## Solving without pencilmarks

Advanced methods and approaches for solving Sudoku puzzles

### Solving without pencilmarks

The more I read posts on this forum, the more I get the feeling that I'm the only person in the world who solve exclusively without pencilmarks, regardless off difficulty. I often read comments like "no other than the easiest puzzles can be solved without pencilmarks" or "techniques that are impossible for human solvers.". Now I wish to show you that none of those are true. First I'll give you some tips that I've found usefull, then I'll take you through a step by step solution of a couple of extreme puzzles, the way I solve them, and hope that you can follow without making any own pencilmarks.

If anybody else has got any good tips on solving without pencilmarks, please post them here and we can try to make a little collection of these techniques.

First of all, whenever I say that I solve without pencilmarks, peoples first reaction is: "then you must have a really good memory". My answer is yes, but I hardly ever use it. The most important thing I've learned so far is to choose what information to memorize. There is simply so much information to be found in any grid, that one can't remember it all. Most of this information is useless. When I started doing Sudoku I often found myself trying to memorize possible values for cells, drawing a mental pencilmarkgrid. When I finally realised that this is pointless, my solving improved significantly. Most puzzles can be solved exclusively with naked or hidden singles, then what's the point in knowing that a cell can hold numbers 1,2,6 and 7? In fact, all numbers are solved by naked or hidden singles, all other techniques only reduce possibilities. So what I’m interrested in is these reduced possibilities. Instead of memorizing possible candidates, I memorize impossible candidates. As most of the impossibles can be read from the column, row and box of each cell, there isn’t really that much to remember. Only when I make a reduction through some other technique, like x-wing, uniqueness etc., I memorize the reduced candidates. Apart from this I memorize locations of pairs and triplets, but the rest of the information about each cell I put aside as soon that I can see that it doesn’t help me at the moment.

I guess most of you use pencilmarks to spot singles, so the most important thing to improve if you want to solve without pencilmarks is not your memory, but your ability to spot singles. The hidden singles are usually easy to spot, just picture horisontal and vertical lines going through all instances of a particular number and find the only empty square. Naked are harder to find. Going through the whole puzzle cell by cell is very timeconsuming. I rather look for them one row or column at a time. I start with the rows/columns that have most numbers already, then I count from 1 to 9, leaving out all the numbers already in the row. Example:
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..| *-----------*`

If I wanted to find naked singles in this puzzle, I would start with row 4 that already has 5 numbers, then count: “. 2 . . . 6 7 8 .” These are the 4 numbers that go in the four empty cells, so now I compare each cell against these numbers. R4c2 has 6, 7 and 8 in the same column, so there’s a naked single (2). This trick is also efficient when looking for hidden singles in rows or columns (try it on row 6). If you systematically go through all rows like this you will find all naked singles in no time.

Another question I often get is ”how can you use advanced techniques like coloring or XYWhatever-wing without pencilmarks?” The answer is very simple, I don’t use them. Techniques like these are in my opinion not at all advanced, but limitations of the real techniques behind them. They are limitations in the sence that they actually apply different trailing techniques, but only on very special patterns that somebody has predefined for you as they can be seen in a pencilmarkgrid. When programming your solver software this might be neccessary, but as a human solver you should be able to see the logic in any situation.

Take for example “multiple colors”. When I first got Simple Sudoku I opened a “multiple colors”-example to see what it was about. First I solved it in my way, then I checked how SS would have done it. I found that it colored all cells with candidate 2 in five different colors, then removed some candidates of number 2. I could see that all these candidates shared an unit with a 2 that I had solved with a short forcing chain. I then checked all the other examples of multiple colors and found that in every case a cell connecting all removed candidates could be solved with a simple forcing chain. So apparently this is a technique that you can use instead of a forcing chain if it considers only one candidate number, if all instances form a specific pattern and if you have a good set of crayons. Well, then I prefer the forcing chain as it applies to endless situations where the crayons won’t help you. Guess you want to see the example:
Code: Select all
` *-----------* |78.|3..|.1.| |615|.8.|.32| |3.9|.1.|8..| |---+---+---| |1.8|..3|.6.| |.73|462|1..| |.6.|1.8|7.3| |---+---+---| |..1|.3.|6..| |85.|.4.|3.1| |.3.|..1|..5| *-----------*`

I used this forcing chain:
if r1c3=2 => r9c5=2 => r4c7=2
if r3c2=2 => r6c1 or r6c3=2 =>r4c7=2

Simple sudokus advanced spread of colors across the grid ended up in removing candidate 2 from r4c2, r6c8 and r9c7, which I obviously also could do after solving r4c7 with the forcing chain.

