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 Post subject: A120 Twin Towers
PostPosted: Fri Sep 12, 2008 10:44 pm 
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Grand Master
Grand Master

Joined: Wed Apr 16, 2008 1:16 am
Posts: 1044
Location: Sydney, Australia
Change of plan. :D I enjoyed this "V2" so much it's now "V1"! [edit: back to being the V2. A120 follows]. I'll give it an estimated rating of "old-style" 1.50 (ie Weekly Assassin level 1.5..and yes, still bothered by the "new" 1.5).

You don't want to know the SSscore. It's so far wrong it's not worth publishing. :oops: Instead, JSudoku, after "recursively solve" says, "17 guesses". I normally look for "16 guesses" to (often) get an SSscore of around 1.25. Hence, didn't throw the "17 guesses" one out. Boy, am I glad.

I have the old one ready to go, so if anyone is desperate for an easier one (though no pushover) I can post that also [Gone: see below]. But only if asked for.

The cage design was inspired by a solving strategy idea, which didn't work for this one. Inadvertently ended up with the Twin Towers! Still have vivid memories of that terrible day 7 years ago.

Assassin 120 V2 [edit: see following puzzle for A120]

Image

Code: copy, select & paste into solver:
3x3::k:4096:4096:4096:5123:5123:5123:3846:3846:3846:3337:4096:2571:2571:6402:6402:6402:3846:1553:3337:5139:5139:5139:6402:5399:5399:5399:1553:5633:5139:2845:4638:4638:4638:2849:5399:6179:5633:5633:2845:4135:4135:4135:2849:6179:6179:5633:6190:2845:2864:2864:2864:2849:4660:6179:1855:6190:6190:6190:5178:4660:4660:4660:3134:1855:5704:5178:5178:5178:2564:2564:6222:3134:5704:5704:5704:2379:2379:2379:6222:6222:6222:

Solution:
+-------+-------+-------+
| 2 9 4 | 7 8 5 | 3 6 1 |
| 6 1 8 | 2 9 3 | 7 5 4 |
| 7 3 5 | 4 6 1 | 8 9 2 |
+-------+-------+-------+
| 1 8 2 | 6 5 7 | 4 3 9 |
| 5 7 6 | 3 4 9 | 2 1 8 |
| 9 4 3 | 8 1 2 | 5 7 6 |
+-------+-------+-------+
| 4 6 9 | 5 3 8 | 1 2 7 |
| 3 2 1 | 9 7 4 | 6 8 5 |
| 8 5 7 | 1 2 6 | 9 4 3 |
+-------+-------+-------+


SS(v3.2.1)score = wrong
JSudoku = "17 guesses"

Gremlins fixed.
Ed


Last edited by Ed on Mon Sep 15, 2008 10:29 am, edited 1 time in total.

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 Post subject: Re: A120 Twin Towers
PostPosted: Sat Sep 13, 2008 8:33 pm 
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Joined: Sat Sep 13, 2008 8:28 pm
Posts: 13
This was a hard one for me. I did have to guess twice. The most important guess was

R1C4:R1C6 is either 479 or 578.


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PostPosted: Sun Sep 14, 2008 9:50 pm 
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Grand Master
Grand Master

Joined: Mon Apr 21, 2008 9:44 am
Posts: 310
Location: MV, Germany
What a challenging Killer!

It took me quite some time to find the breakthrough move (step 8c,d) but this was only because I was looking for a different breakthrough path (combo elimination via Killer subsets) but this led to nothing. After I changed my approach and looked for different moves to progress (e.g. using forcing chains) the first placement came quicky and then it was over.

A120 V2 Walkthrough:

1. R123
a) Outies R1 = 6(2) = {15/24}
b) Innies N3 = 24(3) = {789} locked for N3
c) 21(4) = {1389/1479/1578/2379/2478} <> 6 because R3C78 = (789); R3C6+R4C8 = (12345)
d) 15(4) = 36{15/24} -> 3,6 locked for R1
e) 20(3) = 7{49/58} -> 7 locked for R1+N2
f) 16(4) = 12{49/58} -> 1,2 locked for N1
g) 10(2): R2C3 <> 3 and R2C4 <> 8,9
h) Innies+Outies N2: R2C7 = R23C4+R3C6 -> R3C4 <> 8,9; R23C4+R3C6 cannot be {123} because R3C7 >= 7
i) Hidden Killer triple (123) in 25(4) for N2 since R23C4+R3C6 <> {123} -> 25(4) <> 4
j) 25(4) can only have one of (789) @ N2 since 20(3) must have two of them -> 25(4) <> 1
k) 3 locked in R3C23 @ 20(4) @ N1 for R3; R4C2 <> 3

2. R456
a) Outies R123 = 11(2) = [65/74/83/92]
b) Innies N6 = 10(2) = [28/37/46]
c) Outies R123 = 11(2) <> 6
d) Innies N4 = 12(2) = [75/84/93]
e) 21(4) <> 5 because (15) only possible @ R3C6

3. R123
a) R3C5 <> 8,9 since it sees all 8,9 of N3
b) 25(4) = 9{268/358/367} -> 9 locked for R2
c) 10(2) <> 1
d) 1 locked in R3C46 @ N2 for R3
e) 20(4) = 3{179/269/278/458/467} (from step 1k)
f) Outies R12 = 15(3) = {258/267/456} <> 9 because {249} blocked by Killer triple (124) of
20(4) and R3C6 = (124)
g) 13(2) <> 4
h) 6(2): R2C9 <> 5
i) 3 locked in Innies N1 = 16(3) = 3{49/58/67}: R3C23 <> 4,8 because R2C3 <> 3,5,9

4. N6
a) 9 locked in 24(4) = 9{168/258/267/348/357/456}
b) 24(4) @ N6 <> 9{267/348} because (267,348) are Killer triples of Innies N6
c) 11(3) <> {236} since it's a Killer triple of Innies N6

