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PostPosted: Sat May 31, 2014 6:42 pm 
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Grand Master
Grand Master

Joined: Wed Apr 30, 2008 9:45 pm
Posts: 693
Location: Saudi Arabia
NC ODNC RODNC OL 2

Old Lace
Non Consecutive cell to adjacent cell horizontal and vertical
Ordered = increasing top-left to bottom right (rows first)
Reverse-Ordered = decreasing ditto
Digitised = last digit of cage sum is one of the numbers
NC Killer = no numbers in a cage are consecutive

Six Ordered Digitised NC triple Cages (black)
Three Reverse-Ordered Digitised NC triple Cages (red)

Hopefully very hard - please score it.

Image

Ignore the blue cell - selection mistake when printing.

Solution:
825174963
369528417
147396285
582741639
714963852
936285174
471639528
258417396
693852741


Last edited by HATMAN on Wed Jul 02, 2014 3:02 pm, edited 1 time in total.

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PostPosted: Tue Jun 03, 2014 9:19 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 HATMAN - I found filling out the blank areas as tough as filling the cages!

First some ground rules:
Whole puzzle is orthogonally non-consecutive.
All cages are 3-cells.
Digitization requires each cage to contain one of the pairs (19), (28), (37), (46).
Some cages are 'reversed'.
For those the orders are reversed and meanings exchanged of 'first' and 'last'.
Since the cages are ordered and non-consecutive possible cages are from:

(19): [139], [149], [159], [169], [179]
(28): [248], [258], [268]
(37): [137], [357], [379]
(46): [146], [246], [468], [469]

So: First cell from (1234), middle cell from (34567), last cell from (6789)
First cell = 4 -> middle cell = 6
Last cell = 6 -> middle cell = 4

Optimized Walk through:
1. r4c5 from (1234)
r5c7 from (6789)
Old Lace requires r46c5 to be cloned with r5c37.
-> r4c5 = r5c3 (and r5c7 = r6c5)
But since r5c3 from (34567)
-> r4c5 = r5c3 from (34)
-> r4c4 from (567) and r4c6 from (12)
Also -> r5c5 from (567) and r6c5 and r5c7 from (789)

2. Trying r4c6 = 2
Puts r45c5 = [46]
Puts r4c4 = 7 and no cage can go [*72]
-> r4c6 = 1
-> (Since r4c4 from (567)) -> r3c5 = 9
-> r6c5 = r5c7 from (78)
Also 9 in r5c46 (Old Lace)

3! 8 in n5 in r5c46 or r6c456.
If the former it must also go in r7c5 (Old Lace).
-> 8 in r6c456 or r7c5
-> (NC) 7 cannot go in r6c5!
-> r6c5 = 8
-> r456c5 = [468]
-> r5c37 = [48]
Also r4c4 = 7
Also r5c4 = 9
-> NS r5c6 = 3
-> r7c5 = 3
Also r6c46 = [25]

4. Also r23c6 = [86]
-> HS 7 in n2 -> r1c5 = 7
Also NS r1c4 = 1 (7 at r4c4 prevents it from being a 3)
-> NS r3c4 = 3
-> NS r2c4 = 5
-> NS r2c5 = 2
-> NS r1c6 = 4

-> r789c4 = {468}
r789c5 = [3]{15}
r789c6 = {279}

Now for the blank space!

5! r34c7 only from [25], [26], [46]
7 in c7 only in r789c7
3 in c7 only in r689c7
r6c7 only from (13)
-> cage at r6c7 cannot be [146] (No place for both 3 and 7 in c7)
-> cage at r6c7 from [379] or [1(4567)9]
-> r8c8 = 9!

6. Conclusions from that
-> (HS 9 in c7) -> r1c7 = 9
Also 8 in n9 in r79c9
-> (HS 8 in n3) r3c8 = 8
-> (HS 7 in n3) -> r2c9 = 7
-> (HS 7 in n6) -> r6c8 = 7
-> (HS 4 in n6) -> r6c9 = 4
Also (HS 7 in r5) -> r5c1 = 7
-> (HS 8 in n4) -> r4c2 = 8
Also (HS 5 in r5) -> r5c8 = 5
-> NS r4c7 = 6
Also (HS 9 in n6) -> r4c9 = 9
-> NS r4c3 = 2
-> NS r5c2 = 1
-> NS r5c9 = 2
-> NS r4c8 = 3
-> NS r6c7 = 1
Also NS r4c1 = 5
Also (HS 6 in n3) -> r1c8 = 6
-> (HS 5 in n3) -> r3c9 = 5
Also (HS 2 in n3) -> r3c7 = 2
-> 4 in n3 in r2c78
-> NS r1c9 = 3
-> NP r2c78 = [41]

7 Continuing
NP r23c3 = [97]
-> NS r6c3 = 6
Also NP r3c12 = [14]
-> NP r2c12 = [36]
-> (NS 5 in n1) -> r1c3 = 5
-> NP r1c12 = [82]
etc.

