used an interesting ...

udosuk: Afmob was right that the solving path of this might be a bit shorter than v1, but the critical move ... is definitely more complicated. :idea:
I had a brief look at many walkthroughs posted but don't have time to comment much about them. I don't even have time to write a walkthrough for Ed's attractive A116 Venus. :rambo: Work has been busy, and I have been too excited watching USA getting stomped on the gold medal tally by one of my "parent countries". :king:

Andrew: I eventually found time to have a go at A115V2.
Afmob wrote:
Thanks for this interesting puzzle! Strangely though, I found it a bit easier than V1.
I won't go quite that far ...
There is a lot of similarity between Afmob's walkthrough and my one so there must be a fairly narrow solution path.
I'll also rate A115V2 at 1.5.

Walkthrough by Afmob:
Thanks for this interesting puzzle! Strangely though, I found it a bit easier than V1.

My step 10e (a small forcing chain) finally cracks this Killer but I think the preparing step 10d is more interesting. Also kudos to Andrew, who made me aware of some interesting moves in his walkthrough for V1 (e.g. step 6b in my wt). Now I got the rest of the week to prepare A116. :brickwall:

A115 V2 Walkthrough:

1. R123
a) Innie R1 = R1C5 = 9
b) Innies+Outies N3: -8 = R1C6 - R3C7 -> R1C6 = 1, R3C7 = 9
c) R3C1 = 7, R2C1 = 9
d) 7(3) = {124} locked for N1
e) Innies+Outies N1: R1C4 = R3C3 <> 2,4,7
f) 22(4) = {3568} locked for R1
g) 14(4) = {1247} -> 2,4,7 locked for N3
h) 16(3) = 9{25/34}

2. R1234
a) Outies R123 = 5(2) = [14/23/32]
b) 16(3): R3C6 <> 2
c) Innies R123 = 13(3): R3C34 <> 5 because R3C6 <> 2,6
d) Innies+Outies N1: R1C4 = R3C3 <> 5
e) 7(2) <> {34} since it's a Killer pair of Outies R123
f) Innies R1234 = 20(3) <> 1,2,6 because {569} blocked by Killer pair (56) of 7(2)

3. R789
a) Innie R9 = R9C5 = 1
b) Innies N9 = 26(4) <> 1,3 because {3689} blocked by Killer pair (36) of 9(3)
c) Innies N8 = 27(4) <> 2
d) Outies N9 = 17(3) <> 5 since 5{39/48} blocked by Killer pairs (35,45) of 16(3) @ C6
e) Innies+Outies N9: R9C6 = R7C7 = (4678)
f) Innies+Outies N7: 4 = R9C4 - R7C3 -> R9C4 <> 3,4 and R7C3 <> 6,7,8,9
g) 3 locked in R9C123 @ R9 for N7
h) 9(2) <> 2,6
i) 22(3) = {679} locked for N7 because {589} blocked by Killer pair (58) of 9(2)
j) Innies+Outies N7: 4 = R9C4 - R7C3 -> R9C4 <> 7

4. R789
a) 7 locked in 26(4) @ R9 also locked for N7 because R9C6 = R7C7 (Step 3e)
b) 10(2) <> 3
c) 3 locked in 9(3) @ N9 = 3{15/24}
d) Outies N7 = 20(3) <> 1,2
e) Outies R789 = 10(2) <> 5,9; R6C6 <> 8

5. C456
a) 9 locked in R5C46 @ N5 for R5
b) Innies C6789 = 20(3) <> 2
c) 2 locked in R46C6 @ C6 for N5
d) Outies R123 = 5(2): R4C6 <> 3
e) 16(3) @ C6: R3C6 <> 4
f) 11(3): R3C34 <> 3 because R4C4 = (13) and R3C4 <> 6,8 since 2,4 only possible there
g) Innies+Outies N1: R1C4 = R3C3 <> 3

6. R456
a) 9 locked in R46C3 @ N4 for C3
b) 9 locked in 13(2) + 12(2) = 25(4) @ N6 = {3589/3679/4579}
-> 12(2) <> {48}
c) Outies R789 = 10(2) <> 3,7 because (37) is a Killer pair of 12(2)
d) Killer pair (24) locked in R4C6 + Outies R789 for N5

7. R789 !
a) Outies N7 = 20(3): R79C4 <> 4,6,8 because R6C4 = (468)
b) Innies+Outies N7: 4 = R9C4 - R7C3 -> R7C3 <> 2,4
c) 2 locked in R9C123 @ N7 for R9
d) 6,7 locked in 26(4) @ R9 also locked for N9 (step 4a); 26(4) = 67{49/58}
e) 10(2) <> 4
f) 16(3) = {169/178/358/457} because R7C3 = (15); R7C4 <> 5
g) ! 18(5) = 124{38/56} because {12357} blocked by R79C4 = (3579)
-> 4 locked for N8
h) Innies+Outies N9: R9C6 = R7C7 <> 4
i) 7 locked in R8C23 @ R8 for N7
j) 17(3) <> 9 because [296] blocked by R7C2 = (69)

