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 Post subject: Paper Solvable 5 Zero X
PostPosted: Wed Nov 24, 2010 8:59 am 
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PS 5 - Zero X Killer

This has both types of zero: areas with no cages and cages with no sum.

I do not know how to enter zero cages into SudokuSolver so I am unsure of the score, but I have managed it on paper.

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3x3:d:k:1:3078:3078:5135:18:19:3333:3333:4:3078:1:5135:5135:20:21:22:4:3333:3078:23:1:1:5898:4:4:5134:3333:24:25:1:5898:4361:3852:4:5134:5134:26:27:5898:2827:4361:3852:3852:2321:2321:28:29:2:2827:4361:30:3:5389:5389:3336:31:2:2:32:3:3:5389:3079:3336:2:5136:5136:33:34:35:3:3079:2:3336:3336:5136:36:37:3079:3079:3:


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PostPosted: Thu Nov 25, 2010 4:16 am 
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Thanks HATMAN for another fun puzzle. :D Cages without totals are an interesting concept! They proved useful in the later stages of my solving path.

I solved it by elimination solving although many of the steps are essentially paper solvable ones; for example step 2 since the follow-ups from Prelims (in step 1) can be easily visualised. Also I think that steps 7 and 11 are essentially paper solvable, and also human solvable, steps.

I was going to use one 45 but then spotted the clash in step 16 and found that the 45 wasn't needed.

Rating Comment:
I'll rate Paper Solvable 5 Zero-X at 1.25. I think steps 7 and 11 deserve that rating, as does the clash in step 16.

Here is my walkthrough:
Prelims

a) R56C4 = {29/38/47/56}, no 1
b) R5C89 = {18/27/36/45}, no 9
c) 20(3) cage at R1C4 = {389/479/569/578}, no 1,2
d) 23(3) cage at R3C5 = {689}
e) 20(3) cage at R3C8 = {389/479/569/578}, no 1,2
f) 21(3) cage at R6C8 = {489/579/678}, no 1,2,3
g) 20(3) cage at R8C3 = {389/479/569/578}, no 1,2
h) 12(4) cage in N1 = {1236/1245}, no 7,8,9
i) 13(4) cage in N3 = {1237/1246/1345}, no 8,9
j) 13(4) cage in N7 = {1237/1246/1345}, no 8,9
k) 12(4) cage in N9 = {1236/1245}, no 7,8,9

Steps resulting from Prelims
1a. 12(4) cage in N1 = {1236/1245}, 1,2 locked for N1
1b. 13(4) cage in N3 = {1237/1246/1345}, 1 locked for N3
1c. 23(3) cage at R3C5 = {689}, CPE no 6,8,9 in R3C3 (using D\) + R5C5
1d. 13(4) cage in N7 = {1237/1246/1345}, 1 locked for N7
1e. 12(4) cage in N9 = {1236/1245}, 1,2 locked for N9

2. R5C5 + R6C6 = {12} (hidden pair on D\), locked for N5, clean-up: no 9 in R56C4
2a. R5C5 = 1 (hidden single on D/), R6C6 = 2, clean-up: no 8 in R5C89

3. R5C5 = 1 -> 17(3) cage at R4C5 = {179}, locked for C5 and N5, clean-up: no 4 in R56C4
3a. Killer pair 6,8 in R4C4 and R56C4, locked for C4 and N5

4. 4 in N5 only in R45C6, locked for C6
4a. 15(3) cage at R4C6 = {348/456} -> R5C7 = {68}

5. R5C3 = 9 (hidden single in 23(3) cage at R3C5

6. R37C4 = {12} (hidden pair in C4)
6a. 1 in N6 only in R46C7, locked for C7

7. 20(3) cage at R1C4 and 20(3) cage at R8C3 can only contain one 8 in C3 and one 9 in C4 -> both 20(3) cages = {479/569/578}, no 3, 8 locked in R28C3 for C3

8. 3 in C4 only in R56C4 = {38}, locked for C4 and N5 -> R4C4 = 6, placed for D\, R3C5 = 8

