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 Post subject: Assassin 292
PostPosted: Thu May 15, 2014 11:34 pm 
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Grand Master
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1040
Location: Sydney, Australia
Not my best Assassin. Couldn't crack it by using some interesting features of this cage pattern but just requires regular, common techniques. So, wanted to be able to make A292 Interesting below the v1 puzzle but couldn't solve it. If someone finds a nice way through would love to see it.

Assassin 292 SudokuSolver Score, 1.55

Image
code:paste into solver:
3x3::k:5120:6401:6401:6401:6401:4098:4098:8707:8707:5120:5120:6401:1540:1540:4098:8707:8707:8707:3845:5120:2054:4359:4359:5896:4617:8707:3082:3845:3845:2054:11531:11531:5896:4617:3852:3082:3845:11531:11531:11531:5896:5896:4617:3852:3852:6669:11531:2068:11531:11531:6927:4617:5392:3852:6669:3089:2068:4885:11531:6927:5392:5392:5392:6669:3089:3089:4885:4885:6927:3090:3090:3090:6669:6669:3347:3347:6927:6927:6927:3342:3342:


Assassin 292 Interesting SudokuSolver Score, 1.70
pic and code:
Image
code:paste into solver:
3x3::k:5120:4870:4870:1553:1553:4098:4098:8707:8707:5120:5120:4870:1540:1540:4098:8707:8707:8707:0000:5120:2053:4359:4359:5896:4617:8707:3082:0000:0000:2053:11531:11531:5896:4617:3852:3082:0000:11531:11531:11531:5896:5896:4617:3852:3852:6669:11531:0000:11531:11531:6927:4617:5392:3852:6669:0000:0000:4865:11531:6927:5392:5392:5392:6669:0000:0000:4865:4865:6927:3090:3090:3090:6669:6669:3347:3347:6927:6927:6927:3342:3342:
solution: same for both:
+-------+-------+-------+
| 4 9 2 | 5 1 3 | 6 7 8 |
| 3 6 8 | 2 4 7 | 5 9 1 |
| 1 7 5 | 8 9 6 | 2 4 3 |
+-------+-------+-------+
| 5 2 3 | 1 7 4 | 8 6 9 |
| 7 4 9 | 6 8 5 | 1 3 2 |
| 6 8 1 | 3 2 9 | 7 5 4 |
+-------+-------+-------+
| 2 3 7 | 4 5 8 | 9 1 6 |
| 8 5 4 | 9 6 1 | 3 2 7 |
| 9 1 6 | 7 3 2 | 4 8 5 |
+-------+-------+-------+
Cheers
Ed


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 Post subject: Re: Assassin 292
PostPosted: Sat May 17, 2014 4:16 am 
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Grand Master
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1893
Location: Lethbridge, Alberta, Canada
Thanks Ed for another interesting Assassin.

I hope my key steps weren't OTT; they seem consistent with the SS score.

Here is my walkthrough for Assassin 292:
Thanks Afmob and Ed for correcting typos and suggested clarifications.

Prelims

a) R2C45 = {15/24}
b) R34C3 = {17/26/35}, no 4,8,9
c) R3C45 = {89}
d) R34C9 = {39/48/57}, no 1,2,6
e) R67C3 = {17/26/35}, no 4,8,9
f) R9C34 = {49/58/67}, no 1,2,3
g) R9C89 = {49/58/67}, no 1,2,3
h) 19(3) cage at R7C4 = {289/379/469/478/568}, no 1
i) and, of course, 45(9) cage at R4C4 = {123456789}

1. Naked pair {89} in R3C45, locked for R3 and N2, clean-up: no 3,4 in R4C9

2. 45 rule on R12 2 outies R3C28 = 11 = {47/56}, no 1,2,3

3. 45 rule on N3 3 innie R13C7 + R3C9 = 11 = {137/146/236/245} (cannot be {128} because no 1,2,8 in R3C9), no 8,9
3a. 16(3) cage at R1C6 = {367/457}, no 1,2
3b. 16(3) cage = {367/457}, CPE no 7 in R1C45
3c. 7 in N2 only in R123C6, locked for C6
3d. R13C7 + R3C9 = {137/146/236/245} contains one of 1,2 -> R3C7 = {12}

4. 45 rule on C789 2 innies R19C7 = 10 = {37/46}, no 1,2,5,8,9

5. Deleted, replaced by step 7.

6. 45 rule on C6789 2 outies R59C5 = 11 = {29/38/47/56}, no 1

7. 45 rule on N36 2 innies R1C7 + R6C8 = 11 = [38/47/65/74]
7a. R13C7 + R3C9 (step 3) = {137/146/236} (cannot be {245} = [425] because R34C9 = [57] clashes with R1C7 + R6C8 = [47], killer combo clash), no 5, clean-up: no 7 in R4C9
7b. 6 of {146} must be in R1C7 -> no 4 in R1C7, clean-up: no 7 in R6C8, no 6 in R9C7 (step 4)

