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h3lix
 Post subject: Anti Bishop Killer 2
PostPosted: Sat Jun 12, 2010 1:54 pm 
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Here's another Anti-Bishop killer.
I provided both my standard image type (made by modifying a PerfectSudoku image using a Photoshop template file and used in the book) as well as the JSudoku image. Is there any preference between these types, because the second one is much easier to create.
HARD version:
ImageImage
JSudoku file
3x3::k:3588:3588:1308:4355:4355:4355:4354:2309:2309:30:3598:31:1308:4354:4354:4354:3607:32:4624:33:3598:3097:4122:4122:2578:34:3607:35:4624:36:3097:4122:4122:2578:790:37:2068:38:4624:4359:4359:4359:4369:39:790:40:2068:2323:4635:4635:2584:41:4369:42:2325:43:2323:4635:4635:2584:4111:44:4369:45:2325:4105:4105:4105:2333:46:4111:47:2316:2316:4105:4619:4619:4619:2333:2317:2317:

INSANE version:
ImageImage
JSudoku file
3x3::k:3588:3588:1308:10:11:30:4354:2309:2309:31:3598:32:1308:4354:4354:4354:3607:33:4624:34:3598:3097:4122:4122:2578:35:3607:36:4624:37:3097:4122:4122:2578:790:38:2068:39:4624:4359:4359:4359:4369:40:790:41:2068:2323:4635:4635:2584:42:4369:43:2325:44:2323:4635:4635:2584:4111:45:4369:46:2325:4105:4105:4105:2333:47:4111:48:2316:2316:4105:49:50:51:2333:2317:2317:

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Andrew
 Post subject: Re: Anti Bishop Killer 2
PostPosted: Thu Jun 17, 2010 11:49 pm 
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Thanks Dan for puzzle which was both challenging and fun.

Rating Comment:
I agree with the puzzle title, calling it HARD. It was quite a lot harder than the first Anti-Bishop Killer MEDIUM. As well as CPEs I also used killer pairs on two of the shorter diagonals plus combination analysis on the innies for N7 which used CCC interactions with R67C3.

Here is my walkthrough for Anti-Bishop Killer 2 HARD:
No repeated candidates on the diagonals R1C1-R9C9, R1C3-R7C9, R1C5-R5C9, R3C1-R9C7, R5C1-R9C5, R1C5-R5C1, R1C7-R7C1, R1C9-R9C1, R3C9-R9C3 and R5C9-R9C5.

Prelims (not including any cage interactions)

a) R1C12 = {59/68}
b) 5(2) cage at R1C3 = {14/23}
c) R1C89 = {18/27/36/45}, no 9
d) 14(2) cage at R2C2 = {59/68}
e) 14(2) cage at R2C8 = {59/68}
f) R34C4 = {39/48/57}, no 1,2,6
g) R34C7 = {19/28/37/46}, no 5
h) 3(2) cage at R4C8 = {12}
i) 8(2) cage at R5C1 = {17/26/35}, no 4,8,9
j) R67C3 = {18/27/36/45}, no 9
m) R67C6 = {19/28/37/46}, no 5
n) 9(2) cage at R7C1 = {18/27/36/45}, no 9
o) 16(2) cage at R7C7 = {79}
p) 9(2) cage at R8C6 = {18/27/36/45}, no 9
q) R9C12 = {18/27/36/45}, no 9
r) R9C89 = {18/27/36/45}, no 9

Steps resulting from Prelims
1a. Naked pair {12} in 3(2) cage at R4C8, locked for N6 and R1C5-R5C9 diagonal, clean-up: no 8,9 in R34C7
1b. Naked pair {79} in 16(2) cage at R7C7, locked for N9 and R1C1–R9C9 diagonal, clean-up: no 5 in R1C2, no 5 in 14(2) cage at R2C2, no 3,5 in R3C4, no 1,3 in R7C6, no 2 in R8C6, no 2 in R9C89
[There's are also CPEs from the {79} pair but I only spotted them when I started on the INSANE version of this puzzle. There's another one for the {68} pair in step 2.]

2. Naked pair {68} in 14(2) cage at R2C2, locked for N1 and R1C1–R9C9 diagonal -> R1C1 = 5, placed for R1C1–R9C9 diagonal, R1C2 = 9, clean-up: no 4 in R1C89, no 4,7 in R3C4, no 3 in R6C2, no 2,4 in R7C6, no 4 in R8C2, no 4 in R9C2, no 1,3,4 in R9C8

3. 2 on R1C1–R9C9 diagonal only in R5C5 + R6C6, locked for N5

4. 45 rule on R1 2 innies R1C37 = 5 = {14/23}, no 6,7,8 in R1C7
4a. R1C456 = {278/368/467}, no 1

5. 45 rule on R9 2 innies R9C37 = 9 = {18/36/45}/[72], no 2,9 in R9C3
5a. 9 in R9 only in R9C456, locked for N8, clean-up: no 1 in R6C6