9. Naked pair {45} in R45C6, locked for C6, R5C7 = 6 (step 4a), clean-up: no 3 in R5C89

10. R5C89 = {27} (cannot be {45} which clashes with R5C6), locked for R5 and N6

[Returning to the 20(3) cages at R1C4 and R8C3.]
11. One of the 20(3) cages must contain 4 for C4 and the other must contain 8 for C3 -> 20(3) cages at R1C4 and R8C3 (step 7) = {479/578}, no 6 in R28C3, CPE no 7 in R2C6, no 7 in R8C6
11a. 4 of the cage containing {479} must be in C4 -> no 4 in R28C3
11b. 8 of the cage containing {578} must be in C3 -> no 5 in R28C3
11c. Naked pair {78} in R28C3, locked for C3

12. 21(3) cage at R6C8 = {489/579} (cannot be {678} because 6,7 only in R7C8), no 6, CPE no 9 in R4C8
12a. 7 of {579} must be in R7C8 -> no 5 in R7C8

13. 6 in N9 only in 12(4) cage = {1236}, locked for N9

14. 3 on D\ only in R1C1 + R2C2 + R3C3, locked for N1
14a. 12(4) cage in N1 = {1245} (only remaining combination), locked for N1 -> R3C3 = 3

15. R3C2 = 6 (hidden single in N1)
15a. 9 in N1 only in R1C1 + R2C2, locked for D\
15b. 4,5 on D\ only in R7C7 + R8C8 + R9C9, locked for N9

16. 20(3) cage at R3C8 = {389/578} (cannot be {479} = [749] which clashes with 21(3) cage at R6C8), no 4, 8 locked for R4 and N6
16a. 9 of {389} must be in R3C8, 7 of {578} must be in R3C8 -> R3C8 = {79}

17. 21(3) cage at R6C8 (step 12) = {489/579}
17a. 7,8 only in R7C8 -> R7C8 = {78}
17b. 9 locked for R6 and N6 -> R6C5 = 7, R4C5 = 9

18. R8C7 = 9 (hidden single in N9)

19. 9 in R3 only in R3C68, CPE no 9 in R1C9 + R2C8

20. R3C8 = 9 (hidden single in N3)
20a. 20(3) cage at R3C8 (step 16) = {389} (only remaining combination), 3,8 locked for R4 and N6

21. R6C9 = 9 (hidden single in N6)
21a. R9C1 = 9 (hidden single on D/)

22. 20(3) cage at R8C3 (step 11) = {578} (only remaining combination) -> R8C3 = 8, R89C4 = {57}, locked for C4 and N8, R2C3 = 7, R1C1 = 8, placed for D\, R2C2 = 9, R2C4 = 4, R1C4 = 9

23. R7C8 = 8 (hidden single in N9), R6C8 = 4 (step 12), R4C89 = [38]

24. Naked triple {457} in R7C7 + R8C8 + R9C9, locked for cage at R6C7 -> R6C7 = 1, R4C7 = 5, R4C6 = 4, placed for D/, R5C6 = 5

25. 5 in N3 only in 13(4) cage = {1345} (only remaining combination), locked for N3

26. Naked triple {267} in R1C9 + R2C8 + R3C7, locked for D/, N3 and cage at R1C9 -> R2C7 = 8, R3C6 = 1

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

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


Last edited by Andrew on Wed Dec 01, 2010 12:38 am, edited 1 time in total.

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PostPosted: Sat Nov 27, 2010 8:31 pm 
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Grand Master
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I think cages without totals were started a couple of years ago by Mike-Japan (another lost voice).

Note that your definition of paper-solvable is higher than mine, I'm looking for around 0.75 but definitely below 1.0. Of course we all see different things in a puzzle and I know by know that my "vision" is somewhat different to others.

Apologies for not responding to your PM, I was in Nigeria and workload (for once positive) prevented me checking in regularly.

For the corner lace with Windoku r1c2=r2c4, r2c1= r4c2 and r1c5r5c1 = r2c3r3c2 with these relationships in place both of the lace practice ones rattle out.


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PostPosted: Sat Nov 27, 2010 8:53 pm 
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Grand Master
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Location: Lethbridge, Alberta, Canada
Thanks for those comments.

Assuming that you are referring to the same cage interactions as I did in my rating comment, I think that steps can be Paper Solvable and still rated higher than 1.0. The interactions in my steps 7 and 11 were definitely paper solvable ones. I assume that they were the key steps for this puzzle. If there was something easier which I missed, I'd be interested to know.

Thanks also for the comments about the Lace Windokus. I've also had helpful discussions about those with Ed and Simon by PM. Thanks guys.


