## Solving without pencilmarks

Advanced methods and approaches for solving Sudoku puzzles
DreamCH wrote:
Code: Select all
`+---+---+---+|1..|.6.|..5||.8.|9.3|.1.||..7|...|8..|+---+---+---+|.7.|.4.|.9.||5..|7.1|..6||.6.|.3.|.7.|+---+---+---+|..3|...|9..||.1.|3.9|.4.||9..|.2.|..1|+---+---+---+`
this one is beyond my skills for now

needs X-wing

Pat

Posts: 4055
Joined: 18 July 2005

### Terminology

Being relatively new to Sudoku forums of this caliber, is there a page that lists terminology definitions? I can't seem to follow these:

Uniqueness sqaures?

=> ?

"picture horizontal and vertical lines goind through all instances of a particular number and find only empty square" ?

"nice loops" ?

I thank you for freeing me from the pencilmark rote work. This method seems far more useful to enhance mental exertion.

I use nine colors instead of numbers and find this method easily adapts.
gabriel

Posts: 2
Joined: 26 February 2008

### re: terminology

gabriel wrote:is there a page that lists terminology definitions?

I can't seem to follow these:
"picture horizontal and vertical lines goind through all instances of a particular number and find the only empty square"

Uniqueness squares

nice loops

hi gabriel

in the quote on "hidden singles",
"the only empty square"
should perhaps say
"the only empty cell"
?

the SuDoPedia should help with terminology --

Pat

Posts: 4055
Joined: 18 July 2005

Hi Gabriel!

Glad to see that this ancient thread is still helpful! Yes, it should probably say cell, not square. Uniqueness squares probably refers to unique rectangles. This post was one of my first on these forums and I apparently wasn't all up to date with the basic terminology when I wrote it. Perhaps I should try to find some time to go through it and correct all wrong definitions...

RW
RW
2010 Supporter

Posts: 1010
Joined: 16 March 2006

### Your solution to Tarek's puzzle

The UR you found in the puzzle is a mystery to me.

I understand why 4 & 6 are a hidden double in r1,3: c 5.

But then I'm lost. Where exactly is the UR? Which cells form it?
gabriel

Posts: 2
Joined: 26 February 2008

### Re: Your solution to Tarek's puzzle

Hi,

it would be easier to answer your question if you would specify which puzzle you are talking about...

(Oh, apparently you said it in the subject, I'm not used to look there for hints... Still better to repost the puzzle you're talking about if it's more than a few posts back.)

I suppose you meant this one:
Code: Select all
` 5 . . | 9 . 8 | . 2 1  . . 1 | . 2 . | 9 . .  9 . 2 | . . 1 | 8 . . -------+-------+------ 2 1 4 | 6 9 5 | 3 7 8  6 9 5 | 8 . . | 2 1 4  . 7 . | 2 1 4 | . 9 . -------+-------+------ . . 9 | 1 . 2 | 4 . .  . . 6 | . 8 9 | 1 . 2  1 2 . | 4 . 6 | . . 9`

The UR is in r13c25. Can you see it?

RW
RW
2010 Supporter

Posts: 1010
Joined: 16 March 2006

### Re: Your solution to Tarek's puzzle

gabriel wrote:The UR you found in the puzzle is a mystery to me.

I understand why 4 & 6 are a hidden double in r1,3: c 5.

But then I'm lost. Where exactly is the UR? Which cells form it?

ok if r1c2=4 and r1c5=6 and r3c2=6 and r3c5=4 in the solution to the puzzle, then another equally valid solution would be r1c2=6 and r1c5=4 and r3c2=4 and r3c5=6. If you assume that the puzzle has a unique solution then one of those four cells cannot be 4 or 6. r13c5 is a naked pair in 4 and 6. r13c2 can be either 3,4 or 6. Therefore r1c2 or r3c2 must be 3, to avoid the deadly pattern, if the puzzle has a unique solution. As a result r1c3 must be 7.
ab

Posts: 451
Joined: 06 September 2005

Very useful thread! I never used pencilmarks to begin with and as my skill improved and I moved to ever more difficult sudoku's I tried to maintain that. At a certain point the frustration of not seeing the next step in any reasonable amount of time outweighed the reluctance - quite a euphemism, it's more a feeling of surrender - to use pencilmarks and I started to learn how to effectively use them. This in turn gave me new insights and allowed me to use techniques that I, once excercised a few times, started to apply without pencilmarks. So, lately my skill has caught up a bit and I use pencilmarks less and less again!

