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Ed
 Post subject: Assassin 246
PostPosted: Thu Oct 04, 2012 10:35 pm 
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1044
Location: Sydney, Australia
Do the basics and this one will come out without too much strain though some of them hide! Then I tried HATMANS method of converting it to a zero killer to make the middle more challenging but the score either went down (fewer redundant steps needed by SS) or sky high or not unique. If someone else can work out how to convert it into a good zero killer, please post it.

Assassin 246

note: 1-9 cannot repeat on the diagonals
Image

code: paste into solver:
3x3:d:k:4352:3073:6146:6146:6915:6915:6915:6404:6404:4352:3073:3073:6146:6146:6915:5893:2566:6404:4352:5127:5127:7688:6146:7688:5893:2566:6404:2569:2569:5127:7688:6146:7688:5893:2826:6404:6411:6411:5127:7688:7688:7688:5893:2826:1548:4877:6411:6411:6411:6670:5903:5903:5904:1548:4877:1809:2834:2834:6670:5903:5904:5904:5904:4877:1809:2834:4627:6670:6670:2324:2324:3349:4877:4118:4118:4627:4627:6670:6670:2324:3349:
solution: 
+-------+-------+-------+
| 2 8 5 | 3 9 4 | 6 7 1 |
| 9 1 3 | 7 6 8 | 4 2 5 |
| 6 4 7 | 2 1 5 | 9 8 3 |
+-------+-------+-------+
| 4 6 1 | 8 2 7 | 3 5 9 |
| 5 9 8 | 4 3 1 | 7 6 2 |
| 7 3 2 | 6 5 9 | 8 1 4 |
+-------+-------+-------+
| 3 2 4 | 1 7 6 | 5 9 8 |
| 1 5 6 | 9 8 3 | 2 4 7 |
| 8 7 9 | 5 4 2 | 1 3 6 |
+-------+-------+-------+
SudokuSolver Score = 1.20

Cheerio
Ed
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Pinata
 Post subject: Re: Assassin 246
PostPosted: Thu Oct 04, 2012 11:32 pm 
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Joined: Sat Jul 28, 2012 11:05 pm
Posts: 92
Assassin 246 Zero Killer

Image

(Moderator) SudokuSolver image (thanks Ed); original image no longer available.

Jsudoku Code: 3x3:d:k:4352:0000:6145:6145:6914:6914:6914:6403:6403:4352:0000:0000:6145:6145:6914:5892:2565:6403:4352:5126:5126:7687:6145:7687:5892:2565:6403:2568:2568:5126:7687:6145:7687:5892:2825:6403:6410:6410:5126:7687:7687:7687:5892:2825:1547:4876:6410:6410:6410:0000:5901:5901:0000:1547:4876:1806:2831:2831:0000:5901:0000:0000:0000:4876:1806:2831:4624:0000:0000:2321:2321:3346:4876:4115:4115:4624:4624:0000:0000:2321:3346:

Sudoku Solver Score: 2.00
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Frank
 Post subject: Re: Assassin 246
PostPosted: Sat Oct 06, 2012 3:14 am 
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Hmm - bit of a toughie.
Here is how I did it:
I worked with r9c7.
Because of the outies of c89 it is restricted to {1,2,3,4,5}.
3,4,5 are easy to dismiss.
2 needs a bit more work.
So r9c7 = 1. The 9(3) cage is {2,3,4}. And the rest is a walk in the park. :)

Many thanx Ed - Cheers - Frank
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Andrew
 Post subject: Re: Assassin 246
PostPosted: Sun Oct 07, 2012 2:51 am 
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Thanks Ed for a fun Assassin. I thought that I'd managed to solve it more quickly than usual, although a couple of my steps may be more technically difficult than you intended; step 23 felt like a very useful shortcut. Then on checking my walkthrough I found that I'd done the clean-up after that step incorrectly, in both my walkthrough file and on my worksheet, so I'd accidentally produced another shortcut. I've now re-worked my solving path from a few steps after that typo.

Here is my walkthrough for Assassin 246:
Since there aren’t any cages wholly or significantly along the diagonals, it’s likely that they will only be used in the later stages to give a unique solution.