6. R7C7 = 9 (hidden single in R7), placed for R5C9-R9C5 diagonal, R8C8 = 7, clean-up: no 2 in R1C9, no 2 in R7C1, no 2 in R9C7, no 7 in R9C3 (step 5)

7. 1 on R1C1–R9C9 diagonal only in R5C5 + R9C9, CPE no 1 in R5C9
7a. R5C9 = 2, placed for R5C9-R9C5 diagonal, R4C8 = 1, placed for R3C9 –R9C3 diagonal, clean-up: no 8 in R1C9, no 6 in R6C2, no 8 in R9C7 (step 5), no 1 in R8C6

8. R6C6 = 2 (hidden single on R1C1–R9C9 diagonal), placed for R3C9-R9C3 diagonal, R7C6 = 8, clean-up: no 6 in R5C1, no 1 in R6C3, no 7 in R7C3, no 1 in R8C2, no 1 in R9C7, no 8 in R9C3 (step 5)

9. R9C456 must contain 9 = {279/369/459}, no 1
9a. 7 of {279} must be in R9C5 -> no 7 in R9C46
9b. 1 in N8 only in R7C4 + R8C5, CPE no 1 in R6C5

10. 17(3) cage at R5C7 = {368/458/467}, no 1
10a. Killer pair 3,4 in 5(2) cage at R1C3 and 17(3) cage at R5C7, locked for R1C3-R7C9 diagonal

11. Min R8C45 + R9C3 = 8 -> max R8C3 = 8

12. R8C1 = 9 (hidden single in R8)

13. 2 on R1C9-R9C1 diagonal only in R7C3 + R8C2 + R9C1, locked for N7, clean-up: no 7 in R9C1

14. 45 rule on N7 4 innies R7C23 + R89C3 = 18 = {1368/1458/2367/2457/3456} (cannot be {1278} because R9C3 only contains 3,4,5,6, cannot be {1467/2358} which clash with the 9(2) cages in N7)
14a. 1,8 of {1368/1458} must be in R7C2 + R8C3 (R78C3 cannot be [18] which clashes with R67C3, CCC), no 1 in R78C3, clean-up: no 8 in R6C3
14b. R7C23 + R89C3 = {2367/2457/3456} (cannot be {1368/1458} because R79C3 cannot be {36/45} which clashes with R67C3, CCC), no 1,8

15. 16(4) cage at R8C3 = {1456/2356}, CPE no 5,6 in R8C2, clean-up: no 3,4 in R7C1
15a. 1,2 only in R8C5 -> R8C5 = {12}

16. R7C23 + R89C3 (step 14b) = {2367/2457/3456}
16a. 2,7 of {2367/2457} must be in R7C23 => R89C3 = {36/45}
R89C3 must be {36/45} for {3456} (other combinations give CCC clash with R67C3)
-> R89C3 = {36/45} = 9 -> R8C45 (step 15) = 7 = [52/61], no 3,4 in R8C4
16b. From the above R7C23 = {36/45}/[72]

17. 16(4) cage at R8C3 (step 15) = {1456/2356}, CPE no 5,6 in R7C3 using R5C1-R9C5 diagonal, clean-up: no 3,4 in R6C3, no 3,4 in R7C2 (step 16b)

18. 1 in C3 only in R125C3, CPE no 1 in R3C1
18a. 18(3) cage at R3C1 = {279/369/378/459} (cannot be {189} because no 1,8,9 in R3C1, cannot be {468/567} which clash with 9(2) cage at R8C6 using R3C1-R9C7 diagonal), no 1
18b. 4 of {459} must be in R3C1 -> no 4 in R4C2 + R5C3
18c. 9 of {369/459} must be in R5C3 -> no 5,6 in R5C3

19. 1 in C3 only in R12C3, locked for N1, CPE no 1 in R2C4 using R1C3-R7C9 diagonal, clean-up: no 4 in R1C3, no 1 in R1C7 (step 4)

20. R8C45 = 7 (step 16a)
20a. 45 rule on N8 3 remaining innies R7C45 + R8C6 = 12 = {147/237/345} (cannot be {156} which clashes with R8C4, cannot be {246} which clashes with R9C456), no 6, clean-up: no 3 in R9C7, no 6 in R9C3 (step 5), no 3 in R8C3 (step 16a)
20b. 7 of {147/237} must be in R7C5 -> no 7 in R7C4
20c. 7 in N8 only in R79C5, locked for C5

21. 16(4) cage at R8C3 (step 15) = {1456/2356}, 6 locked for R8

22. 7 in N7 only in R7C12 + R9C2, CPE no 7 in R6C2 using R1C7-R7C1 diagonal, clean-up: no 1 in R5C1

23. 18(3) cage at R3C1 (step 18a) = {279/369/459} (cannot be {378} which clashes with R5C1) -> R5C9 = 9, placed for R1C7-R7C1 and R3C1-R9C7 diagonals
23a. R3C1 + R4C2 = {27}/[36/45], no 3,8 in R4C2
23b. 9 in N6 only in R46C9, locked for C9, clean-up: no 5 in R2C8

