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How to solve Sudoku

 
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keith



Joined: 19 Sep 2005
Posts: 3333
Location: near Detroit, Michigan, USA

PostPosted: Mon Feb 13, 2017 7:34 pm    Post subject: How to solve Sudoku Reply with quote

This thread is about systematic and effective ways to solve Sudoku.

A beginner friend asked me for advice, and I realized that most tutorials on how to solve puzzles are structured around applying solution tricks sequentially, in the order of their perceived difficulty. IMO, that does not make an efficient or effective way to approach most puzzles.

I propose to make three posts:

1. The basics. This will enable you to solve virtually every newspaper puzzle published in the USA.

2. Extended single-digit techniques.

3. Two-digit techniques.

At the end of this, you should be able to solve any puzzle that would be of interest to an armchair, pencil and paper, human solver. That would be me and perhaps two of my friends.

What I will say is a little unusual, for it does not follow the established path. It does not help that many published puzzles are screened, based on the techniques required to solve them.

Keith
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keith



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Location: near Detroit, Michigan, USA

PostPosted: Mon Feb 13, 2017 7:35 pm    Post subject: Solving Sudoku: The AP Class Reply with quote

Based on feedback, I intend to revise this part - Keith May 17, 2018

Part 1: Solving Sudoku: The AP Class

A friend asked for advice on solving newspaper Sudoku puzzles. I looked around, and discovered that most “How to solve” tutorials are built around a hierarchy of solution tactics. For example,
http://angusj.com/sudoku/hints.php

That’s not how I solve puzzles, so I decided to write my methodology down. As an example, we have the Sunday puzzle from the Arizona Republican. (For no reason other than that is where my friend is, and it is the newspaper he has in hand.)

This post covers “the basics”, which is a set of single-digit solving techniques that will enable you to solve virtually any newspaper puzzle published in the USA, or nearly every puzzle in the books they sell at your local dollar store. (These books are a great source of puzzles for beginners.)
Sudoku AP: Arizona Republican 12 Feb 2017 *****
Code:

+-------+-------+-------+
| . . . | . 7 . | . . 8 |
| . 1 . | . . 9 | . 4 6 |
| 9 . . | . 8 . | 5 7 . |
+-------+-------+-------+
| . . . | 7 . . | . . . |
| 2 . 8 | . 4 . | 7 . 1 |
| . . . | . . 3 | . . . |
+-------+-------+-------+
| . 8 4 | . 3 . | . . 5 |
| 5 6 . | 8 . . | . 3 . |
| 7 . . | . 1 . | . . . |
+-------+-------+-------+

Play this puzzle online at the Daily Sudoku site

Or, paste the following in your browser:

http://sudoku.saueregger.at/HSH/HSH.htm?p=*EN*HSF*6*2/13/2017*0:0:0:0:7:0:0:0:8:0:1:0:0:0:9:0:4:6:9:0:0:0:8:0:5:7:0:0:0:0:7:0:0:0:0:0:2:0:8:0:4:0:7:0:1:0:0:0:0:0:3:0:0:0:0:8:4:0:3:0:0:0:5:5:6:0:8:0:0:0:3:0:7:0:0:0:1:0:0:0:0:*undefined

Here is the puzzle with all the possibilities for the unsolved cells. Ignore the possibilities (pencil marks) for now.

Code:
+----------------------+----------------------+----------------------+
| 346    2345   2356   | 123456 7      12456  | 1239   129    8      |
| 38     1      2357   | 235    25     9      | 23     4      6      |
| 9      234    236    | 12346  8      1246   | 5      7      23     |
+----------------------+----------------------+----------------------+
| 1346   3459   13569  | 7      2569   12568  | 234689 25689  2349   |
| 2      359    8      | 569    4      56     | 7      569    1      |
| 146    4579   15679  | 12569  2569   3      | 24689  25689  249    |
+----------------------+----------------------+----------------------+
| 1      8      4      | 269    3      267    | 1269   1269   5      |
| 5      6      129    | 8      29     247    | 1249   3      2479   |
| 7      239    239    | 24569  1      2456   | 24689  2689   249    |
+----------------------+----------------------+----------------------+

Here is some terminology:
Rows: 1 thru 9, top to bottom.
Columns: 1 thru 9, left to right.
Together, rows and columns are called “Lines”.
Boxes: 1 to 9, as in
Code:
123
456
789


So, for example, B1 contains the cells common in R123C123.