8. R789 !
a) Hidden Single: R5C6 = 9 @ C6
b) ! Innies R789 = 23(4) = {3578} because R7C2 = (69) blocks 69{17/35} and
(15) only possible @ R7C3
-> R7C3 = 5; {378} locked for R7 and 3 also locked for N8
c) 9(2) = {18} -> R7C1 = 1, R8C1 = 8
d) 10(2) = {19} -> R7C9 = 9, R8C9 = 1
e) R7C2 = 6, R8C3 = 7, R8C2 = 9
f) 18(4) = {2349} -> R9C4 = 9, 4 locked for R9
g) Hidden Single: R5C2 = 7 @ C2, R1C2 = 8 @ C2
h) R1C4 = 6, R1C3 = 3, R1C1 = 5, R3C3 = 6

9. N46+C4
a) 12(2): R6C8 <> 3
b) 5 locked in both 7(2) @ N4 -> 2 locked there for N4
c) 31(5) must have 3 xor 6 -> R5C1 = (36)
d) 5 locked in Innies C1234 = 14(3) = 5{18/27}

10. R456 !
a) Innies R6789 = 16(3): R6C7 <> 6 because R6C3 <> 2,3,7 and R6C5 <> 1,2,9
b) Innies R6789 = 16(3): R6C7 <> 3 because 3{58/67} blocked by
Killer pairs (35,37) of 12(2) and R6C5 <> 4,9
c) 1,2 locked in 20(5) @ N6 = 12{368/458/467}
d) ! 20(5): R5C789 <> 3 because R5C1 = (36) blocks {12368}
e) ! Consider placement of 3 in N6 -> R4C8 <> 9
- i) R4C7 = 3 -> Innies R1234 = {389} -> R4C8 <> 9
- ii) R6C9 = 3 -> R6C8 = 9 -> R4C8 <> 9
f) Hidden Single: R6C8 = 9 @ N6 -> R6C9 = 3

11. N3456
a) 11(2) = {56} -> R3C9 = 5, R2C9 = 6
b) 7(2) @ R6 <> 4
c) R3C6 = 3 -> R4C6 = 4
d) 17(3) = {278} -> R6C6 = 2
e) R6C1 = 6, R4C1 = 2 -> R4C2 = 5
f) 13(2) = {67} -> R4C9 = 7, R4C8 = 6

12. Rest is singles.

Rating: 1.5. I used an interesting cage blocker and a very small forcing chain.
Walkthrough by udusok:
This is my walkthrough for v2

0. Prelims

16/2 @ r2c1={79} (NP @ c1,n1)
7/3 @ r2c2={124} (NT @ n1)
11/2 @ r2c9={29|38|47|56}
7/2 @ r4c1 & r6c1={16|25|34}
13/2 @ r4c8={49|58|67}
12/2 @ r6c8={39|48|57}
9/2 @ r7c1={18|36|45} (not {27} for 16/2 @ r2c1)
22/3 @ r7c2={589|679} (9 @ n7 locked)
9/3 @ r7c8={126|135|234}
10/2 @ r7c9={19|28|37|46}


1. r1239

a. Innies @ r1: r1c5=9

b. Innies @ r9: r9c5=1

c. Innie-outies @ n3: r3c7=r1c6+8
=> r1c6=1, r3c7=9
=> 16/2 @ r2c1=[97]

d. Innie-outies @ n1: r1c4=r3c3
=> r1c4 & r3c3 must be both from {3568}
=> 22/4 @ r1c1={3568} (NQ @ r1)
=> r1c789={247} (NT @ n3)


2. c456 (!)

a. Outies @ n7: r679c4=20 from {3456789}

b. Innies @ n8: r79c46=27 from {3456789}

c. Innies @ c6789: r258c6=20 from {3456789}

d. Outies @ r123: r4c46=5=[14|23|32]
=> 16/3 @ r3c6: r34c6=16-9=7=[34|43|52]

e. 2 @ c6,n5 locked @ r46c6
=> r4c46=[14|32]
=> r34c6=[34|52]

f. Innies @ n2: r1c4+r3c46=13 can't be {355}
Now r3c6 from {35}
=> r13c4 can't have {35}
=> r1c4 & r3c3 must be both from {68} (1d)
=> 11/3 @ r3c3=[623|641|821]

g. Outies @ r789: r6c46=10=[37|46|64|73|82]
=> 9 @ r5,n5,30/5 locked @ r5c46
=> 9 @ c2,n7,22/3 locked @ r78c2


h. Innies @ n5: r46c46=15 with 2 locked @ r46c6
=> r46c46 from {1234678} must be {1248|2346} (4 @ n5 locked)
=> r46c46=[1482|3246|3264]


3. n79,r9,c4 (!)