24. R5C456 = {368/458/467}, no 1
24a. Naked pair {34} in R4C4 + R5C5, locked for N5 and R1C1-R9C9 diagonal -> R9C9 = 1, R9C8 = 8, clean-up: no 6 in R3C9

25. R5C2 = 1 (hidden single in R5), R6C2 = 5, placed for R1C7-R7C1 and R5C1-R9C5 diagonals, R5C1 = 3, placed for R1C5-R5C1 and R5C1-R9C5 diagonals, R5C5 = 4, placed for R1C9-R9C1 diagonal, R4C4 = 3, placed for R1C7-R7C1 diagonal, R3C4 = 9, R7C3 = 2, R6C3 = 7, R9C1 = 6, placed for R1C9-R9C1 diagonal, R79C2 = [73], R8C2 = 8, placed for R1C9-R9C1 diagonal, R2C8 = 9, placed for R1C9-R9C1 diagonal, R3C9 = 5, placed for R3C9-R9C3 diagonal

26. R6C4 + R7C5 = [13] = 4 -> R6C5 + R7C4 = 14 = [95]

and the rest is naked singles, without needing to use diagonals.

Solution:
5 9 1 8 6 3 4 2 7
2 6 3 4 5 7 1 9 8
7 4 8 9 2 1 3 6 5
4 2 6 3 8 5 7 1 9
3 1 9 7 4 6 8 5 2
8 5 7 1 9 2 6 3 4
1 7 2 5 3 8 9 4 6
9 8 5 6 1 4 2 7 3
6 3 4 2 7 9 5 8 1

The INSANE version will probably be a lot harder because step 5 isn't available.


Last edited by Andrew on Sat Jun 26, 2010 10:27 pm, edited 1 time in total.

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h3lix
 Post subject: Re: Anti Bishop Killer 2
PostPosted: Fri Jun 18, 2010 8:32 am 
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Joined: Sun May 18, 2008 6:22 pm
Posts: 47
The innies resulting from the 17/3 and 18/3 cages are so crucial to the solve path that it indeed took me quite a while to find ways around them for the INSANE version. I'll get a walkthrough going for it.

Your solve path is certainly more direct than my design path. I used complex innies and outies (right term?) to determine that the sum of R2C7 and R8C3 is 6:

(all cages within R19+C456) + (17 + 18) [overlapping regions]- 45*5 = 6

This was then used to determine the maximum values for these cells.

teaser for my next project:
like Anti-Chess Killer Sudoku, this project's starting off with a true INSANE. JSudoku took about 2-3 minutes to solve it using the following techniques:

Hidden Text:
The grid is solved!
Techniques used:
76 Naked Singles
5 Hidden Singles
7 Unique Pairs
2 Naked Pairs
4 Hidden Pairs
29 Intersections
2 LoL Synchronization
2 Law of Leftovers up to 3 cells
1 Unique Triplets
4 Law of Leftovers up to 5 cells
6 Odd Pairs
11 Odd Triplets
6 Mandatory Inclusions
4 Odd Quads
2 Triple Innies & Outies
2 Double Outies minus Innies
1 Complex Naked Pairs
4 Conflicting Pairs
7 Quadruple Innies & Outies
2 Triple Outies minus Innies
1 Odd Combinations
1 Conflicting Triplets
1 Grouped XY-Chains up to 3 links
4 Conflicting Partial Pairs
15 Multiple Innies & Outies
32 Multiple Outies minus Innies
9 Multiple Overlaps
12 Complex XY-Chains up to 3 links


SudokuSolver gave it a rating of 6.63:

Hidden Text:
Puzzle Completed. Step Analysis:
10 Find Hidden Cages
68 Naked Singles
18 Naked Pairs
10 Naked Triples
5 Naked Quads
4 Hidden Singles
15 Locked Candidates (Box/Line)
7 Jigsaw Law of Leftovers
5 Cage Blockers
1 Cage Placement 5
1 Cage Placement 4
1 Cage Placement 3
2 Cage Placement 2
3 Cage Placement 1
2 Cage Placement
2 Killer Pairs
13 Cage Combinations
6 Cage Blockers Extended 9
2 Cage Placement Extended 5
1 Cage Placement Extended 2
1 Cage Placement Insane 2
1 45 Rule Single Innies/Outies
3 45 Rule Two Innies/Outies
5 45 Rule Extended 5
4 Common Peer Elimination Extended 3
2 Bowmans Bingo Complex 21
1 Bowmans Bingo Complex 18
1 Bowmans Bingo Complex 17
1 Bowmans Bingo Complex 13
1 Bowmans Bingo Complex 12
1 Bowmans Bingo Complex 11
2 Bowmans Bingo Complex 9
1 Bowmans Bingo Complex 6
67 Cage Cleanup
1 45 Rule Extended Single House 5
2 45 Rule Extended Multiple Houses 6
1 Grouped X-Cycle Simple 2
1 Cage Placement Complex 10
1 Cage Placement Complex 8
3 Bowmans Bingo Extended (Lite) 4
10 Bowmans Bingo Extended (Lite) 3
1 Bowmans Bingo Extended (Lite) 2
3 Hidden Killer Pairs
3 Locked Candidates (House/Cage)
4 Forced Cage Candidates
2 45 Rule Extended Multiple Houses Innies&Outies 6
1 45 Rule Extended Innies&Outies 7
2 45 Rule Extended Innies&Outies 6
Total solving time (seconds): 41.57
Calculated score: 6.63

**** Uses Extended Trial and Error Steps ****


I did not use trial and error. I have yet to verify the validity of my solve path (which I do for every puzzle), I used some interesting techniques, but I need to make sure I didn't skip any possibilities.

