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[Sudoku123456789]

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[JC Van Hay]

Code: Select all

+--------------+------------------+----------------+
| 69    28  69 | 28   7     1     | 3   4     5    |
| 28    5   13 | 238  4     9     | 7   1(2)  6    |
| 4     7   13 | 236  56-2  56(2) | 8   9     1(2) |
+--------------+------------------+----------------+
| 2689  28  5  | 1    3     6(2)  | 4   7     89   |
| 7     3   4  | 5    9     8     | 12  6     12   |
| 2689  1   69 | 7    6-2   4     | 5   3     89   |
+--------------+------------------+----------------+
| 1     9   7  | 26   58    3     | 26  58    4    |
| 3     4   2  | 9    1568  56    | 16  58    7    |
| 5     6   8  | 4    1(2)  7     | 9   1(2)  3    |
+--------------+------------------+----------------+

Your analysis of the 2s is incomplete : you excluded 2r3c5, but there is only 1 solution for the 2s!
A simple interpretation : the solutions of {2R9, 2B3, 2C6} exclude 2r36c5; stte

[Leren]

The puzzle solved cell status in line format : ....71345.5..497.647....89...513.47.734598.6..1.7.453.197..3..43429....75684.79.3

The simplest solution from your position is to use a couple of Skyscrapers, both on digit 2.

Code: Select all

*----------------------------------------*
| 69   a28 69 | 8-2 7     1   | 3  4  5  |
| 28    5  13 | 238 4     9   | 7  12 6  |
| 4     7  13 | 236 256  d256 | 8  9  12 |
|-------------+---------------+----------|
| 2689 b28 5  | 1   3    c26  | 4  7  89 |
| 7     3  4  | 5   9     8   | 12 6  12 |
| 2689  1  69 | 7   26    4   | 5  3  89 |
|-------------+---------------+----------|
| 1     9  7  | 26  58    3   | 26 58 4  |
| 3     4  2  | 9   1568  56  | 16 58 7  |
| 5     6  8  | 4   12    7   | 9  12 3  |
*----------------------------------------*

At your position there is a Skyscraper in cells a-b-c-d which eliminates 2 in r2c4 and solves 6 cells with follow on singles.

This brings you to here :

Code: Select all

*--------------------------------------*
| 69  2 69 | 8   7    1   | 3   4   5  |
| 8   5 13 |b23  4    9   | 7  a12  6  |
| 4   7 13 | 236 256  256 | 8   9   12 |
|----------+--------------+------------|
| 26  8 5  | 1   3    26  | 4   7   9  |
| 7   3 4  | 5   9    8   | 12  6   12 |
| 269 1 69 | 7   26   4   | 5   3   8  |
|----------+--------------+------------|
| 1   9 7  |c26  58   3   |d26  58  4  |
| 3   4 2  | 9   1568 56  | 16  58  7  |
| 5   6 8  | 4   12   7   | 9   1-2 3  |
*--------------------------------------*

Another Skyscraper in cells a-b-c-d removes 2 from r9c8 and the puzzle solves in singles from there.

If you are unfamiliar with Skyscrapers you can read about them here, or if you like I can give you a more detailed explanation.

Leren

PS I see that JC Van Hay and I cross-posted, so you can choose your preferred solution, Leren

[Sudoku123456789]

Code: Select all

+--------------+------------------+----------------+
| 69    28  69 | 28   7     1     | 3   4     5    |
| 28    5   13 | 238  4     9     | 7   1(2)  6    |
| 4     7   13 | 236  56-2  56(2) | 8   9     1(2) |
+--------------+------------------+----------------+
| 2689  28  5  | 1    3     6(2)  | 4   7     89   |
| 7     3   4  | 5    9     8     | 12  6     12   |
| 2689  1   69 | 7    6-2   4     | 5   3     89   |
+--------------+------------------+----------------+
| 1     9   7  | 26   58    3     | 26  58    4    |
| 3     4   2  | 9    1568  56    | 16  58    7    |
| 5     6   8  | 4    1(2)  7     | 9   1(2)  3    |
+--------------+------------------+----------------+

Your analysis of the 2s is incomplete : you excluded 2r3c5, but there is only 1 solution for the 2s!
A simple interpretation : the solutions of {2R9, 2B3, 2C6} exclude 2r36c5; stte

Thanks JC,

What is the technique behind this? Why don't the 2's, for example in B5, don't contribute to this?