Prelims

a) R23C8 = {19/28/37/46}, no 5
b) R4C12 = {19/28/37/46}, no 5
c) R45C8 = {29/38/47/56}, no 1
d) R56C9 = {15/24}
e) R78C2 = {16/25/34}, no 7,8,9
f) R89C9 = {49/58/67}, no 1,2,3
g) R9C23 = {79}
h) 23(3) cage at R6C6 = {689}
i) 11(3) cage at R7C3 = {128/137/146/236/245}, no 9
j) 9(3) cage at R8C7 = {126/135/234}, no 7,8,9
k) 27(4) cage at R1C5 = {3789/4689/5679}, no 1,2

Steps resulting from Prelims
1a. Naked pair {79} in R9C23, locked for R9 and N7, clean-up: no 4,6 in R8C9
1b. 27(4) cage at R1C5 = {3789/4689/5679}, CPE no 9 in R1C4

2. 45 rule on C89 2 outies R78C7 = 7 = {16/25/34}, no 7,8,9

3. 45 rule on N7 2(1+1) outies R6C1 + R7C4 = 8 = {17/26/35}/[44], no 8,9

4. 45 rule on C1 2 innies R45C1 = 9 = {18/27/36}/[45], no 9, no 4 in R5C1, clean-up: no 1 in R4C2

5. 45 rule on N9 1 outie R6C8 = 1 innie R9C7, no 7,9 in R6C8

6. 9 in C1 only in 17(3) cage at R1C1, locked for N1
6a. 17(3) cage = {179/269/359}, no 4,8

7. 24(6) cage at R1C3 must contain 1,2,3, CPE no 3 in R1C5
7a. 26(6) cage at R6C5 must contain 1, CPE no 1 in R9C5

8. 18(3) cage at R8C4 = {189/369/378/459/468/567} (cannot be {279} because 7,9 only in R8C4), no 2
8a. 9 of {189} must be in R8C4 -> no 1 in R8C4
8b. 7 of {378/567} must be in R8C4, 9 of {369/459} must be in R8C4 -> no 3,5 in R8C4

9. 45 rule on C789 3 innies R169C7 = 15 = {159/168/249/258/267/348} (cannot be {357} because R6C7 only contains 6,8,9, cannot be {456} which clashes with R78C7)
9a. 9 of {159/249} must be in R6C7 -> no 9 in R1C7
9b. 1,2 only in R9C7, 8 of {348} must be in R9C7 -> R9C7 = {1234}, clean-up: R6C8 = {1234} (step 5)

10. 27(4) cage at R1C6 = {3789/4689/5679}, 9 locked for N2

11. Hidden killer triple 7,8,9 in R7C89 and R89C9 for N9, R89C9 contains one of 7,8,9 -> R7C89 must contain two of 7,8,9 -> R7C89 = {789}

12. 45 rule on C9 1 innie R7C9 = 1 outie R1C8 + 1, R1C8 = {678}

13. Killer quad 6,7,8,9 in R1C8, R23C8, R45C8 and R7C8, locked for C8

14. 3 in C9 only in 25(5) cage at R1C8 = {13579/13678/23479} (cannot be {13489/23569/23578/34567} which clash with R56C9)
14a. 8 of {13678} must be in R1C8 (because 6{1378}/7{1368} clash with R1C8 + R7C9 = [67/78], step 12), no 6 in R1C8, no 8 in R1234C9, clean-up: no 7 in R7C9 (step 12)
14b. 8 in C9 only in R789C9, locked for N9

15. Combined cage 25(5) cage at R1C8 (step 14) + R56C9 contains 2,4, locked for C9, clean-up: no 9 in R8C9
15a. 9 in N9 only in R7C89, locked for R7
15b. 9 in 23(3) cage at R6C6 only in R6C67, locked for R6

16. 45 rule on R6789 4 innies R6C2349 = 15 = {1248/1257/1347/1356/2346}
16a. Combined cage R6C2349 + R6C8 must contain 1, locked for R6, clean-up: no 7 in R7C4 (step 3)

17. 18(3) cage at R8C4 (step 8) = {189/369/378/459/567} (cannot be {468} which clashes with R7C6)
17a. 7,9 only in R8C4 -> R8C4 = {79}

18. 23(4) cage at R6C8 = {1589/1679/2489/…}
18a. 1 of {1589/1679} must be in R6C8 -> no 1 in R7C7, clean-up: no 6 in R8C7 (step 2)

19. 9(3) cage at R8C7 = {135/234}, 3 locked for N9, clean-up: no 3 in R6C8 (step 5), no 4 in R8C7 (step 2)

20. R6C2349 (step 16) = {1257/1347/1356/2346} (cannot be {1248} which clashes with R6C8), no 8

21. Hidden killer pair 7,9 in 26(6) cage at R6C5 and R8C4 for N8, R8C4 = {79} -> 26(6) cage must contain one of 7,9 in N8 (this cage may also contain 7 in R6C5) = {123479/123569/123578/134567} (cannot be {124568} which doesn’t contain 7 or 9), CPE no 1 in R9C4, no 3 in R9C5
21a. Min R8C4 + R9C5 = 11 -> no 8 in R9C4

