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HATMAN
 Post subject: CNC 27 and 39X
PostPosted: Tue Sep 17, 2013 8:23 am 
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CNC 39 X

These puzzles work best with overlapping cages, however that makes them visually a bit difficult.

Killer cages are non-consecutive i.e. no consecutive entries within the cage.

Twin killer cages are remban groups so fully consecutive entries in any order.

All corner nonets have three vertical remban groups and three horizontal non-consecutive groups.

Nonets 1, 3, 7 also have a diagonal NC triplet and nonet 9 has a remban doublet in r7c7r8c8.

Reasonably hard - my solution path was narrow - almost HS.

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CNC 27

I'm posting this one as it only has two cages with numbers - which I think is the minimum.
I do not think it has a satisfactory solution - my approach was unsatisfactory: a four path elimination, three paths elinated quite quickly, which left an easy solution of the fourth path.

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wellbeback
 Post subject: Re: CNC 27 and 39X
PostPosted: Sat Sep 21, 2013 8:16 pm 
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Here is a WT for 39X. Another ingenious puzzle from HATMAN ;clapclap;

BTW - Where do the terms "Twin killer" and "Remban Group" come from?

Hidden Text:
1. Remban cages in corner nonets must in each case contain {123}, {456}, {789}
-> vertical columns in n4 and n6 must also contain {123}, {456}, {789}

n4 and n6 each contain three 2-cell killer cages which are non-consecutive and because of their placement:
-> r6c123 = {258} and r4c789 = {258}
-> r5c456 = {258}

Remban cage in n4 must be {34} or {67}
-> 5 in n4 in r6c23
-> r6c1 from (28) and r6c4 from (39)

Similarly Remban cage in n6 must be {34} or {67}
-> 5 in n6 in r4c89
-> r4c7 from (28)

2. Given r4c7 from (28) and r6c123 = {258}
-> Remban cage at r6c67 from [43] or [67]

HT in D/ in n5 = +15
Given r6c4 from (39) and r5c456 = {258}
-> Possible solutions for HT are [159], [429], [483], and [753]
In no case can 1 go in r6c5 (Remban cage in n5)
-> HS 1 in r6 -> r6c8 = 1.

-> r456c8 = [231]
-> r456c9 = [564]
-> r456c7 = [897]
-> r6c6 = 6
-> r6c45 = {39}

-> r4c456 = {147}
-> r4c123 = {369}
-> r5c123 = {147}

3. Since r4c7 = 8 -> Max r4c6 = 4
-> r6c5 cannot be 9
-> r6c45 = [93]
-> r6c123 = [8{25}]
-> Remban cage in n4 must be [34]
-> r456c2 = [312]
-> r456c3 = [645]
-> r456c1 = [978]

4. r5c5 from (25)
-> One of the NC 15/3 cages in n3 and n7 contains an 8 and can only be {168}
-> D/ in n5 must be [429]
-> r5c456 = [825]
Also -> r3c6 = 1

5. 2-cell Remban cage in n9 must be {34} or {67}
But since 6 already in D\ and 3 already in c8 -> it must be [34]
-> r789c7 = [312] or [321]
-> r789c8 = [546] or [645]
-> r789c9 = {789}

-> r123c7 = {456} with 4 not in r3c7
-> r123c8 = {789} with 9 not in r2c8
-> r123c9 = {123} with 2 not in r1c9

6. Innies D\ -> r4c4+r9c9 = +15
-> r4c4 = 7 and r9c9 = 8
-> r4c5 = 1
Also -> HS 1 in D\ -> r1c1 = 1
-> r123c1 = [1{23}]
-> r123c2 = [456] or [654]
-> r123c3 = [{78}9]

Also HS1 in n3 -> r2c9 = 1
-> 15/3@r1c9 = [375]

-> r789c1 = [456]
-> r789c2 = [789] (NC cage in r9c123)
-> r789c3 = [1{23}]

Just clean up from here
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Andrew
 Post subject: Re: CNC 27 and 39X
PostPosted: Sun Sep 22, 2013 4:54 am 
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"Twin Killer" refers to a cage pattern which is made up from two overlapping cages patterns. The first example I came across was Nasenbaer's Assassin 114 Twin Killer.