My last tip before getting on with the actual puzzles is this: UNIQUENESS!! Uniqueness patterns are everywhere and they often provide a very easy solution to extreme puzzles. The best thing with them is that they are very easy to spot. Again, don’t limit yourself to the reductions explained in different forums, be creative. My approach is that whenever I see two corners of a uniqueness square in place, I put in a third corner and read ahead to see if it gives me the forth. Actually, whenever I could make any move that results in three corners of the square in place I read ahead to see if the forth falls in place. It often does, so these are very good spots to start your trails from if you're stuck.

Now let’s have a look at some extreme puzzles. I recommend that you copy them into some sudoku program (without pencilmarks) and follow as we go along. To help you follow without pencilmarks I’ll use these abbreviations on how I solve each number:

Ce = only possible number that can go in the cell
Co = only possible cell for number in column
R = only possible cell for number in row
B = only possible cell for number in box

First an extreme from Simple Sudoku:

Puzzle 1
Code: Select all
`*-----------* |.2.|1.8|..5| |1..|.76|.3.| |7..|4..|...| |---+---+---| |..7|...|2..| |48.|...|.59| |..2|...|6..| |---+---+---| |...|..9|..1| |.5.|74.|..6| |6..|8.1|.7.| *-----------*`

First the obvious ones:
r7c2=7 (B)
r1c7=7 (B)
r6c9=7 (B)
r5c6=7 (B)
r8c3=1 (B)

I can’t see anymore obvious cells by drawing lines, but I can see two pairs: r78c1={2,8} (R) and r23c3={5,8} (B). Next I start checking rows, starting with row 1. Two pairs there also: c1&5={3,9} and c3&8={4,6}. This is enough to solve the next number:

[edit: typo there, the first pair should read r78c1={2,8} (Co) = the two numbers cannot go anywhere else in the column.]

r9c3=9 (Co)

This gives me another pair: r7c3 and r9c2={3,4} – so far 5 pairs to remember. Now I see a very obvious trail:

if r1c1=9 => r3c2=3 (B) => r9c2=4 (Co) => r1c3=4 (B) => nowhere to place 6 in box 1. This gives me:

r1c1=3 (Ce)
r1c5=9 (Ce)

At this point I can tell that 5 cannot go in r4c4 or r6c4 ({9,5} uniqueness square with c1), always worth to notice.

I decide to check the other pair of row 1 and find:
if r1c3=6 => r1c8=4 => r4c9=4 (B) and
if r1c3=6 => r2c2=4 => r9c2=3 => nowhere to place 3 in column 9. This gives me:

r1c3=4 (Ce)
r1c8=6 (Ce)
r3c2=6 (B)
r5c3=6 (B)
r7c3=3 (Ce)
r9c2=4 (B)
r2c2=9 (B)

Now I have:
Code: Select all
` *-----------* |324|198|765| |19.|.76|.3.| |76.|4..|...| |---+---+---| |..7|...|2..| |486|..7|.59| |..2|...|6.7| |---+---+---| |.73|..9|..1| |.51|74.|..6| |649|8.1|.7.| *-----------*`

Still can’t see any obvious so I have a look at column 9, missing numbers 2,3,4 and 8. This gives me an idea that I have to check out and this is what I see:

if r3c5 or r3c6=2 =>r2c9=2 (B) and
if r2c4=2 => r2c3=5 (R) => r3c3=8 => r3c9=2 (Ce)

Anyway, r9c9 cannot be 2. This gives me a 2 in either r7c8 or r8c8 that not only forms an x-wing with c1 but also lets me exclude 8 from both squares (uniqueness). Thanks to this I can go on:

r8c6=3 (Ce)
r9c5=2 (B)
r5c4=2 (B)
r3c6=2 (B)
r2c9=2 (B)

and so on... rest of the puzzle is only singles so I guess you can do it yourself. There the puzzle was solved with a few short trails, max 4 steps. Could you follow without pencilmarks? If you could, then you can also do this yourself. If you try enough short trails you will find lots of reductions like the ones I made here. As you could see the only neccessary memorizing I did was the 5 pairs, that soon could be forgotten as they were solved (or made obvious by filling in the rest of the boxes), and then i memorized max 4 numbers at a time while trailing.