8. 16(3) cage at R1C6 (step 3a) = {367} (only remaining combination, cannot be {457} which clashes with R2C45), no 4,5
8a. 16(3) cage = {367}, CPE no 3,6 in R1C45
8b. R123C6 = {367} (hidden triple in N2), locked for C6
8c. Naked quad {1245} in R12C45. CPE no 1,2,4,5 in R2C3

9. 45 rule on N2, using R123C6 = {367} = 16, 3 remaining outies R1C23 + R2C3 = 19 = {289/379/469/478/568}, no 1

10. 27(6) cage at R6C6 = {123489/123579/124578/134568} (cannot be {123678/234567} because 3,6,7 only possible in R9C57, cannot be {124569} because R9C57 = [64] clashes with the pair of 13(2) cages in R9), 1 locked for C6

11. Naked triple {367} in 16(3) cage at R1C6 and in R123C6 -> R1C7 = R3C6
11a. R13C7 + R3C9 (step 7a) = R3C679 = 11 = {137/236} (cannot be {146} which clashes with R3C28), no 4, 3 locked for R3 and N3, clean-up: no 5 in R4C3, no 8 in R4C9
11b. Killer pair 6,7 in R3C28 and R3C679, locked for R3, clean-up: no 1,2 in R4C3
11c. 45 rule on R3, 2 remaining innies R3C13 = 6 = [15/42/51], no 2 in R3C1

12. 18(4) cage at R3C7 = {1269/1278/2358} (cannot be {1368/1467/2349/2367/2457/3456} which clash with R19C7, cannot be {1359} which clashes with R4C9, cannot be {1458} which clashes with R6C8), no 4, 2 locked for C7

13. R59C5 = 11 (step 6), 1,2,4,5,8,9 in C6 only in R456789C6
13a. 27(6) cage at R6C6 (step 10) = {123489/124578/134568} (cannot be {123579} because R45C6 = {48} clashes with R59C5 = [47/83], killer combo clash)
13b. 27(6) cage = {123489/134568} (cannot be {124578} = {458}[217], other permutations clash with the pair of 13(2) cages in R9, because R45C6 = {29} clashes with R59C5 = [92], killer combo clash), no 7, 3 locked for R9, clean-up: no 3 in R1C7 (step 4), no 3 in R3C6 (step 11), no 8 in R6C8 (step 7)

14. R3C9 = 3 (hidden single in R3) -> R4C9 = 9, clean-up: no 4 in R9C8
14a. 18(4) cage at R3C7 (step 12) = {1278/2358}, no 6, 8 locked for C7 and N6
14b. 6 in N6 only in 15(4) cage at R4C7 = {1356/2346}, no 7, 3 locked for N6
14c. 18(4) cage = {1278} (only remaining combination), locked for C7 -> R1C7 = 6, R9C7 = 4 (step 4), R3C6 = 6 (step 11), R6C8 = 5 (step 7), clean-up: no 5 in R3C2 (step 2), no 3 in R7C3, no 9 in R9C34, no 9 in R9C8, no 8 in R9C9
14d. 15(4) cage at R4C8 = {2346} (only remaining combination), locked for N6, 3 also locked for C8
[Cracked. The rest is straightforward.]

15. Naked pair {47} in R3C28, locked for R3, clean-up: no 2 in R3C3 (step 11c), clean-up: no 6 in R4C3
15a. Naked pair {15} in R3C13, locked for R3 and N1 -> R3C7 = 2

16. 4 in C6 only in R45C6, locked for N5
16a. 23(4) cage at R3C6 contains 4,6 = {4568} (only remaining combination), 5,8 locked for N5
16b. R9C5 = 3 (hidden single in R9), R5C5 = 8 (step 6), R3C45 = [89], clean-up: no 5 in R9C3
16c. Naked pair {45} in R45C6, 5 locked for C6

17. 19(3) cage at R7C4 = {469} (only remaining combination), locked for N8, 9 also locked for C4, clean-up: no 7 in R9C3
17a. Naked triple {128} in R789C6, locked for C6 and N8 -> R6C6 = 9

18. 45(9) cage at R4C4 = {123456789} -> R6C2 = 8, R5C23 = {49}, locked for R5 and N4

19. 5 in N4 only in R4C12 + R5C1, locked for 15(4) cage at R3C1 -> R3C1 = 1, R3C3 = 5, R4C3 = 3
19a. 15(4) cage = {1257} (only remaining combination), 2,7 locked for N4 -> R6C1 = 6, R6C3 = 1 -> R7C3 = 7, R7C5 = 5, R9C4 = 7 -> R9C3 = 6, R9C89 = [85]
19b. Naked pair {39} in R78C7, locked for C7 and N9 -> R2C7 = 5, clean-up: no 1 in R2C45
19c. Naked pair {24} in R2C45, locked for R2 and N2 -> R1C45 = [51]

20. R1C23 + R2C3 (step 9) = {289}, (only remaining combination, cannot be {379} because 3,7 only in R1C2, cannot be {478} which clashes with R3C2), locked for N1, 8 also locked for C3

21. 9 in R9 only in R9C12, locked for N7
21a. R9C1 = 9 (hidden single in C1)
21b. R69C1 = [69] -> 26(5) cage at R6C1 = {12689} (cannot be [64592] which clashes with R12C1 + R45C1, ALS block) -> R9C2 = 1, R78C1 = {28}, locked for C1 and N7

22. Naked pair {57} in R45C1, locked for C1 and N4 -> R12C1 = [43], R4C2 = 2

23. R8C3 = 4 -> R78C2 = [35], R7C7 = 9
23a. R8C7 = 3 -> R8C89 = 9 = {27}, locked for R8 and N9 -> R8C1 = 8

and the rest is naked singles.