22. 45 rule on N1 2 outies R45C3 = 1 innie R1C3 + 4, IOU no 4 in R45C3

23. R9C7 = {124} “sees” all of its value in N8 except for R7C4 -> R7C4 = R9C7 = {124}, clean-up: R6C1 = {467} (step 3)

24. R7C9 = R1C8 + 1 (step 12)
24a. R1C8 + R7C9 = [78/89] -> R17C8 = [79/87], 7 locked for C8, clean-up: no 3 in R23C8, no 4 in R45C8

25. R6C8 = R9C7 (step 5) -> R689C8 = R89C8 + R9C7, R89C8 + R9C7 cannot total 9 because that would clash with 9(3) cage at R8C7, CCC -> R689C8 cannot total 9
25a. 45 rule on C8 5 innies R16789C8 = 24 contains 7, R689C8 cannot total 9 -> R7C8 = 9, R7C9 = 8, R1C8 = 7 (step 12), R8C9 = 7 (hidden single in N9), R9C9 = 6, placed for D\, R7C6 = 6, R8C4 = 9, clean-up: no 1 in R23C8, no 2 in R45C8, no 1 in R8C2, no 1 in R8C7 (step 2)
25a. Naked pair {89} in R6C67, locked for R6
25b. R8C4 = 9 -> R9C45 = 9 = {45}, locked for R9 and N8, clean-up: no 4 in R6C1 (step 3), no 4 in R6C8 (step 5)
25c. Killer pair 1,2 in R56C9 and R6C8, locked for N6
25d. Killer pair 1,2 in 9(3) cage at R8C7 and R9C7, locked for N9, clean-up: no 5 in R8C7 (step 2)

26. R7C5 = 7 (hidden single in R7)
26a. R7C5 + R8C56 + R9C67 = {12378} = 21 -> R6C5 = 5, clean-up: no 1 in R5C9
26b. 1 in N6 only in R6C89, locked for R6

27. 3 in R6 only in R6C234, locked for 25(5) cage at R5C1, no 3 in R5C12, clean-up: no 6 in R4C1 (step 4), no 4 in R4C2
27a. 6,7 in R6 only in R6C1234, CPE no 6,7 in R5C12, clean-up: no 2,3 in R4C1 (step 4), no 7,8 in R4C2

28. 3 in R7 only in R7C123, locked for N7, clean-up: no 4 in R7C2

29. 1,2 in R7 only in R7C1234, CPE no 1,2 in R8C3

30. 9(3) cage at R8C7 = {135/234} (step 19)
30a. 4,5 only in R8C8 -> R8C8 = {45}
30b. 1 in N9 only in R9C78, locked for R9

31. 45 rule on C8 1 outie R8C7 = 1 innie R6C8 + 1 -> R6C8 + R8C7 = [12/23], CPE no 2 in R9C8
31a. 2 in 9 only in R89C7, locked for C7

32. R169C7 (step 9) = {159/168/249/258} (cannot be {348} because 3,4 only in R1C7)
32a. 4,5,6 only in R1C7 -> R1C7 = {456}
32b. 27(4) cage at R1C5 = {4689/5679} (cannot be {3789} because R1C7 only contains 4,5,6), no 3, 6 locked for R1
32c. 7 of {5679} must be in R2C6 -> no 5 in R2C6

33. 6 in C7 must be in R169C7 = {168} (step 32) or in 23(4) cage at R2C7
33a. 7 in C7 only in 23(4) cage at R2C7 = {1679/3479} (cannot be {3578} which clashes with R169C7 = {168}, locking-out cages), no 5,8, 9 locked for C7 -> R6C67 = [98]
33b. R6C7 = 8 -> R169C7 (step 32) = {168/258}, no 4

34. R1C5 = 9 (hidden single in N2)
34a. 27(4) cage at R1C5 (step 32b) = {4689/5679} -> R1C7 = 6, R9C7 = 1 (step 33b), R9C8 = 3, R8C7 = 2, R8C8 = 4 (cage sum), R7C7 = 5 (both placed for D\), R6C8 = 1 (step 5), clean-up: no 3 in R7C2 (rest of step 34a replaced by step 34e)
34b. Naked pair {28} in R23C8, locked for C8 and N3
34c. Naked pair {56} in R45C8, locked for N6
34d. Naked pair {24} in R56C9, locked for C9 and N6
34e. Naked pair {38} in R8C56, locked for R8 and N8 -> R9C6 = 2, R9C1 = 8, placed for D/, R7C4 = 1, R8C1 = 1 (hidden single in R8), clean-up: no 2,9 in R4C2


35. R7C2 = 2, R8C2 = 5, placed for D/, R8C3 = 6, R7C3 = 4 (cage sum), placed for D/, R7C1 = 3, R6C1 = 7 (cage sum), R4C1 = 4, R5C1 = 5 (step 4), R4C2 = 6, R45C8 = [56], R6C23 = [32], R56C9 = [24], R6C4 = 6, R1C1 = 2, placed for D\

36. Naked pair {13} in R1C9 + R5C5, locked for D/ -> R3C7 = 9, R4C6 = 7

37. 1 in C6 only in R35C6, locked for 30(7) cage at R3C4 -> R5C5 = 3, placed for both diagonals, R1C9 = 1, R4C4 = 8, placed for D\, R5C46 = [41], R3C6 = 5, R3C4 = 2 (cage sum)

and the rest is naked singles, without using the diagonals.