Let’s have a look at one more puzzle, this is a Solo extreme, that Simple Sudoku cannot provide a logical solution for. Let’s find out how hard it really is:

Puzzle 2
Code: Select all
`*-----------* |1..|.32|5..| |5..|...|.92| |7..|5..|4..| |---+---+---| |...|.2.|71.| |...|4.8|...| |.21|.5.|...| |---+---+---| |..8|..5|..7| |31.|...|..6| |..7|84.|..9| *-----------*`

First the singles:
r3c9=1 (Co)
r5c2=7 (B)
r1c8=7 (B)
r1c9=8 (Ce)
r2c6=4 (B)
r4c1=8 (B)
r3c3=2 (B)
r5c5=1 (B)
r2c4=1 (B)
r2c5=7 (B)
r2c2=8 (R)
r3c5=8 (B)
r9c6=1 (B)
r7c7=1 (B)
r7c5=6 (R)
r8c5=9 (Ce)
r7c4=3 (B)
r8c4=2(B)
r8c6=7 (Ce)
r6c4=7 (B)
r8c7=8 (Ce)
r6c8=8 (B)

This is the situation:
Code: Select all
` *-----------* |1..|.32|578| |58.|174|.92| |7.2|58.|4.1| |---+---+---| |8..|.2.|71.| |.7.|418|...| |.21|75.|.8.| |---+---+---| |..8|365|1.7| |31.|297|8.6| |..7|841|..9| *-----------*`

At this point I can’t see any more obvious cells, so I start reading through rows and columns. In column 1 I react on r5c1 that has to hold 6 or 9. This tells me that if r5c7=9 => r5c1=6 (Ce) => r6c7=6 (B). That’s three corners of the uniqueness square so I decide to follow the trail a bit further. And, not to my suprise, I find three steps ahead:

if r5c7=9 => r5c1=6 (Ce) => r6c7=6 (B) => r3c8=6 (B) => r3c6=9 (Ce) => r6c1=9 (R) => Double solution. I can remove that option and go on:

r6c7=9 (B)
r5c1=9 (Ce)
r7c2=9 (B)
r1c3=9 (B)
r3c6=9 (B)
r4c4=9 (B)
r1c4=6 (Ce)
r1c2=4 (Ce)

Current situation:
Code: Select all
` *-----------* |149|632|578| |58.|174|.92| |7.2|589|4.1| |---+---+---| |8..|92.|71.| |97.|418|...| |.21|75.|98.| |---+---+---| |.98|365|1.7| |31.|297|8.6| |..7|841|..9| *-----------*`

Now this trail immediately jumps into my face:

if r4c2=3 => r6c6=3 (B) => r6c1=6 (R) => no way to fit numbers 3 and 6 into column 3. I can remove that option and continue:

r3c2=3 (Co)
r2c3=6 (Ce)

...you may fill in the remaining singles. That wasn’t so hard, was it? This time I didn’t actually memorize anything except the two short trails.

I hope I have showed by this that memory isn’t the biggest issue when solving harder puzzles without pencilmarks, but learning to see patterns and short trails. Don’t focus on trails you’ve read about in technique guides, because (a) you won’t see them (they are defined by possible candidates) and (b) they are very limited. With pure logic and the ability to read 5 steps ahead you can solve almost any puzzle this way.

regards, RW

PS. Seems my trails have a lot in common with what you call nice loops. If so, feel free to translate them into the "correct" language that I don't really master yet.
Last edited by RW on Wed Sep 06, 2006 4:35 pm, edited 3 times in total.
RW
2010 Supporter

Posts: 1000
Joined: 16 March 2006

Great article, RW! Thanks for sharing!