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Last edited by h3lix on Sat Jun 19, 2010 4:16 am, edited 1 time in total.

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Ed
 Post subject: Re: Anti Bishop Killer 2
PostPosted: Sat Jun 19, 2010 1:14 am 
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Joined: Wed Apr 16, 2008 1:16 am
Posts: 1044
Location: Sydney, Australia
h3lix wrote:
Is there any preference between these types, because the second one is much easier to create.
I used the second picture since the diagonals were easier for me to fixate on while doing the puzzle with SudokuSolver as an editor.

The Insane version was really insane for me. :D Wonderful puzzle. Thanks a lot h3lix. So many areas to explore before finding the crucial bits (for me, steps 6,7 & 9).

Walkthrough for Anti Bishop Killer 2 Insane
NOTE: this is an optimised solution so some obvious eliminations are left out. However, I try and do all the clean-up as I go. Please let me know if anything is wrong or could be clearer.

Prelims
i. both 14(2) in n1 = {59/68}
ii. 5(2)n1 = {14/23}
iii. 9(2)n3: no 9
iv. 14(2)n3 = {59/68}
v. 12(2)n2: no 1,2,6
vi. 10(2)n3: no 5
vii. 3(2)n6 = {12}
viii. 8(2)n4: no 4,8,9
ix. 9(2)n4: no 9
x. 10(2)n5: no 5
xi. both 9(2) in n7: no 9
xii. 16(2)n9 = {79}
xiii. 9(2)n8: no 9
xiv. 9(2)n9: no 9

1. 16(2)n9 = {79}: both locked for n9 and D\r1c1
1a. and no 7 or 9 in r6c8 and no 7 in r8c6 (CPE with D/r5c9)
1b. no 2 in 9(2)n8
1c. no 2 in 9(2)n9
1d. no 1,3 in r7c6
1e. no 3,5 in r3c4

2. 14(2)r2c2 = {68}: both locked for n1 and D\r1c1
2a. r1c1 = 5 (placed for D\r1c1)
2c. r1c2 = 9
2d. no 6,8 in r4c2 (CPE on 14(2)r2c2 and D/r1c5)
2e. r9c8 = (568)
2f. no 2,4 in r7c6
2g. no 4 in 9(2)n3
2h. no 3 in r6c2
2i. no 4 in r89c2
2j. no 4,7 in r3c4

3. 3(2)n6 = {12}: both locked for n6 and D\r1c5
3a. no 8,9 in 10(2)n3

4. 2 on D\r1c1 only in n5: 2 locked for n5
4a. no 2 in r7c5 (CPE with D/r3c9)

5. each cell in 3(2)n6 sees one cell with 2 in D\r1c1 through r5 and D/r3c9
5a.-> generalized x-wing on 2: 2 locked for r5 and D/r3c9
5b. and no 2 in r1c9
5c. no 7 in r1c8
5d. no 6 in r6c2

6. each cell in 3(2)n6 sees at least one cell containing 1 in D\r1c1 through r5, c9 and D/r3c9
6a. ->Grouped turbot fish on 1 -> no 1 in r4c56, r9c5, r1c9 and 1 locked for D/r3c9. Like this.
6b. 1 in r5c9 (note: ->no 1 in r9c5 from D/r5c9) -> 1 on D\r1c1 only in r6c6 -> 1 locked for D/r3c9 and n5
6c. 1 in r4c8 -> 1 locked for D/r3c9 and no 1 in r4c56 and 1 in D\r1c1 only in r5c5 or r9c9 -> no 1 in r9c5 or r1c9
6d. no 8 in r1c8

7. "45" on n5: 4 outies r3c56 + r7c45 - 6 = 2 innies r4c4 + r6c6
7a. max 2 innies = 7 -> max. 4 outies = 13
7b. -> max. r7c45 = 10 -> no 8,9 in r7c4
7c. min. r7c45 = 4 -> max. r3c56 = 9; no 9 in r3c56

8. 9 in r9 only in r9c3456 -> no 9 in r8c45 (CPE same cage)
8a. and no 9 in r7c5 (CPE from D/r3c9)

9. no 9 in r8c8 because of 9 in r3. Like this.
9a. 9 in r3c9 -> no 9 in r9c3 (from D/r3c9) -> 9 in n7 in r8 -> no 9 in r8c8
9b. 9 in r3c4 -> 9 on D/r1c9 in r2c8 (->no 9 in r8c8) or r4c6 (-> 9 in r7 in r7c7 -> no 9 in r8c8)
9c. 9 in r3c8 -> no 9 in r8c8