I've no idea about the origin of "Remban group". Possibly someone's name?
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HATMAN
 Post subject: Re: CNC 27 and 39X
PostPosted: Mon Sep 23, 2013 3:14 pm 
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I sent an early one of these to Para and he pointed out that they use the term Remban group for consecutive cages. Not sure what the origin is - but if someone is already using a term I like to stick with it.

Considering that the NC cages are the opposite I'll call them "Nabmer Groups" from now on.

I checked and found that JC posted one in 2006. I then realised that he had implemented it in JSudoku however there appears to be a bug in it - shame.
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Andrew
 Post subject: Re: CNC 27 and 39X
PostPosted: Sat Oct 05, 2013 11:22 pm 
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Thanks HATMAN for another interesting and enjoyable puzzle.

I hope I'm not giving too much of a hint by saying that once the first step has been spotted this is a fairly easy puzzle.

HATMAN suggested a fairly narrow solving path. wellbeback and I started in similar ways but diverged in the way we used D/.

Here is my walkthrough for CNC 39X:
Red border cages contain consecutive numbers, not necessarily in order; black border cages cannot contain consecutive numbers.

1. Remban groups in N1, N3, N7 and N7 must be {123}/{456}/{789} -> there must be Remban groups with these combinations in the columns in N4 and N6
1a. Killer cages R45C1, R45C2 and R45C3 must be {13}/{46}/{79} -> R6C123 must contain 2,5,8, locked for R6
1b. 16(3) cage at R6C2 must contain two of 2,5,8 and have non-consecutive numbers = {259/268/358} (cannot be {169} which doesn’t contain any of 2,5,8) -> R6C4 = {369}
1c. Killer cages R56C7, R56C8 and R56C9 must be {13}/{46}/{79} -> R4C789 must contain 2,5,8, locked for R4
1d. R5C456 = {258} (hidden triple in R5)

2. 13(3) cage at R3C6 = {148/157/247} (cannot be {139} because R4C7 only contains 2,5,8, other combinations contain consecutive numbers), no 3,6,9
2a. R4C7 = {258} -> no 2,5,8 in R3C6

3. Remban group R4C2 + R5C3 = {34}/{67} -> killer cage {46} must be either R45C2 or R45C3, no 4,6 in R45C1
3a. R4C2 + R5C3 = {34}/{67} -> R4C3 + R5C2 = {1469}, no 3,7

4. Remban group R5C8 + R6C9 = {34}/{67} -> killer cage {46} must be either R56C8 or R56C9, no 4,6 in R56C7
4a. R5C8 + R6C9 = {34}/{67} -> R5C9 + R6C8 = {1469}, no 3,7

5. R45C1 = {13}/{79} -> hidden Remban group R456C1 (step 1) = {123}{789}, no 5
5a. 5 in R6 only in 16(3) cage at R6C2 (step 1b) = {259/358}, no 6 in R6C4

7. R56C7 = {13}/{79} -> hidden Remban group R456C7 (step 1) = {123}/{789}, no 5

8. 13(3) cage at R3C6 (step 2) = {148/247}, 4 locked for C6

9. Remban group R6C67 must contain an even and an odd number -> R6C6 = 6, placed for D\, R6C7 = 7, R45C7 = [89] (hidden Remban group R456C7, step 1, and killer group R56C7)
9a. Remban group R5C8 + R6C9 (step 4) = {34} (only remaining combination), locked for N6 -> R6C8 = 1, R5C9 = 6, R45C8 = [23] (hidden Remban group R456C8, step 1, and killer group R56C8)
9b. Naked pair {39} in R6C45, locked for N5
9c. Naked pair {147} in R4C456, locked for R4
9d. Killer pair R45C2 = {13}/{46} -> hidden Remban group R456C2 (step 1) = {123}/{456}, no 8 in R6C2
9e. Killer pair R45C3 = {46}/{79} -> hidden Remban group R456C4 (step 1) = {456}/{789}, no 2 in R6C3

