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How to solve Sudoku

 
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keith



Joined: 19 Sep 2005
Posts: 3288
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



Joined: 19 Sep 2005
Posts: 3288
Location: near Detroit, Michigan, USA

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

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 Tue Feb 14, 2017 9:26 am; edited 8 times in total
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keith



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

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PostPosted: Mon Feb 13, 2017 7:36 pm    Post subject: Post 3 Reply with quote

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Dennis



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PostPosted: Thu Apr 13, 2017 1:58 pm    Post subject: Re: How to solve Sudoku Reply with quote

keith wrote:
This thread is about systematic and effective ways to solve Sudoku.
Next post?
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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View user's profile Send private message
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