5. R456 !
a) ! 21(4): R3C6 <> 4 because (478) is a Killer triple of Outies R12
b) 21(4) must have 3 xor 4 -> R4C8 <> 2
c) Outies R123 = 11(2) <> 9
d) Innies N4 = 12(2) <> 3
e) Innies N6 = 10(2) <> 8
f) 11(3) @ N4, 11(3) @ N5 <> 5 because {245} blocked by R6C2 = (45)
g) 11(3) @ N5 <> 4 since (46) is a Killer pair of Outies R789
h) Innies N46 = 12(2) + 10(2) = 22(4) = [7456/8347]
-> CPE: R4C13+R6C79 <> 4, R4C79+R6C13 <> 7
i) 4 locked in R6C123 @ R6 for N4
j) 18(3) <> 2 because (27) is a Killer pair of 11(3) @ N5

6. C789 !
a) 6(2) + 12(2) = 18(4) = {1359/1458/2349/2457}
b) ! 24(4) @ N6 <> 2,4 because combined cage 18(4) has two of (2589, 4569) @ C9
c) 2 locked in 11(3) @ N6 = 2{18/45}; 2 locked for C7
d) 24(4) @ N6: R5C8 <> 3,7 because (59) is a Killer pair of combined cage 18(4)
e) 24(4) @ N6: R456C9 <> 5 because (35) is a Killer pair of combined cage 18(4)

7. R456 !
a) 9 locked in Innies R6789 = 23(4) = 9{158/167/248/257/347/356}
b) Innies R6789 = 23(4) <> 7 because {1679} blocked by R6C8 = (67), {2579} blocked by
Killer pair (57) of Outies R789 and {3479} unplaceable since R6C7 = (1258)
c) ! Consider placement of 7 in R6 -> R4C456+R6C9 <> 3
- i) 7 in 11(3) @ N5 = {137} locked for R6+N5
- ii) 7 in Innies N6 = [37] -> R4C8 = 3
d) 24(4) @ N6: R5C9 <> 3 because 7 only possible there
e) 3 locked in R4C89 @ N6 for R4

8. R456 !
a) Hidden Killer pair (69) in Innies R1234 because 18(3) cannot be {369}
-> 2 locked in Innies R1234 = 16(4) = 2{149/167/356} <> 8
b) Innies R1234 = 16(4): R4C7 <> 5 because 3 must be in R4C9 @ 24(4) = {3579} -> R4C7 <> 5
c) ! Innies N46 = 22(4) = [7456/8347] -> Either R6C2 = 5 xor R4C8 = 3
d) ! Innies R1234 = 16(4) = 12{49/67} because [5623] leaves neither 5 in R6C2 nor 3 in R4C8
-> 1 locked for R4
e) Hidden Single: R4C8 = 3 @ N6
f) Innies N46 = 21(4) = [8347] -> R4C2 = 8, R6C2 = 4, R6C8 = 7
g) 24(4) @ N6 = {1689} locked for N6
h) Naked triple (245) locked in R456C7 for C7
i) 21(4) = 39{18/27} -> 9 locked for R3+N3

9. R123
a) 9 locked in R2C56 @ R2 for N2
b) 20(3) = {578} locked for R1+N2
c) 16(4) = {1249} -> 4 locked for R1+N1
d) 20(4) = 38{27/45}
e) Hidden Single: R3C6 = 1 @ N2
f) 21(4) = {1389} -> 8 locked for R3+N3
g) R2C7 = 7
h) 25(4) = {3679} -> R3C5 = 6; 3 locked for R2
i) 13(2) = [67/85]
j) Hidden Single: R2C8 = 5 @ R2

10. R789
a) Innies N9 = 9(3) = {126} -> R7C8 = 2; {16} locked for C7+N9
b) 24(4) @ N9 = {3489} locked for N9 because R89C8+R9C7 = (3489); 3 also locked for R9
c) 9(3) = {126} locked for R9+N8
d) 18(4) = 27{18/36} -> R7C6 = (38)
e) 24(4) @ N7 = 49{38/56} because R7C9 = (57) blocks 4{578} -> 9 locked for R7
f) 4,7 locked in Outies R89 = 14(3) = {347} -> R7C9 = 7, {34} locked for R7
g) R8C9 = 5, R7C6 = 8 -> R7C7 = 1, R8C7 = 6 -> R8C6 = 4
h) 20(4) = {1379} since R7C5+R8C45 = (379) -> R7C5 = 3, R8C3 = 1; {79} locked for R8

11. N145
a) 11(3) @ N4 = {236} locked for C3+N4
b) R2C3 = 8 -> R2C4 = 2, R3C4 = 4, R2C1 = 6 -> R3C1 = 7
c) 11(3) @ N5 = {128} because {236} blocked by R6C3 = (236) -> R6C6 = 2; {18} locked for R6+N5

12. Rest is singles.

Rating: (Hard) 1.5. I used lots of Killer triples and a small forcing chain.


Last edited by Afmob on Thu Jul 22, 2010 5:23 am, edited 8 times in total.

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 Post subject: Re: A120 Twin Towers
PostPosted: Mon Sep 15, 2008 10:23 am 
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Grand Master
Grand Master

Joined: Wed Apr 16, 2008 1:16 am
Posts: 1044
Location: Sydney, Australia
:bigoops: Sorry guys. Goofed badly. Did a double-check after seeing Afmob needed forcing chains and found a really silly mistake. Back to plan A! I'll make the first one the V2. This time the Twins are not quite identical.

Estimated rating of 1.25.

[edit: forgot to say welcome to herschko!]