10. r8c8 = 7
10a. no 2 in r7c1

11. r7c7 = 9 (placed for D/r5c9)
11a. no 1 in r6c6

12. no 9 in r5c46. Like this. (another Fish)
12a. 9 on D/r1c9 in r2c8 -> 9 in c5 only in n5: 9 Locked for n5
12b. 9 on D/r1c9 in n5: 9 locked for n5

13. 17(3)n5 = {278/368/458/467}(no 1)

14. r9c9 = 1 (hidden single on D\r1c1)
14a. no 8 in 9(2)r9c1
14b. no 8 in r8c6

15. r9c8 = 8
15a. no 6 in r3c9
15b. no 1 in r8c6

16. r5c9 = 2 (placed for D/r5c9)
16a. r4c8 = 1
16b. no 8 in r1c9

17. naked pair {34} in n5: locked for n5

18. r6c6 = 2
18a. no 6 in r5c1
18b. no 7 in r7c3

19. r7c6 = 8
19a. no 1 in r8c2
19b. no 1 in r6c3

20. 1 in n8 only in r7c4 & r8c5 -> no 1 in r6c5 (CPE)
20a. r6c4 = 1 (hsingle n5); placed for D/r1c9
20b. no 4 in r1c3
20c. no 7 in r5c1
20d. no 8 in r6c3

21. 17(3)n5; {368} can only be [836] but this clashes with [8/3..] in 12(2)n2
21a. 17(3) = {458/467}; must have 4
21b. -> r5c5 = 4; placed for D/r1c9
21c. no 5 in r9c2
21d. no 5 in r6c3
21e. no 6 in r4c7

much easier now.
solution:
+-------+-------+-------+
| 5 9 1 | 8 6 3 | 4 2 7 |
| 2 6 3 | 4 5 7 | 1 9 8 |
| 7 4 8 | 9 2 1 | 3 6 5 |
+-------+-------+-------+
| 4 2 6 | 3 8 5 | 7 1 9 |
| 3 1 9 | 7 4 6 | 8 5 2 |
| 8 5 7 | 1 9 2 | 6 3 4 |
+-------+-------+-------+
| 1 7 2 | 5 3 8 | 9 4 6 |
| 9 8 5 | 6 1 4 | 2 7 3 |
| 6 3 4 | 2 7 9 | 5 8 1 |
+-------+-------+-------+

Cheers
Ed
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Andrew
 Post subject: Re: Anti Bishop Killer 2
PostPosted: Sat Jun 26, 2010 4:55 am 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
I finished Anti-Bishop Killer 2 INSANE a few days ago and then had another try at the first Anti-Bishop Killer INSANE so that I could post my walkthroughs in the correct order.

These two INSANE puzzles were very different. The first one started with quite a few fairly easy placements before the hard work started. The second one gave two easy placements at the start, then there was a lot of hard work to make further placements.

Congratulations to Ed for finding a more direct solving path. His step 9 led to a significantly easier solution than mine which relied to a large extent to combination analysis in N7, similar to what I had used for the HARD version.

Here is my walkthrough for Anti-Bishop Killer 2 INSANE:
No repeated candidates on the diagonals R1C1-R9C9, R1C3-R7C9, R1C5-R5C9, R3C1-R9C7, R5C1-R9C5, R1C5-R5C1, R1C7-R7C1, R1C9-R9C1, R3C9-R9C3 and R5C9-R9C5.

Prelims (not including any cage interactions)

a) R1C12 = {59/68}
b) 5(2) cage at R1C3 = {14/23}
c) R1C89 = {18/27/36/45}, no 9
d) 14(2) cage at R2C2 = {59/68}
e) 14(2) cage at R2C8 = {59/68}
f) R34C4 = {39/48/57}, no 1,2,6
g) R34C7 = {19/28/37/46}, no 5
h) 3(2) cage at R4C8 = {12}
i) 8(2) cage at R5C1 = {17/26/35}, no 4,8,9
j) R67C3 = {18/27/36/45}, no 9
m) R67C6 = {19/28/37/46}, no 5
n) 9(2) cage at R7C1 = {18/27/36/45}, no 9
o) 16(2) cage at R7C7 = {79}
p) 9(2) cage at R8C6 = {18/27/36/45}, no 9
q) R9C12 = {18/27/36/45}, no 9
r) R9C89 = {18/27/36/45}, no 9

Steps resulting from Prelims
1a. Naked pair {12} in 3(2) cage at R4C8, locked for N6 and R1C5-R5C9 diagonal, clean-up: no 8,9 in R34C7
1b. Naked pair {79} in 16(2) cage at R7C7, locked for N9 and R1C1–R9C9 diagonal, CPE no 7,9 in R6C8 + R8C6 using R5C9-R9C5 diagonal, clean-up: no 5 in R1C2, no 5 in 14(2) cage at R2C2, no 3,5 in R3C4, no 1,3 in R7C6, no 2 in 9(2) cage at R8C6, no 2 in R9C89