What I am going to describe is a very effective recipe I have developed over the past 15 years or so.

In the margin of the puzzle, write this down:

Code:
123
456
789


This is your list of "solved" digits.

Step 1: Sweep the floors and the chimneys

Note that 1 is the only possibility in R7C1. Ignore that, it will come out in the wash. It is a "forced" cell.

The first step is to "sweep" the floors. Look at R123 which are B123. Which candidates occur in two of the boxes? Where might they be in the third?

There is a 7 in R1B2 and in R3B3. So, where is the 7 in B1. Clearly it is R2C3, because there is already a 7 in C1. Similarly, there is an 8 in B2 and B3. The 8 in B1 must be R2C1.

Code:
+----------------------+----------------------+----------------------+
| 346    2345   2356   | 123456 7      12456  | 1239   129    8      |
| 8      1      7      | 235    25     9      | 23     4      6      |
| 9      234    236    | 12346  8      1246   | 5      7      23     |
+----------------------+----------------------+----------------------+
| 1346   3459   13569  | 7      2569   12568  | 234689 25689  2349   |
| 2      359    8      | 569    4      56     | 7      569    1      |
| 146    4579   1569   | 12569  2569   3      | 24689  25689  249    |
+----------------------+----------------------+----------------------+
| 1      8      4      | 269    3      267    | 1269   1269   5      |
| 5      6      129    | 8      29     247    | 1249   3      2479   |
| 7      239    239    | 24569  1      2456   | 24689  2689   249    |
+----------------------+----------------------+----------------------+


Now you can sweep floor 2. Because of the 7 you solved in B1. R6C2 must be 7.

Proceed along to sweep the "chimneys" and you will see that R8C9 must be 7, as must R7C6.

So, now all the 7s are solved, and you can black out 7 in your "solved" diagram.

Code:
+----------------------+----------------------+----------------------+
| 346    2345   2356   | 123456 7      12456  | 1239   129    8      |
| 8      1      7      | 235    25     9      | 23     4      6      |
| 9      234    236    | 12346  8      1246   | 5      7      23     |
+----------------------+----------------------+----------------------+
| 1346   3459   13569  | 7      2569   12568  | 234689 25689  2349   |
| 2      359    8      | 569    4      56     | 7      569    1      |
| 146    7      1569   | 12569  2569   3      | 24689  25689  249    |
+----------------------+----------------------+----------------------+
| 1      8      4      | 269    3      7      | 1269   1269   5      |
| 5      6      129    | 8      29     24     | 1249   3      7      |
| 7      239    239    | 24569  1      2456   | 24689  2689   249    |
+----------------------+----------------------+----------------------+

Step 2: Scan the lines

The next step is to look at each row and then each column. What are the possible values in the unsolved cells?

If N cells are solved, the remaining cells comprise 9-N candidates. So, R2 has 146789 solved, the remaining cells are 235. Look for solved cells in the relevant column and box. Not much luck in R2.

When I do this it is usually only for cells with four or more solved candidates. With practice, it is easy to carry the unsolved digits in your head. At this stage, there is no need to use pencil marks.

In R5 we see the solved cells are 12478 and the unsolved candidates are 3569. Looking at each of these candidates in the row in the unsolved cells, the only place for 3 is R5C2.

Then you find 3 in the row R9C3 and in the column R1C1. And, 8 in the column R4C6.

Code:
+-------------------+-------------------+-------------------+
| 3     245   256   | 12456 7     12456 | 129   129   8     |
| 8     1     7     | 235   25    9     | 23    4     6     |
| 9     24    26    | 12346 8     1246  | 5     7     23    |
+-------------------+-------------------+-------------------+
| 146   459   1569  | 7     2569  8     | 23469 2569  2349  |
| 2     3     8     | 569   4     56    | 7     569   1     |
| 146   7     1569  | 12569 2569  3     | 24689 25689 249   |
+-------------------+-------------------+-------------------+
| 1     8     4     | 269   3     7     | 1269  1269  5     |
| 5     6     129   | 8     29    24    | 1249  3     7     |
| 7     29    3     | 24569 1     2456  | 24689 2689  249   |
+-------------------+-------------------+-------------------+


Note that we have not yet found the "gimmee", 1 in R7C1. My point is that examining each cell for all 9 candidates is an inefficient way to approach this.