Rating Comment:
I'll rate my walkthrough at Hard 1.5; I used killer combo clashes.


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PostPosted: Sat May 17, 2014 4:44 pm 
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Grand Master
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Joined: Mon Apr 21, 2008 9:44 am
Posts: 310
Location: MV, Germany
I found this Killer quite hard as the rating shows and I tried to avoid using a chain-type move for a long time.
A292 Walkthrough:
1. R123
a) 17(2) = {89} locked for R3+N2
b) 16(3): R1C7 <> 1,2 since R12C6 <= 13
c) Innies N3 = 11(3) <> 9 must have one of (12) -> R3C7 = (12), R1C7 <> 8
d) Outies R12 = 11(2) = {47/56}

2. C6789
a) Innies C789 = 10(2) = {37/46}
b) Innies N36 = 11(1+1): R6C8 = (4578)
c) Innies+Outies N6: -12 = R3C7 - (R4C9+R6C8): R4C9+R6C8 <> 7 since R3C7 = (12) and R4C9+R6C8 <> 6
d) Innies N36 = 11(1+1): R1C7 <> 4

3. R123
a) 16(3) = {367} since {457} blocked by Killer pair (45) of 6(2) -> CPE: R1C45 <> 3,6,7
b) Hidden triple (367) in R123C6 @ N2 locked for C6; R3C6 = (367)
c) Using R123C6 = (367): Innies N12 = 6(2) = {15/24}; R3C1 <> 2
e) Killer pair (45) locked in Innies N12 + Outies R12 for R3
f) 12(2) = [39/75]

4. C789
a) Outies C89 = 17(3)
b) Hidden Killer pair (89) in Outies C89 + 18(4) for C7 since each can have at most one of (89)
c) Innies+Outies N6: -12 = R3C7 - (R4C9+R6C8) = 1+[58/94] / 2+[95]
d) Killer pair (89) locked in 18(4) + R4C9+R6C8 for N6
e) Killer pair (45) locked in 15(4) + R4C9+R6C8 for N6
f) 18(4) = 12{69/78} since {1368} blocked by Killer pair (36) of Innies C789 -> 1,2 locked for C7

5. R789+N4 !
a) R5C23+R7C5 <> 3,6,7 since it sees all 3,6,7 of N5
b) Innies+Outies C12: -17 = R8C3 - R156C2 -> R8C3 <> 8,9
c) ! Hidden Killer pair (67) in R9C45 for N8 since 19(3) can only have one of (67)
d) 13(2) @ N9 <> {67} since it's a Killer pair of R9C45

6. C789 !
a) ! Innies+Outies N6: -12 = R3C7 - (R4C9+R6C8): R4C9+R6C8 can't be [94] since it forces R1C7 = 7 (Innies N36) which is blocked by Killer pair (79) of 18(4) -> R4C9+R6C8 = 5{8/9} -> 5 locked for N6
b) Innies N36 = 11(1+1): R1C7 <> 7
c) 16(3) = {367} -> 7 locked for C6
d) Innies C789 = 10(2) = [37/64]

7. C4567+R9 !
a) ! Killer triples (457,478) locked in both 13(2) + R9C7 for R9
b) Outies C6789 = 11(2) <> 1: R5C5 = (2589)
c) 23(4) = {4568} since R3C6 = (36) and they are only possible there -> R3C6 = 6; 4,5,8 locked for N5; 4 also locked for C6
d) Hidden Single: R3C9 = 3 @ R3 -> R4C9 = 9
e) Innies C789 = 10(2) = [64] -> R1C7 = 6, R9C7 = 4
f) 13(2) @ N9 = {58} locked for R9+N9
g) 13(2) @ N8 = {67} locked for R9
h) 27(6) = {123489} -> R9C5 = 3; 8 locked for C6+N8
i) 19(3) = {469} locked for N8
j) Hidden Single: R5C5 = 8 @ N5, R6C2 = 8 @ 45(9)

8. C123+N9
a) R9C4 = 7, R9C3 = 6, R6C8 = 5
b) Both 8(2) <> 2
c) Innies N12 = 6(2) = {15} locked for R3
d) 8(2) = [17/53]
e) 4 locked in 45(9) for R5+N4
f) Killer pair (37) locked in 15(4) + R4C3 for N4
g) R6C3 = 1 -> R7C3 = 7
h) 12(3) = {345} locked for N7 since {129} blocked by R9C12 = (129)
i) 7 locked in 12(3) @ N9 = {237} locked for R8+N9; R8C7 = 3

9. Rest is singles.

Rating:
1.5. I used a small contradiction move.