I'm satisfied when pencilmarks reveal the next step to be a longer XY-chain or harder. So the trick is to be fairly certain the next step does indeed involve this before I pencil in the marks. In time I hope to become better in chaining without pencilmarks, but for now this is where I draw the line. How do I determine that I have a arrived at that point in the solving process? Well, my scanning usually gives me the pairs, triples or quads, including the ones that come from elimations caused by pointing and claiming (box-line interactions). I don't really use any structured approach there, I just scan.

When I encounter a uniqueness pattern I file it in memory to be used as a last resort to prevent pencilmarks. I consider them to be superiour in elegance to the longer chains that I find to be a bit too much trial-and-error. This is of course a matter of personal taste.

When I feel this leads me nowhere I try to find the additional eliminations caused by skyscrapers, 2-string kites and x-wings. I rarely encouter other fish varieties, let alone in an unmarked grid. For this I do use a structured method: On a number-by-number basis I try to find 4 boxes that adhere to the following rules:

- the number must not already be placed in any of the 4 boxes
- the only remain possibilities for that number in every of the 4 boxes must not be in the same row or column
- the 4 boxes must be arrayed in a rectangular pattern.

I then try to apply the technique on those boxes! This is much quicker than scanning each row/column in a structured way. When If I'm satisfied the patterns are not possible I get out my pencil.
Ardnas

Posts: 3
Joined: 30 July 2008

### Re: Solving without pencilmarks

So true! One can solve any puzzle without using pencilmarks.

Back in the day, when I was a beginner, I also thought one could not solve a hard sudoku without pencilmarks. After some time though, sudoku puzzles began to feel like a chore (cause I had to find the candidates first, before even trying to solve the puzzle, and when I found a number, I first had to wipe the candidates, rinse and repeat, etc. etc.), and I believe that's not what a game should feel like.

One should play sudoku for the fun of it, it may not turn into a chore. That's when I stopped using pencilmarks. I have found the right techniques now and up until now, I've never been stuck on a hard sudoku.
teroxec

Posts: 1
Joined: 08 February 2011

### Re: Solving without pencilmarks

teroxec wrote:So true! One can solve any puzzle without using pencilmarks.

Back in the day, when I was a beginner, I also thought one could not solve a hard sudoku without pencilmarks. After some time though, sudoku puzzles began to feel like a chore (cause I had to find the candidates first, before even trying to solve the puzzle, and when I found a number, I first had to wipe the candidates, rinse and repeat, etc. etc.), and I believe that's not what a game should feel like.

One should play sudoku for the fun of it, it may not turn into a chore. That's when I stopped using pencilmarks. I have found the right techniques now and up until now, I've never been stuck on a hard sudoku.

It's conceivable that your view of a 'hard' Sudoku is different than mine.
DonM
2013 Supporter

Posts: 483
Joined: 13 January 2008

### Re: Solving without pencilmarks

Solving without pencil marks sounds like fun, but I'm having a lot of trouble doing it. I understand the method, but I don't quite understand where to start when looking for forced chains. Exactly how do you find them? How do certain forced chains "jump out" at you?

Could you explain this one to me using forced chains? This one came from Mensa's Absolutely Nasty Sudoku Level 2 Book, #156. (Yeah, I know, Level 2... Eye iz teh s00p37 n00b.)