Step 3: Examine the boxes

So now we are to phase 3, examining the boxes. It's the same logic as for the rows and columns (lines), except we are looking at the unsolved cells in each box. This is where I might start to add pencil marks for unsolved cells.

Look at B5. R6C4 must be 1. Then, R3C6 must be 1. And in B7, R7C2 is not 29, it must be 1. (Yay!)

This opens up other cells that can be solved by the methods already described. After this, you can black out 1 on the list of solved digits. So, we are here:

Code:
+-------------------+-------------------+-------------------+
| 3     245   256   | 2456  7     256   | 9     1     8     |
| 8     1     7     | 235   25    9     | 23    4     6     |
| 9     24    26    | 2346  8     1     | 5     7     23    |
+-------------------+-------------------+-------------------+
| 46    459   1     | 7     2569  8     | 2346  2569  2349  |
| 2     3     8     | 569   4     56    | 7     569   1     |
| 46    7     569   | 1     2569  3     | 2468  25689 249   |
+-------------------+-------------------+-------------------+
| 1     8     4     | 269   3     7     | 26    269   5     |
| 5     6     29    | 8     29    4     | 1     3     7     |
| 7     29    3     | 2569  1     256   | 2468  2689  249   |
+-------------------+-------------------+-------------------+


At this point, I will not usually have put any pencil marks in the puzzle. If I have, it is only for cells that have only two candidates.

Step 4: Look for naked or hidden subsets

The next stage is to look for subsets, or pairs and triples, etc.

Look at C1. The only unknowns are 46, both in B4. Therefore, the other unsolved cells in B4 cannot contain 46. Not much help here, with this puzzle.

Code:
+-------------------+-------------------+-------------------+
| 3     245   256   | 2456  7     256   | 9     1     8     |
| 8     1     7     | 235   25    9     | 23    4     6     |
| 9     24    26    | 2346  8     1     | 5     7     23    |
+-------------------+-------------------+-------------------+
| 46    59    1     | 7     2569  8     | 2346  2569  2349  |
| 2     3     8     | 569   4     56    | 7     569   1     |
| 46    7     59    | 1     2569  3     | 2468  25689 249   |
+-------------------+-------------------+-------------------+
| 1     8     4     | 269   3     7     | 26    269   5     |
| 5     6     29    | 8     29    4     | 1     3     7     |
| 7     29    3     | 2569  1     256   | 2468  2689  249   |
+-------------------+-------------------+-------------------+


Step 5: Box-line intersections

The final basic strategy is called "box-line intersections".

Look at B1. The candidate 5 only occurs in R1, so 5 can be eliminated as a candidate in R1C46.

Look at B5. The candidate 2 only appears in C5, so 2 can be eliminated in R18C5.

Code:
+-------------------+-------------------+-------------------+
| 3     245   256   | 246   7     26    | 9     1     8     |
| 8     1     7     | 235   5     9     | 23    4     6     |
| 9     24    26    | 2346  8     1     | 5     7     23    |
+-------------------+-------------------+-------------------+
| 46    59    1     | 7     2569  8     | 2346  2569  2349  |
| 2     3     8     | 569   4     56    | 7     569   1     |
| 46    7     59    | 1     2569  3     | 2468  25689 249   |
+-------------------+-------------------+-------------------+
| 1     8     4     | 269   3     7     | 26    269   5     |
| 5     6     29    | 8     9     4     | 1     3     7     |
| 7     29    3     | 2569  1     256   | 2468  2689  249   |
+-------------------+-------------------+-------------------+


The puzzle is then solved by going back to Step 2 etc., the basics of looking at candidates in each row, column, and box. As the experts will say, STTE, singles to the end. (The experts are wrong in their terminolology.)

These techniques will solve virtually all the newspaper Sudoku puzzles published in the USA, except the Detroit Free Press and LA Times Friday and possibly their Thursday and Sunday puzzles. (The Freep and the LATimes publish the same puzzle.)

The technique I start with, sweeping the floors and then sweeping the chimneys, is very powerful. I find it avoids much drudgery in the initial stage of solving puzzles.