Rating Comment:
I'll rate my walkthrough for A246 at 1.5. I used a fairly easy to spot "clone", a couple of variable combined cages and a locking-out cages step.
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Ed
 Post subject: Re: Assassin 246
PostPosted: Tue Oct 09, 2012 9:58 am 
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This Assassin went down better than expected. Glad you enjoyed it and found it tough enough. I used (technically) much simpler moves than Andrew needed though I can see why Andrew enjoyed his solve with some fun stuff in there. Looks like my steps 2 & 10 were the keys with 14a helping.

A246 Walkthrough
21 steps:
Please let me know of any corrections or clarifications needed. I want a newbie to killers to be able to follow it easily.

1. 16(2)N7 = {79}: Both locked for r9 and n7

2. 23(3)r6c6 = {689} only: r7c7 sees each of 6,8,9 through D\ -> no 6,8,9 in r7c7 (this no 6 in r7c7 is a key elimination which is why X killers are cool!)

3. "45" on c89: 2 outies r78c7 = 7 (no 7,8,9; no 1 in r8c7)

4. 27(4)r1c5 = {3789/4689/5679}(no 1,2)
4a. "45" on c789: 3 innies r169c7 = 15
4b. min. r16c7 = [36] = 9 -> max. r9c7 = 5 [can't be 366](no 6,8 in r9c7)

5. 9(3)n9 = {126/135/234}(no 7,8,9)
5a. 13(2)n9 = {58}[76/94](no 1,2,3; no 4,6 in r8c9) = [7/8/9..]

6. Hidden killer triple 7,8,9 in n9: 13(2) has one of 7,8,9 -> r7c89 = {789} only

7. "45" on c9: 1 innie r7c9 - 1 = 1 outie r1c8 (IODc9=-1)
7a. r1c8 = (678)

8. 10(2)n3 = {19/28/37/46}(no 5) = [6/7/8/9..]
8a. 11(2)n6 = {29/38/47/56}(no 1) = [6/7/8/9..]

9. Killer quad 6,7,8,9 in c8 in r1c8+10(2)+11(2)+r7c8: no 6,7,8,9 in r689c8

Another hard one to see
10. "45" on c89: 1 outie r7c7 + 2 = 2 innies r89c8
10a. -> no 2 in r89c8 since it would force the one outie in n9 to equal the other innie in n9 (IOU)

11. 9(3)n9: {126} blocked since 2 & 6 are only in r8c7
11a. = {135/234}(no 6)
11b. must have 3 -> 3 locked for n9
11c. no 4 in r8c7 (h7(2)r78c7)
11d. no 1 in r7c7 (h7(2)r78c7)

12. Hidden single 6 in n9 -> r9c9 = 6: Placed for D\
12a. r8c9 = 7
12b. no 6 in r1c8 (IODc9=-1)

13. Naked pair {89} at r7c89: both locked for r7
13a. r7c6 = 6

14. From step 7, IODc9=-1->r7c9+r1c8 = [87/98]
14a. must have 8 -> no 8 in r7c8 since it sees both r7c9 & r1c8
14b. r7c8 = 9, r7c9 = 8 -> r1c8 = 7 (IODc9=-1)
14c. no 2, 4 in 11(2)n6
14d. no 1, 3 in 10(2)n3

15. 2 in n9 only in c7: locked for c7

16. 23(4)r2c7: must have 7 for c6
16a. but {3578} blocked by h7(2)r78c7 = {25}/[34] = [3/5,...]
16b. = {1679/3479}(no 5)
16c. must have 9: 9 locked for c7

17. r6c67 = [98]; 9 placed for D\
17a. 27(4)r1c5 must have 9 -> r1c5 = 9
17b. Hidden single 9 in n8: r8c5 = 9
17c. 11(2)n6 = {56} only: both locked for c8 and n6
17d. 10(2)n3 = {28} only: 2 locked for c8
17e. 6(2)n6 = {24} only: both locked for c9 and n6

18. "45" on n9: 1 outie r6c8 = 1 innie r9c7: only common number is 1 -> both = 1
18a. h15(3)r169c7: r69c7 = 9 -> r1c7 = 6