Assassin 120
Image
"Code: select, copy & paste into solver":
3x3::k:5376:5376:5376:4355:4355:4355:4614:4614:4614:2057:5376:3595:3595:5122:5122:5122:4614:2577:2057:3603:3603:3603:5122:5143:5143:5143:2577:3585:3603:5405:3614:3614:3614:5153:5143:4643:3585:3585:5405:3879:3879:3879:5153:4643:4643:3585:4654:5405:4144:4144:4144:5153:5428:4643:4415:4654:4654:4654:4154:5428:5428:5428:1086:4415:4424:4154:4154:4154:3332:3332:5198:1086:4424:4424:4424:4939:4939:4939:5198:5198:5198:

Solution:
+-------+-------+-------+
| 6 3 4 | 7 8 2 | 1 5 9 |
| 7 8 9 | 5 4 1 | 6 3 2 |
| 1 2 5 | 6 9 3 | 4 7 8 |
+-------+-------+-------+
| 2 1 8 | 4 3 7 | 9 6 5 |
| 3 4 6 | 9 1 5 | 8 2 7 |
| 5 9 7 | 8 2 6 | 3 1 4 |
+-------+-------+-------+
| 8 6 2 | 1 5 4 | 7 9 3 |
| 9 7 3 | 2 6 8 | 5 4 1 |
| 4 5 1 | 3 7 9 | 2 8 6 |
+-------+-------+-------+

SS(V3.2.1)score = 1.27
JSudoku = "15 guesses"

Enjoy this time!
Ed


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 Post subject: Re: A120 Twin Towers
PostPosted: Tue Sep 16, 2008 3:33 am 
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Grand Master
Grand Master

Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Thanks Ed. Great timing. I'd reached the stage with what is now the V2 where after prelims and 18 steps I was looking for contradiction moves to make progress. I don't like using those moves or forcing chains, which I rarely find, so early in a puzzle.

The SS score looks spot on. I'll rate A120 at 1.25 because I used combined cages which were partly formed from hidden cages and then for the final breakthrough I used 4 innies.

Here is my walkthrough. Thanks Afmob for pointing out that step 18 was incorrect because of an elimination that I'd made earlier. I've re-worked step 18 and edited the remaining steps. I've also corrected typos in steps 18c and 25.

Prelims

a) R23C1 = {17/26/35}, no 4,8,9
b) R2C34 = {59/68}
c) R23C9 = {19/28/37/46}, no 5
d) R78C1 = {89}, locked for C1 and N7
e) R78C9 = {13}, locked for C9 and N9, clean-up: no 7,9 in R23C1
f) R8C67 = {49/58/67}, no 1,2,3
g) R456C3 = {489/579/678}, no 1,2,3
h) R456C7 = {389/479/569/578}, no 1,2
i) R9C456 = {289/379/469/478/568}, no 1
j) 14(4) cage at R3C2 = {1238/1247/1256/1346/2345}, no 9
k) 14(4) cage at R4C1 = {1238/1247/1256/1346/2345}, no 9

1. 45 rule on R1 2 outies R2C28 = 11 = {29/38/47/56}, no 1

2. 45 rule on R123 2 outies R4C28 = 7 = {16/25/34}, no 7,8,9

3. 45 rule on R9 2 outies R8C28 = 11 = [29/38]/{47/56}, no 1 in R8C2, no 2 in R8C8

4. 45 rule on R789 2 outies R6C28 = 10 = {19/28/37/46}, no 5

5. 45 rule on N4 2 innies R46C2 = 10 = [19/28/37/46/64], no 5 in R4C2, no 1,2,3 in R6C2, clean-up: no 2 in R4C8 (step 2), no 7,8,9 in R6C8 (step 4)
5a. 1 in N6 locked in R456C8, locked for C8

6. 45 rule on R12 3 outies R3C159 = 18 = {189/279/369/378/459/468/567}
6a. 9 of {189} must be in R3C5 -> no 1 in R3C5
6b. 2 of {279} must be in R3C9 -> no 2 in R3C15, clean-up: no 6 in R2C1
6c. 9 of {369} must be in R3C5, 6 of {468} must be in R3C1 -> no 6 in R3C5

7. 45 rule on N9 3 innies R7C78 + R8C7 = 21 = {489/579/678}, no 2
7a. 2 in N9 locked in R9C789, locked for R9

8. 1 in R9 locked in R9C123, locked for N7
8a. 17(4) cage in N7 = {1367/1457}, no 2, 7 locked for N7, clean-up: no 9 in R8C8 (step 3)

9. R9C456 = {379/469/478/568}
9a. Hidden killer pair 8,9 in R9C456 and R9C789 for R9 -> R9C789 must have one of 8,9
9b. 20(4) cage at N9 = {2459/2468} (cannot be {2567} which doesn’t contain 8 or 9), no 7, 4 locked for N9, clean-up: no 4 in R8C2 (step 3), no 9 in R8C6
9c. 8,9 must be in R9C789 -> no 8 in R8C8, clean-up: no 3 in R8C2 (step 3)

10. 45 rule on R89 3 outies R7C159 = 16 = {169/178/349/358} (cannot be {259/268/457} because R7C9 only contains 1,3, cannot be {367} because R7C1 only contains 8,9)
10a. 4,5,6,7 only in R7C5 -> R7C5 = {4567}

11. Combined cage R8C2678 using R8C28 (step 3) = 24 = {4569/4578}, 4,5 locked for R8
11a. 4,9 of {4569} must be in R8C67, 5,8 of {4578} must be in R8C67 -> no 6,7 in R8C67
11b. Killer pair 8,9 in R8C1 and R8C67, locked for R8
11c. 7 in N9 locked in R7C78, locked for R7

12. Combined cage R2C2348 using R2C28 (step 1) = 25 = {2689/3589/4579/4678} (cannot be {3679} which clashes with R2C34)
12a. 5 or 6 must be in R2C34 -> no 5,6 in R2C28