If your goal is to conquer the USA Today or Arizona Republic types of puzzles, this is all you need to know.

I plan three more posts: One on more advanced single-digit techniques, one on two-digit techniques, and one to sum it up and point to more online resources.

Best wishes,

Keith


Last edited by keith on Thu May 17, 2018 4:21 pm; edited 10 times in total
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keith



Joined: 19 Sep 2005
Posts: 3333
Location: near Detroit, Michigan, USA

PostPosted: Mon Feb 13, 2017 7:35 pm    Post subject: Post 3 Reply with quote

Content replaced on 13 May 2018, 20:00 Eastern

Part 2: Advanced Single-Digit Solving Techniques

With the “basic” methods of the previous post, you will be able to solve the vast majority of newspaper puzzles printed in the United States. But, you may wish to progress to doing harder puzzles. If so, this is for you.

Please note, these are techniques for human solvers using pencil and paper. If you are a computer programmer, there is likely to be little for you here.

You might do well to review Havard’s classic “Strong Links for Beginners”
http://forum.enjoysudoku.com/strong-links-for-beginners-t3326.html

The Single-Digit Strong Link

This is the most useful concept you will ever learn for solving Sudoku: The Single-Digit Strong Link (SDSL).

An SDSL is when a candidate digit occurs only twice in a house. (A house is a line or a box. A line is a row or a column.) The logic is, one of the occurrences is false, the other is true.

That’s it. All you need to know. Pretty simple concept, no?

Let’s look at a few examples:

In the following,
A-a, B-b, C-c are SDSLs.
X is a cell where the candidate does occur.
/ is a cell where the candidate does not occur.
. is a cell that does not matter (some may need to be X)
* is a target cell where the candidate may be eliminated.

Two SDSLs:
Code:
+---------+---------+---------+
| *  *  / | .  .  . | .  Xb . |
| .  .  / | .  .  . | .  /  . |
| .  .  Xa| .  .  . | *  /  * |
+---------+---------+---------+
| .  .  / | .  .  . | .  /  . |
| .  .  / | .  .  . | .  /  . |
| .  .  / | .  .  . | .  /  . |
+---------+---------+---------+
| .  .  XA| .  .  . | .  XB . |
| .  .  / | .  .  . | .  /  . |
| .  .  / | .  .  . | .  /  . |
+---------+---------+---------+

Figure 1:  A Skyscraper

Now, the logic is: One or both of A and B are false, since they are in the same row. Therefore, one or both of a and b are true.

If this is new to you, think about it carefully. Note we are not claiming one of A and B is true: A-B is not necessarily an SDSL, though it may be.

This pattern is usually called a Skyscraper. It can be tilted on its side, so the SDSLs are in the rows.

Here is another:
Code:
+---------+---------+---------+
| .  .  Xb| .  .  . | *  .  . |
| .  .  / | .  .  . | .  .  . |
| .  .  / | .  .  . | .  .  . |
+---------+---------+---------+
| .  .  / | .  .  . | .  .  . |
| XA /  / | /  /  / | Xa /  / |
| .  .  XB| .  .  . | .  .  . |
+---------+---------+---------+
| .  .  / | .  .  . | .  .  . |
| .  .  / | .  .  . | .  .  . |
| .  .  / | .  .  . | .  .  . |
+---------+---------+---------+

Figure 2:  A Kite

The same logic applies. This pattern is called a Kite. Together, Skyscrapers and Kites are Turbot Fish.

There is one more 2 SDSL pattern, the X-wing:
Code:
+---------+---------+---------+
| .  .  / | .  .  . | .  /  . |
| .  .  / | .  .  . | .  /  . |
| *  *  Xa| *  *  * | *  Xb * |
+---------+---------+---------+
| .  .  / | .  .  . | .  /  . |
| .  .  / | .  .  . | .  /  . |
| .  .  / | .  .  . | .  /  . |
+---------+---------+---------+
| *  *  XA| *  *  * | *  XB * |
| .  .  / | .  .  . | .  /  . |
| .  .  / | .  .  . | .  /  . |
+---------+---------+---------+

Figure 3:  An X-wing

I leave it to you to consider the logic that leads to all the X-wing eliminations.