I actually like solving without pencilmarks, too, although I'm not too great at it yet. The simple reason is that I have a handheld PC (Nokia 770) with a sudoku program on, and while the screen is very good, pencilmarks would have to be awkwardly small. The program doesn't do anything to help you solve the puzzle, by the way; no highlighting, filtering, candidates or anything, so I regard it perfectly equivalent to paper and pencil solving. So this technique just might be what I need to crank up the difficulty a bit.

I do enjoy working with pencilmarks, too, by the way, but I rarely get around to it, except on my computer. The bookkeeping overhead of working with candidates on paper gets a bit tedious at times. And IMHO, it provides a rather different solving approach. When you work with candidates, naked singles really stand out and hidden singles often stay exactly that (hidden). Without, though, hidden singles are much easier to spot. IMHO.

Anyway, thanks for a very interesting read, RW.

Vidar
vidarino

Posts: 295
Joined: 02 January 2006

Hi RW, thanks for your article. Its a new POV for me. I am a lazy person, and whenever i was stuck solving a puzzle on paper i always thought, filling in the candidates is more effective as trying to find more sophisticated ways without pm's. But maybe thats one reason that i am rather slow with hard rated newspaper puzzles. As an example, i tried one of these nice puzzles on paper - #011 (Ruud) - and came here without full pm's (but i always put in pairs). It was a bit frustrating, because it took me 5 minutes to fill them in, and then 2 minutes to see how to solve the puzzle.
Code: Select all
` *-----------* |652|.8.|..4| |739|...|8..| |841|2.7|...| |---+---+---| |.8.|...|2..| |297|.4.|.58| |.16|...|.4.| |---+---+---| |.6.|1.3|.8.| |..3|...|4..| |1.8|.5.|.36| *-----------*`

How would you proceed?
ravel

Posts: 998
Joined: 21 February 2006

ravel wrote:Hi RW, thanks for your article. Its a new POV for me. I am a lazy person, and whenever i was stuck solving a puzzle on paper i always thought, filling in the candidates is more effective as trying to find more sophisticated ways without pm's. But maybe thats one reason that i am rather slow with hard rated newspaper puzzles. As an example, i tried one of these nice puzzles on paper - #011 (Ruud) - and came here without full pm's (but i always put in pairs). It was a bit frustrating, because it took me 5 minutes to fill them in, and then 2 minutes to see how to solve the puzzle.
Code: Select all
` *-----------* |652|.8.|..4| |739|...|8..| |841|2.7|...| |---+---+---| |.8.|...|2..| |297|.4.|.58| |.16|...|.4.| |---+---+---| |.6.|1.3|.8.| |..3|...|4..| |1.8|.5.|.36| *-----------*`

How would you proceed?

Code: Select all
` *-----------* |652|981|374| |739|564|821| |841|237|695| |---+---+---| |485|719|263| |297|346|158| |316|825|749| |---+---+---| |964|173|582| |523|698|417| |178|452|936| *-----------*`

done without pincelmarks in Simple Sudoku
barneyzhen

Posts: 3
Joined: 26 March 2006

Newbies always seem to think it's cute to post answers to puzzles.

Kinda like first-time deer hunters getting "buck fever", I guess.
ronk
2012 Supporter

Posts: 4764
Joined: 02 November 2005
Location: Southeastern USA

ravel wrote:
Code: Select all
`*-----------*  |652|.8.|..4|  |739|...|8..|  |841|2.7|...|  |---+---+---|  |.8.|...|2..|  |297|.4.|.58|  |.16|...|.4.|  |---+---+---|  |.6.|1.3|.8.|  |..3|...|4..|  |1.8|.5.|.36|  *-----------* `

How would you proceed?

I should maybe specify that I usually don't turn to trailing techniques before it is absolutely neccessary. In a case like this, where it is specified that the puzzle can be solved with pairs, triplets and X-wing I would first try to find these, because I want to improve my eye for these common patterns. Only if I can't find the "simple" solution I start trailing, and usually after solving the puzzle the hard way I enter it into Simple Sudoku to see what I had missed.