13. 2 in R8 locked in R8C345, 16(4) cage at R7C5 = {2347/2356} (cannot be {1267} because 6,7 of {1267} must both be in N8 clashing with R9C456), no 1, 2,3 locked in R8C345, locked for R8 -> R78C9 = [31]
13a. 4,5 only in R7C5 -> R7C5 = {45}

14. 2 in N7 locked in R7C23 + R8C3
14a. 45 rule of N7 3 innies R7C23 + R8C3 = 11 = {236} (only remaining combination, cannot be {245} which clashes with R7C5) -> R8C3 = 3, R7C23 = {26} locked for R7, N7 and 18(4) cage at R6C2, clean-up: no 4 in R4C2 (step 5), no 3 in R4C8 (step 2), no 4 in R6C8 (step 4), no 8 in R7C78 + R8C7 (step 7), no 5 in R8C6, no 5 in R8C8 (step 3)

15. R7C23 = {26} = 8 -> R6C2 + R7C4 = 10 = [91], R4C2 = 1 (step 5), R4C8 = 6 (step 2), R6C8 = 1 (step 4), R8C8 = 4, R8C2 = 7 (step 3), R8C6 = 8, R78C1 = [89], R8C7 = 5, R7C5 = 5 (step 13), R7C6 = 4 (hidden single in R7), clean-up: no 4,5 in R456C3 (prelim g), no 6 in R9C456 (step 9)
15a. Naked triple {678} in R456C3, locked for C3 and N4 -> R7C23 = [62], clean-up: no 6,8 in R2C4
15b. Naked pair {59} in R2C34, locked for R2, clean-up: no 3 in R3C1
15c. Naked triple {379} in R9C456, locked for R9

16. 45 rule on N1 2 remaining outies R23C4 = 11 = [56/92]

17. 1 in R5 locked in R5C56 -> R5C456 = {159/168}, no 2,3,4,7

18. R456C7 = {389} (only remaining combination, cannot be {479} which clashes with R7C7), locked for C7 and N6 -> R7C78 = [79]
18a. 4 in N6 locked in R456C9, locked for C9, clean-up: no 6 in R23C9
18b. Naked pair {28} in R23C9, locked for C9 and N3 -> R9C9 = 6, R9C78 = [28]
18c. R1C9 = 9 (hidden single in C9), R5C8 = 2 (hidden single in C8)

19. 14(4) cage at R3C2 = {1256/1346} (cannot be {1238} because 3,8 only in R3C2) -> R3C4 = 6, R3C23 = [25/34], R8C45 = [26], clean-up: no 2 in R2C1

20. R1C1 = 6 (hidden single in C1), 8 in N1 locked in R12C2 -> 21(4) cage = {2568/3468}, no 1
20a. Naked pair {45} in R13C3, locked for C3 and N1 -> R29C3 = [91], R2C4 = 5, clean-up: no 3 in R2C1
20b. Naked triple {238} in R123C2, locked for C2

21. R5C456 (step 17) = {159/168}
21a. R5C4 = {89} -> R5C5 = 1, R5C6 = {56}

22. R6C456 = {268/358/367/457}
22a. 5,6 only in R6C6 -> R6C6 = {56}
22b. Naked pair {56} in R56C6, locked for N5

23. Deleted

24. Deleted
24a. Deleted
24b. R2C7 = 6 (hidden single in C7)

25. 18(4) cage in N3 = {1359} (only remaining combination) -> R1C7 = 1, R12C8 = [53], R3C78 = [47], R13C3 = [45], R23C1 = [71], R3C6 = 3 (cage sum), R123C2 = [382], R23C9 = [28], R2C56 = [41], R3C5 = 9

26. R4C456 = {239/248/347}
26a. 7 of {347} must be in R4C6 -> no 7 in R4C45
26b. 4 of {248} must be in R4C4 -> no 8 in R4C4

27. R6C456 = {268/367/457} (cannot be {358} which clashes with R6C7)
27a. 2 of {268} must be in R6C5 -> no 8 in R6C5

28. 45 rule on R4 4 remaining innies R4C1379 = 24 = {2589/3579/4578} (cannot be {3489} which clashes with R4C456)
28a. 9 of {3579} must be in R4C7 -> no 3 in R4C7
28b. 7 of {3579/4578} must be in R4C3 -> no 7 in R4C9

29. 45 rule on R6 4 remaining innies R6C1379 = 19 = {2467/3457} (cannot be {2368} because R6C9 only contains 4,5,7, cannot be {2458} because 2,4,5 only in R6C19), no 8, 4,7 locked for R6
29a. R6C7 = 3, R6C139 = {457} -> R6C3 = 7, R6C19 = {45}, locked for R6

and the rest is naked singles

With hindsight, I've realised that I didn't need to use combined cages in steps 11 and 12 although they seemed obvious and attractive at the time; I like using combined cages, particularly pairs of 2-cell cages. 6,7 can be eliminated from R8C67 in step 11 because of the clash with R8C28, leading to killer pair 4,5 in R8C28 and R8C67. Step 12 was just a simple clash with R2C34 that I ought to have spotted at step 1 :oops: . I also missed that in step 27 {457} clashes with R6C9. That additional elimination would have made the final steps simpler.


Last edited by Andrew on Thu Sep 18, 2008 10:06 pm, edited 2 times in total.

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PostPosted: Tue Sep 16, 2008 7:15 am 
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Grand Master
Grand Master

Joined: Mon Apr 21, 2008 9:44 am
Posts: 310
Location: MV, Germany
It seems that the most important move is step 1e which Andrew also found quite early but afterwards I took a different path to finish this Killer.

BTW, welcome herschko!