2. Naked pair {68} in 14(2) cage at R2C2, locked for N1 and R1C1–R9C9 diagonal, CPE no 6,8 in R4C2 using R1C5-R5C1 diagonal, clean-up: no 4 in R3C4, no 2,4 in R7C6, no 1,3 in R9C8
2a. R1C1 = 5, placed for R1C1–R9C9 diagonal, R1C2 = 9, clean-up: no 4 in R1C89, no 7 in R3C4, no 3 in R6C2, no 4 in R8C2, no 4 in R9C2, no 4 in R9C8

3. 2 on R1C1–R9C9 diagonal only in R5C5 + R6C6, locked for N5
3a. Generalised X-Wing for 2 in 3(2) cage at R4C8 and R5C5 + R6C6, no other 2 in R5 and R3C9-R9C3 diagonal, clean-up: no 6 in R6C2
3b. 2 in R5 only in R5C59, CPE no 2 in R1C9 using R1C9-R9C1 diagonal, no 2 in R9C5 using R5C9-R9C5 diagonal, clean-up: no 7 in R1C8

4. 9 in R9 only in R9C3456, CPE no 9 in R7C5 + R8C4 using R3C9-R9C3 diagonal
4a. 9 in N7 only in R8C13 + R9C3, CPE no 9 in R8C5

5. 17(3) cage at R5C7 = {269/278/359/368/458/467} (cannot be {179} because 7,9 only in R5C7), no 1

6. 18(3) cage at R3C1 = {279/378/459/567} (cannot be {189/369/468} because 6,8,9 only in R5C3), no 1
6a. 6,8,9 only in R5C3 -> R5C3 = {689}
6b. 5 of {459} must be in R4C2 -> no 4 in R4C2

7. R5C456 = {179/269/278/359/368/458/467}
7a. R5C5 = {1234} -> no 1,3,4 in R5C46

8. Consider placements for R5C9
R5C9 = 1 => R4C8 = 2, R6C6 = 1 (hidden single on R1C1-R9C9 diagonal)
R5C9 = 2 => R4C8 = 1, R6C6 = 2 (hidden single on R1C1-R9C9 diagonal)
-> R6C6 = {12}, clean-up: no 6,7 in R7C6
8a. Naked pair {12} in R4C8 + R6C6, locked for R3C9-R9C3 diagonal, CPE no 1 in R4C56
8b. Min R4C56 = {35} (cannot be {34} which clashes with R4C4) = 8 -> max R3C56 = 8, no 8,9 in R3C56
8c. Min R8C45 + R9C3 = 8 -> max R8C3 = 8

9. Consider placements for R5C9
R5C9 = 1 => no 1 in R1C9, no 1 in R9C5 using R5C1-R9C5 diagonal
R5C9 = 2 => R4C8 = 1, R6C6 = 2 => 1 on R1C1–R9C9 diagonal only in R5C5 + R9C9, CPE no 1 in R1C9 using R1C9-R5C5 diagonal, no 1 in R9C5
-> no 1 in R1C9 + R9C5, clean-up: no 8 in R1C8

10. Consider placements for 9 in N7
R8C1 = 9 => R8C8 = 7 => R7C7 = 9 => no 9 in R5C7
R9C3 = 9, placed for R3C9-R9C3 diagonal => no 9 in R5C7
-> no 9 in R5C7

11. Consider placements for 9 in N7
R8C1 = 9 => R8C8 = 7 => R7C7 = 9 => R7C6 = 8 => R6C6 = 2 => R5C9 = 2 (hidden single in R5) => no 2 in R7C9
R9C3 = 9 => R8C8 = 9 (hidden single in R8) => R7C7 = 7 => no 7 in R5C7
11a. 17(3) cage at R5C7 (step 5) = {368/458/467} (cannot be {278} because no 2 in R7C9 or no 7 in R5C7), no 2
11b. Killer pair 3,4 in 5(2) cage at R1C3 and 17(3) cage at R5C7, locked for R1C3-R7C9 diagonal

12. Consider placements for 9 in N7
R8C1 = 9 => R8C8 = 7 => no 7 in R8C345
R9C3 = 9 => R8C345 = 7 = {124}
-> no 1,2,4 in R8C1, no 7 in R8C345

13. Consider placements for 1 in R9
1 in R9C12 = {18} => no 8 in R9C7 => no 1 in R8C6
1 in R9C467, CPE no 1 in R8C6
1 in R9C9 => R9C8 = 8 => no 8 in R9C7 => no 1 in R8C6
-> no 1 in R8C6, clean-up: no 8 in R9C7

14. Consider placements for 9 in N7
R8C1 = 9 => R8C8 = 7 => no 7 in R8C2
R9C3 = 9, placed for R3C9-R9C3 diagonal => R8C8 = 9 (hidden single in R8) => 14(2) cage at R2C8 = {68}, locked for N3 => R1C9 = 7, placed for R1C9-R9C1 diagonal -> no 7 in R8C2
-> no 7 in R8C2, clean-up: no 2 in R7C1
14a. R8C18 = {79} (hidden pair in R8)