Code: Select all
` . . 5 | . . . | . 6 .   . . 6 | . . 3 | . . 2   8 4 . | 2 6 . | . . .  -------+-------+------  . . . | . 4 . | 7 . 3   . 2 . | . 3 . | . 5 .   1 . 4 | . 5 . | . . .  -------+-------+------  . . . | . 2 6 | . 7 4   4 . . | 1 . . | 6 . .   . 5 . | . . . | 9 . .`

Singles:

R1C1 = 8
R1C4 = 4
R1C7 = 3
R2C8 = 4
R3C3 = 3
R3C8 = 9
R4C1 = 5
R4C2 = 6
R4C6 = 2
R4C8 = 1
R5C6 = 1
R5C7 = 4
R6C2 = 3
R6C7 = 2
R6C8 = 8
R7C1 = 3
R8C3 = 2
R8C8 = 3
R9C1 = 6
R9C4 = 3
R9C6 = 4
R9C8 = 2

Which gives you...

Code: Select all
` 2 . 5 | 4 . . | 3 6 .   . . 6 | . . 3 | . 4 2   8 4 3 | 2 6 . | . 9 .  -------+-------+------  5 6 . | . 4 2 | 7 1 3   . 2 . | . 3 1 | 4 5 .   1 3 4 | . 5 . | 2 8 .  -------+-------+------  3 . . | . 2 6 | . 7 4   4 . 2 | 1 . . | 6 3 .   6 5 . | 3 . 4 | 9 2 .`

(There might be typo in the solution, so just use the Post-singles Sudoku directly above. I'm pretty sure that one is right.)

On another note... Let's say you wanted to make this even more applicable to speedsolving... Would it be better to use PMs in moderation solely for the purpose of finding forced chains, or would it be better to practice memory skills and keep the board clear for visual purposes? If the former, then how many PMs are too many PMs?
DaKrazedKyubizt

Posts: 1
Joined: 15 March 2011

### Re: Solving without pencilmarks

Hi KrazedKyubizt,

Welcome to the forums!

DaKrazedKyubizt wrote:Exactly how do you find them? How do certain forced chains "jump out" at you?

It very much depends on the puzzle. How fast they "jump out" also depends on how lucky I am at choosing starting points for chains. And it depends on how stubborn I am continuing to follow chains that don't lead anywhere. Mostly I choose a starting point where there are two different options and both options immediately advances the puzzle. If I can eliminate one of those options, I will have advanced the puzzle.

I gave your example a shot without pencilmarks:

Code: Select all
` 2 . 5 | 4 . . | 3 6 .   . . 6 | . . 3 | . 4 2   8 4 3 | 2 6 . | . 9 .  -------+-------+------  5 6 . | . 4 2 | 7 1 3   . 2 . | . 3 1 | 4 5 .   1 3 4 | . 5 . | 2 8 .  -------+-------+------  3 . . | . 2 6 | . 7 4   4 . 2 | 1 . . | 6 3 .   6 5 . | 3 . 4 | 9 2 .`

First I check for unique rectangles (these are often not noticed by puzzle makers and provide easy backdoors), seeing potential deadly patterns 89 in r45c34 and 69 in r56c79, which tells me r5c34 and r56c4<>9. A third one in r12c25 tells me that r2c5<>79. Note that unique rectangles always appear under very specific conditions and are therefore very easy to find. Finding those three rectangles was a matter of seconds. This is why I look for them first. But unfortunately this time none of them advanced the puzzle.

Next I check for digits whose remaining unsolved instances can be placed in only two ways, hoping that if I can eliminate one of those options, I will solve a lot of the puzzle. Digit 5 catches my eye, unsolved in four boxes with two possible layouts. Either r2c4, r8c6, r7c7 and r3c9=5 or r3c6, r2c7, r8c9 and r7c4=5. Each layout is easy to memorize as a pattern, so I start with the first option. Note that I don't think of any particular cell as the starting point of the chain. The starting point is that all four cells=5, which they are if any one of them is 5. Looking at the first option I can see that it solves a lot of cells. Starting from boxes 3 and 9, then moving left I can rather quickly see that most of the puzzle would be solved without contradictions, so I suspect that this option is not false and move on to the other option.