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PostPosted: Sun Nov 28, 2010 8:34 am 
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To me, "Paper Solvable" means you can solve it without using any pencilmarks. When I went through Andrew's walkthrough (briefly), I really could not see how a player can follow it without listing out the pencilmarks.

Below I will include a walkthrough, which I think one can follow it without listing pencilmarks. However one may need to jot down some (very few) small notes on the side to keep track of some tiny details.

I would urge the readers to try to follow this walkthrough by actually create a blank 9x9 grid on paper (you can draw it by hand if you don't want to use the printer), and try to fill in the 81 cells one by one while following this walkthrough. You can also have the image of this puzzle on the computer screen as a reference.

Please don't list any pencilmark, and if you find any particular gap in the walkthrough that needs more elaboration, please do post a comment, I will try to fill in the gap.

Here is my walkthrough (9 steps with a big step 5):
1:
{1} of N19 locked in 12(4) cages
{1} of N37 locked in 13(4) cages
--> N5 must include 1 of D\,D/ --> R5C5=1

2:
17(3): R46C5 =17-1 =16 ={79}
--> 23(3)={689} with 9 only possible in R5C3
--> R5C3=9, R3C5+R4C4={68}
{2} of N19 locked in 12(4) cages
--> Hidden single D\: R6C6=2

3:
20(3) cages cannot include {12}
Hidden pair C4: R37C4={12}
Outies C4: R28C3+R3C5 =20+23+11+20+1+2-45-9 =23 =7+8+8
--> R28C3={78}, R3C5=8 --> R4C4=6

4:
Innies N5: R45C6=9={45} --> 15(3): R5C7 =15-4-5 =6
--> 9(2)<>{18/36/45}, ={27}
--> 21(3): R7C8<>{6} (R6C89<>{78})
--> 12(4) in N9 must include {6} of N9, ={1236}
--> D\123 must include {3} of D\
--> 12(4) in N1={1245}
Hidden single N1: R3C2=6

5a:
D\12 must include {9} of N1
--> D\789, R3C134<>{9}
13(4) cages cannot include {9}
--> R3C678 must include {9} of R3
--> D/12<>{9}
--> R46C9 must include {9} of C9
--> R46C8<>{9}
--> R37C8 must include {9} of C8

5b:
20(3)+21(3) of C89: R37C8+R46C89 =20+21 =41
R46C89 must come from {134589} (R5C789={267})
max R37C8 =8+9 =17 --> min R46C89 =41-17 =24
--> R46C89<>{1} (or max =1+5+8+9 =23<24)
max R46C89 =4+5+8+9 =26 --> min R37C8 =41-26 =15
But R37C8<>{6} and must include {9}
--> R37C8<>15 --> min R37C8=16 --> max R46C89 =41-16 =25
--> R46C89 must include {3} (or min =4+5+8+9 =26>25)

5c:
But 21(3) cannot include {3} --> R4C89 must include {3}
--> R3C8 20(3)={389} --> R3C8<>{38}, =9
Hidden single C9: R6C9=9 --> 17(3)=[917]
Hidden single N9: R8C7=9
Hidden single D/: R9C1=9
Hidden single N1: R2C2=9
Hidden single N8: R7C6=9
Hidden single N2: R1C4=9

6:
11(2)={38} --> R1C4 20(3)<>{389/569}, ={479}
--> R2C34=[74] --> D\13=[83], R8C3=8
Hidden single N9: R7C8=8 --> 21(3)=[498]
--> R3C8 20(3)=[938]
Hidden single N3: R2C7=8
Hidden single D/: R6C4=8 --> 11(2)=[38]
Hidden singles N48: R5C2=R9C6=8

7:
D\789={457} --> R6C7<>{5}
Hidden singles N6: R46C7=[51] --> 15(3)=[456]
Hidden single N4: R5C1=4
Hidden single R3: R3C9=4
Hidden singles D\,R8: R7C7=R8C5=4

8:
Innies N3: D/123=15<>{14589}={267} --> D/78=[53]
--> R6C123=[356]
Innies N7: R7C2=7
Hidden single N4: R4C1=7

9:
R3C6=1 (only possibility) --> R37C4=[21]
--> R3C17=[57], R4C23=[21]
Hidden single N2: R1C6=7 --> R28C6=[36]
Hidden singles N3: R1C7=3 --> R9C7=2
--> N7 13(4)=[6214] --> N9 12(4)=[3126] --> N3 13(4)=[3154]
--> N1 12(4)=[4215], R1279C5=[5623], D/12=[62]
--> 9(2)=[72], D\89=[57] --> R8C3 20(3)=[875]
(Typos corrected thanks to Andrew)

(Revised step 5b for better min-max analysis)

Note that when solving without pencilmarks it is not very often to use naked singles or subsets. In the puzzle I just used one naked single, which should be easy to spot on paper at that stage.