My rating:
Hard 1.25


Last edited by wellbeback on Sat Jul 12, 2014 6:01 pm, edited 1 time in total.

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PostPosted: Tue Jul 01, 2014 3:25 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 HATMAN for another interesting "total zero" killer. It was definitely harder than the chequerboard 8x8 puzzle.

I loved the first part of step 3 and the last part of step 5 (which cracked the puzzle) in wellbeback's walkthrough. I found that breakthrough in a somewhat similar way after I'd eliminated more candidates.

Here is my walkthrough for Ordered and Reverse Ordered 2:
Non-consecutive (NC) horizontally and vertically in cage pattern. Old Lace. Black cages are Ordered (increasing top to bottom, then left to right), Non-consecutive, Digitised (last digit of cage total must be same as one of the cell values). Red cages are Reverse Ordered (decreasing top to bottom, then left to right), Non-consecutive, Digitised. Note that the cage at R3C5 decreases in order R3C5 – R4C4 – R4C6.

Digitised cages must contain two values totalling 10 with any other value with isn’t consecutive with these two values.
Valid Permutations for ordered in either direction
1,9: <139>, <149>, <159>, <169> and <179>
2,8: <248>, <258> and <268>
3,7: <137>, <357> and <379>
4,6: <146>, <246>, <468> and <469>
First cells of cages (or last cells of reversed cages) = {1234}, middle cells {34567} and last cells (first of reversed cages) = {6789}
Thus R1C4, R3C4, R3C7, R4C5, R4C6, R5C2 and R6C7 = {1234}
R2C4, R3C3, R3C6, R4C4, R4C7, R5C3, R5C5 and R7C7 = {34567}
R2C3, R2C6, R3C5, R5C4, R5C7, R6C3, R6C5 and R8C8 = {6789}

Old Lace property. R37C5 must exactly equal R5C46 and R46C5 must exactly equal R5C37.

1. Old Lace R46C5 must exactly equal R5C37, only common candidates in R4C5 and R5C3 are 3,4 -> R4C5 = R5C3 = {34}
1a. R4C5 = {34} -> no 3,4 in R4C46 + R5C5 (NC)
1b. R4C6 = {12} -> no 1,2 in R5C6 (NC)
1c. R5C3 = {34} -> no 3,4 in R4C3 + R5C2 (NC)
1d. R5C2 = {12} -> no 1,2 in R46C2 + R5C1 (NC)
1e. Old Lace R37C5 must exactly equal R5C46, no 1,2 in R5C46 -> no 1,2 in R7C5

2. Reverse-ordered cage at R2C6, R4C5 = {34} -> no 3,4 in R3C6 (NC)
2a. R3C6 = {567} -> no 6 in R2C6 + R3C5 (NC)

3. R5C5 = {567} -> no 6 in R5C46 + R6C5 (NC)
3a. Old Lace R5C7 = R6C5, no 6 in R6C5 -> no 6 in R5C7

4. Reverse-ordered cage at R3C5, R3C5 + R4C6 must total 10 = [82/91], R4C56 = [31/41/42] (cannot be [32], NC)
4a. Reverse-ordered cage at R2C6 = [753/864/964] (cannot be [973] which clashes with R3C5 + R4C6 = [91] when R4C5 = 3), no 7 in R3C6
4b. Similarly ordered cage at R4C5 = [357/468/469], no 7 in R5C5

5. R3C5 + R4C6 must total 10 = [82/91], R4C56 = [31/41/42] (step 4)
5a. Reverse-ordered cage at R3C5 = [951/961/971] (cannot be [852] because R4C45 cannot be [54], NC, cannot be [862] which clashes with ordered cage at R4C6 = [469]) -> R3C5 = 9, R4C6 = 1
5b. Old Lace R37C5 must exactly equal R5C46, R3C5 = 9 -> R5C46 must contain 9, locked for R5 and N5

6. R5C5 = {56} -> no 5 in R5C6 (NC)
6a. R5C7 = {78} -> no 7,8 in R4C7 + R5C68 (NC)
6b. R6C5 = {78} -> no 7,8 in R6C46 + R7C5 (NC)

7. Old Lace R37C5 must exactly equal R5C46, no 7,8 in R37C5 -> no 7,8 in R5C4 -> R5C4 = 9, R5C6 = {34} -> R7C5 = {34}
7a. Naked pair {34} in R4C5 + R5C6, locked for N5
7b. Naked pair {34} in R47C5, locked for C5

8. R4C4 + R5C7 = [78] (hidden pair in Old Lace)

9. R6C5 = 8 -> ordered cage at R4C5 = [468], R5C36 = [43], R7C5 = 3
9a. R5C6 = 3 -> R6C6 = 5 (cannot be 2, NC), R6C4 = 2, R3C6 = 6, R2C6 = 8 (cannot be 7, NC)
9b. R2C6 = 8 -> no 7,9 in R1C6 + R2C57 (NC)
9c. R1C5 = 7 (hidden single in N2)