10. R4C7 = 8 -> R34C6 = 5 = {14}, locked for C6

11. Remban group at R4C6 = [123/453] -> R6C5 = 3, R6C4 = 9, placed for D/
11a. R6C4 = 9 -> R6C23 = 7 = [25], R6C1 = 8, R45C2 = [31] (step 9d), R45C1 = [97], R45C3 = [64]

12. 15(3) cages at R1C9 and R7C3 = {168/258/357} (other combinations contain consecutive numbers)
12a. 15(3) cages at R1C9 and R7C3 = {168/357} (cannot be {258} which clashes with {168} and {357}), locked for D/
12b. R4C6 = 4, R3C6 = 1, R5C5 = 2, placed for D\, R5C6 = 5, R5C4 = 8
12c. 15(3) cages at R1C9 = {168/357}
12d. {168} can only be [186] -> no 8 in R1C9, no 6 in R2C8

13. 15(3) cage at R1C1 = {159/357} (other remaining combinations contain consecutive numbers), 5 locked for N1 and D\
13a. Remban group R7C7 + R8C8 = [34] (only possible permutation), 3,4 placed for D\
13b. 15(3) cage = {159} (only remaining combination) -> R3C3 = 9, placed for D\, R2C2 = 5, R1C1 = 1, placed for D\, R4C4 = 7, placed for D\, R4C5 = 1, R9C9 = 8
[I could have got step 13a first, but I’d forgotten about the vertical Remban groups in N9.]

14. 15(3) cages at R1C9 (step 12c) = {357} (only remaining combination) -> R3C7 = 5, R2C8 = 7, R1C9 = 3, all placed for D/, R9C1 = 6, R8C2 = 8, R7C7 = 1

15. R3C7 = 5, R3C9 = 2 -> R3C8 = 8 (cannot be 6 because of killer cage at R3C7)

16. R9C9 = 8 -> R9C8 = 5 (cannot be 9 because of killer cage at R9C7)

17. R1C9 = 3 -> R1C7 = 6 (cannot be 4 because of killer cage at R1C7), R1C8 = 9, R2C79 = [41], R7C7 = 6, R7C9 = 9 (cannot be 7 because of killer cage at R7C7), R8C9 = 7

18. R1C1 = 1 -> R1C3 = {78} (cannot be 2 because of killer cage at R1C1)
18a. 2 in N1 only in R2C13, locked for R2

19. R3C2 = 6 (hidden single in C2), R3C5 = 7 (hidden single in R3)

20. R7C3 = 1 -> R7C1 = {45} (cannot be 2 because of killer cage at R7C1)

21. R9C1 = 6 -> no 7 in R9C23 (because of killer cage at R9C1)
21a. R7C2 = 7 (hidden single in N7), R1C2 = 4, R3C1 = 3, R2C13 = [28], R1C3 = 7, R3C4 = 4

22. R3C4 = 4 -> Remban group at R1C5 = [534]

and the rest is naked singles.

Rating Comment:
It was hard to know what rating to give my walkthrough, but definitely no higher than 1.25 and probably less, after finding the first step. I can't give a rating for the first step, which is a HS step. I would imagine that, even if software solvers were programmed for Remban and non-consecutive cages, they would find this very hard to solve as they probably wouldn't be programmed to spot step 1.
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Andrew
 Post subject: Re: CNC 27 and 39X
PostPosted: Wed Oct 09, 2013 10:37 pm 
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CNC 27

A really challenging CNC. There are some easy steps to get one started and draw one into the puzzle.

I hope my solution is correct and that I didn't make any mistakes in my solving path.

HATMAN wrote "I'm posting this one as it only has two cages with numbers - which I think is the minimum.
I do not think it has a satisfactory solution - my approach was unsatisfactory":
a four path elimination, three paths eliminated quite quickly, which left an easy solution of the fourth path.
I'll be interested to know what they were; I didn't notice a point in my walkthrough when this applied.

Here is my walkthrough for CNC 27:
HATMAN wrote “I'm posting this one as it only has two cages with numbers - which I think is the minimum.
I do not think it has a satisfactory solution - my approach was unsatisfactory: a four path elimination, three paths eliminated quite quickly, which left an easy solution of the fourth path.”