a. Innie-outies @ n7: r9c4=r7c3+4
=> r7c3 from {12345}, r9c4 from {56789}

b. Innies @ n9: r7c7+r9c789=26
But 10/2 @ r7c9={19|28|37|46} must have 3|6|8|9
=> r7c7+r9c789=26 can't be {3689}, can't have 3

c. Innie-outies @ n9: r7c7=r9c6
=> r7c7 & r9c6 must be both from {45678}

d. 3 @ r9,n7 locked @ r9c123
=> 9/2 @ r7c1={18|45} must have 5|8
=> 22/3 @ r7c2 cant be {589}, must be {679} (NT @ n7)
=> r7c3 from {1245}
=> r9c4=r7c3+4 must be from {5689}

e. r679c4=20 (2a) can have at most one from {468}
Now r6c4 from {468}
=> r79c4 can't have {468}
=> r7c4 from {3579}, r9c4 from {59}
=> r7c3=r9c4-4 must be from {15}


4. r79,n79 (!)

a. Now r7c3+r9c4=[15|59] form Pointing Pair of 5
=> r7c456,r9c123 can't have 5
=> r7c7=r9c6 (3c) can't both be 5
=> r7c7 & r9c6 must be both from {4678}

b. Innies @ r789: r7c3467=23
But r7c2 from {679}
=> r7c467 from {346789} can't be {589|679}, can't sum to 22
=> r7c3 can't be 23-22=1
=> r7c3+r9c4=[59]
=> r7c467=23-5=18 from {34678} with r7c4 from {37}
=> r7c467={378} (NT @ r7)

c. 9/2 @ r7c1 from {148} must be [18]
=> r9c123={234} (NP @ r9)
=> r7c7+r9c789={5678} (NQ @ n9)
=> 10/2 @ r7c9 from {12349}=[91]
=> 22/3 @ r7c2=[697]

(Cleanup)
d. HS @ c2: r5c2=7
=> HS @ c2: r1c2=8
=> 22/4 @ r1c1=[5836], r3c3=6


5. n46,r46 (!!)

a. 31/5 @ r4c3: r456c3+r5c1=31-7=24
{89} @ c3,n4 locked @ r456c3
=> r456c3+r5c1 from {1234689} must be {1689|3489}
=> r456c3 from {1489}
=> r5c1 can't be 4
=> r5c1 from {36}

b. Innies @ r1234: r4c357=20 from {3456789}
But 13/2 @ r4c8={49|58|67} has 5|6|9
=> r4c357=20 can't be {569}, can't have 6

c. r6c4 from {48}
=> 12/2 @ r6c8 can't be {48}, must be [57|75|93]

d. Innies @ r6789: r6c357=16
Now r6c3 from {1489}, r6c5 from {35678}
But 12/3 @ r6c8=[57|75|93] must have 3|5
=> r6c357=16 can't be [466|853]
=> r6c7 can't be 3|6

e. {12} @ n6 locked @ 20/5 @ r4c7

=> 20/5 @ r4c7 from {12345678}={12368|12458|12467}
But r5c1 from {36} & r46c7 have no 6
=> 20/5 @ r4c7 can't be {12368} with {36} @ r5c789
=> r5c789 can't have 3

f. 3 @ n6 locked @ r4c7+r6c9
But r4c46=[14|32] (2e) & 12/2 @ r6c8=[57|75|93]
=> r4c467+r6c89=[14357|14375|14?93|32?93]
=> r4c6+r6c8=[45|47|49|29] must have 4|9
=> 13/2 @ r4c8, seeing r4c6+r6c8, can't be {49}
=> 13/2 @ r4c8={58|67} must have 5|7
=> 12/2 @ r6c8 can't be {57}, must be [93]

(Cleanup)
g. 11/2 @ r2c9 from {568} must be [65]
=> r34c6=[34] (2e)
=> r4c4=1 (2e)
=> r3c4=4 (2f)
=> 7/2 @ r4c1 from {2356} must be [25]
=> 13/2 @ r4c8 from {678} must be [67]
=> 7/2 @ r6c1 from {146} must be [61]


All naked singles from here.


Afmob was right that the solving path of this might be a bit shorter than v1, but the critical move (5f) (and the leading up to it) is definitely more complicated. :idea:


I had a brief look at many walkthroughs posted but don't have time to comment much about them. I don't even have time to write a walkthrough for Ed's attractive A116 Venus. :rambo: Work has been busy, and I have been too excited watching USA getting stomped on the gold medal tally by one of my "parent countries". :king:
Walkthrough by Andrew:
I eventually found time to have a go at A115V2.

Afmob wrote:
Thanks for this interesting puzzle! Strangely though, I found it a bit easier than V1.
I won't go quite that far; having gone as far as I could without using a chain I took quite a long time to find it.

There is a lot of similarity between Afmob's walkthrough and my one so there must be a fairly narrow solution path. Afmob's final breakthrough with a short forcing chain was simpler than my contradiction chain; both were in the same area N6 and used interaction with R4C357.

I'll also rate A115V2 at 1.5 because I used a short contradiction chain.

Here is my walkthrough.