Also I used a lot of cage sums min-max analysis, which I believe HATMAN would consider appropriate as a Paper Solving techniques because the cage sums are all we have recorded on the grid when solving without pencilmarks.


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PostPosted: Wed Dec 01, 2010 2:20 am 
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Well done Simon for posting a genuinely "Paper Solvable" walkthrough. I assume that steps where one combination is listed for a cage are written as notes outside the grid, what I call pencilmarks (see below), and only actual placements are written in the cells of the grid.

I liked the 3rd line of step 3. I wish I'd spotted that step; it would have simplified my solving path a bit.

Further Rating Comment:
After going through Simon's walkthrough, I'll stick to my rating for this puzzle. While steps like minmax, must include and don't include are technically simpler, in my opinion the way that step 5a (for example) builds up a long chain of must include and don't include is at least a 1.25 step.

I'm inclined to think that this puzzle is easier to solve by elimination solving, but not a lot easier.

Simon wrote:
To me, "Paper Solvable" means you can solve it without using any pencilmarks. When I went through Andrew's walkthrough (briefly), I really could not see how a player can follow it without listing out the pencilmarks.
I take a broader view of "Paper Solvable". While it means that it can be solved without using pencilmarks (in your use of that term) I take it as meaning an easier puzzle which can be solved that way if one wishes. I use a sort of "paper solvable" approach on another sudoku site although not as rigorously as in Simon's walkthrough.

If HATMAN, or anyone else posting "Paper Solvable" puzzles, would prefer that they should only be solved that way then I'll limit myself to brief posts about them, without a walkthrough, unless I'm able to find a "paper solvable" solving path.


BTW How did the term pencilmarks become a word meaning some/all candidates written in the cells of a sudoku grid? Candidates seems to me a much more appropriate term. When I first saw the phrase pencilmarks I assumed it referred to notes written outside the grid; I was therefore very surprised when I later discovered that many people use it to mean candidates.

Also how did the term player get introduced in the context of sudokus? In my opinion solver is far more appropriate for sudukos, crossword puzzles, etc. Player implies playing, either individually or in a team, against opponent(s). I consider myself to be a solver so I trust that nobody will ever refer to me as a player.


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PostPosted: Wed Dec 01, 2010 4:11 am 
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I think the term "pencilmark" originated from the Sudoku Players' Forum, used to be in http://www.sudoku.com and now in http://forum.enjoysudoku.com. Obviously the term "players" also came from there. Other similar forums such as the Sudoku Programmers Forum (in http://www.setbb.com) and another one (which no longer exists) used to have a certain group of common visitors, and they helped formulate a "standard terminology" in various places.


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PostPosted: Thu Dec 02, 2010 2:18 pm 
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Simon

I started these Paper Solvable for interesting puzzles that I can do on paper (including pencil-marks, but hopefully minimal). I am rather messy and for harder puzzles I have to do them on spreadsheet (developed on Udosuk's basis) or on JSudoku.

For stronger solvers doing them without pencilmarks (Andrew note the sale of pencils with rubbers has jumped manifold since the re-introduction of Sudoku to the English speaking world) seems eminently sensible.

Andrew

A few posts ago I talked of using dots and dashes as pencilmarks but did not make myself fully clear. The approach is to only partially mark-up and works well with non-consecutive or anti-chess. When you derive non-obvious placement possibilities: put dots. When you derive non-obvious placement eliminations: put dashes.

I'm just back in England for a few days awaiting my Nigerian visa. As you can imagine I am not enjoying the cold, however yesterday I had the pleasure of going to my grandson's second birthday. The difficulty was that I had to drive from London to Leeds and back which in snowstorms on English roads was not a pleasure. Tonight I go back to Africa; thankfully.

Maurice


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