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 Post subject: Re: Assassin 292
PostPosted: Mon May 19, 2014 10:08 am 
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Grand Master
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1040
Location: Sydney, Australia
Sorry to butt in on wellbeback but quite proud of this solution so busting to get it out there! But first, confession time. I posted the wrong puzzle as the V1! I had a slightly different cage formation to make the start a bit easier. But Afmob didn't have any ! for his early steps and Andrew used some neat steps early on so no real damage. I forgot about my couple of interesting differences to the other guys so overall, this one turned out better than my intro suggested. It goes without saying that wellbeback will have a completely different solution again!

Start to A292
18 steps:
[edit: fixed up step counts. Thanks Afmob! And some typos - thanks Andrew!]
Prelims courtesy of SudokuSolver
Cage 17(2) n2 - cells ={89}
Cage 6(2) n2 - cells only uses 1245
Cage 8(2) n47 - cells do not use 489
Cage 8(2) n14 - cells do not use 489
Cage 12(2) n36 - cells do not use 126
Cage 13(2) n78 - cells do not use 123
Cage 13(2) n9 - cells do not use 123
Cage 19(3) n8 - cells do not use 1

1. 17(2)n2 = {89} only: both locked for r3 and n2
1a. no 3,4 in r4c9
1b. Max. r12c6 = 13 -> min. r1c7 = 3

2. "45" on r12: 2 outies r3c28 = 11 = {47/56}(no 1,2,3)

3. "45" on n3: 3 innies r13c7+r3c9 = 11
3a. can't be {128} because no 1,2,8 in r3c9
3b. = {137/146/236/245}(no 8,9)
3c. must have 1 or 2 which are only in r3c7 -> r3c7 = (12)

4. "45" on c789: 2 innies r19c7 = 10 (no 5; no 1,2,8,9 in r9c7)

5. "45" on n36: 2 innies r1c7+r6c8 = 11 (no 1; no 2,3,6,9 in r6c8)

Andrew (step 7a) and Afmob (step 2c) both did this next elimination differently and seems to be one of the really crucial ones.
6. The h11(2)r3c28 and 11(2) innies n36 must have different combos since r1c7+r6c8 both see r3c8 -> [47] in innies n36 must have {56} in r3c28 -> [425] blocked from h11(3)n3 since it forces two 5s into r3 (Blocking cages)
6a. -> h11(3)n3 = {137/146/236}(no 5)
6b. 4 in {146} must be in r3c9 -> no 4 in r1c7
6c. no 7 in r6c8 (innies n36=11)
6e. no 6 in r9c7 (h10(2)r19c7)
6e. no 7 in r4c9

7. 16(3)r1c6 = {367/457}
7a. but {45}[7] blocked by 4 or 5 in 6(2)n2
7b. 16(3)r1c6 = {367} only
7c. r1c45 sees all of the 16(3) -> no 3,6,7 in r1c45 (CPE)

8. Naked quad {1245} in r12c45: all locked for n2

9. Naked triple {367} in r123c7: all locked for c6

10. r1c45 must be {15/24} to avoid clashing with 6(2)n2: ie h6(2)r1c45 (no elims yet)
10a. "45" on n1: 2 outies r1c45 = 2 innies r3c13
10b. ->r3c13 = 6 = {15/24}(no 3,6,7; no 2 in r3c1)

11. h11(2)r3c28 = {47/56} = 4 or 5 -> Killer pair 4,5 with r3c13: 4 locked for r3
11a. no 8 in r4c9

12. h10(2)r19c7 = {37}/[64]= 6/7 -> 18(4)r3c7 can't have two of 6 and 7
12a. -> 15(4)n6 must have at least one of 6 or 7 for n6 = {1257/1347/1356/2346}(no 8 or 9)

13. 15(4)n6 can't have both 6 & 7
13a. -> 18(4)r3c7 must have exactly one of 6 or 7 for n6 = {1269/1278/1368/2457/3456}
13b. but {1368/2457/3456}} blocked by h10(2)r19c7 = {37}/[64]
13c. 18(4)r3c7 = {1269/1278}(no 3,4,5)
13d. must have 1 and 2: both locked for c7

14. "45" on n9: 1 outie r6c8 - 1 = 1 innie r9c7; also note that 8 in n6 only in 18(4)r3c7 or r6c8
14a. 18(4)r3c7 = {1269/1278}: ie, if it has 8 must also have 7
14b. or 8 in r6c8 -> 7 in r9c7
14c. -> 7 in r4569c7: locked for c7 (Locking cages)
14d. no 3 in r9c7 (h10(2)r19c7)
14e. no 4 in r6c8 (innies n36=11)

15. 16(3)r1c6 must have 7 which is only in n2, locked for n2

16. The two 13(2) cages in r9 can't be {49/67} because r9c7 = (47) -> one of the 13(2) must be {58}: both locked for r9
16a. whichever 13(2) doesn't have {58} must have {49/67}, ie, must have one of 4 or 7 -> Killer pair 4,7 with r9c7: both locked for r9

17. "45" on c6789: 2 outies r59c5 = 11 = {29}/[83/56](no 1,4,7) r5c5 = (2589)