A120 Walkthrough:

1. R789 !
a) 17(2) = {89} locked for C1+N7
b) 4(2) = {13} locked for C9+N9
c) Outies R9 = 11(2) <> 1; R8C8 <> 2
d) 20(4) = 2{459/468/567} -> 2 locked for R9+N9
e) ! Hidden Killer pair (89) in 20(4) for R9 since 19(3) cannot be {289}
-> 20(4) = 24{59/68} -> 4 locked for N9; R8C8 <> 8,9
f) Outies R9 = 11(2) = [56/65/74]
g) 17(4) = 17{36/45} -> 1,7 locked for N7
h) 13(2) <> {67} because it's a Killer pair of Outies R9
i) Killer pair (89) locked in R8C1 + 13(2) for R8
j) Killer pair (45) locked in Outies R9 + 13(2) for R8
k) 13(2): R8C6 <> 9

2. R789
a) Outies R89 = 16(3) = {169/178/349/358} <> 2 because R7C1 = (89)
and R7C9 = (13); R7C5 = (4567)
b) 2 locked in 16(4) @ R8 = 2{167/347/356}
c) Innies N9 = 21(3) = 7{59/68} -> 7 locked for R7+21(4)
d) Innies N9 = 21(3): R7C78 <> 8 because 6,7 only possible there

3. R456+C1
a) Innies N6 = 7(2) <> 7,8,9
b) Outies R789 = 10(2) <> 5; R6C2 <> 1,2,3
c) Innies N4 = 10(2): R4C2 <> 5,7,8
d) Innies N6 = 7(2): R4C8 <> 2
e) 14(4) <> 7 because (47) is a Killer pair of 21(3)
f) Innies+Outies C1: -6 = R5C2 - R19C1 -> R5C2 <> 8 because R19C1 <= 13
g) Innies N4 = 10(2) <> {46} since it's a Killer pair of 14(4)
h) Outies R789 = 10(2) <> 4,6
i) Innies N6 = 7(2): R4C8 <> 1,3

4. R789
a) 18(4): R7C4 <> 8,9 because R6C2+R7C23 >= 12
b) Outies N9 = 13(2+1): R7C6 <> 9 because R8C6 >= 4
c) 9 locked in 19(3) @ N8 for R9 -> 19(3) = 9{37/46}
d) 20(4) = {2468} locked for N9
e) 13(2): R8C6 <> 5
f) 5 locked in R7C456 @ N8 for R7
g) Hidden Single: R8C7 = 5 @ N8 -> R8C6 = 8
h) Outies R9 = 11(2) = {47} -> R8C2 = 7, R8C8 = 4
i) 16(4) = {2356} -> R7C5 = 5; 3 locked for R8
j) R8C9 = 1, R7C9 = 3
k) 18(4) = {1269} -> R6C2 = 9, R7C4 = 1; {26} locked for R7+N7

5. R456
a) Innie N4 = R4C2 = 1
b) Outie R789 = R6C8 = 1
c) Innie N6 = R4C8 = 6
d) 21(3) = {678} locked for C3+N4
e) 20(3) = {389} locked for C7+N6 since {479} blocked by R7C7 = (79)
f) 18(4) @ N6 = {2457} -> 4 locked for C9

6. R123
a) 10(2) = {28} locked for C9+N3
b) Hidden Single: R1C9 = 9 @ C9
c) 18(4) = {1359} -> R1C7 = 1, {35} locked for C8
d) R3C8 = 7, R3C7 = 4 -> R3C6 = 3
e) 14(2) = {59} locked for R2
f) Outies R1 = 11(2) = {38} -> R2C8 = 3, R2C2 = 8

7. R456
a) 1 locked in 15(3) @ N5 = 1{59/68}
b) 16(3) <> 4 because {457} blocked R6C9 = (457)
c) 4 locked in 14(3) @ N5 for R4 -> 14(3) = 4{28/37}
d) 9 locked in 15(3) @ N5 = {159} locked for R5+N5

8. Rest is singles.

Rating: Easy 1.25. I used a Hidden Killer pair.


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 Post subject: Re: A120 Twin Towers
PostPosted: Tue Sep 23, 2008 7:41 pm 
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1044
Location: Sydney, Australia
Afmob's WT for A120 V2 has a number of really neat moves, all of which I either missed or never got near (eg 3a :oops: ; 3f and 5h). But two more of the stand-outs deserve some pics. The first of these is what Afmob refers to as a "forcing chain", a name which has a very negative connotation for me. But Afmob's one is actually very elegant!

End of step 7b. here
Code:
.-------------------------------.-------------------------------.-------------------------------.
| 124589    124589    124589    | 45789     45789     45789     | 13456     123456    123456    |
| 5678      1245      4678      | 2346      235689    235689    | 789       1245      124       |
| 5678      35679     35679     | 12456     256       12        | 789       789       245       |
:-------------------------------+-------------------------------+-------------------------------:
| 12356789  78        123678    | 13456789  13456789  13456789  | 12458     34        13689     |
| 12356789  12356789  123678    | 123456789 123456789 123456789 | 12458     15689     136789    |
| 12345689  45        123468    | 123678    123678    123678    | 1258      67        13689     |
:-------------------------------+-------------------------------+-------------------------------:
| 123456    123456789 123456789 | 123456789 123456789 123456789 | 13456789  123456789 345789    |
| 123456    123456789 123456789 | 123456789 123456789 12346789  | 1346789   123456789 345789    |
| 123456789 123456789 123456789 | 123456    123456    123456    | 13456789  123456789 123456789 |
'-------------------------------.-------------------------------.-------------------------------'
Afmob wrote:
7c) ! Consider placement of 7 in R6 -> R4C456+R6C9 <> 3
- i) 7 in 11(3) @ N5 = {137} locked for R6+N5
- ii) 7 in Innies N6 = [37] -> R4C8 = 3

Image


Afmob tells me this "forcing chain" is called "Locked Cages" on sudopedia. I personally prefer SudokuSolver's "Dependant Cages" as a name. It is really neat!