Now, just as a teaser, what might you do with three SDSLs?
Code:
+---------+---------+---------+
| .  /  . | .  .  . | /  .  . |
| .  Xa . | .  *  . | /  .  . |
| .  /  . | .  .  . | /  .  . |
+---------+---------+---------+
| .  /  . | .  .  . | Xb .  . |
| .  /  . | .  .  . | /  .  . |
| /  /  / | /  Xc / | /  /  XC|
+---------+---------+---------+
| .  /  . | .  .  . | /  .  . |
| .  /  . | .  .  . | /  .  . |
| .  XA . | .  .  . | XB .  . |
+---------+---------+---------+

Figure 4:  A Three-SDSL Creature

What is this? I don’t know, I just made it up. But, the logic is pretty simple: One or both of A and B are false; one or both of a and b are true. If b is true, C is false, c is true. a and c are pincers.

So, the method is this: Identify two SDSLs that have their base in the same house (here, A,B). Then the other ends of the SDSLs are pincers (here, a, b).

You never need to visit the aquarium again! You can forget about Turbot Fish, Swordfish, etc., and all their mutant, kraken and franken variants. Also, coloring, multi-coloring, hinges, and empty rectangles. (If anyone has a counter-example for this assertion, I’d love to see it.)

How to Find Them

How do you know where to look? Do you have to find all the SDSLs for all the digits? The answer to that is “No!”

In a previous post I (unfortunately) defined a Type C box. That is a box where a candidate occurs both in at least one row and at least one column, like this:
Code:
+---------+
| X  .  . |
| X  .  .X|
| .  .  . |
+---------+

Figure 5: A Type C Box

For an SDSL to be useful in making an elimination, it must start and end in a Type C box. Further, for there to be an elimination, there must be at least four Type C boxes arranged in a rectangle.

The original post is here: http://forum.enjoysudoku.com/finding-the-possibility-of-single-candidate-eliminations-t32748.html
In the following diagrams,
O is a box that does not contain the candidate digit.
C is a Type C box that contains the candidate.
Code:
COC
OOO
COC

Figure 6: A box configuration for SDSL eliminations

Code:
COC
OCC
CCO

Figure 7: Another box configuration for SDSL eliminations

To make SDSL eliminations there must be four Type C boxes in a rectangle (Figure 6) or six Type C boxes in a swordfish type pattern (Figure 7). This is not a universal truth, note Figure 4.

In practice, this is a very easy evaluation. Look at the boxes where a digit has not been solved. Are they Type C? Are there four or more? If so, this is a place to pay attention. In any puzzle there will usually be only two or three digits that merit closer scrutiny.

Examples

Example 1

This example is good to illustrate the process. I found it in a booklet of puzzles from a dollar store.
Code:
Puzzle: SDSL Process
+-------+-------+-------+
| 1 . . | . . 2 | . . . |
| 7 6 . | . . . | 8 2 . |
| 3 . . | . 7 8 | 1 . . |
+-------+-------+-------+
| . . . | 2 . . | . 7 4 |
| . . . | . 1 . | . . . |
| 4 9 . | . . 5 | . . . |
+-------+-------+-------+
| . . 3 | 5 6 . | . . 8 |
| . 1 8 | . . . | . 4 3 |
| . . . | 8 . . | . . 1 |
+-------+-------+-------+
Figure 8:  Example 1

After basics:
Code:
 

+----------------+----------------+----------------+
| 1    8    9    | 6    5    2    | 4    3    7    |
| 7    6    45   | 1    34   349  | 8    2    59   |
| 3    25   245  | 49   7    8    | 1    56   569  |
+----------------+----------------+----------------+
| 8    35   1    | 2    9    6    | 35   7    4    |
| 56   2357 2567 | 347  1    47   | 9    8    256  |
| 4    9    267  | 37   8    5    | 36   1    26   |
+----------------+----------------+----------------+
| 2    4    3    | 5    6    1    | 7    9    8    |
| 56   1    8    | 79   2    79   | 56   4    3    |
| 9    57   567  | 8    34   34   | 2    56   1    |
+----------------+----------------+----------------+

Figure 9: Example 1, after basics


Now, the job is to identify digits which might yield SDSL simplifications. You can draw diagrams like Figure 6 above for each digit, but I suggest the following: Make a simple tic-tac-toe grid as in the figure below, and list in each box the digits for which it is Type C.