18. 23(4)r3c6 must have exactly one of 3 or 6 for r3c6 = {4568} only
18a. -> r3c6 = 6, r45c6+r5c5 = {458} only: all locked for n5 and 4 for c6
18b. r9c5 = (36)(h11(2)r59c5)

Cracked, as the other two WTs show.
Cheers
Ed


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 Post subject: Re: Assassin 292
PostPosted: Tue May 20, 2014 8:51 pm 
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Grand Master
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Joined: Tue Jun 16, 2009 9:31 pm
Posts: 280
Location: California, out of London
Thanks Ed for the comments, but my solution path was not as good as any of the others posted. For what it's worth here are my starting steps.
I also did the 'interesting' version, but to be honest I didn't find it any more so than the V1. Lots of combination analysis. (I only noticed later that it had split up the 25/5 cage at r1c2 - unnecessarily I think.)

Corrections thanks to Andrew.
Hidden Text:
0. The '45/9'
Whatever goes in r5c23 must go in r46c6 in n5
-> Whatever goes in r7c5 also goes in r5c6
-> Whatever goes in r6c2 also goes in r5c5

1. 17/2@r3c4 = {89}
-> Max r12c6 = +13
-> Min r1c7 = 3
Also Min r3c9 = 3
Innies n3 -> r13c7 + r3c9 = +11
-> r3c7 from (12)
Also Max r1c7 = 7

From here on mostly working on r1c7 ... currently has value from (34567)

2. Innies c789 -> r19c7 = +10. (No 5)
-> r1c7 from (3467)

3. Innies n36 -> r1c7 + r6c8 = +11

Putting r1c7 = 4
Puts r3c79 = [25]
Puts 7 in both r4c9 and r6c8 in n6.

-> r1c7 from (367)

4. Since r1c7 from (367) - whatever value it is can only go in n2 in r3c6
-> r123c6 = +16
-> r1c45 = +6
-> r123c6 = {367}
Also -> r3c13 = +6

5. Where do (367) go in n8?
None can go in r7c5 (which is cloned to r5c6)

The 3 is either in r9c5 or in the 19/3 which makes that {379}

if 3 in r9c5
Puts r5c5 = 8 (Outies c6789 -> r59c5 = +11)
Puts 8 in n8 in r789c6
puts 19/3 in n8 = {469}

-> 19/3 in n8 is either {379} or {469}

Also one of (67) in r9c45
Also 9 in n5 in r4c6 or r6c6.

6. Some combination work - not the most elegant.
Putting r19c7 = [73]
Puts r3c79 = [13], r6c8 = 4, and r4c9 = 9
Leaves no solution for 18/4@r3c7

-> r1c7 from (36), r9c7 from (74)

7. Some more combination work - even less elegant.
Putting r19c7 = [37]
Puts r3c6 = 3
Also puts r9c5 = 6 (One of (67) in r9c45)
Puts r5c5 = 5 (Outies c6789 = +11)
Leaves no solution for 23/4@r3c6

-> r1c7 = 6

Straightforward from here


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 Post subject: Re: Assassin 292
PostPosted: Wed May 21, 2014 2:56 am 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1893
Location: Lethbridge, Alberta, Canada
Ed wrote:
I posted the wrong puzzle as the V1! I had a slightly different cage formation to make the start a bit easier.
I assume you meant with the 25(5) cage split as in the Interesting version. That would have taken away the interesting start, so Assassin 292 was a better puzzle as posted.

wellbeback wrote:
I only noticed later that it had split up the 25/5 cage at r1c2 - unnecessarily I think.
Thanks for pointing that out. I solved this version with the 25(5) cage, then re-worked with it split as in Ed's diagram.

Thanks Ed for the interesting and challenging version. Your choice of which cages you removed meant that when one had reached the position where the original A292 was cracked, one had to start thinking of it as a new puzzle and, in my case, to use harder steps to finish it.

Here is my walkthrough for A292 Interesting:
I originally used as many of my steps from Assassin 292 as possible, before working on the area where cages have been removed, since in the simpler version I didn’t work in that area until toward the end of my solving path.

wellbeback pointed out that the 25(5) cage at R1C2 had been split into 19(3) cage at R1C2 and 6(2) cage at R1C4 for this version. I hadn’t noticed when I first solved this version; however when I started checking my walkthrough I realised that it simplifies some of the early steps so I’ve re-worked my steps with the correct cage pattern.

Prelims

a) R1C45 = {15/24}
b) R2C45 = {15/24}
c) R34C3 = {17/26/35}, no 4,8,9
d) R3C45 = {89}
e) R34C9 = {39/48/57}, no 1,2,6
f) R9C34 = {49/58/67}, no 1,2,3
g) R9C89 = {49/58/67}, no 1,2,3
h) 19(3) cage at R1C2 = {289/379/469/478/568}, no 1
i) 19(3) cage at R7C4 = {289/379/469/478/568}, no 1
j) and, of course, 45(9) cage at R4C4 = {123456789}

Steps resulting from Prelims
1a. Naked pair {89} in R3C45, locked for R3 and N2, clean-up: no 3,4 in R4C9
1b. Naked quad {1245} in R12C45, locked for N2
1c. Naked triple {367} in R123C6, locked for C6
1d. R123C6 = {367} = 16, 16(3) cage at R1C6 -> R1C7 = R3C6 = {367}