That move is essential to get to Afmob's even cooler breakthrough. But first, a couple more steps in-between.
7d) 24(4) @ N6: R5C9 <> 3 because 7 only possible there
e) 3 locked in R4C89 @ N6 for R4

8. R456 !
a) Hidden Killer pair (69) in Innies R1234 because 18(3) cannot be {369}
-> 2 locked in Innies R1234 = 16(4) = 2{149/167/356} <> 8
b) Innies R1234 = 16(4): R4C7 <> 5 because 3 must be in R4C9 @ 24(4) = {3579} -> R4C7 <> 5

Now ready for the puzzle breaker move.
Image
Afmob wrote:
8c) ! Innies N46 = 22(4) = [7456/8347] -> Either R6C2 = 5 xor R4C8 = 3
d) ! Innies R1234 = 16(4) = 12{49/67} because [5623] leaves neither 5 in R6C2 nor 3 in R4C8
A really hidden hidden-cage blocker!

ALT way to see this move
I think I'd have a better chance of being able to find that elimination this way:

From step 8c, r6c2 & r4c8 cannot BOTH be 4, so anything that forces both to 4 cannot be true.
(i) If h16(4)r4 = {2356} -> r4c8 = 4.
(ii) {2356} must have 5 in r4c1. This forces r6c2 = 4
(iii) -> {2356} blocked since it forces both r4c8 & r6c2 to 4.

Well done Afmob!

Personally, I have no hesitation in giving this puzzle a rating of 1.75. Because of the number of key solving techniques needed (none of which are, in themselves, too complicated), no way would I consider it as the Weekly Assassin (and very glad I changed it to the V2!). It reminds me a lot of the A60 RP-Lite.

Cheers
Ed


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 Post subject: Re: A120 Twin Towers
PostPosted: Thu Jul 22, 2010 5:10 am 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Almost two years ago I wrote:
Thanks Ed. Great timing. I'd reached the stage with what is now the V2 where after prelims and 18 steps I was looking for contradiction moves to make progress. I don't like using those moves or forcing chains, which I rarely find, so early in a puzzle.
That was when Ed posted what is now the V1.

At the time Ed and I discussed my first 18 steps and discovered that I'd missed a CPE which I've now added to step 6; that led to some changes to step 13. Then after two or three more steps I moved on to other puzzles and didn't come back to A120 V2 until this week. It took me some time to find the first key breakthrough in step 29a (Afmob's step 6b) but after that I was making progress although I still needed a contradiction move for my final breakthrough.

Afmob wrote:
What a challenging Killer!
It certainly was!

Rating Comment:
I'll rate my walkthrough for A120 V2 at 1.75 because I found this puzzle almost as hard as A108 V2 which I finished recently. For A120 V2 I used combination and permutation analysis, a "ruudiculous" ALS block using a combined cage for the first key breakthrough and a fairly short contradiction move for my final breakthrough.

Here is my walkthrough for A120 V2.

Prelims

a) R23C1 = {49/58/67}, no 1,2,3
b) R2C34 = {19/28/37/46}, no 5
c) R23C9 = {15/24}
d) R78C1 = {16/25/34}, no 7,8,9
e) R78C9 = {39/48/57}, no 1,2,6
f) R8C67 = {19/28/37/46}, no 5
g) R1C456 = {389/479/569/578}, no 1,2
h) R456C3 = {128/137/146/236/245}, no 9
i) R456C7 = {128/137/146/236/245}, no 9
j) R6C456 = {128/137/146/236/245}, no 9
k) R9C456 = {126/135/234}, no 7,8,9

1. 45 rule on R1 2 outies R2C28 = 6 = {15/24}

2. 45 rule on R123 2 outies R4C28 = 11 = {29/38/47/56}, no 1

3. 45 rule on R789 2 outies R6C28 = 11 = {29/38/47/56}, no 1

4. 45 rule on R9 2 outies R8C28 = 10 = {19/28/37/46}, no 5

5. 45 rule on N4 2 innies R46C2 = 12 = {39/48/57}, no 2,6
5a. 45 rule on N6 2 innies R46C8 = 10 = {28/37/46}, no 5,9

6. 45 rule on N3 3 innies R2C7 + R3C78 = 24 = {789}, locked for N3, CPE no 7,8,9 in R3C5
6a. 15(4) cage in N3 = {1356/2346}, 3,6 locked for R1
[I’ve added the CPE to step 6 after Ed pointed it out in discussions about my first 18 moves, at the time I originally got stuck.]

7. R1C456 = {479/578}, 7 locked for R1 and N2, clean-up: no 3 in R2C3
7a. 3 in R2 locked in R2C456, locked for N2

8. 21(4) cage at R3C6 contains two of 7,8,9 in R3C78 = {1389/1479/2379/2478} (cannot be {1578} because 1,5 only in R3C6, other combinations only have one of 7,8,9), no 5,6, clean-up: no 5 in R4C2 (step 2), no 7 in R6C2 (step 5), no 4 in R6C8 (step 5a)
8a. R3C78 = {789} -> no 7,8,9 in R3C6 + R4C8, clean-up: no 3,4 in R4C2 (step 2), no 8,9 in R6C2 (step 5), no 2,3 in R6C8 (step 5a)

9. 45 rule on N9 3 innies R7C78 + R8C7 = 9 = {126/135/234}, no 7,8,9, clean-up: no 1,2,3 in R8C6

10. 45 rule on N9 3 outies R6C8 + R78C6 = 19, max R6C8 + R8C6 = 17 -> min R7C6 = 2
10a. R6C8 + R8C6 cannot total 13 (because R7C6 “sees” both of R6C8 + R8C6 and there’s no 4,5,9 in R6C8 and no 5 in R8C6) -> no 6 in R7C6