Code:
     |     |     
     |     |     
     |     |     
-----+-----+-----
     |     |     
     |     |     
     |     |     
-----+-----+-----
     |     |     
     |     |     
     |     |     

Figure 10: A stencil to log Type C boxes

Then we have the following:

Code:

5    |4 9  |5     
     |     |     
     |     |     
-----+-----+-----
5 6 7|7    |5 6   
     |     |     
     |     |     
-----+-----+-----
5 6  |     |5 6   
     |     |     
     |     |     

Figure 11: Digits with Type C boxes

How you do this is not important. With a little practice, you will be able to just look at the grid and not need ancillary diagrams.

Clearly, only 5 and 6 are worth looking at. Here are the unsolved possibilities for 5:

Code:
      |     |       
      5|     |     5
   5  5|     |   5 5
-------+---- -+------
   5*  |     |5b   
5a 5  5|     |     5*
       |     |     
-------+-----+-------
       |     |       
5A     |     |5B     
   5  5|     |   5   

Figure 12: Example 1, unsolved possibilities for 5

There are SDSLs in C17, making the eliminations shown, which solves the puzzle.

The story for digit 6 is very similar; I leave it as an exercise for the reader.

Example 2

This example is partly solved: I do not know what the original puzzle is.
But, it enables some good points.

http://forum.enjoysudoku.com/simple-colouring-t34415.html#p263218
Code:
+-------------+-------------+-------------+
| 459 589 1   | 3   78  6   | 578 479 2   |
| 345 58  7   | 1   2   9   | 6   34  358 |
| 369 2   36  | 47  5   48  | 378 1   389 |
+-------------+-------------+-------------+
| 2   4   39  | 5   89  1   | 38  6   7   |
| 36  1   5   | 47  678 48  | 9   2   38  |
| 8   7   69  | 2   69  3   | 4   5   1   |
+-------------+-------------+-------------+
| 57  3   4   | 9   1   2   | 57  8   6   |
| 1   59  8   | 6   4   7   | 2   39  359 |
| 79  6   2   | 8   3   5   | 1   79  4   |
+-------------+-------------+-------------+

Figure 13: Example 2

Basics are done, so let’s look at the unsolved digits.

1 is solved
2 is solved
3 has four Type C boxes in a rectangle, boxes 1, 3, 4, 6. There are SDSLs in C37 (an X-wing), eliminating 3 in R3C19.
4 does not have any Type 4 boxes. Nothing to find there.
5 has four Type C boxes in a rectangle, boxes 1, 3, 7, 9. There are lots of options here.
Code:
+---------+---------+---------+
| X  X  . | .  .  . | X  .  . |
| X  X  . | .  .  . | .  .  X |
| .  .  . | .  .  . | .  .  . |
+---------+---------+---------+
| .  .  . | .  .  . | .  .  . |
| .  .  . | .  .  . | .  .  . |
| .  .  . | .  .  . | .  .  . |
+---------+---------+---------+
| X  .  . | .  .  . | X  .  . |
| .  X  . | .  .  . | .  .  X |
| .  .  . | .  .  . | .  .  . |
+---------+---------+---------+

Figure 14: The digit 5 in Example 2

The SDSLs in R7 and C9 are a Kite grounded in B8; they take out 5 in R2C1.
But notice also that C2 and C9 are links grounded in R8. The logic is, one of R12C2 is 5 and/or R2C9 is 5. C2 is a grouped link. Since R2C1 sees all three cells, 5 can be eliminated there.

There is another grouped link in C1 to go with the SDSL in C7, eliminating 5 in R1C2.
With the elimination of these two candidates 5, 5 has only two candidates in each unsolved house. No more single-digit eliminations are possible. Move on.
6 has no Type C boxes.
7 has 3 Type C boxes, not enough. Move on.
8 has SDSLs in R4 and C6, eliminating 8 in R3C7. Nothing more that I can see.
9 has SDSLs in R9 and C9, taking out 9 in R3C1 and solving the puzzle.