Prelims

a) R23C1 = {79}, locked for C1 and N1
b) R23C9 = {29/38/47/56}, no 1
c) R4C12 = {16/25/34}, no 7,8,9
d) R4C89 = {49/58/67}, no 1,2,3
e) R6C12 = {16/25/34}, no 7,8,9
f) R6C89 = {39/48/57}, no 1,2,6
g) R78C1 = {18/36/45}, no 2
h) R78C9 = {19/28/37/46}, no 5
i) 7(3) cage in N1 = {124}, locked for N1
j) 11(3) cage at R3C3 = {128/137/146/236/245}, no 9
k) 22(3) cage in N7 = {589/679}, 9 locked for N7
l) 9(3) cage in N9 = {126/135/234}, no 7,8,9
m) R1C6789 = {1238/1247/1256/1346/2345}, no 9
n) 31(5) cage in N2 = {16789/25789/34789/35689/45679}, 9 locked for N2
o) 18(5) cage in N8 = {12348/12357/12456}, no 9, 1,2 locked for N8
p) R9C6789 = {2789/3689/4589/4679/5678}, no 1

1. 45 rule on R1 1 innie R1C5 = 9

2. 45 rule on R9 1 innie R9C5 = 1

3. 45 rule on N3 1 innie R3C7 = 1 outie R1C6 + 8 -> R1C6 = 1, R3C7 = 9, R23C1 = [97], clean-up: no 2,4 in R2C9, no 2 in R3C9

4. 45 rule on N1 1 outie R1C4 = 1 innie R3C3 -> R1C4 = {3568}
4a. Naked quad {3568} in R1C1234, locked for R1
4b. Naked triple {247} in R1C789, locked for N3

5. R3C7 = 9 -> R34C6 = 7 = {25/34}, no 6,7,8

6. 45 rule on R123 2 outies R4C46 = 5 = [14/23/32], clean-up: no 2 in R3C6
6a. R4C12 = {16/25} (cannot be {34} which clashes with R4C46), no 3,4
6b. Killer pair 1,2 in R4C12 and R4C46, locked for R4

7. 45 rule on N2 3 remaining innies R1C4 + R3C46 = 13 = {238/256/346}
7a. 2 of {238} must be in R3C4 -> no 8 in R3C4
7b. 5 of {256} must be in R3C6 -> no 5 in R13C4, clean-up: no 5 in R3C3 (step 4)

8. 45 rule on N7 1 outie R9C4 = 1 innie R7C3 + 4, no 6,7,8 in R7C3, no 3,4 in R9C4

9. 45 rule on N9 1 outie R9C6 = 1 innie R7C7, no 1,2 in R7C7, no 9 in R9C6

10. 45 rule on R789 2 outies R6C46 = 10 = [19]/{28/37/46}, no 5, no 9 in R6C4

11. 45 rule on N9 4 innies R7C7 + R9C789 = 26 = {2789/4589/4679/5678} (cannot be {3689} which clashes with 9(3) cage), no 3, clean-up: no 3 in R9C6 (step 9)

12. 3 in R9 locked in R9C123, locked for N7, clean-up: no 6 in R78C1, no 7 in R9C4 (step 8)
12a. 45 rule on N7 4 innies R7C3 + R9C123 = 14 = {1238/2345} (cannot be {1346} which clashes with R78C1, no 6,7

13. 22(3) cage in N7 = {679} (hidden triple in N7)

14. 7 in R9 locked in R9C6789
14a. R7C7 = R9C6 (step 9) -> 7 locked in R7C7 + R9C789, locked for N9, clean-up: no 3 in R78C9
14b. 3 in N9 locked in 9(3) cage = {135/234}, no 6

15. 45 rule on R1234 3 innies R4C357 = 20 = {389/479/578} (cannot be {569} which clashes with R4C12), no 6
15a. 9 of {389/479} must be in R4C3 -> no 3,4 in R4C3

16. 45 rule on N7 3 outies R679C4 = 20 = {389/479/569/578}, no 1,2, clean-up: no 8,9 in R6C6 (step 10)
16a. 6 of {569} must be in R6C4 -> no 6 in R79C4, clean-up: no 2 in R7C3 (step 8)

17. 2 in N7 locked in R9C123, locked for R9
17a. 6 in R9C6789, R7C7 = R9C6 (step 9) -> 6 locked in R7C7 + R9C789, locked for N9, clean-up: no 4 in R78C9

18. 9 in N5 locked in R5C46, locked for R5
18a. 9 in N4 locked in R46C3, locked for C3

19. 9 in N6 locked in R46C89
19a. Combined cage R4689 = 25 = {3589/3679/4579} -> R6C89 = {39/57} (cannot be {48} which is blocked by combined cage combinations), no 4,8

20. R6C46 (step 10) = {28/46} (cannot be {37} which clashes with R6C89), no 3,7
20a. Killer pair 2,4 in R4C46 and R6C46, locked for N5

21. R679C4 (step 16) = {389/479/569/578}
21a. R6C4 = {468} -> no 4,8 in R79C4, clean-up: no 4 in R7C3 (step 8)
21b. 16(3) at R6C4 = {169/178/358/457} (cannot be {349/367} because R7C3 only contains 1,5)
21c. R7C3 = {15} -> no 5 in R7C4