2. 45 rule on R12 2 outies R3C28 = 11 = {47/56}, no 1,2,3

3. 45 rule on N3 3 innie R13C7 + R3C9 = 11 = {137/146/236/245}
3a. 1,2 only in R3C7 -> R3C7 = {12}

4. 45 rule on C789 2 innies R19C7 = 10 = [37/64/73]

5. 45 rule on N1 2 innies R3C13 = 6 = [15/42/51], clean-up: no 1,2,5 in R4C3
5a. Killer pair 4,5 in R3C13 and R3C28, locked for R3, clean-up: no 7,8 in R4C9
5b. R13C7 + R3C9 (step 3) = {137/236}, 3 locked for N3

6. 45 rule on C6789 2 outies R59C5 = 11 = {29/38/47/56}, no 1

7. 45 rule on N36 2 innies R1C7 + R6C8 = 11 = [38/65/74]

8. 18(4) cage at R3C7 = {1269/1278/2358} (cannot be {1368/1467/2349/2367/2457/3456} which clash with R19C7, cannot be {1359} which clashes with R4C9, cannot be {1458} which clashes with R6C8), no 4, 2 locked for C7

9. 27(6) cage at R6C6 = {123489/123579/124578/134568} (cannot be {123678/234567} because 3,6,7 only possible in R9C57, cannot be {124569} because R9C57 = [64] clashes with the pair of 13(2) cages in R9), 1 locked for C6

10. R59C5 = 11 (step 6), 1,2,4,5,8,9 in C6 only in R456789C6
10a. 27(6) cage at R6C6 (step 9) = {123489/124578/134568} (cannot be {123579} because R45C6 = {48} clashes with R59C5 = [47/83], killer combo clash)
10b. 27(6) cage = {123489/134568} (cannot be {124578} = {458}[217], other permutations clash with the pair of 13(2) cages in R9, because R45C6 = {29} clashes with R59C5 = [92], killer combo clash), no 7, 3 locked for R9, clean-up: no 3 in R1C7 (step 4), no 3 in R3C6 (step 1d), no 8 in R6C8 (step 7)

11. R3C9 = 3 (hidden single in R3) -> R4C9 = 9, clean-up: no 4 in R9C8
11a. 18(4) cage at R3C7 (step 8) = {1278/2358}, no 6, 8 locked for C7 and N6
11b. 6 in N6 only in 15(4) cage at R4C7 = {1356/2346}, no 7, 3 locked for N6
11c. 18(4) cage = {1278} (only remaining combination), locked for C7 -> R1C7 = 6, R9C7 = 4 (step 4), R3C6 = 6 (step 1d), R6C8 = 5 (step 7), clean-up: no 5 in R3C2 (step 2), no 9 in R9C34, no 9 in R9C8, no 8 in R9C9
11d. 15(4) cage at R4C8 = {2346} (only remaining combination), locked for N6, 3 also locked for C8

12. Naked pair {47} in R3C28, locked for R3, clean-up: no 2 in R3C3 (step 5), clean-up: no 6 in R4C3
12a. Naked pair {15} in R3C13, locked for R3 and N1 -> R3C7 = 2

13. 4 in C6 only in R45C6, locked for N5
13a. 23(4) cage at R3C6 contains 4,6 = {4568} (only remaining combination), 5,8 locked for N5
13b. R9C5 = 3 (hidden single in R9), R5C5 = 8 (step 6), R3C45 = [89], clean-up: no 5 in R9C3
13c. Naked pair {45} in R45C6, 5 locked for C6

14. 19(3) cage at R7C4 = {469} (only remaining combination), locked for N8, 9 also locked for C4, clean-up: no 7 in R9C3
14a. Naked triple {128} in R789C6, locked for C6 and N8 -> R6C6 = 9

15. 45(9) cage at R4C4 = {123456789} -> R6C2 = 8, R5C23 = {49}, locked for R5 and N4 -> R45C6 = [45]
15a. 45(9) cage = {123456789} -> R7C5 = 5, R9C4 = 7 -> R9C3 = 6, R9C89 = [85], clean-up: no 1 in R1C4, no 1 in R2C4
15b. 9 in R9 only in R9C12, locked for N7

16. R2C7 = 5 (hidden single in C7), clean-up: no 1 in R2C5
16a. Naked pair {24} in R2C45, locked for R2 and N2 -> R1C45 = [51]

17. R78C7 = {39} (hidden pair in C7), locked for N9
17a. 12(3) cage at R8C7 = {129/237}, no 6, 2 locked for R8 and N9
17b. 21(4) cage at R6C8 = {1569/3567}, 6 locked for R7

18. 19(3) cage at R1C2 = {289/379} (cannot be {478} which clashes with R3C2), no 4, 9 locked for N1
18a. R9C1 = 9 (hidden single in C1)
18b. R4C7 = 8, R6C9 = 4 (hidden singles in N6)