In this case I immediately spotted some uniqueness reductions, quite fast I had all the pairs picked out and eventually I found the X-wing in r15c47 that leaves us a {7,9} pair in r69c7, and some singles. That should be enough to solve the puzzle, right?

It did take me some time to find this X-wing, so I then tried how long it would take me to find some other solution through trailing. This forcing chain came up in a few minutes:

if r5c7=3 => r5c4=6 => r4c8=6
if r5c4=3 => r3c5=3 => r2c5=6 => r4c5=1 => r5c6=6 => r4c8=6

That should also be enough to solve the puzzle. I'm not at all suprised that it turned out that the trailing was quicker. When I looked for the "simple" solution I was looking for the only one possible X-wing pattern in the puzzle. When looking for trailing solutions there are usually lots of different available patterns to solve through, the amount depending on how long trails you are ready to make. I usually follow a trail at least 10-15 steps before trying another one, that tends to give result at almost every second trail.

The thread you mentioned is by the way great for training solving without pm:s. Another good thread, with puzzles that will teach you to spot singles, can be found here.

RW
RW
2010 Supporter

Posts: 1000
Joined: 16 March 2006

Thanks, guess i do need some mind training and a lot of practice to be able to solve such puzzles without pm's.
To see the x-wing is probably not that difficult, but to realise that it opens a pair, that gives a number, was beyond my skills.
I can follow your trailing without candidates, but doing it is very different from how i look for forcing chains in candidate grids.
But this way looks a bit easier for me, because i dont have to keep almost pairs in mind.
ravel

Posts: 998
Joined: 21 February 2006

RW This was a great article, thank you very much for taking the time to writre this. I am sorry that I missed it when you first posted, but I was away on a trip with only limited access to this site.

Solving without pencilmarks has become an important subject for me too. When I started doing sudoku puzzles, I did them too without using pencilmarks too, but a few of the Fiendish ones from the Times (UK) proved beyond my then current techniques. Last year, when I entered the UK Times championships, I actually did the qualifiers without pencilmarks, but I realised that if I wanted to do well at the competiion, then I would have to learn to do pencilmarks, which I did. After being beaten by some truely amazing competitors at the competition ((summary given in another thread), I took the time to revise my techniques. My breakthrough came when I had to write down the steps to solve a Fiendish puzzle for the husband of a friend, and I realised just how solving most puzzles was actually straightforward. Since then, I have been doing the Fiendish ones from the newspapers without using pencilmarks, with the exception of the Superior ones in the UK Sunday Times.

Solving the Superior ones has been discussed separately (http://forum.enjoysudoku.com/viewtopic.php?t=3373 and
http://forum.enjoysudoku.com/viewtopic.php?t=3087).

It looks like your article has provided me with another two methods. I am now searching every puzzle that I do for uniqueness situations, and it has surprised me how often possible uniqueness squares appear. Though there might be other/better tecnniques to apply, the uniqueness one certainly reveals a few shortcuts to the final solving. I now need to practice some more so that I can follow the trails too.

Thanks once again.
Heuresement

Posts: 54
Joined: 19 August 2005

Thanks Heuresement, glad you liked it.

Heuresement wrote:I am now searching every puzzle that I do for uniqueness situations, and it has surprised me how often possible uniqueness squares appear. Though there might be other/better tecnniques to apply, the uniqueness one certainly reveals a few shortcuts to the final solving.

They certainly do appear all the time, and very often they reveal shortcuts. Especially computergenerated puzzles often don't notice these when they judge the level of difficulty, so they can really make "extreme" puzzles very easy. Some puzzles that you can practise your eye for uniqueness patterns on has been posted here.

Not quite as common, but still quite easy to spot when they appear are the three-pair uniqueness patterns. When you get confident in applying uniqueness reductions on simple rectangles it's also worth trying to look for larger patterns. I'll post some of them to the thread mentioned above as soon as I have time to make some good examples.

As for the other/better techniques to apply... I don't think techniques can be rated as good or bad in a particular situation. I often find that I solve numbers as result of a uniqueness reduction, when they would in fact have been naked singles already before my reduction. In a non-pencilmarkgrid this happens, because a uniquenesspattern is often easier to spot than a naked single. Does it matter in this case what technique I use? A human being don't need to iterate through the puzzle tehcnique by technique like a computerized solver, but can scan all techniques simultaneously. I say use whatever technique you see first.