19. r8c4 = 9 -> r9c45 = 9 = {45} only: both locked for r9 and n8
19a. r9c8 = 3, r8c8 = 4 (Placed for D\), r8c7 = 2 (cage sum)

20. Naked pair {38} in r8c56: both locked for r8 and n8

21. Hidden single 1 in n8 -> r7c4 = 1
21a. -> r78c3 = 10 = [46] only: 4 placed for D/


on from there. Don't forget the diagonals!!
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Andrew
 Post subject: Re: Assassin 246
PostPosted: Wed Oct 10, 2012 2:36 am 
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Thanks Ed for showing the important step I missed in my solving path:
Your CPE elimination of 6 using the 23(3) cage and D\ proved to be powerful because it quickly led to a hidden single 6 in N9, leading to more placements. I had to work harder, using an interesting CCC, to get those placements.

It was strange that I spotted one IOU but not your one. Fortunately missing that one wasn't important; the way I reduced the 9(3) cage to two combinations wasn't any more difficult.
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Andrew
 Post subject: Re: Assassin 246
PostPosted: Sat Oct 13, 2012 7:36 pm 
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Thanks Pinata for creating the Zero Killer version of Ed's A246. I had another look at it this week and managed to find how to "get into" this puzzle.

Ed will appreciate this Zero Killer. It made the U-shaped 30(7) cage at R3C4 much more important.

I think there are only three steps in my solving path which can be described as "heavy".

Removing the 23(4) cage at R6C8:
was the most important change, taking away the easy ways into A246. Removing the 26(6) cage at R6C5 also removed the "clone" which I found when solving A246. The 12(3) cage in N1 could probably have been kept; its absence didn't make any significant difference to my solving path.

Here is my walkthrough for A246 Zero Killer:
If you find any errors, including typos, please tell me by PM.

Prelims

a) R23C8 = {19/28/37/46}, no 5
b) R4C12 = {19/28/37/46}, no 5
c) R45C8 = {29/38/47/56}, no 1
d) R56C9 = {15/24}
e) R78C2 = {16/25/34}, no 7,8,9
f) R89C9 = {49/58/67}, no 1,2,3
g) R9C23 = {79}
h) 23(3) cage at R6C6 = {689}
i) 11(3) cage at R7C3 = {128/137/146/236/245}, no 9
j) 9(3) cage at R8C7 = {126/135/234}, no 7,8,9
k) 27(4) cage at R1C5 = {3789/4689/5679}, no 1,2

Steps resulting from Prelims
1a. Naked pair {79} in R9C23, locked for R9 and N7, clean-up: no 4,6 in R8C9
1b. 27(4) cage at R1C5 = {3789/4689/5679}, CPE no 9 in R1C4
[And the step I missed in Assassin 246 but Ed found …]
1c. 23(3) cage at R6C6 = {689}, CPE no 6,8,9 in R7C7 using D\

2. 45 rule on N7 2(1+1) outies R6C1 + R7C4 = 8 = {17/26/35}/[44], no 8,9

3. 45 rule on C1 2 innies R45C1 = 9 = {18/27/36}/[45], no 9, no 4 in R5C1, clean-up: no 1 in R4C2

4. 9 in C1 only in 17(3) cage at R1C1, locked for N1
4a. 17(3) cage = {179/269/359}, no 4,8

5. 24(6) cage at R1C3 must contain 1,2,3, CPE no 3 in R1C5

6. 18(3) cage at R8C4 = {189/369/378/459/468/567} (cannot be {279} because 7,9 only in R8C4), no 2
6a. 9 of {189} must be in R8C4 -> no 1 in R8C4
6b. 7 of {378/567} must be in R8C4, 9 of {369/459} must be in R8C4 -> no 3,5 in R8C4

7. 45 rule on C9 1 innie R7C9 = 1 outie R1C8 + 1, no 9 in R1C8, no 1 in R7C9

8. 45 rule on N36 3(2+1) innies R1C7 + R6C78 = 15
8a. Min R16C7 = 9 -> max R6C8 = 5 (R1C7 + R6C78 cannot be [36]6)
8b. Min R6C78 = 7 -> max R1C7 = 8

9. 27(4) cage at R1C5 = {3789/4689/5679}, 9 locked for N2
9a. 4 of {4689} must be in R12C6 (R12C6 cannot be {689} which clashes with R67C6, ALS block), no 4 in R1C57
9b. R1C7 + R6C78 = 15 (step 8)
9c. R16C7 cannot total 10 -> no 5 in R6C8

10. R7C9 = R1C8 + 1 (step 7)
10a. Consider placement for 3 in C9
3 in R1234C9 => no 3 in R1C8
or 3 in R7C9 => R1C8 = 2
-> no 3 in R1C8, clean-up: no 4 in R7C9