11. 45 rule on N89 2 outies R6C8 + R8C3 = 1 innie R7C4 + 3
11a. Min R6C8 + R8C3 = 7 -> min R7C4 = 4
11b. Max R7C4 = 9 -> max R6C8 + R8C3 = 12, max R8C3 = 6

12. 45 rule on N1 3 outies R23C4 + R4C2 = 14
12a. R4C2 = {789} -> R23C4 = 5,6,7, no 8,9, clean-up: no 1,2 in R2C3

13. 25(4) cage at R2C5 can only contain 7 if it also contains 3 (because R2C34 = [73] is the only other way to place 3 in R2) -> 25(4) cage = {2689/3589/3679} (cannot be {1789/4579/4678} which contain 7 but not 3), no 1,4, 9 locked in R2C567, locked for R2, clean-up: no 1 in R2C4, no 4 in R3C1
13b. 1 in N2 only in R3C46, locked for R3, clean-up: no 5 in R2C9
13c. 5 of {3589} must be in R3C5 -> no 5 in R2C56
[Before I added the CPE to step 6, I used hidden killer quint 1,2,3,4,5 in R2C156, R2C2, R2C34, R2C8 and R2C9 for R2 -> R2C156 can only contain one of 1,2,3,4,5 to make step 13c work. This step is simpler after the CPE. Also I think that 9 locked ... in step 13 and the whole of step 13b require that there to be no 9 in R3C5.]

14. 16(4) cage in N1 = {1249/1258}, 2 locked for N1

15. 45 rule on N2 3 innies R23C4 + R3C6 = 1 outie R2C7
15a. R2C7 = {789} -> R23C4 + R3C6 (must contain 1, step 13b) = 7,8,9 = {124/125/134/126} (cannot be {135} because R23C4 must total 5,6,7 (step 12a) and 3 only in R2C4)
15b. 4 of {124/134} must be R23C4 because R23C4 = 5,6,7 (step 12a) -> no 4 in R3C6
15c. 2 of {126} must be in R3C6 (R23C4 cannot be {26} because R23C4 = 5,6,7) -> {126} must be [612], no 6 in R3C4

16. 25(4) cage at R2C5 (step 13a) = {2689/3589/3679}
16a. R23C4 + R3C6 (step 15a) = {124/134/126} (cannot be {125} which clashes with {2689/3589} and for {3679} R2C7 = 7 => R23C4 + R3C6 = {124} using step 15) -> no 5 in R3C4
[Ed suggested that simpler is R2C7 = 8 => R23C4 + R3C6 (step 15) = 8. R2C7 = 8 => no 8 in R2C3, no 2 in R2C4 => min R2C4 = 3 => max R3C46 = 5, no 5.
Alternatively R2C7 = 8 => R23C4 + R3C6 (step 15) = 8. R2C7 = 8 => no 8 in R2C3, no 2 in R2C4 => R23C4 + R3C6 cannot be {125}, no 5.]

17. 45 rule on R12 3 outies R3C159 = 15 = {258/267/456} (cannot be {249} which clashes with R3C46, ALS block), no 9, clean-up: no 4 in R2C1

18. 21(4) cage at R3C6 (step 8) = {1389/1479/2379/2478}
18a. R3C6 = {12} -> no 2 in R4C8, clean-up: no 9 in R4C2 (step 2), no 3 in R6C2 (step 5), no 8 in R6C8 (step 5a)
18b. Max R6C2 = 5 -> min R7C234 = 19, no 1
18c. R6C8 + R78C6 = 19 (step 10), max R6C8 = 7 -> min R78C6 = 12, no 2 in R7C6

19. R456C3 = {128/137/146/236} (cannot be {245} which clashes with R6C2), no 5

20. R6C456 = {128/137/236} (cannot be {146} which clashes with R6C28, cannot be {245} which clashes with R6C2), no 4,5
20a. R4C456 = {189/369/459/567} (cannot be {279} which clashes with R6C456, cannot be {378} which clashes with R4C2, cannot be {468} which clashes with R4C28), no 2

21. 3 in R3 only in R3C23 -> 20(4) cage at R3C2 = {1379/2378/3458/3467} (cannot be {2369} because R4C2 only contains 7,8)
21a. 4 of {3458/3467} must be in R3C4 -> no 4 in R3C23

22. 3 in R3 only in R3C23
22a. 45 rule on N1 3 innies R2C3 + R3C23 = 16 = {349/358/367}
22b. 8 of {358} must be in R2C3 -> no 8 in R3C23

23. 45 rule on C89 4 innies R3467C8 = 2 outies R19C7 + 9
23a. R46C8 = 10 (step 5a) -> R19C7 = R37C8 + 1
23b. Min R37C8 = 8 -> min R19C7 = 9, no 1,2 in R9C7
23c. Min R37C8 = 8 = [71] => R78C7 = {26/35} (step 9) -> min R19C7 = 9 but cannot be [63] which clashes with R78C7 -> no 3 in R9C7
23d. Max R19C7 = [69] = 15 -> max R37C8 = 14 = [95] (because 9 in R9C7 forces 9 in R3C8)
Max R19C7 without using 9 = [68] = 14 -> max R37C8 = 13 = [85] (9 in N3 must be in R23C7 and R27C8 cannot be [76] which clashes with R6C8)
-> no 6 in R7C8

24. R4C28 (step 2) = [74/83], R46C2 (step 5) = [75/84] -> R4C28 + R6C2 = [745/834], CPE no 4 in R4C13 + R6C79

25. R4C28 (step 2) = [74/83], R46C8 (step 5a) = [37/46] -> R4C28 + R6C8 = [746/837], CPE no 7 in R4C79 + R6C13

26. 9 in R6 only in R6C19
26a. 45 rule on R6789 4 innies R6C1379 = 23 = {1589/2489/3479/3569} (cannot be {1679} which clashes with R6C8, cannot be {2579} which clashes with R6C28)
26b. 4 of {2489/3479} must be in R6C13
{1589/3569}, 5 locked for R6 -> R6C2 = 4
-> 4 locked in R6C123, locked for N4