10. R3C5 = 9 -> ordered cage at R1C4 = [139/149/159] -> R1C4 = 1
10a. R3C4 = {34} -> no 3,4 in R2C4 -> R2C4 = 5, R2C5 = 2, R1C6 = 4, R3C4 = 3

11. R3C4 = 3 -> reverse-ordered cage at R2C3 = [753/973] -> R2C3 = {79}, R3C3 = {57}, 7 locked for C3 and N1

12. R5C3 = 4 -> ordered cage at R5C2 = [146/149] -> R5C2 = 1
12a. 2 in R5 only in R5C89, locked for N6

13. R5C7 = 8 -> ordered cage at R3C7 = [258/268/468] -> R3C7 = {24}, R4C7 = {56}
13a. R3C7 = {24} -> no 3 in R2C7 (NC)
13b. R4C7 = {56} -> no 5,6 in R4C8 (NC)

14. Now a few more NCs. No 2,8 in R1C3, no 3,5 in R1C7, no 6 in R4C1, no 5,6,8 in R4C3, no 6 in R6C1, no 4 in R6C7, no 4 in R7C4, no 2 in R7C6
14a. 1 in N1 only in R23C1, locked for C1
14b. 8 in C3 only in R789C3, locked for N7
14c. 7 in C7 only in R789C7, locked for N9

15. Ordered cage at R6C7 = [146/149/159/169/179] (cannot be [379] which clashes with R4C8) -> R6C7 = 1, R8C8 = {69}

16. Ordered cage at R3C7 = [258/268] (cannot be [468] which clashes with R2C7) -> R3C7 = 2
16a. R3C7 = 2 -> no 1 in R3C8 (NC)
16b. 4 in R6 only in R6C89 -> no 3 in R6C89 (NC)
16c. 3 in N6 only in R4C89, locked for R4
16d. 7 in N6 only in R5C9 + R6C89 -> no 6 in R6C9 (NC)
16e. 3 in C7 only in R89C7, locked for N9
16f. 3 in C7 only in R89C7 -> no 4 in R89C7 (NC)

17. 8 in R4 only in R4C12 -> no 9 in R4C12 (NC)
17a. Consider placements for 8 in R4
R4C1 = 8 => R4C3 = 2 (hidden single in R4)
or R4C2 = 8 => R4C3 = 2 (not 9, NC)
-> R4C3 = 2
[Alternatively R4C123 cannot be [289] (NC), no 2 in R4C1 -> R4C3 = 2 (hidden single in R4).]
17b. Naked triple {568} in R4C127, locked for R4
17c. Naked pair {39} in R4C89, locked for N6

18. Consider placements for R8C8 = {69}
R8C8 = 6 => R4C7 = 6 (hidden single in N6) => R1C7 = 9
or R8C8 = 9 => R1C7 = 9 (hidden single in C7)
-> R1C7 = 9
18a. R1C7 = 9 -> no 8 in R1C8 (NC)

19. 2 in R1 only in R1C12 -> no 3 in R1C12 (NC)
19a. Consider placements 2 in R1C12
R1C1 = 2 => no 1,3 in R2C1 (NC) => R3C1 = 1 (hidden single in N1)
or R1C2 = 2 => no 3 in R1C3 + R2C2 (NC) => R23C1 = [31] (hidden pair in N1)
-> R3C1 = 1

20. Ordered cage at R6C7 (step 15) = [149/159/169/179] (cannot be [146] which clashes with R8C7 = {357}, NC) -> R8C8 = 9, R4C89 = [39]
20a. R4C8 = 3 -> R5C8 = 5 (not 2, NC), R1C8 = 6, R2C7 = 4, R4C7 = 6, R5C19 = [72]
20b. R6C8 = 7 (not 4, NC), R6C9 = 4, R23C8 = [18]
20c. R3C8 = 8 -> R3C9 = 5 (not 7, NC), R12C9 = [37], R12C3 = [59], R3C23 = [47], R6C3 = 6
20d. R3C2 = 4 -> R4C2 = 8 (not 5, NC), R4C1 = 5, R1C12 = [82]
20e. R1C2 = 2 -> R2C2 = 6 (not 3, NC), R2C1 = 3, R6C12 = [93]
20f. R8C8 = 9 -> no 8 in R8C9 (NC)

21. R7C18 = {24} (hidden pair in R7)
21a. R8C2 = {57} -> no 6 in R8C1 (NC)
21b. R9C1 = 6 (hidden single in C1)
21c. R9C1 = 6 -> no 5,7 in R9C2 (NC) -> R9C2 = 9
21d. R9C2 = 9 -> no 8 in R9C3 (NC)

22. R9C4 = 8 (cannot be 4, NC, because R9C35 cannot both be 1; alternatively consider placements for R9C3 = {13} …)

and the rest is naked singles, without using NC.

Rating Comment:
I'll rate my walkthrough at 1.5. Maybe this is a bit on the high side, the hardest steps were Easy 1.5, but I've taken into account difficulty in finding some of the later steps after I'd finished the central area.

BTW Why is cell R2C4 blue?


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