With r3c6, r2c7, r8c9 and r7c4=5 I happen to look at digit 8 next, trying to see how they would place. The trail of 8s takes me through r1c9, r7c7, r8c2, r9c5 and r2c4 at which point I cannot place an 8 in box 5, in other words, the initial assumption of this chain was false and I can place 5 in r2c4, r8c6, r7c7 and r3c9.

So from the point where I decided to see what happens if the second layout of 5s is true, it was once again a matter of seconds, because I happened to look at the 8s first. If I instead had started to look for contradiction in some certain box, it could have taken me a lot longer. Or then I might have noticed that r3c7=1 as it was also with the other layout of 5s, which would have solved that cell. For a pencilmark solver this would have been immediately obvious as a naked single after finding out that digit 5 can be solved only in two ways, candidate 5 eliminated from r3c7 by an X-wing. Without pencilmarks I didn't think of the pattern as an X-wing, but I did notice that r3c7<>5 while looking at the possible layouts for digit 5. However, I did not follow up on that lead by looking at the implications for that cell, which would have solved the puzzle much faster for me. This is of course the problem with not using pencilmarks and not being a computer that does full searches for each and every technique at every step - Sometimes the solution path is a lot longer and more complicated than necessary.

On another note... Let's say you wanted to make this even more applicable to speedsolving... Would it be better to use PMs in moderation solely for the purpose of finding forced chains, or would it be better to practice memory skills and keep the board clear for visual purposes? If the former, then how many PMs are too many PMs?

I don't know how PMs affect speedsolving, because I still never have used them while solving manually. But I suppose that they at least don't hurt and slow down your progress. If I wanted to become a professional speed solver, then I suppose I would develop some markup system that lets me quickly write down the essential information without cluttering up the board too much.

RW
RW
2010 Supporter

Posts: 1010
Joined: 16 March 2006

### Re: Solving without pencilmarks

Hello.

Yes we can resolve this puzzle without pencilmarks.

Code: Select all
` *-----------* |..1|..2|35.| |...|...|2..| |.8.|4..|.91| |---+---+---| |5.9|31.|4..| |...|7.4|...| |..7|.25|8.3| |---+---+---| |86.|..9|.3.| |..3|...|...| |.74|1..|5..| *-----------*`

Listed below are the resolution and the method given by the solver online:
http://en.top-sudoku.com

Make:
Click:Enter a grid
Enter the statement.
Click: Solution of the grid
Click : move list