27. 9 in N6 only in 24(4) cage = {1689/2589/3579/4569} (cannot be {2679} which clashes with R6C8, cannot be {3489} which clashes with R4C8)
27a. R23C9 = [15]/{24}, R78C9 = {39/48/57} -> combined cage R2378C9 = [15]{39}/[15]{48}/{24}{39}/{24}{57}
27b. 24(4) cage = {1689/3579} (cannot be {2589/4569} which clash with R2378C9, ALS block), no 2,4
27c. Killer pair 6,7 in 24(4) cage and R6C8, locked for N6

28. 2 in N6 only in R456C7, locked for C7, clean-up: no 8 in R8C6
28a. R456C7 = {128/245}, no 3

29. R7C78 + R8C7 (step 9) = {126/135/234} -> R7C78 = {12/15/23/24/26/35} (cannot be {13/16/34} because no 2,5 in R8C7)
29a. 18(4) cage at R6C8 = {1269/1278/2367/2457/3456} (cannot be {1359/1458/2349/2358} because R6C8 only contains 6,7, cannot be {1368/1467} which don’t contain any combinations for R7C78)
29b. R7C78 = {12/23/24/26/35} (cannot be {15} because no combinations for 18(4) cage contain both of 1,5)
29c. 2 of {12/24} must be in R7C8 -> no 1,4 in R7C8

30. R7C78 + R8C7 (step 9) = {126/135/234}
30a. 5 of {135} must be in R7C8 (R78C7 cannot be [51] which clashes with R456C7), no 5 in R7C7
30b. R78C7 = {13/16/34}
30c. Killer pair 1,4 in R456C7 and R78C7, locked for C7

31. R78C7 (step 30b) = {13/16/34}
31a. 24(4) cage in N6 (step 27b) = {1689/3579} cannot be {3579}, here’s how
{3579} => R78C9 = {48} (cannot be {39/57} which clash with {3579}, ALS block) => R78C7 = {13/16} => R456C7 (step 28a) = {245} clashes with {3579}
[Almost a complete loop around N69. ;) ]
31b. -> 24(4) cage = {1689}, locked for N6 -> R456C7 = {245}, locked for C7 and N6, R46C8 = [37], R4C2 = 8 (step 2), R6C2 = 4 (step 3), clean-up: no 2 in R2C8 (step 1), no 3,7 in R8C2 (step 4), no 6 in R8C6, no 2,6 in R8C8 (step 4)
31c. 7 in N3 only in R23C7, locked for C7

32. 15(4) cage in N3 (step 6a) = {1356/2346}
32a. 4 of {2346} must be in R2C8 -> no 4 in R1C89

33. 18(4) cage at R6C8 (step 29a) = {1278/2367} (cannot be {2457} because R7C7 only contains 1,3,6) -> R7C8 = 2, clean-up: no 5 in R8C1
33a. R7C67 = [36/81], no 4,5,9 in R7C6, no 3 in R7C7

34. R7C78 + R8C7 (step 9) = {126} (only remaining combination) -> R78C7 = {16}, locked for C7 and N9 -> R1C7 = 3, clean-up: no 9 in R8C2 (step 4), no 7 in R8C6

35. R23C4 + R4C2 = 14 (step 12), R4C2 = 8 -> R23C4 = 6 = {24}, locked for C4 and N2 -> R3C6 = 1, clean-up: no 9 in R1C456 (step 7), no 4,7 in R2C3

36. Naked triple {578} in R1C456, locked for R1 and N2 -> R3C5 = 6, clean-up: no 7 in R2C1

37. 25(4) cage at R2C5 (step 13) = {3679} (only remaining combination) -> R2C7 = 7
37a. Naked pair {89} in R3C78, locked for R3, clean-up: no 5 in R2C1

38. 16(4) cage in N1 (step 14) = {1249} (only remaining combination), no 5, clean-up: no 1 in R2C8 (step 1)

39. R2C8 = 5 (hidden single in R2), R2C2 = 1 (step 1), clean-up: no 9 in R8C8 (step 4)
39a. Naked pair {24} in R23C9, locked for C9, clean-up: no 8 in R78C9

40. Naked triple {489} in R389C8, locked for C8
40a. Naked triple {489} in R8C8 + R9C78, locked for N9, clean-up: no 3 in R78C9

41. R9C9 = 3 (hidden single in N9)

42. R9C456 = {126} (only remaining combination), locked for R9 and N8

43. R6C456 (step 20) = {128/236}, 2 locked for R6 and N5 -> R6C7 = 5

44. 24(4) cage at R6C2 = 4{569} (only remaining combination, cannot be 4{389} which clashes with R7C6, cannot be 4{578} which clashes with R7C9), 5,6,9 locked for R7, 6 also locked for N7 -> R78C9 = [75], R7C7 = 1, R8C7 = 6, R8C6 = 4, R7C6 = 8 (step 33a), R7C5 = 3, R7C1 = 4, R8C1 = 3, R8C2 = 2, R8C3 = 1, R8C8 = 8

45. R7C4 = 5 (hidden single in N8)

46. R456C3 (step 19) = {236} (only remaining combination), locked for C3 and N4 -> R2C3 = 8, R2C4 = 2, R2C1 = 6, R3C1 = 7

47. 7 in R4 only in R4C456, locked for N5
47a. R4C456 (step 20a) = {567} (only remaining combination), locked for R4 and N5 -> R4C3 = 2

48. R6C456 (step 43) = {128} (only remaining combination) -> R6C6 = 2, R6C45 = {18}, locked for R6 and N5 -> R6C1 = 9

and the rest is naked singles.


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