Regards
RW
RW
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Posts: 1000
Joined: 16 March 2006

As many solvers who do not use PMs spot hidden subsets before naked subsets....

As I'm trying to find how difficult a technique is in rating a puzzle, I was wondering how would different solvers approach this....

here is gsf's #012 from the superior 100 list advanced to the point in question........
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 `

What technique Ultimately solves the puzzle above (forgetting singles), or to put it into context, what is your last non-single move?

As too many techniques can be utilised at first(including what I'm after), here is the exact PM'd postion, feel free to use either, however start with the non-PM'd one first if you usually solve without pencilmarks
Code: Select all
`*--------------------------------------------------------*| 5     346   37   | 9     46    8    | 67    2     1    || 478   468   1    | 357   2     37   | 9     3456  3567 || 9     46    2    | 357   46    1    | 8     3456  3567 ||------------------+------------------+------------------|| 2     1     4    | 6     9     5    | 3     7     8    || 6     9     5    | 8     37    37   | 2     1     4    || 38    7     38   | 2     1     4    | 56    9     56   ||------------------+------------------+------------------|| 378   358   9    | 1     357   2    | 4     3568  3567 || 347   345   6    | 37    8     9    | 1     35    2    || 1     2     378  | 4     35    6    | 57    358   9    |*--------------------------------------------------------*`

When answering, could you specify if you used PMs or not..

tarek

tarek

Posts: 2699
Joined: 05 January 2006

Here's the solution I found in the non pencilmarkgrid:

Hidden pair: r13c5=46
Only number except 4&6 that can go in any of cells r13c2=3 => UR: r1c2 or r3c2=3 => r1c3=7

I didn't solve the rest of the puzzle, but I suppose this is what you asked for.

RW
RW
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Thanx RW, What I was after was the last non-single step.......

Anyway, your answer was interesting enough because it shows how uniqueness patterns can be easily identied (or is it just you RW )...

How would other members tackle this puzzle without PMs

tarek

tarek

Posts: 2699
Joined: 05 January 2006

tarek wrote:What I was after was the last non-single step.......

I finished the puzzle and the uniqueness rectangle was the last required non-single step. After that I solved only singles. I'm not sure if I understand what you're asking for?

RW
RW
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Hi Tarek! I solved this one without pencilmarks too. It took me quite a while to find an opening, but once I had spotted a uniqueness rectangle, and used it to place the 3 in r1c2, everything else fell quickly into place which was all singles if I remember correctly.

I then tried the puzzle again without using the uniqueness square or pencilmarks, and it was certainly a more lengthy procedure, and I needed to make use of two x-wings, and some doubles too, but the solution eventually came.

In order to improve my solving techniques, I am currently concentrating on using uniqueness squares wherever possible, following RW's lead. I know that when I am able to spot one, it can provide a substantial and often suprising shortcut. However, often I am not able to spot any, and I then just have to use xwings, doubles and triples to solve the superior puzzles.

As your example puzzle contained such a good uniqueness rectangle (UR), which substantially shortened the otherwise difficult solving process, may I suggest that you pose another test example which doesn't have a good UR?
Heuresement

Posts: 54
Joined: 19 August 2005

Heuresement wrote:As your example puzzle contained such a good uniqueness rectangle (UR), which substantially shortened the otherwise difficult solving process, may I suggest that you pose another test example which doesn't have a good UR?

Who said it wasn't intentional............

However for the non PM grid, I would have thought the spotting the hidden double in column 5 would be easiest........

which RW & yourself confirmed.....(which is easier than spotting its naked triple counterpart)....

I was counting on the fact that you would perform a few box-line eliminations then spot either the hidden triple or the naked double in row 3....

but then the UR was easier (& the x-wing is not needed in any case).......

for the PM'd grid you have a choice between all three.....

UR, naked double & the hidden triple....(The naked double probably being very evident IMO)...........

The RW & Heuresement for that feedback....

I think I have some other puzzle with an interesting perspective....

tarek

tarek

Posts: 2699
Joined: 05 January 2006

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