19. 4,6 in N1 only in 20(4) cage at R1C1 = {2468/3467}
19a. 2 of {2468} must be in R1C1 -> no 8 in R1C1
19b. {2468} can only be [2864]

20. 9 in 26(5) cage at R6C1 = {12689/13589/13679/14579/23489/23579/24569}
20a. Hidden killer pair 4,8 in C1 20(4) cage at R1C1 and 26(5) cage for C1, 20(4) cage cannot contain both of 4,8 in C1 (step 19b) -> 26(5) cage must contain at least one of 4,8 in C1 = {12689/13589/14579/23489/24569} (cannot be {13679/23579} which don’t contain 4 or 8)
20b. 6 of {12389} must be in R6C1, 1,2 of other combinations must be in R9C2 -> no 1,2 in R6C1
20c. 3 of {13589/23489} must be in R6C1 -> no 3 in R78C1

21. R3C13 = {15}, R34C3 = [17/35] -> R3C1 + R4C3 = [13/57]
21a. 26(5) cage at R6C1 (step 20a) = {12689/14579/23489/24569} (cannot be {13589} = [38591] which clashes with R3C1 + R4C3)

22. 26(5) cage at R6C1 (step 21a) = {12689/14579/23489/24569}
22a. Consider combinations for 19(3) cage at R1C2 (step 18) = {289/379}
R1C23 + R2C3 = {289} with 2 in R1C2 = 2{89} => R9C2 = 1 => 26(5) cage = {12689/14579}
or R1C23 + R2C3 = {289} with 2 in R1C3 = [928] => R5C2 = 4, R3C2 = 7 => R1C1 = 4 (hidden single in N1) => 26(5) cage = {12689}
or R1C23 + R2C3 = {379} => R2C1 = 8 (hidden single in N1) => 26(5) cage = {14579/24569} => 26(5) cage = {12689/14579/24569}, no 3
22b. R6C1 = {67} -> no 7 in R78C1
22c. 5 of {14579/24569} must be in R8C1 -> no 4 in R8C1
[An alternative way to eliminate that combination is 26(5) cage = {23489} => R3C2 = 4 (hidden single in N1), R5C2 = 9 => R1C23 + R2C3 cannot be {289} so cannot place 2 in N1.]

23. 26(5) cage at R6C1 (step 22a) = {12689/14579/24569}, 5 in N4 only in R4C12
23a. Consider placements for R3C1 = {15}
R3C1 = 1 with 5 in R4C1 => 26(5) cage => {12689} => R6C1 = 6
or R3C1 = 1 with 5 in R4C2 => R6C3 = 1 (hidden single in N4) => R6C7 = 7 => R6C1 = 6
or R3C1 = 5 => 26(5) cage => {12689} => R6C1 = 6
-> R6C1 = 6
23b. R2C2 = 6 (hidden single in N1)

[Returning to the theme of step 21.]
24. R6C1 = 6 -> 26(5) cage at R6C1 (step 22a) = {12689/24569}, 20(4) cage at R1C1 (step 19) = {2468/3467}
Consider permutations for R3C1 + R4C3 (step 21) = [13/57]
R3C1 + R4C3 = [13] => 3 in C1 only in 20(4) cage = {3467} => 8 in C1 only in 26(5) cage = {12689}
or R3C1 + R4C3 = [57] => 26(5) cage = {12689}
-> 26(5) cage = {12689}, 1,2,8 locked for N7, 8 also locked for C1
24a. 20(4) cage = {3467} (only remaining combination), locked for N1, 3 also locked for C1

25. Naked pair {18} in R8C16, locked for R8
25a. Naked pair {27} in R8C89, locked for R8 and N9
25b. R8C89 = {27} = 9 -> R8C7 = 3 (cage sum), R7C7 = 9, R7C4 = 4

26. R1C1 = 4 (hidden single in C1), R3C2 = 7, R7C23 = [37], R4C3 = 3 -> R3C3 = 5

and the rest is naked singles.

Rating Comment:
Since I gave a fairly high rating to my walkthrough for A292, I've had to go even higher for this version. I'll rate my walkthrough for A292 Interesting at 1.75. I used several forcing chains, two of which also used the position of a candidate as a secondary variable.


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PostPosted: Fri May 23, 2014 5:50 pm 
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Grand Master
Grand Master

Joined: Mon Apr 21, 2008 9:44 am
Posts: 310
Location: MV, Germany
I tried solving A292 V2 last Saturday but I gave up after looking for hours for some resourceful forcing chains without any success. Andrew's walkthrough inspired me to try again, especially since I missed his relatively easy step 20a. My hardest step (9f) is just a different wording of Andrew's step 22 but after that our walkthroughs differ.