Red border cages contain consecutive numbers, not necessarily in order; black border cages cannot contain consecutive numbers.

1. 19(3) killer cage at R6C1 = {379/469} (other combinations contain consecutive numbers), no 1,2,5,8, 9 locked for R6 and N4
1a. Remban group at R6C6 = {123/456}, no 7,8
1b. Combined cage 19(3) cage + Remban group = {379}{456}/{469}{123}, 3,4,6 locked for R6

2. Killer cage at R1C4 = {159} (only way to fit the killer cage and the two Remban groups at R1C1 and R1C7 into R1), locked for R1 and N2
2a. Remban group at R2C4 = {234/678}
2b. Remban group at R1C5 must contain an even and an odd number -> R2C5 = {2468}

3. Killer cage at R2C7 = {159} (only way to fit the killer cage and the two Remban groups at R2C1 and R2C4 into R2), locked for R2 and N3
3a. Remban group at R2C4 = {234/678} -> Remban group at R2C1 = {234/678}
3b. R3C123 = {159} (hidden triple in N1)
3c. Remban group at R3C3 = {123/345/567/789}
3d. Remban group at R3C2 must contain an even and an odd number -> R4C2 = {2468}

4. Remban group at R1C7 = {234/678} -> R3C789 = {234/678}
4a. Killer cage at R3C7 must contain {24} or {68} in R3C78 -> R3C9 = {37}
4b. R3C78 = {24}/{68} -> no 5 in R4C8

5. 13(3) killer cage at R7C9 = {139/148/157} (cannot be {247} which clashes with Remban group at R3C9, other combinations contain consecutive numbers), no 2,6, 1 locked for C9 and N9
5a. Remban group at R3C9 = {234/567/678} (cannot be {456} because R3C9 only contains 3,7, other Remban combinations clash with 13(3) cage), no 9
5b. R3C9 = {37} -> no 3,7 in R45C9
5c. Combining the combinations in C9 -> R16C9 = [32/42/68] -> R1C9 = {346}, R6C9 = {28}
5d. Remban group at R1C7 = {234/678}
5e. 6 of {678} must be in R1C9 -> no 6 in R1C78

6. 2 in C9 only in R456C9, locked for N6
6a. Remban group at R6C6 (step 1a) = {123/456}
6b. 2 of {123} must be in R6C6 -> no 1,3 in R6C6

7. Remban group at R3C1 = {123/456/789} (cannot be {234/678} because R3C1 only contains 1,5,9, cannot be {345/567} which clash with 19(3) killer cage at R6C1)
7a. R3C1 = {159} -> no 1,5 in R45C1
7b. Combining Remban group with 19(3) killer cage at R6C1 -> R45C1 + 19(3) cage = {23}{469}/{46}{379}/{78}{469}, 4,6 locked for N4
7c. R4C2 = {28} -> Remban group R34C2 = [12/98], no 5

8. 1 in N4 only in killer cage at R4C3 -> no 2 in killer cage
8a. 2 in N4 only in Remban group at R3C1 = {123} or in Remban group R34C2 = [12] -> 1 in R3C12, locked for R3
8b. R3C3 = {59} -> Remban group at R3C3 (step 3c) = {345/567/789}, no 2 in R3C45
8c. Remban group at R2C4 = {234/678} -> R3C456 = {234/678}
8d. 2 of {234} must be in R3C6 -> no 3,4 in R3C6

9. Remban group at R3C9 (step 5a) = {234/567/678}
9a. Consider placements for 2 in C9
2 in R45C9 => Remban group at R3C9 = {234}
or 2 in R6C9 => Remban group at R6C6 (step 1a) = {456} => Remban group at R3C9 = {234/678} (cannot be {567} = 7{56} which clashes with Remban group at R6C6)
-> Remban group at R3C9 = {234/678}, no 5
9b. 2 in R45C9 => Remban group at R3C9 = 3{24} => Remban group at R6C6 = 2{13}/4{56}
or 2 in R6C9 => Remban group at R3C9 = 7{68} => Remban group at R6C6 = 6{45}
-> 4 in R45C9 + R6C78, locked for N6
and R6C6 = {246}
9c. 2 in R45C9 => Remban group at R3C9 = 3{24} => Remban group at R6C6 = 2{13} => killer cage at R4C7 = {579}
or 2 in R45C9 => Remban group at R3C9 = 3{24} => Remban group at R6C6 = 4{56}, 6 locked for N6
or 2 in R6C9 => Remban group at R3C9 = 7{68}, 6 locked for N6
-> no 6 in killer cage at R4C7

