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Nataraj Diagrams

 
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keith



Joined: 19 Sep 2005
Posts: 3179
Location: near Detroit, Michigan, USA

PostPosted: Fri Jul 04, 2008 9:09 pm    Post subject: Nataraj Diagrams Reply with quote

This thread really starts here:

http://www.dailysudoku.com/sudoku/forums/viewtopic.php?t=2630

You can see how nataraj developed the idea. Then, come back here.

=================================

nataraj,

Perhaps you should patent / copyright your idea?

Here is a puzzle that might be a good example:
Code:
Puzzle: M5844552sh(23)
+-------+-------+-------+
| . . . | . . . | . . . |
| . 4 5 | . 1 . | . . 7 |
| . 6 1 | 8 . . | 2 . . |
+-------+-------+-------+
| . . . | 9 4 1 | 3 . . |
| . 8 . | 5 . 2 | . 4 . |
| . . 9 | 3 8 7 | . . . |
+-------+-------+-------+
| . . 4 | . . 8 | 1 5 . |
| 1 . . | . 3 . | 6 2 . |
| . . . | . . . | . . . |
+-------+-------+-------+
After basics, it becomes
Code:
+-------------------+-------------------+-------------------+
| 8     239   237   | 27    5     39    | 4     1     6     |
| 239   4     5     | 26    1     369   | 8     39    7     |
| 379   6     1     | 8     79    4     | 2     39    5     |
+-------------------+-------------------+-------------------+
| 256   25    26    | 9     4     1     | 3     7     8     |
| 37    8     37    | 5     6     2     | 9     4     1     |
| 4     1     9     | 3     8     7     | 5     6     2     |
+-------------------+-------------------+-------------------+
| 269   29    4     | 67    279   8     | 1     5     3     |
| 1     7     8     | 4     3     5     | 6     2     9     |
| 23569 2359  236   | 1     29    69    | 7     8     4     |
+-------------------+-------------------+-------------------+
There are a couple of UR's that I have not explored, but also two different four-link chains that eliminate <9> in R9C1 and <2> in R1C2. (Full disclosure: I do not wish to imply that I have seen these chains in the diagrams.)

(I have not yet solved the puzzle, but I do not see any conventional moves.)

Would your diagrams help to find these short chains?

Best wishes,

Keith


Last edited by keith on Mon Jul 07, 2008 10:49 pm; edited 2 times in total
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keith



Joined: 19 Sep 2005
Posts: 3179
Location: near Detroit, Michigan, USA

PostPosted: Sat Jul 05, 2008 3:22 am    Post subject: Reply with quote

This may not be a good example. Perhaps, I was trying to solve a very difficult situation, whereas Nataraj is trying to unify techniques. Anyway, here is (what I think is) the Nataraj diagram for this puzzle:



Any insights are greatly appreciated!

Thank you,

Keith
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Asellus



Joined: 05 Jun 2007
Posts: 865
Location: Sonoma County, CA, USA

PostPosted: Sat Jul 05, 2008 8:08 am    Post subject: Reply with quote

nataraj's diagrams are interesting. However, it seems to me that what this is doing is a sort of
"fractured" Medusa coloring and that straightforward Medusa coloring is easier.

I use dots within a cell, arranged in a 3x3 grid, to mark candidates. It takes some getting used
to. But, with practice I think that at least some folks can adapt to it. (I know others who do it
this way.) An advantage is that it makes Medusa color marking easier provided that you use a
sufficiently large grid. (Blank templates are, as always, useful.) I use upper and lower case
letters for the color pairs (avoiding letters such as Cc and Oo and Pp where upper and lower
case are too similar).

Below is an example of what I mean. I don't have a scanner so used Excel to create the
example. Instead of dots, I've used x's. But, with pencil, a simple dot suffices. It is easy to
change a dot into a letter by over-writing it.



I have marked conjugate pairs using Aa. (Ignore Bb for now.) It should be easy to see that the
<3> at r2c1, circled in red, is trapped by the 2A in the same cell and the 3a in r5c1.

Now, at this point, if we'd used that 39 UR in r23c18, we'd know that the <9> in r2c1 is also toast,
leaving only the 2A. So, all the "A" values are true (and all "a" values false), which I assume
solves the puzzle. However, we will ignore the UR.

With the <3> gone from r2c1, we change the <9> "dot" to "a" and trap the <9> in r7c1. In turn,
we change the <2> dot in r7c1 to "a" and trap the <2>s in r49c1.