11. From step 10, 25(5) cage at R1C8 must contain 3 or must contain 2 in R1C8 = {12589/12679/13489/13579/13678/23479/23569/23578/34567} (cannot be {14569/14578} which don’t contain 2 or 3, cannot be {24568} which clashes with R56C9)
[Note that when 25(5) cage contains 3, it cannot contain 2 in R1C8 because of step 7. Also when 25(5) cage doesn’t contain 1 then R56C9 = {15}, only other place for 1 in C9, so any 5 must be in R1C8.]
11a. 25(5) cage = {12589/12679/13489/13579/13678/23479/23578} (cannot be {23569/34567} because 5{2369}/5{3467} clash with R1C7 + R7C9 = [56] because the 5 in these combinations must be in R1C8, as in above note)
11b. 6 of {12679/13678} must be in C9 (cannot be 6{1279/1378} which clash with R1C7 + R7C9 = [67]) -> no 6 in R1C8, clean-up: no 7 in R7C9 (step 7)

12. R1C7 + R6C78 = 15 (step 8)
12a. 25(5) cage at R1C8 (step 11a) = {12589/12679/13489/13579/13678/23479/23578}
Consider combinations for 25(5) cage
25(5) cage = {12589/12679} => R1C8 = 2, 1,9 locked for C9 => R7C8 = 9 (hidden single in N9), R56C9 = {24}, locked for N6 => R45C8 = {38/56}, then R23C8 = {46/37} => R1C7 + R6C78 cannot be [393] which clashes with R45C8 = {38} or R23C8 = {37}
or 25(5) cage = {13489/13579/13678/23479/23578}, 3 must be R1234C9 => R1C7 + R6C78 cannot be [393]
-> no 3 in R6C8 (because [393] was only permutation of R1C7 + R6C78 with 3 in R6C8)

13. 45 rule on C7 (using R1C7 + R6C78 = 15, step 8) 3 innies R789C7 = 1 outie R6C8 + 7
13a. R6C8 = {124} -> R789C7 = 8,9,11 = {125/134/126/234/146/245} (cannot be {135} which clashes with 9(3) cage at R8C7 = {135} and R6C8 + R789C7 = 2{135} clashes with 9(3) cage at R8C7 = {126/234}, cannot be {128/137/236} because R789C7 must contain 4 when R6C8 = 4), no 7,8
13b. 6 of {126} must be in R9C7 (R8C7 cannot be 6 because R79C7 = {12} clashes with R89C8 = {12}), 6 of {146/236} must be in R9C7 because R79C7 = {14/23} clash with R89C8 = {12}), no 6 in R8C7

14. Hidden killer pair 7,8,9 in R7C89 and R89C9 for N9, R89C9 contains one of 7,8,9 -> R7C89 = {89}, clean-up: R1C8 = {78} (step 7)
14a. Killer quad 6,7,8,9 in R17C8, R23C8 and R45C8, locked for C8
14b. 9(3) cage at R8C7 = {135/234}, 3 locked for N9
14c. 6 in N9 only in R9C79, locked for R9

15. R1C8 + R7C9 (step 7) = [78/89], CPE no 8 in R1234C9
15a. R1C8 + R7C9 = [78/89] -> R7C89 = [98/79], 9 locked for R7 and N9, clean-up: no 4 in R9C9
15b. R7C89 = [98/79] -> R7C8 = {79}
15c. 23(3) cage at R6C6 = {689}, 9 locked for R6

16. R1C8 + R7C9 (step 15) = [78/89] -> R17C8 = [79/87], 7 locked for C8, clean-up: no 3 in R23C8, no 4 in R45C8

17. 18(3) cage at R8C4 (step 6) = {189/378/459} (cannot be {468} which clashes with R7C6, cannot be {369/567} because 6,7,9 only in R8C4)
17a. 7,9 only in R8C4 -> R8C4 = {79}

18. 45 rule on N9 4 innies R7C789 + R9C7 = 23 = {1589/1679/2489} (cannot be {2579/2678/4568} which clash with R89C9)
18a. R789C7 (step 13a) = {125/126/234/146/245} (cannot be {134} because R7C789 + R9C7 doesn’t contain both of 1,4)
18b. 1,5 of {125} must be in R79C7, 1,6 of {126/146} must be in R79C7 (because of combinations for R7C789 + R9C7) -> no 1 in R8C7

19. R1C7 + R6C78 = 15 (step 8)
19a. Consider placements for R6C8 = {124}
R6C8 = 1 => R789C7 = 8 (step 13a) = {125} => R16C7 = 14 = {68}
or R6C8 = {24} => R6C7 = {68} (cannot be 9 because no 2,4 in R1C7)
-> R6C7 = {68}