10. 13(3) killer cage at R7C9 = {139/157} (cannot be {148} which clashes with Remban group at R3C9), no 4,8
[Alternatively hidden killer pair 5,9 in R2C9 and 13(3) cage at R7C9 …]
10a. Killer pair 3,7 in R3C9 and 13(3) cage, locked for C9
10b. 8 in C9 only in R456C9, locked for N6
10c. Remban group at R1C7 (step 4) = {234/678}
10d. R1C9 = {46} -> no 4 in R1C78

11. Remban group R8C89 must contain an even and an odd number -> R8C8 = {2468}

12. 5 in C1 only in R3C1 => Remban group at R3C1 (step 7) = {456}, locked for C1
or 5 in 15(3) killer cage at R7C1 which cannot also contain 4 or 6
-> no 4,6 in 15(3) killer cage at R7C1

13. Consider combinations for Remban group at R3C1 (step 7) = {123/456/789}
Remban group at R3C1 = {123} => 5 in C1 only in 15(3) killer cage at R7C1 = {579}
or Remban group at R3C1 = {456} => 1 in C1 only in 15(3) killer cage at R7C1 => no 2 in 15(3) killer cage
or Remban group at R3C1 = {789} => 15(3) killer cage at R7C1 = {135}
-> no 2 in 15(3) cage at R7C1

14. 19(3) killer cage at R6C1 = {379/469}, R3C789 (step 4) = {234/678}, Remban group at R3C9 (step 9a) = 3{24}/7{68}
14a. Consider combinations for Remban group at R6C6 (step 1a) = {123/456}
Remban group at R6C6 = 2{13}
or Remban group at R6C6 = 4{56} => Remban group at R3C9 = 3{24} => R3C789 = {234}, locked for R3 => no 2 in R3C6
or Remban group at R6C6 = 6{45} => R45C9 = {68} => R6C9 = 2, 1(3) killer cage = {379} => R6C45 = {18} => no 2 in R3C6 (killer cage at R3C6 cannot contain both of 1,2 and {2468} clashes with R6C6)
-> no 2 in R3C6

15. 2 in R3 only in R3C789 (step 4) = {234} (only remaining combination) -> R3C9 = 3, R3C78 = {24}, locked for R3 and N3 -> R1C9 = 6, R1C78 = {78}, locked for R1
15a. Remban group at R2C4 = {234} (hidden triple in N2), locked for R2
15b. Remban group of R12C5 = [12/54], no 9
15c. R3C78 = {24} -> no 1,3 in R4C8 (killer cage at R3C7)
15d. 13(3) killer cage at R7C9 (step 10) = {157} (only remaining combination), locked for C9 and N9 -> R2C9 = 9

16. R3C9 = 3 -> Remban group at R3C9 = 3{24}, locked for N6 -> R6C9 = 8
16a. 2 in R6 only in R6C456, locked for N5

17. Remban group at R6C6 (step 1a) = {123/456}
17a. 2,4 only in R6C6 -> R6C6 = {24}

18. Remban group at R3C3 (step 3c) = {567/789}, 7 locked for R3
18a. R3C6 = {68} -> no 7 in R45C6 + R6C5 (killer cage at R3C6)
18b. 7 in C9 only in R789C6, locked for N8

19. Consider combinations for Remban group at R6C6 (step 1a) = {123/456}
Remban group = 2{13} => R6C45 = [75]
or Remban group at R6C6 = 4{56} => R6C45 = [21] (hidden pair in R6, cannot be [12] which clashes with Remban group at R5C4 = {123}
-> R6C45 = [21/75]
19a. Remban group at R5C4 = {234/678/789} (cannot be {345/456} because R6C4 only contains 2,7, cannot be {123} which clashes with R6C45 = [21], cannot be {567} which clashes with R6C45 = [75]), no 1,5
19b. R6C4 = {27} -> no 2,7 in R5C4 + R7C5
19c. {234} must be [324] (cannot be [423] which clashes with R6C6), no 4 in R5C4, no 3 in R7C5
19d. Naked pair {15} in R16C5, locked for C5