You might think that this is the end of the Aa road. But, if you are able to "extend" things in your
head, there is critical progress to be made. But first, a comment about extended Medusa coloring:
I've been avoiding it lately unless I can do it in my head since I don't want to deal with
"sub-color" marks that later have to be erased. If you absolutely must, then use a separate grid
for those extended markings.

Now, for the next step: Note that all of the <9>s in b7 can "see" a 9a (recalling that we have
a 9a in r2c1 now) except for the <9> in r7c2. So, it must be an "extended" 9a. Due
to the conjugate <9>s in r1, we then have 9a in r1c6. But, this results in two 9a's in c6, which is a
color wrap! So, all of the "a"s are false and all of the "A"s are true.

I don't believe that this is so difficult to do with pencil and paper using such a grid, and that if you
do the marking, that final "extended" step isn't so hard to do without marking. (Just noticing it
is the hard part!)

Now, about the Bb stuff: I wanted to provide an example of how you accommodate multi-coloring
in a grid such as this. So, going back to the beginning, I have exploited the weak link between <2>
and <7> in r7c5 (or between the <9>s in r9c56, or between the <9>s in r39c5) to create an "ab"
weak link "bridge" between that Aa and Bb clusters. The strong pair is therefore AB. The <2> in
r9c1 is trapped in this way. (This approach clearly isn't necessary as this <2> was easily removed
by the Aa cluster only, as described above.)

Considering that a grid such as this puts everything into one place, with all of the relationships/links
relatively easy to see (with practice), and allows color marking for single digits, Medusa, and
multi-coloring forms of both, without requiring erasures (provided no "extensions" are marked),
I believe that this is a highly workable approach to the most difficult puzzles.
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nataraj



Joined: 03 Aug 2007
Posts: 1029
Location: Vienna, Austria

PostPosted: Sun Jul 06, 2008 12:23 pm    Post subject: Reply with quote

No patenting or copyrighting when it comes to pure science Smile but thank you for the encouragement, Keith !

I've tried the puzzle and this is what I came up with:
Code:


+--------------------------+--------------------------+--------------------------+
| 8       239     237      | 27      5       39       | 4       1       6        |
| 239     4       5        | 26      1       369      | 8       39      7        |
| 379     6       1        | 8       79      4        | 2       39      5        |
+--------------------------+--------------------------+--------------------------+
| 256     25      26       | 9       4       1        | 3       7       8        |
| 37      8       37       | 5       6       2        | 9       4       1        |
| 4       1       9        | 3       8       7        | 5       6       2        |
+--------------------------+--------------------------+--------------------------+
| 269     29      4        | 67      279     8        | 1       5       3        |
| 1       7       8        | 4       3       5        | 6       2       9        |
| 23569   2359    236      | 1       29      69       | 7       8       4        |
+--------------------------+--------------------------+--------------------------+



"2"                    "3"
++++    ++++
  o o7               * o    9     
     |               |      |     
*-----6             o |      *  9 
                     |         | 
                   *-------------9 
++++    +|+++
o 5 6                |            
                     |            
                   7---7          
                     |            
                     |            
++++    +|+++
o 9    *             |            
       |             |            
       |             |            
       |             |            
o o o  9           o * o          
++++    ++++

"6"                    "7"                    "9"
++++    ++++    ++++
                       *-2               *-------3     
                      /||\                           
     2---*           / |||            o        o  3 
     |   |         /  ||\                        | 
     |   |         *-------9           o      7    3 
++||++    +||+||++    ++++
*---2|   |         |   || |                          
    ||   |         |   || |                          
    ||   |         3---3| |                          
    ||   |              | |                          
    ||   |              | |                          
+|+||++    ++||++    ++++
*-----7   |              6-*           o 2    o       
    | \  |                                           
    |  \ |                                           
    |   \|                                           
o   *    9                            o o    2 6     
++++    ++++    ++++




Diagram for "2":

possible m-wings:
for 9 we have (2):r7c2-r7c5=r9c5 but no strong link (9) from there

Diagram for "3":

possible UR (39) r23c18. check, using strong link 3 in r3 => r2c1 <> 9
possible m-wings:
start with 9: (3):r2c8-r2c6=r1c6, yes there is a strong link in 9 to r1c2.
=> m-wing" (9) pincers r1c2 and r2c8 => r2c1 <> 9

another 9: (3):r2c8-r3c8=r3c1 , no strong link from there
and 9 again: (3):r1c6-r1c2=r9c2, no strong link from there