A292 V2 (a.k.a. A292 Interesting) Walkthrough:
1. R123
a) 17(2) = {89} locked for R3+N2
b) Hidden triple (367) locked in R123C6 = (367) @ N2 for C6
c) 16(3) = {367} since R12C6 = (367)
d) Innies N1 = 6(2) = {15}/[42]
e) Outies R12 = 11(2) = {47/56}
f) Killer pair (45) locked in Innies N1 + Outies R12 for R3
g) 12(2) = [39/75]
h) Innies N3 = 11(3) must have one of (12) -> R3C7 = (12)

2. C6789
a) Innies C789 = 10(2) = {37}/[64]
b) Innies N36 = 11(1+1): R6C8 = (458)
c) Outies C89 = 17(3)
d) Hidden Killer pair (89) in Outies C89 + 18(4) for C7 since each can have at most one of (89)
e) Innies+Outies N6: -12 = R3C7 - (R4C9+R6C8) = 1+[58/94] / 2+[95]
f) Killer pair (89) locked in 18(4) + R4C9+R6C8 for N6
g) Killer pair (45) locked in 15(4) + R4C9+R6C8 for N6
h) 18(4) = 12{69/78} since {1368} blocked by Killer pair (36) of Innies C789 -> 1,2 locked for C7

3. R789+N4
a) R5C23+R7C5 <> 3,6,7 since it sees all 3,6,7 of N5
b) Hidden Killer pair (67) in R9C45 for N8 since 19(3) can only have one of (67)
c) 13(2) @ N9 <> {67} since it's a Killer pair of R9C45

4. C789 !
a) ! Innies+Outies N6: -12 = R3C7 - (R4C9+R6C8): R4C9+R6C8 can't be [94] since it forces R1C7 = 7 (Innies N36) which is blocked by Killer pair (79) of 18(4) -> R4C9+R6C8 = 5{8/9} -> 5 locked for N6
b) Innies N36 = 11(1+1): R1C7 <> 7
c) 16(3) = {367} -> 7 locked for C6
d) Innies C789 = 10(2) = [37/64]

5. C4567+R9 !
a) ! Killer triples (457,478) locked in both 13(2) + R9C7 for R9
b) Outies C6789 = 11(2) <> 1: R5C5 = (2589)
c) 23(4) = {4568} since R3C6 = (36) and they are only possible there -> R3C6 = 6; 4,5,8 locked for N5; 4 also locked for C6
d) Hidden Single: R3C9 = 3 @ R3 -> R4C9 = 9
e) Innies C789 = 10(2) = [64] -> R1C7 = 6, R9C7 = 4
f) 13(2) @ N9 = {58} locked for R9+N9
g) 13(2) @ N8 = {67} locked for R9
h) 27(6) = {123489} -> R9C5 = 3; 8 locked for C6+N8
i) 19(3) = {469} locked for N8
j) Hidden Single: R5C5 = 8 @ N5, R6C2 = 8 @ 45(9)

6. R123+N4
a) R9C4 = 7, R9C3 = 6, R6C8 = 5
b) 8(2) = [17/53]
c) Innies N1 = 6(2) = {15} locked for R3+N1
d) 19(3) = 9{28/37} -> 9 locked for N1 since {478} blocked by R3C2 = (47)
e) Hidden Single: R7C5 = 5 @ N8, R2C7 = 5 @ C7, R1C4 = 5 @ N2 -> R1C5 = 1, R6C6 = 9 @ C6
f) Hidden pair (49) in R5C23 = (49) for 45(9) locked for N4+R5

7. R789+N6
a) 12(3) = 2{19/37} -> 2 locked for R8+N9
b) 9 locked in 26(5) @ R9 for N7
c) Hidden Single: R9C1 = 9 @ C1

8. R123+N4
a) 6(2) @ R2 = {24} locked for R2
b) 4,6 locked in 20(4) @ N1 = 46{28/37}
c) 26(5): R6C1 <> 1,2 since R9C2 = (12) and R78C1 <> 6

9. C123 !
a) 20(4): R12C1 cannot be {48} since R23C2 <> 2
b) Hidden Killer pair (48) in 26(5) for C1 since 20(4) can only have one of them in C1 -> 26(5) = 9{1268/1358/1457/2348/2456}
c) 26(5): R78C1 <> 3,7 since R6C1 = (367)
d) Innies+Outies N1: 2 = R4C3 - R3C1 -> R3C1+R4C3 = [13/57]
e) 26(5) <> {13589} since it's blocked by Killer pair (35) of Innies+Outies N1
f) ! Consider placement of 2 in N1 -> 26(5) <> 3
- i) R1C1 = 2 -> 20(4) = [2864]
- ii) R1C2 = 2 -> R9C2 = 1
- iii) R1C3 = 2 -> 19(3) = [928] -> R5C2 = 4 -> R1C1 = 4 (HS @ N1)

10. C123 !
a) Hidden Killer pair (48) in R12C1 for C1 since 26(5) can only have one of (48)
b) ! 20(4) = 46{28/37}: R1C1 <> 3,7 since R1C2 <> 4 but R12C1 must have one of (48)
c) 20(4): R2C12 <> 7 since 3,6 only possible there
d) 7 locked in R456C1 @ C1 for N4
e) R4C3 = 3 -> R3C3 = 5
f) Hidden Single: R2C1 = 3 @ C1
g) 20(4) = {3467} -> R1C1 = 4, R2C2 = 6, R3C2 = 7
h) 26(5) = {12689} -> R6C1 = 6

11. Rest is singles.

Rating:
(Hard ?) 1.75. I used a small contradiction move and a 3-way forcing chain.


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