15. Consider placements for 9 in N7
R8C1 = 9 => R8C8 = 7 => R7C7 = 9 => R7C6 = 8 => R6C6 = 2 => no 2 in R6C3 => no 7 in R7C3
R9C3 = 9 => R8C1 = 7 => no 7 in R7C3
-> no 7 in R7C3, clean-up: no 2 in R6C3

16. Consider placements for 9 in N7
R8C1 = 9 => R8C8 = 7 => R7C7 = 9 => R7C6 = 8 => no 8 in R7C3
R9C3 = 9 => R35C3 = {68}, locked for C3
-> no 8 in R7C3, clean-up: no 1 in R6C3

17. Consider placements for 9 in N7
R8C1 = 9 => R8C8 = 7 => R7C7 = 9 => no 9 in R7C4
R9C3 = 9, placed for R3C9-R9C3 diagonal => R8C8 = 9 (hidden single in R8) => 14(2) cage at R2C8 = {68} => naked pair {68} in R3C39 => R3C4 = 9 => no 9 in R7C4
-> no 9 in R7C4

18. Consider placements for 9 in N7
R8C1 = 9 => R8C8 = 7 => R7C7 = 9, placed for R5C9-R9C5 diagonal => no 9 in R9C5
R9C3 = 9 => no 9 in R9C5
-> no 9 in R9C5

19. 9 in N7 only in R8C1 + R9C3
45 rule on N7 5 innies R7C23 + R8C13 + R9C3 = 27 = {12789/13689/14589/23679/24579/34569} (cannot be {14679/23589} which clash with the 9(2) cages in N7)
19a. Consider placements for 9 in N7
R8C1 = 9 => R7C23 + R89C3 = 18 must consist of R7C23 = 9, R89C3 = 9 (other combinations cause CCC clash with R67C3) => R8C45 = 7
R9C3 = 9 => R8C345 = 7 = {124}, locked for R8 => R7C8 = 2 (hidden single in R7), R8C1 = 7 => no 2 in 9(2) cages at R7C1 and R9C1 => R8C3 = 2 (hidden single in N7) => R89C3 = [29] => R8C45 = [41]
19b. -> R89C3 = 9 or [29], no 8 in R8C3
-> R8C45 = 7 or R8C45 = [41] -> R8C45 = [34/41/43/52/61], no 8 in R8C4, no 5,6,8 in R8C5
19c. R8C1 = 9 => R7C23 = 9
R9C3 = 9 => R8C13 = [72] = 9 => R7C23 = 9
-> R7C23 = 9 = {36/45}/[72/81], no 1,2 in R7C2
-> R7C23 = R67C3 -> R7C2 = R6C3
[With hindsight step 19 could have been done after step 14a.]

20. Consider placements for 9 in N7
R8C1 = 9 => R8C8 = 7 => R7C7 = 9 => R7C6 = 8 => no 8 in R7C2
R9C3 = 9 => R8C3 = 2 (step 19a), R35C3 = {68}, locked for C3 => R67C3 = {45} => R7C23 = {45} (step 19c)
-> no 8 in R7C2, clean-up: no 1 in R7C3 (step 19c), no 8 in R6C3

21. R9C3 cannot be 9, here’s how
R9C3 = 9 => R8C345 = [241] (step 19a), R7C23 = {45} (step 20) => 4 in R9 only in R9C79
Now consider placements for 4 in R9C79
R9C7 = 4 => R8C6 = 5 => cannot place 5 in R9
R9C9 = 4 => R8C8 = 5, no 4 in R9C7 => no 5 in R8C6 => cannot place 5 in N8
-> no 9 in R9C3

22. R8C1 = 9 (hidden single in N7), R8C8 = 7, R7C7 = 9, R7C6 = 8, R6C6 = 2, placed for R3C9-R9C3 diagonal, R4C8 = 1, R5C9 = 2, clean-up: no 8 in R1C9, no 6 in R5C1, no 1 in R8C2, no 1 in R9C7
22a. 1 on R1C1-R9C9 diagonal only in R5C5 + R9C9, CPE no 1 in R9C1 using R1C9-R9C1 diagonal, clean-up: no 8 in R9C2
22b. 8 in R1 only in R1C457, CPE no 8 in R2C5

23. 18(3) cage at R3C1 (step 6) = {279/378/459} (cannot be {567} which clashes with 9(2) cage at R8C6 using R3C1-R9C7 diagonal), no 6
23a. {378} must be [738] (cannot be [378] which clashes with 8(3) cage at R5C1), no 3 in R3C1