Diagram for "6":

possible m-wings:
2: r2c4-r7c4=r7c1 but no strong link from there
same with r2c4-r7c4=r9c6
9: r9c6-r7c4=r7c1, no strong link

Diagram for "7":

2: r1c4-r1c3=r3c1, no strong link
same with (9) r3c5-r3c1=r1c3, but
2: r1c4-r7c4=r7c5 has a strong link and there is an m-wing pincers (2) r9c5 and r1c4.
This m-wing is "useless" but r1c4 can be transported to r2c1 and now we get r9c1<>2
3: r5c3-r5c1=r3c1, strong link to r3c8 and we got another useless m-wing (3) pincers r5c3 and r3c8

Diagram for "9":
that 39 UR again, but not much else.

-----

So all in all, what we have is
one UR
one regular m-wing, that says r2c1<>9
one useless but transportable m-wing that says r9c1<>2

which is not what I expected (and definitely not what Keith and Asellus found).

But after these two eliminations I got a few xy- and xyz-wings and the puzzle was solved. Very Happy

So it seems that the method does help in spotting m-wings and stuff but does not go beyond, like into Medusa coloring, for example.
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Dorothyn



Joined: 16 Jul 2008
Posts: 1
Location: Boulder Colorado

PostPosted: Wed Jul 16, 2008 11:48 pm    Post subject: Reply with quote

My diagrams are not nearly so sophisticated, but what I find useful is to slip the puzzle into a sheet protector and then use dry erase markers. More than one color can be used at a time and if the diagram doesn't help it can be quickly erased and I can move one to another solving possibility.
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keith



Joined: 19 Sep 2005
Posts: 3179
Location: near Detroit, Michigan, USA

PostPosted: Sat Jul 19, 2008 7:41 pm    Post subject: Reply with quote

There is more from Nataraj on this idea:

http://www.dailysudoku.com/sudoku/forums/viewtopic.php?p=12413

Keith
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nataraj



Joined: 03 Aug 2007
Posts: 1029
Location: Vienna, Austria

PostPosted: Wed Oct 01, 2008 7:58 pm    Post subject: Reply with quote

QUARTERLY REPORT

on using "nataraj diagrams"


I've been using the diagrams for 3 months now. A good time to report on success/failures and other sundry experiences with the method.

A) The Beginning / Learning Curve

Moving from simple dot + line drawings like I did before (dots for the occurrences of a candidate, lines to show strong links), the problem I had in the first week was to learn to see simple x-wings, kites and skyscrapers, again. Introducing numbers (the bi-value cells) into the grid was so extremely distracting that it took a slow process of re-training my brain to easily recognize the familiar patterns again. Maybe it was not just the fact that there were now numbers in addition to dots/circles, but also that the information content in the drawings had increased dramatically and I tried to see everything at once.

B) The development of a routine

Things improved when I found another kind of routine:
- look for URs in the grid before looking at the drawings
- realize that xy(z) wings cannot be spotted in the diagrams
- look at the simple/multi- coloring chains first (Ignore the numbers)
- look for w-wings second (those can be found by looking at one single diagram)
- then look for "bi-value sees strong link" patterns and cross over to the other diagram to spot m-wings (if the other diagram does not exist, create it now)
- take "useless" xy-wings, w-wings and m-wings and draw them as additional (sometimes oblique/cross house) strong links. Look for "coloring" eliminations including the new links

C) usefulness / side effects

The method proved extremely successful with slightly advanced puzzles (beyond vh-level but not requiring Medusa, ALS/APE or more exotic techniques).

Reliable detection of w-wing and m-wing, the latter even in generalized patterns. Before, I never used w-wing or m-wing except in very rare cases. I never found the buggers. Detection of wing-plus-transport or wing-plus-"pincer coloring" solutions. So much easier if presented graphically ...

ER, swordfish, and finned creatures are just as easy or hard to find as before - no change.

Because these diagrams tend to help to identify so many different patterns, I draw them at an earlier stage than I used to before. This adds considerable "fixed cost" to the process, especially if one does the drawings in sequence (1,2,3...) and the breakthrough "move" is a w-wing on (9) ... or, worse yet, a simple xy-wing. Therefore I mix the techniques: do a few of the diagrams, search one "floor" for xy-wings, do another few diagrams, search another floor ....

Summary

Overall, using the "nataraj diagrams" has definitely improved the "sudoku experience" for me. And I had lots of fun developping a new method Smile.

Thanks to all for your comments and especially thanks to Keith for your encouragement and support!
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