Â 1)Â Â lin:4Â Â col:2Â Â val:2Â Â by inclusion
Â 2)Â Â lin:6Â Â col:4Â Â val:9Â Â by exclusion
Â 3)Â Â lin:2Â Â col:6Â Â val:1Â Â by exclusion
Â 4)Â Â lin:5Â Â col:3Â Â val:8Â Â by exclusion
Â 5)Â Â lin:5Â Â col:5Â Â val:6Â Â by inclusion
Â 6)Â Â lin:4Â Â col:6Â Â val:8Â Â by inclusion
Â 7)Â Â lin:7Â Â col:7Â Â val:1Â Â by exclusion
Â 8)Â Â lin:5Â Â col:7Â Â val:9Â Â by inclusion
Â 9)Â Â lin:6Â Â col:1Â Â val:6Â Â by exclusion
10)Â Â lin:6Â Â col:8Â Â val:1Â Â by inclusion
11)Â Â lin:6Â Â col:2Â Â val:4Â Â by inclusion
12)Â Â lin:1Â Â col:2Â Â val:9Â Â by inclusion
13)Â Â lin:5Â Â col:8Â Â val:2Â Â by inclusion
14)Â Â lin:5Â Â col:9Â Â val:5Â Â by inclusion
15)Â Â lin:2Â Â col:5Â Â val:9Â Â by exclusion
16)Â Â lin:8Â Â col:7Â Â val:6Â Â multiple choice
17)Â Â lin:9Â Â col:8Â Â val:8Â Â by inclusion
18)Â Â lin:9Â Â col:5Â Â val:3Â Â by inclusion
19)Â Â lin:9Â Â col:6Â Â val:6Â Â by inclusion
20)Â Â lin:8Â Â col:6Â Â val:7Â Â by inclusion
21)Â Â lin:3Â Â col:6Â Â val:3Â Â by inclusion
22)Â Â lin:8Â Â col:8Â Â val:4Â Â by inclusion
23)Â Â lin:3Â Â col:7Â Â val:7Â Â by inclusion
24)Â Â lin:3Â Â col:1Â Â val:2Â Â by inclusion
25)Â Â lin:9Â Â col:1Â Â val:9Â Â by inclusion
26)Â Â lin:8Â Â col:1Â Â val:1Â Â by inclusion
27)Â Â lin:2Â Â col:8Â Â val:6Â Â by inclusion
28)Â Â lin:3Â Â col:5Â Â val:5Â Â by inclusion
29)Â Â lin:7Â Â col:5Â Â val:4Â Â by inclusion
30)Â Â lin:8Â Â col:5Â Â val:8Â Â by inclusion
31)Â Â lin:1Â Â col:5Â Â val:7Â Â by inclusion
32)Â Â lin:2Â Â col:4Â Â val:8Â Â by inclusion
33)Â Â lin:1Â Â col:4Â Â val:6Â Â by inclusion
34)Â Â lin:4Â Â col:8Â Â val:7Â Â by inclusion
35)Â Â lin:4Â Â col:9Â Â val:6Â Â by inclusion
36)Â Â lin:1Â Â col:1Â Â val:4Â Â by inclusion
37)Â Â lin:8Â Â col:2Â Â val:5Â Â by inclusion
38)Â Â lin:8Â Â col:4Â Â val:2Â Â by inclusion
39)Â Â lin:7Â Â col:3Â Â val:2Â Â by inclusion
40)Â Â lin:7Â Â col:4Â Â val:5Â Â by inclusion
41)Â Â lin:8Â Â col:9Â Â val:9Â Â by inclusion
42)Â Â lin:7Â Â col:9Â Â val:7Â Â by inclusion
43)Â Â lin:2Â Â col:2Â Â val:3Â Â by inclusion
44)Â Â lin:2Â Â col:1Â Â val:7Â Â by inclusion
45)Â Â lin:5Â Â col:1Â Â val:3Â Â by inclusion
46)Â Â lin:9Â Â col:9Â Â val:2Â Â by inclusion
47)Â Â lin:1Â Â col:9Â Â val:8Â Â by inclusion
48)Â Â lin:2Â Â col:9Â Â val:4Â Â by inclusion
49)Â Â lin:5Â Â col:2Â Â val:1Â Â by inclusion
50)Â Â lin:3Â Â col:3Â Â val:6Â Â by inclusion
51)Â Â lin:2Â Â col:3Â Â val:5Â Â by inclusion

Cordially
Alex
Alex22

Posts: 2
Joined: 14 May 2011

### Re: Solving without pencilmarks

An interesting topic ;

With/without pencil marks

What will be the border line that one can solved without pencil markings ?
In general.... puzzle rating ? Up to certain techniques ?
7b53
2012 Supporter

Posts: 156
Joined: 01 January 2012
Location: New York

### Re: Solving without pencilmarks

That very much depends on your exact definition of "without pencil marks". Some can keep the entire set of pencil marks in their head and see the solution that way, no pencil marks are ever written down, but still in some sense they are used.

JasonLion
2017 Supporter

Posts: 642
Joined: 25 October 2007
Location: Silver Spring, MD, USA

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