20. R6C45 (step 19) = [21/75] -> R6C456 = [214/752], no 4 in R45C6 (using killer cage at R3C6)
20a. Killer cage at R3C6 must contain 1 (because no 7 in R45C6), locked for N5

21. 2 in C1 only in R1C1 = 2 or Remban group at R3C1 (step 7) = 1{23} -> no 3 in R1C1 (locking-out cages)

22. Remban group at R3C1 (step 7) = {123/456/789}
22a. Remban group = {456} = 5{46} => R4C2 = 2 (hidden single in N4) => R45C9 = [42] -> no 4 in R4C1, no 6 in R5C1
22b. 6 in R5 only in R5C456, locked for N5

23. R6C45 (step 19) = [21/75]
23a. Killer cage at R3C6 cannot be [8361], which clashes with Remban group at R5C4 = [324] -> no 6 in R5C6
23b. 6 in R5 only in R5C45, CPE no 6 in R7C5 using Remban group at R5C4

24. Remban group at R5C4 (step 19a) = {234/678/789}
24a. Consider combinations for R6C45 (step 19) = [21/75]
R6C45 = [21] => Remban group at R5C4 = [324]
or R6C45 = [75] => R3C6 = 8, no 9 in R45C6 (because of killer cage at R3C6) => 8,9 in N5 only in R45C45 but killer cage at R4C4 cannot contain both of 8,9 => R5C4 = {89}
-> no 6 in R5C4
24b. R5C5 = 6 (hidden single in N5)
24c. R5C5 = 6 -> no 5,7 in R4C45 (because of killer cage at R4C4)

25. R6C4 = 7 (hidden single in N5), R6C5 = 5 (step 19), R6C6 = 2 (hidden single in R6), R6C78 = {13} (step 1a), locked for R6 and N6
25a. Naked triple {469} in 19(3) killer cage at R6C1, locked for N4
25b. R4C8 = 6 (hidden single in N6)

26. R6C5 = 5 -> killer cage at R3C6 = 8{13}5 (only remaining combination) -> R3C6 = 8, R45C6 = {13}, locked for C6 and N5

27. 4 in N5 only in R4C45, locked for R4 -> R45C9 = [24], R4C2 = 8 -> Remban group R34C2 = [98], R3C13 = [15]
27a. Remban group at R3C1 (step 7) = {123} (only remaining combination) -> R45C1 = [32], R1C1 = 4, R45C6 = [13], R45C3 = [71], R5C2 = 5

28. Killer cage at R7C1 = {579} (only remaining combination), locked for C1 and N7 -> R6C1 = 6, R2C1 = 8, R2C23 = [76]

29. R5C4 = 8 (hidden single in N5), R6C4 = 7 -> R7C5 = 9 (Remban group at R5C4, step 19a), R4C45 = [94], R1C456 = [519], R2C456 = [324], R3C45 = [67]

30. Remban group at R8C4 = {123/234} -> R8C5 = 3, R9C5 = 8
30a. Remban group = {123/234}, CPE no 2 in R8C23

31. Killer cage at R7C3 = {248} (only remaining combination), locked for C3 and N7 -> R1C23 = [23], R6C23 = [49]

32. Remban group at R7C2 = {123/234}, 2,3 locked for R7
32a. R8C3 = 8 (hidden single in N7)

33. 8 in R7 only in R7C78 -> Remban group at R7C6 = {678} (only remaining combination) = [768], R7C1 = 5, R7C9 = 1

34. Remban group R8C89 = [45] (only remaining permutation)

and the rest is naked singles, without using the killer cages and Remban groups.

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

Rating Comment:
I'll rate my walkthrough for CNC 27 at 1.75. I used a lot of forcing chains; some of them fairly long and using the properties of Remban groups.
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