22. Killer pair 1,5 in R78C1 and R7C3, locked for N7

23. 45 rule on R6789 3 innies R6C357 = 16 = {169/178/349/457} (cannot be {259/358/367} which clash with R6C89, cannot be {268} which clashes with R6C46), no 2
23a. 3,6 of {169/349} must be in R6C5 -> no 3,6 in R6C37

24. 2 in N6 locked in R5C789, locked for R5
24a. 2 in N4 locked in R46C12 -> R46C12 must also contain 5 (to make 7(2) cage sum in R4C12 or R6C12), locked for N4

25. 45 rule on R789 4 innies R7C3467 = 23 = {1589/3569/3578} (cannot be {1679} which clashes with R7C2, cannot be {3479} because R7C3 only contains 1,5, cannot be {4568} because R7C4 only contains 3,7,9), no 4, clean-up: no 4 in R9C6 (step 9)
25a. 5,6 of {3569} must be in R7C37 -> no 6 in R7C6

26. 4 in N8 locked in 18(5) cage = {12348/12456}, no 7

27. 45 rule on C6789 3 innies R258C6 = 20 = {389/479/569/578}, no 2
27a. 4 in {479} must be in R8C6, no 4 in R2C6
27b. 9 of {389/569} must be in R5C6, no 3,6 in R5C6

28. 2 in C6 locked in R46C6, locked for N5, clean-up: no 3 in R4C6 (step 6), no 4 in R3C6 (step 5)

29. 45 rule on N1 3 outies R134C4 = 11 = {128/146/236}
29a. 3 of {236} must be in R4C3 -> no 3 in R13C4, clean-up: no 3 in R3C3 (step 4)
29b. 2,4 of {146/236} must be in R3C4 -> no 6 in R3C4

30. 45 rule on N9 3 outies R679C6 = 17 = {269/278/368/467} (cannot be {359/458} which clash with R34C6), no 5, clean-up: no 5 in R7C7 (step 9)

31. R7C3467 (step 25) = {3569/3578} (cannot be {1589} because 1,5 only in R7C3) -> R7C3 = 5, 3 locked in R7C46, locked for R7 and N8, R9C4 = 9 (step 8), clean-up: no 6 in R6C4 (step 21c), no 4 in R6C6 (step 20), no 4 in R78C1
31a. R7C3467 = {3578} (only remaining combination), locked for R7 -> R78C1 = [18], clean-up: no 6 in R46C2, no 2 in R7C9, no 2,9 in R8C9
31b. R78C9 = [91], R7C2 = 6, R8C23 = [97], clean-up: no 4 in R4C8, no 3 in R6C8, no 5 in R8C78 (step 14b), no 6 in R9C6 (step 9)
31c. Naked triple {234} in R9C123, locked for R9

32. R5C6 = 9 (hidden single in C6)
32a. R258C6 (step 27) = {479/569} (cannot be {389} because 3,8 only in R2C6), no 3,8

33. R5C2 = 7 (hidden single in C2)
33a. 7 in N5 locked in R46C5, locked for C5

[I had the following nice step, which has been available for a few moves
34. 45 rule on N5 4 innies R46C46 = 15 = {1248/2346}
34a. R4C46 = [14/32], R6C46 = [46/82] -> R46C4 = [18/34]
34b. R134C4 (step 29) = {146/236} (cannot be {128} which clashes with R46C4 = [18]) -> R1C4 = 6, R3C3 = 6 (step 4), clean-up: no 5 in R2C9, no 5 in R8C6 (step 32a)
34c. Killer pair 3,4 in R34C4 and R67C4, locked for C4
34d. 5 in C6 locked in R23C6, locked for N2
Then I realised that there is a hidden single which makes this unnecessary]

34. R1C2 = 8 (hidden single in C2), R1C4 = 6, R1C13 = [53], R3C3 = 6, clean-up: no 5 in R2C9, no 2 in R46C2, no 5 in R8C6 (step 32a)
34a. R3C3 = 6 -> R34C4 = 5 = [23/41]
34b. R7C3 = 5 -> R67C4 = 11 = [47/83]
34c. Killer pair 3,4 in R34C4 and R67C4, locked for C4
34d. 5 in C6 locked in R23C6, locked for N2
34e. 2 in N4 locked in R46C1, locked for C1

35. 45 rule on C1234 3 innies R258C4 = 14 = {158/257}
35a. 1 of {158} must be in R5C4 -> no 8 in R5C4
35b. 7,8 only in R2C4 -> R2C4 = {78}

36. 31(5) cage in N4 = 789{16/34}
36a. 3,6 only in R5C1 -> R5C1 = {36}

37. 31(5) cage in N2 = {25789/34789}
37a. Killer pair 2,4 in R23C5 and R7C5, locked for C5

38. 20(5) cage in N6 = {12368/12458/12467}
38a. 6 of {12368} must be in R5C789 -> no 3 in R5C789 because R5C1 = {36}