24. R8C1 = 9 -> R89C3 (step 19a) = 9 = [27]/{36/45} (cannot be [18] because cannot place 1 in N9), no 1,8
24a. 8 in N7 only in R8C2 + R9C1, locked for R1C9-R9C1 diagonal, clean-up: no 6 in R3C9
24b. 8 in R9 only in R9C18 -> R9C12 = [81] or R9C89 = [81] (locking cages), 1 locked for R9
24c. 1 in N8 only in R7C4 + R8C5, CPE no 1 in R6C5

25. 1 in N7 only in R7C1 + R9C2, CPE no 1 in R6C2 using R1C7-R7C1 diagonal, clean-up: no 7 in R5C1

26. 1 in C3 only in R12C3, locked for N1, CPE no 1 in R2C4 using R1C3-R7C9 diagonal, clean-up: no 4 in R1C3

27. 2 on R1C9-R9C1 diagonal only in R7C3 + R8C2 + R9C1, locked for N7, clean-up: no 7 in R9C1, no 7 in R9C3 (step 24)

28. 16(4) cage at R8C3 = {1456/2356}, CPE no 5,6 in R7C3 + R8C2 using R5C1-R9C5 diagonal, clean-up: no 3,4 in R6C3, no 3,4 in R7C1, no 3,4 in R7C2 (step 19c)
28a. 1,2 only in R8C5 -> R8C5 = {12}
28b. R89C3 = 9 (step 24) -> R8C45 = 7 = [52/61], no 3,4

29. 2 on R1C9-R9C1 diagonal only in R7C3 + R8C2 + R9C1 -> R67C3 = [72] or 9(2) cage at R7C1 = [72] or 9(2) cage at R9C1 = [27] (locking cages), CPE no 7 in R6C2 using R1C7-R7C1 diagonal
29a. R6C2 = 5, placed for R1C7-R7C1 diagonal, R5C1 = 3, placed for R1C5-R5C1 diagonal, clean-up: no 2 in R1C3, no 4 in R7C3, no 4 in R9C1, no 6 in R9C2
[I overlooked that R5C1 was also placed for the R5C1-R9C5 diagonal. Since it’s so close to the finish, I haven’t re-worked the remaining steps.]

30. R89C3 = {45} (hidden pair), locked for C3 and 16(4) cage at R8C3 -> R8C4 = 6, placed for R3C9-R9C3 diagonal, R8C5 = 1 (step 28b), R5C5 = 4, placed for R1C1-R9C9 and R1C9-R9C1 diagonals, R4C4 = 3, placed for R1C1-R9C9 and R1C7-R7C1 diagonals, R3C4 = 9, R9C9 = 1, R9C8 = 8, clean-up: no 5 in R2C8, no 3 in R9C7

31. 9(2) cage at R7C1 = [18] (hidden pair in N7), 1 placed for R1C7-R7C1 diagonal, R2C2 = 6, R3C3 = 8, R2C8 = 9, placed for R1C9-R9C1 diagonal, R3C9 = 5, placed for R3C9-R9C3 diagonal, R89C3 = [54], R7C2 = 7, R9C2 = 3, R7C3 = 2, R6C3 = 7, R9C1 = 6, placed for R1C9-R9C1 diagonal, R45C3 = [69], R5C2 = 1, R4C2 = 2, placed for R1C5-R5C1 diagonal, R3C12 = [74], R2C1 = 2, R2C4 = 4, placed for R1C3-R7C9 diagonal, R1C3 = 1, R2C3 = 3, R3C7 = 3

32. R6C4 = 1, R7C45 = [53], R6C5 = 9 (cage sum)

and the rest is naked singles without using diagonals.

Solution:
5 9 1 8 6 3 4 2 7
2 6 3 4 5 7 1 9 8
7 4 8 9 2 1 3 6 5
4 2 6 3 8 5 7 1 9
3 1 9 7 4 6 8 5 2
8 5 7 1 9 2 6 3 4
1 7 2 5 3 8 9 4 6
9 8 5 6 1 4 2 7 3
6 3 4 2 7 9 5 8 1
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Andrew
 Post subject: Re: Anti Bishop Killer 2
PostPosted: Sat Jun 26, 2010 5:05 am 
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Joined: Wed Apr 23, 2008 6:04 pm
Posts: 1895
Location: Lethbridge, Alberta, Canada
Dan commented above that when solving the HARD version he had used overlapping 45s for R19+C456 to get
(all cages within R19+C456) + (17 + 18) [overlapping regions] - 45*5 = 6 -> R2C7+R8C3 = 6, thus setting maximum values for these two cells.

I noticed while working on the INSANE version that there is still something similar, although it's not really useful so I'm just posting this as a separate observation.

45 rule on R19+C456
45*5 - (all cages within R19+C456) = 64
Whatever values are in the overlapping non-cage regions R19C456 must be counted twice so must be even -> R2C7+R8C3 must be even.
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h3lix
 Post subject: Re: Anti Bishop Killer 2
PostPosted: Sat Jun 26, 2010 5:04 pm 
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Joined: Sun May 18, 2008 6:22 pm
Posts: 47
Thanks for the walkthroughs! I still need to make mine to compare, hopefully I'll have time this weekend and early next week to get to it.

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