Example 3

Take a look at the original post on how to look for single-digit eliminations:
http://forum.enjoysudoku.com/finding-the-possibility-of-single-candidate-eliminations-t32748.html

Example 4

This discussion on DailySudoku shows how thinking in terms of SDSLs reduces the clutter caused by named patterns.
http://www.dailysudoku.com/sudoku/forums/viewtopic.php?t=8431&sid=ed82068886fe3cd318fd10fb2cf1a7a5

Closure

In this post I have described the idea of Single-Digit Strong Links. This enables one to abandon a litany of named patterns (the "Aquarium") and to use a single logical framework to solve such patterns.

Also, I have given simple rules to identify which digits might (or which digits cannot) yield SDSL reductions. In most puzzles, this focuses attention on at most three digits, saving much time in searching for possibilities.

Keith
May, 2018


Last edited by keith on Tue May 15, 2018 5:11 pm; edited 5 times in total
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PostPosted: Mon Feb 13, 2017 7:36 pm    Post subject: Post 3 Reply with quote

Updated 14 May 2018 07:30 Eastern

Part 3: Other Advanced Solving Techniques

As I have mentioned, Sudoku puzzles that are published are selected. Authors generate thousands of puzzles, and then subject them to a hierarchy of solving methods. Only a handful of puzzles make it through. Their difficulty is graded by the solution methods needed to solve them.

The single-digit techniques I have described above will get you a long way, but it is unlikely they will solve many “Very Hard”, “Advanced”, or “Fiendish” puzzles you encounter. For those you need to learn a handful of techniques that are not single-digit. I list some of those techniques below, with my comments.

This is not a tutorial on the methods. There are plenty of resources available. Just search on
Sudoku {technique name}
And you will find plenty of information.

1: XY-wing

The XY-wing is probably the most common “advanced” move. Many puzzles can be solved by basics plus an XY-wing. I find them tiresome.

2: XYZ-wing

Not very common, but if you’re going to learn XY-, you may as well learn XYZ-wings.

3: Unique Rectangle

This is a technique based on an assumption the puzzle has a single unique solution. This was very controversial when first proposed, but it is now agreed that a “valid” Sudoku has but a single solution.

4: BUG+n

A BUG is a non-unique pattern where every unsolved cell has only two candidates. In a BUG+n (n is usually 1), n is the number of cells that have more than two candidates. The logic to defeat a BUG can be quite entertaining.

5: Single-Digit Techniques

These are discussed above in Part 2. An X-wing is a popular “difficult” move in newspapers in the UK. The Friday puzzle in the Detroit Free Press and the LA Times (they are the same) can often be solved with a Skyscraper or Kite.

6: W-wing

The W-wing was observed by George Woods and documented by me.
http://www.dailysudoku.com/sudoku/forums/viewtopic.php?t=2008

7: M-wing

The M-wing was proposed by me as the simplest case of Medusa coloring.
http://www.dailysudoku.co.uk/sudoku/forums/viewtopic.php?p=9090&sid=9422d0a4ddfe3baa4565811a1706cd23#9090

Others, particularly Helmut Saueregger (Nataraj) and Asellus have generalized and greatly expanded the idea.
http://www.dailysudoku.co.uk/sudoku/forums/viewtopic.php?t=2908
Re’born pointed out that M-wings often are cycles, opening additional eliminations.

The idea for W- and M-wings comes from a pair: Suppose you have two cells in a grid which each have possibilities XY, but do not “see” each other. How might they be connected? Here is a discussion:

Remote Pairs http://www.dailysudoku.co.uk/sudoku/forums/viewtopic.php?p=8277#8277

7: Extended wings or pincers

This is a very useful technique, used only by a few people. Suppose you have a “flightless” or “useless” XY-wing: XZ – XY – YZ. XZ and YZ are pincers on Z, but there are no eliminations to be made, or you have already made the eliminations. If you can find an SDSL that lines up with either XZ or YZ, you can extend the pincers to a new cell, perhaps to make additional eliminations.

http://www.dailysudoku.com/sudoku/forums/viewtopic.php?p=41349&sid=411f678295fb840f21e8b93e4e642024#41349

Closure

In my opinion, these are the non-single-digit advanced methods that are most useful. You can use this as a checklist of methods to learn.


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PostPosted: Mon Feb 13, 2017 7:37 pm    Post subject: Post 4 Reply with quote

Part 4: Discussion and Resources

While “How to Solve Sudoku” may seem presumptuous, I have summarized the methods I have adopted to solve Sudoku over the past 14 years or so. I solve puzzles in an armchair with pencil and paper. The only time I use software is to check my work and to format puzzles for posting.