39. 45 rule on C9 5 innies R14569C9 = 24 = {23478/24567}
39a. R6C9 of {24567} must be {57} making R6C89 naked pair {57} -> no 5 in R45C9, clean-up: no 8 in R4C8
39b. 4 of {24567} must be in R4C9 (only way to place 9 in N6 for this combination) -> no 6 in R4C9; clean-up: no 7 in R4C8
39c. 3 of {23478} must be in R6C9 and R4C9 must then be {78} (step 19a) making naked pair {78} in R49C9 -> no 8 in R5C9

40. R4C89 cannot be [94]
R4C89 = [94] => R4C357 = {578} (step 15) -> cannot be place 3 in N6 because no 3 in R4C7 and combined cage R46C89 (step 19a) cannot contain both of 3,4
40a. R6C8 = 9 (hidden single in N8), R6C9 = 3, clean-up: no 8 in R23C9, no 4 in R6C12
40b. R23C9 = [65]

41. R5C1 = 3 (hidden single in N5), R456C3 = {489} (step 36), locked for C3 -> R9C123 = [432], R2C3 = 1

42. Naked pair {38} in R2C78, locked for R2 and N3 -> R3C8 = 1

and the rest is naked singles and a cage sum.


Last edited by Andrew on Fri Jun 17, 2011 8:54 pm, edited 1 time in total.

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PostPosted: Wed Jun 15, 2011 4:06 am 
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Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Messy One #5 by Nasenbaer (August 2008) here
Puzzle Diagram:
Image
Images with "udosuk style Killer Cages" by Børge:
Image     Image
Code: Select, Copy & Paste into solver:
3x3::k:3328:3328:3074:2307:2307:6917:6917:6917:6917:3328:3074:3074:7180:7180:2318:3599:6917:6917:5394:5394:3604:3604:7180:2318:3599:3097:3097:3611:5394:3604:4382:7180:2848:2848:5154:3097:3611:2597:4382:4382:3368:3368:2848:5154:3097:6189:2597:3887:3376:3376:3368:4659:5154:7733:6189:2597:3887:3887:3887:4659:4659:7733:7733:6189:3136:3136:3136:7491:1860:1860:3398:7733:6189:6189:6189:7491:7491:7491:1860:3398:7733:
Solution:
+-------+-------+-------+
| 1 9 4 | 6 3 5 | 7 2 8 |
| 3 2 6 | 9 7 8 | 5 1 4 |
| 7 8 5 | 2 4 1 | 9 6 3 |
+-------+-------+-------+
| 5 6 7 | 4 8 2 | 3 9 1 |
| 9 4 8 | 5 1 3 | 6 7 2 |
| 2 1 3 | 7 6 9 | 8 4 5 |
+-------+-------+-------+
| 8 5 9 | 1 2 6 | 4 3 7 |
| 6 7 2 | 3 5 4 | 1 8 9 |
| 4 3 1 | 8 9 7 | 2 5 6 |
+-------+-------+-------+
Quote:
Nasenbaer: So, here's my donation for Messy One round 2. :cheesey: Actually it's Messy One #5v6 ;) . Tuning it for the right rating (and removing a uniqueness problem I overlooked :brickwall: ) took a lot of time. I think it's really messy, and the rating now is 1.02 although for me it feels more like a 0.75. There is also a nice shortcut (which isn't found by SSolver).
Have fun!

Afmob: Thanks for this fun Killer, Nasenbaer! I guess I didn't found the shortcut since SudokuSolver's path is not much different from mine though certainly longer. :cheesey:
Rating: 0.75 - (Hard) 0.75.

Jean-Christophe: I went the same way as Afmob. But after solving, I found this:
Hidden Text:
Innies @ r12 -> r2c4..7 = 29 = {5789} (NQ @ r2)

Nasenbaer: Actually I was thinking about this:
Hidden Text:
Outies of n7 = h(3+4) = 16 -> r5c2 r6c123 = {1234} and r7c45 r8c4 = {123}
Not an obvious move, but since I knew what to look for... :whistle:

Andrew: Thanks Nasenbaer for a fun puzzle. I think this was the easiest Messy One so far.
I was amazed when I went through Afmob's walkthrough to find that I hadn't used his first move. I'm not sure how I missed that one. Maybe it's because of the format I use on my Excel worksheet with coloured cages. However when I now look at my worksheet it's obvious. As a result my walkthrough is very different but probably not significantly slower or harder.
I also missed the shortcuts posted by J-C and Nasenbaer; I feel I ought to have seen J-C's one. Nasenbaer's is one that I think is much easier for the puzzle creator to see than for solvers to spot.
I'll rate my walkthrough at Hard 0.75 because a few of my steps are slightly harder than a normal 0.75.

azpaull: I think my breakthrough - and first placement - finally came from analyzing the
Hidden Text:
outies on Nonet 9, and that helped me work Column 7.
(Sorry, I'll have to work on my walk-thru skills - I have a tendency to do these puzzles in little bits and pieces in various places, instead of sitting at a computer or with a second sheet of paper.)

I'll give it a 1.0.