20. 23(3) cage at R6C6 = {689} -> R6C6 = 9, placed for D\
20a. R1C5 = 9 (hidden single in N2)
20b. R8C4 = 9 (hidden single in N8), R9C45 = 9 = {18/45}, no 3

21. 27(4) cage at R1C5 = {3789/4689/5679}
21a. 3 of {3789} must be in R1C67 (R1C67 cannot be {78} which clashes with R1C8), no 3 in R2C6

22. 30(7) cage at R3C4 = {1234578} (only remaining combination), no 6

23. R1C7 + R6C78 = 15 (step 8) = [384/681/762/861] (cannot be [582] because R12C6 = {67} clashes with R7C6 = 6), no 5 in R1C7

24. 24(6) cage at R1C3 and 30(7) cage at R3C4 both contain 3 -> no 3 in R1C6

25. 3 in C9 only in 25(5) cage at R1C8 (step 11a) = {13579/13678/23479} (cannot be {13489/23578} which clash with R56C9)
25a. Consider combinations for 25(5) cage
25(5) cage = {13579/13678} => no 2
or 25(5) cage = {23479} => R1C8 = 7, R7C89 = [98], R7C6 = 6, R6C7 = 8, 8 in N3 only in R23C8 = {28}, locked for N3 => no 2 in R123C9
-> no 2 in R123C9

26. 2 in C9 only in R456C9, locked for N6, clean-up: no 9 in R45C8

27. R6C8 = {14} -> R789C7 (step 13a) = 8,11 -> R789C7 (step 18a) = {125/146/245}, no 3
27a. 3 in N9 only in R89C8, locked for C8, clean-up: no 8 in R45C8
27b. Naked pair {56} in R45C8, locked for C8 and N6 -> R6C7 = 8, R7C6 = 6, clean-up: no 4 in R23C8, no 1 in R56C9, no 2 in R6C1 (step 2), no 1 in R8C2
27c. Naked pair {24} in R56C9, locked for C9 and N6 -> R6C8 = 1, clean-up: no 9 in R23C8, no 7 in R7C4 (step 2)
27d. Naked pair {28} in R23C8, locked for C8 and N3 -> R1C8 = 7, R7C89 = [98]

28. R8C9 = 7 (hidden single in N9), R9C9 = 6, placed for D\

29. Naked pair {34} in R89C8, locked for N9, R8C7 = 2 (cage sum), clean-up: no 5 in R7C2
29a. Naked pair {15} in R79C7, locked for C7
29b. Killer pair 1,5 in R9C45 and R9C7, locked for R9

30. 7,9 in C7 only in 23(4) cage at R2C7 = {3479} (only remaining combination), locked for C7

31. R1C7 = 6 -> R12C6 = 12 = [48/57/84], no 5 in R2C6

32. R7C5 = 7 (hidden single in R7)

33. 3,4 in R7 only in R7C1234, CPE no 3,4 in R8C3

34. 11(3) cage at R7C3 = {128/146/236/245}
34a. 5 of {245} must be in R8C3 -> no 5 in R7C34, clean-up: no 3 in R6C1 (step 2)
34b. 5,6,8 only in R8C3 -> R8C3 = {568}

35. 24(6) cage at R1C3 = {123468/123567}
35a. 7 of {123567} must be in R2C4 -> no 5 in R2C4
35b. 6 in N2 only in 24(6) cage, no 6 in R4C5

36. 6 in N5 only in R6C45, locked for R6, clean-up: no 2 in R7C4 (step 2)
36a. R9C6 = 2 (hidden single in N8)

37. 45 rule on N25 2 outies R1C37 = 2 remaining innies R6C45
37a. R1C7 = 6, R6C45 contains 6 (step 36) -> R1C3 must be the same as one of R6C45, no 1,8 in R1C3, no 7 in R6C4

38. 7 in N5 only in 30(7) cage at R3C4, no 7 in R3C46

39. 7 in R6 only in R6C789, locked for N4, clean-up: no 3 in R4C12, no 2 in R4C1 (step 3), no 8 in R4C2, no 2,6 in R5C1 (step 3)

40. R4C6 = 7 (hidden single on D/), R5C7 = 7 (hidden single in C7), clean-up: no 5 in R1C6 (step 31)
40a. Naked pair {39} in R4C79, locked for R4, clean-up: no 1 in R4C1, no 8 in R5C1 (step 3)

41. Naked pair {48} in R12C6, locked for C6 and N2

42. R2C4 = 7 (hidden single in N2) -> 24(6) cage at R1C3 = {123567} (only remaining combination), no 4,8, 6 locked for C5, clean-up: no 4 in R6C45 (step 37a)
42a. R1C26 = {48} (hidden pair in R1)