There are only a couple of Sudoku discussion forums remaining: The New Sudoky Players’ Forum and the DailySudoku Forum. Both of these are lightly used, compared to a few years ago. But, they both remain good places to get help and advice.

New Sudoku Players’ Forum

http://forum.enjoysudoku.com/

This forum includes some vey knowledgeable people. It is not geared towards pencil and paper human solvers; sometimes it seems to be a contest between computer programmers.

By all means, post questions there, but be prepared for the occasional snippy response.

DailySudoku Forum

This site has a forum where, history shows, if you post a query you will always get a courteous response. If you wish to learn, there is always a discussion thread for the site’s “very hard” puzzles.

The site publishes one puzzle a day. The easy, medium, and hard puzzles are solvable by basics. The very hard puzzles may require X-wings, XY-wings, or XYZ-wings. The discussion thread often includes other methods to solve these puzzles, such as W-wings and skyscrapers.

Sources for Puzzles

I get most of my puzzles from the local newspaper (Detroit Free Press, only the Thu, Fri, and Sun puzzles are of interest to me). The same puzzles appear in the LA Times. Also, we buy booklets of puzzles at the local dollar store. My wife starts at Level 3, and works puzzles until she runs out of steam. So, I usually get the last half-dozen Level 4 puzzles. The remaining easier Level 1 and Level 2 puzzles go to the “free” table at our recreation center, where they are taken in no time at all.

So far as online sources are concerned, I use only two:

Helmut’s Sudok-o-rama

http://www.saueregger.at/sudoku/

Helmut Saueregger (web name Nataraj) publishes five puzzles a day: Easy, Mild, Hard, (Very hard or Advanced), and (X-Sudoky, Squiggly, or Compass). The puzzles themselves are very nice.

The site also includes HSH, Helmut’s Sudoku Helper. This is a very interesting tool. At any point in a puzzle, it will tell you the moves available. You choose one, and it will take that next step. Helmut’s Sudoku Helper is highly recommended. It includes diagrams that show SDSLs, and also information that helps to construct multi-digit chains.

HSH (and, indeed, the whole site) is a little idiosyncratic, but with a bit of practice, you’ll get it quite quickly.

Menneske

http://www.menneske.no/sudoku/eng/

Menneske is a curious site. It is a database of millions of puzzles, sorted by type and graded by difficulty. The difficulty rating is not based on the techniques needed to solve the puzzle. I have my suspicions, but I don’t really knowwhat the grading is based on.

I have had very good luck with selecting Very Hard (0) puzzles, which turn out to be challenging enough, but you have no idea in advance which techniques might apply. Great fun.

Closing Remarks

So, there you have it. This is most of what I know about Sudoku, and my recommendations on how to solve the puzzles.

As a final comment, I think Sudoku is sort of like yoga. Hopefully you have a good instructor and resources, but the choice of level is up to you. Find the place where you get most personal benefit and most enjoyment.

Keith
May, 2018


Last edited by keith on Mon May 14, 2018 5:55 pm; edited 1 time in total
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PostPosted: Mon May 14, 2018 9:49 pm    Post subject: Reply with quote

Updated: How to solve Sudoku

I have added the content I originally intended. You need to read the thread from the beginning.

Comments and criticisms will be gratefully received.

Best wishes,

Keith
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arupbiswas



Joined: 16 May 2018
Posts: 1

PostPosted: Wed May 16, 2018 2:12 am    Post subject: Reply with quote

Always play online sudoku (www.sudokuiq.com) to help your patience and skills improvement
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Olinelooo



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PostPosted: Fri Jul 06, 2018 12:15 pm    Post subject: Reply with quote

One of the greatest aspects of Sudoku is that the game offers engaging challenges to both the novice, as well as the seasoned puzzle player. Whenever they play a puzzle tailored for their level of competence, both the beginner and the experienced Sudoku solver will have to put a good amount of thought and technique into completing the task. Their approach, though, may not be the same. Solving a hard Sudoku puzzle will require quite a different set of techniques compared to an easy one. This article presents nine such techniques; in increasing difficulty.

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