(P.S. I know I probably didn't need to use *hide", but I never have had the chance to use it, and I wanted the practice. :) )

Walkthrough by Afmob:
Thanks for this fun Killer, Nasenbaer! I guess I didn't found the shortcut since SudokuSolver's path is not much different from mine though certainly longer. :cheesey:

Messy One #5 Walkthrough:

1. C6789
a) Outies C89 = 12(2) <> 1,2,6
b) Outies N3 = 8(2+1) <> 7,8,9; R45C9 <> 5,6 because R1C6 >= 3
c) Killer pair (56) locked in 14(2) + 12(4) for N3
d) Outies C89 = 12(2): R1C7 <> 3,4
e) 7 locked in 27(6) @ N3 = 1257{39/48} -> 1,2 locked for N3
f) 12(4) = 12{36/45} -> R45C9 = {12} locked for C9+N6
g) Outies N3 = 8(3+1): R1C6 = 5 since R45C9 = {12}
h) Outies C89 = R1C7 = 7

2. C456
a) Hidden triple (123) in R157C5 for C5 -> R157C5 = {123}
b) 9(2) @ R1 = [63/81]
c) 28(4) = {4789} since {5689} blocked by R1C4 = (68)
d) Hidden Single: R6C5 = 6 @ C5 -> R6C4 = 7
e) 28(4) = {4789} -> 7 locked for C5+N2
f) 9(2) @ C6 = {18/36}
g) Naked quad (1368) locked in both 9(2) for N2
h) 28(4) = {4789} -> R4C5 = 8 and 4 locked for N2
i) 29(4) = {5789} -> R9C6 = 7, R9C4 = 8; {59} locked for C5+N8
j) R1C4 = 6 -> R1C5 = 3
k) 13(3) = {139} -> R5C5 = 1, {39} locked for C6+N5

3. N245
a) 17(3) = {458} because R45C4 = (245) -> R5C3 = 8; 4 locked for C4+N5
b) R2C4 = 9, R3C4 = 2
c) 14(3) = 2{39/57}
d) 14(2) = {59} locked for C1+N4
e) 21(3) = 8{49/67} since (59) only possible @ R3C2 -> 8 locked for R3+N1

4. R123
a) R3C6 = 1, R2C6 = 8
b) 14(2) = {59} -> R2C7 = 5, R3C7 = 9
c) 12(4) = {1236} -> R5C9 = 2, R4C9 = 1, {36} locked for R3+N3
d) 21(3) = {678} -> R4C2 = 6, {78} locked for R3+N1
e) R3C3 = 5 -> R4C3 = 7

5. N178
a) 10(3) = 5{14/23} because R5C2 = (34) -> R7C2 = 5; R6C2 <> 3,4
b) 24(6) = {123468} -> 6,8 locked for N7 and 8 also locked for C1
c) Hidden Single: R8C2 = 7 @ N7
d) 12(3) = 7[23/41]
e) Hidden Single: R7C6 = 6 @ N8
f) 18(3) = {468} -> 4,8 locked for C7
g) Hidden Single: R1C2 = 9 @ C2
h) 13(3) = {139} -> R1C1 = 1, R2C1 = 3

6. N9
a) 13(2) = [49/85/94]

7. Rest is singles.

Rating: 0.75 - (Hard) 0.75.
Walkthrough by Andrew:
Thanks Nasenbaer for a fun puzzle. I think this was the easiest Messy One so far.

I was amazed when I went through Afmob's walkthrough to find that I hadn't used his first move. I'm not sure how I missed that one. Maybe it's because of the format I use on my Excel worksheet with coloured cages. However when I now look at my worksheet it's obvious. As a result my walkthrough is very different but probably not significantly slower or harder.

I also missed the shortcuts posted by J-C and Nasenbaer; I feel I ought to have seen J-C's one. Nasenbaer's is one that I think is much easier for the puzzle creator to see than for solvers to spot.

I'll rate my walkthrough at Hard 0.75 because a few of my steps are slightly harder than a normal 0.75. I originally used combined cage R2367C7 to lock 8 for C7 but that didn't help much and would have pushed my rating up to at least Easy 1.0 so I removed it.

Here is my walkthrough, a short one for me.

Prelims

a) R1C56 = {18/27/36/45}, no 9
b) R23C6 = {18/27/36/45}, no 9
c) R23C7 = {59/68}
d) R45C1 = {59/68}
e) R6C45 = {49/58/67}, no 1,2,3
f) R89C8 = {49/58/67}, no 1,2,3
g) 21(3) cage at R3C1 = {489/579/678}, no 1,2,3
h) 11(3) cage at R4C6 = {128/137/146/236/245}, no 9
i) R456C8 = {389/479/569/578}, no 1,2
j) R567C2 = {127/136/145/235}, no 8,9
k) 7(3) cage at R8C6 = {124}, CPE no 1,2,4 in R8C89, clean-up: no 9 in R9C8
l) 28(4) cage at R2C4 = {4789/5689}, no 1,2,3, CPE no 8 in R1C5, clean-up: no 1 in R1C4
m) 12(4) cage at R3C8 = {1236/1245}, no 7,8,9, CPE no 1,2 in R12C9
n) 29(4) cage in N8 = {5789}, locked for N8