43. R6C4 = 6 (hidden single in N5), placed for D/, clean-up: no 1 in R7C2

44. 25(5) cage at R5C1 contains 6 = {12679/13678/14569/23569/24568} (cannot be {34567} which clashes with R6C1)
44a. 8,9 only in R5C2 -> R5C2 = {89}
44b. 25(5) cage = {12679/14569/23569} (cannot be {13678} which clashes with R45C1, CCC, cannot be {24568} which clashes with R6C9) -> R5C2 = 9, R9C23 = [79]
44c. 25(5) cage = {14569/23569} (cannot be {12679} which clashes with R4C12 = [82] using step 3), no 7, 5 locked for N4
44d. Killer pair 2,4 in R6C23 and R6C9, locked for R6

45. R6C1 = 7 (hidden single in R6), R7C4 = 1 (step 2), R7C7 = 5, placed for D\, R9C7 = 1, clean-up: no 1 in 17(3) cage at R1C1 (step 4a), no 8 in R9C45 (step 20b)
45a. R3C3 = 7 (hidden single in C3)

46. Naked pair {45} in R9C45, locked for R9 and N8 -> R8C56 = [83], R8C8 = 4, placed for D\, R9C8 = 3, R9C1 = 8, R8C2 = 5, both placed for D/, R7C2 = 2, R8C3 = 6, R7C3 = 4 (cage sum), placed for D/, R78C1 = [31], R5C1 = 5, R4C1 = 4 (step 3), R4C2 = 6, R6C23 = [32]

47. R3C2 = 4 (hidden single in R3), R1C2 = 8, R2C2 = 1, placed for D\

48. R1C1 = 2, placed for D\, R5C5 = 3, placed for D/, R1C9 = 1, R3C7 = 9

and the rest is naked singles, without using the diagonals.

Rating Comment:
I'll rate my walkthrough for A246 Zero Killer at Hard(?) 1.75, based on steps 11 to 13. I also used some forcing chains.
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wellbeback
 Post subject: Re: Assassin 246
PostPosted: Sun Oct 14, 2012 4:07 am 
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Grand Master

Joined: Tue Jun 16, 2009 9:31 pm
Posts: 282
Location: California, out of London
Thanks Ed! I found it interesting that you and I followed a similar path at the beginning - but we used different proofs for many of the steps. Also, like Andrew, I didn't use the diagonals until much later in the solution.

Hidden Text:
1. 16/2 @r8c2 = {79}
23/3@r6c6 = {689}

2. Outies c89 -> r78c7 = +7 (no 789)
Innies c789 -> r169c7 = +15

Now, max r1c5+r1267c6 = 9+30 = +39
-> Min r16c7 = (27+23-39) = +11
-> Max r9c7 = 4 (no 789) (or 56)

9/3@r8c7 (no 789)
-> 789 in n9 locked in r7c89 + one of them in r89c9.

3. Innies - Outies n9 = 0 -> r6c8 = r9c7

Since r78c7 = +7 -> r79c7 cannot = +7
-> r7c7+r6c8 cannot = +7
-> r7c89 cannot = +16 - i.e., cannot be {79} -> must include an 8.
-> 13/2@r8c9 cannot be {58}

This next section reworked slightly to clarify.
Innies - Outies c9 -> r7c9 = r1c8+1
-> r1c8 from (678)
Whatever it is must go in r89c9 in c9
-> 13/2@r8c9 cannot be {49}

-> 13/2@r8c9 = [76]
-> r7c89 = {89}

Since r1c8 is one less than r7c9 -> r7c89 = [98]

Also r7c6 = 6
Also r1c8 = 7

4. r7c7+r6c8 = +6
-> r7c7+r9c7 = +6 (since r6c8 = r9c7)
-> r1c7+r9c7 cannot = +6
-> (Innies c789 r169c7 = +15) r6c7 cannot = 9.

-> r6c67 = [98]
-> 10/2@r2c8 = {28}
-> 11/2@r4c8 = {56}
-> 6/2@r5c9 = {24}
-> r6c8,r7c7 = [15]
-> r8c7 = 2
Also r9c7 = 1
-> r89c8 = {34}
Also r1c7 = 6

5. 27/4@r1c5 must contain a 9. Only place for it -> r1c5 = 9
-> HS 9 in n8 -> r8c4 = 9
-> HS 1 in n8 -> r7c4 = 1
-> Outies n7 r6c1 = 7

Also (16) in r8 both in n7 -> r8c1 = 1, r8c3 = 6
-> r7c3 = 4
-> 7/2@r7c2 = [25]
-> r9c6 = 2
Also (Outies n8) r6c5 = 5
-> r9c45 = [54]
-> (since 30/7@r3c4 must contain a 5) -> r3c6 = 5
-> r12c6 = {48}

Pretty easy from here
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