1. 45 rule on C6789 1 innie R9C6 = 1 outie R5C5 + 6 -> R5C5 = {123}, no 5 in R9C6

2. 45 rule on N1 1 outie R4C2 = 1 innie R3C3 + 1, no 1,2,9 in R3C3

3. 45 rule on C123 4 outies R3C4 + R7C45 + R8C4 = 1 innie R5C3
3a. Min R3C4 + R7C45 + R8C4 = 7 -> min R5C3 = 7
3b. Min R7C45 + R8C4 = 6 -> max R3C4 = 3
3c. Max R7C45 + R8C4 = 8 = {123/124/134}, no 6, 1 locked in R7C45 + R8C4, locked for N8
3d. R7C6 = 6 (hidden single in N8), clean-up: no 3 in R23C6
3e. R7C6 = 6 -> R67C7 = 12 = {39/48/57}, no 1,2
3f. 1 in 7(3) cage at R8C6 locked in R89C7, locked for C7 and N9
3g. Min R45C7 = 5 -> no 7,8 in R4C6

4. 45 rule on N9 3 remaining outies R6C79 + R8C6 = 17, max R8C6 = 4 -> min R6C79 = 13, no 1,2,3, clean-up: no 9 in R7C7 (step 3e)
4a. 1 in N6 locked in R45C9, locked for 12(4) cage at R3C8, no 1 in R3C89

5. 1 in N3 locked in R12C8, locked for 27(6) cage, no 1 in R1C6
5a. 45 rule on C8 4 innies R1237C8 = {1236/1245}, no 7,8,9

6. 45 rule on N3 3 outies R1C6 + R45C9 = 8
6a. Min R45C9 = 3 -> max R1C6 = 5
6b. Min R1C6 = 2 -> max R45C9 = 6, no 6

7. 45 rule on C789 3 remaining outies R148C6 = 11 = {245} (only remaining combination), locked for C6, clean-up: no 7 in R23C6
7a. Naked pair {18} in R23C6, locked for C6 and N2

8. 3 in C6 locked in R56C6 for N5
8a. 13(3) cage in N5 = {139} (only remaining combination) -> R5C5 = 1, R56C6 = {39}, locked for C6 and N5 -> R9C6 = 7, clean-up: no 4 in R6C45, no 6 in R8C8
8b. R4C9 = 1 (hidden single in C9)

9. R4C5 = 8 (hidden single in 28(4) cage at R2C4), clean-up: no 6 in R5C1, no 5 in R6C45
9a. R9C4 = 8 (hidden single in C4), clean-up: no 5 in R8C8
9b. Naked pair {59} in R89C5, locked for C5, clean-up: no 4 in R1C4
9c. R2C4 = 9 (hidden single in C4), clean-up: no 5 in R3C7
9d. 28(4) cage at R2C4 = {4789} (only remaining combination) -> R23C5 = {47}, locked for C5 and N2 -> R6C45 = [76], clean-up: no 2,5 in R1C4, no 2 in R1C5, no 5 in R7C7 (step 3e)
9e. R1C45 = [63], R3C4 = 2, R1C6 = 5, R7C5 = 2, R8C6 = 4, R4C6 = 2
9f. R5C9 = 2 (hidden single in 142(4) cage at R3C8), R3C89 = {36/45}

10. R3C4 = 2, R78C4 = {13} = 4, R7C5 = 2 -> R5C3 = 8 (step 3), clean-up: no 6 in R4C1
10a. Naked pair {59} in R45C1, locked in C1 and N4
10b. X-Wing on 5 in R45C14, 5 locked for R45

11. R3C4 = 2 -> R34C3 = 12 = [57], clean-up: no 4 in R3C89 (step 9f)
11a. Naked pair {36} in R3C89, locked for R3 and N3, clean-up: no 8 in R23C7
11b. R23C7 = [59], clean-up: no 3,7 in R7C7 (step 3e)
11c. Naked pair {48} in R67C7, locked for C7

12. R45C7 = {36} (hidden pair in C7), locked for N6
12a. R1C7 = 7 (hidden single in C7)
12b. Naked pair {48} in R12C9, locked for C9 and N3

13. R456C8 = {479} (only remaining combination, cannot be {578} because 5,8 only in R6C8) = R5C8 = 7, R46C8 = {49}, locked for C8 and N6 -> R67C7 = [84], R6C9 = 5, R7C8 = 3, R3C89 = [63], R89C8 = [85], R89C5 = [59], R9C9 = 6, R7C34 = [91], R78C9 = [79], R7C12 = [85], R8C4 = 3

14. Naked pair {47} in R3C15, locked for R3 -> R3C2 = 8, R23C6 = [81], R12C9 = [84], R23C5 = [74], R3C1 = 7, R4C2 = 6 (prelim g), R45C7 = [36]

15. R1C2 = 9 (hidden single in R1), R12C1 = 4 = [13]

16. 24(6) cage at R6C1 must contain 1 -> 1 locked in R9C23, locked for R9 and N7

and the rest is naked singles


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