[Sudoku123456789]

The puzzle solved cell status in line format : ....71345.5..497.647....89...513.47.734598.6..1.7.453.197..3..43429....75684.79.3

The simplest solution from your position is to use a couple of Skyscrapers, both on digit 2.

Code: Select all

*----------------------------------------*
| 69   a28 69 | 8-2 7     1   | 3  4  5  |
| 28    5  13 | 238 4     9   | 7  12 6  |
| 4     7  13 | 236 256  d256 | 8  9  12 |
|-------------+---------------+----------|
| 2689 b28 5  | 1   3    c26  | 4  7  89 |
| 7     3  4  | 5   9     8   | 12 6  12 |
| 2689  1  69 | 7   26    4   | 5  3  89 |
|-------------+---------------+----------|
| 1     9  7  | 26  58    3   | 26 58 4  |
| 3     4  2  | 9   1568  56  | 16 58 7  |
| 5     6  8  | 4   12    7   | 9  12 3  |
*----------------------------------------*

At your position there is a Skyscraper in cells a-b-c-d which eliminates 2 in r2c4 and solves 6 cells with follow on singles.

This brings you to here :

Code: Select all

*--------------------------------------*
| 69  2 69 | 8   7    1   | 3   4   5  |
| 8   5 13 |b23  4    9   | 7  a12  6  |
| 4   7 13 | 236 256  256 | 8   9   12 |
|----------+--------------+------------|
| 26  8 5  | 1   3    26  | 4   7   9  |
| 7   3 4  | 5   9    8   | 12  6   12 |
| 269 1 69 | 7   26   4   | 5   3   8  |
|----------+--------------+------------|
| 1   9 7  |c26  58   3   |d26  58  4  |
| 3   4 2  | 9   1568 56  | 16  58  7  |
| 5   6 8  | 4   12   7   | 9   1-2 3  |
*--------------------------------------*

Another Skyscraper in cells a-b-c-d removes 2 from r9c8 and the puzzle solves in singles from there.

If you are unfamiliar with Skyscrapers you can read about them here, or if you like I can give you a more detailed explanation.

Leren

PS I see that JC Van Hay and I cross-posted, so you can choose your preferred solution, Leren

One question about the skyscraper: is it true that a-b-c-d all have to be in different boxes, but c+d have to be in the same chute in order to obtain exclusions?

I thought there should be a difference in height of c-d, for example: a+b in R8 (base) and C in R2 + D in R3. How does exclusion of R2C4 follows out of the skyscraper you mentioned?

[Leren]

OK, This is how Skyscrapers work.

Look at the first one. Note that there are exactly two 2's in Columns 2 and 6.

Suppose cell a was False, then cell b would have to be True (two 2's in Column 2) so cell c would have to be False, so cell d would have to be True (two 2's in Column 6).

Now suppose cell d was False and follow a similar chain of logic via cells d-b-c-a and you would have to conclude that cell a would have to be True.

What all this proves is that at least one of cells a and d must be True. They might both be True but they can't both be False.

Since cell r1c4 can see both cells a and d it can't be 2. QED.

The second Skyscraper works the same way as the first one except that it is "on its side" and there are exactly two 2's in Rows 2 and 7.

I thought all this was explained in the link I gave, but perhaps not in so many words.

Leren

[JC Van Hay]

What is the technique behind this? Why don't the 2's, for example in B5, don't contribute to this?

Technique : "colouring from each candidate of a well-defined set of constraints". See, for example, here.
Application to the set of candidates of a single digit here.

Here is the possibility matrix for the 2s.

Code: Select all

+----------+----------+----------+
| .  2gG.  | 2  .  .  | .  .  .  |
| 2  .  .  | 2  .  .  | .  2d .  |
| .  .  .  | 2h 2  2  | .  .  2D |
+----------+----------+----------+
| 2  2  .  | .  .  2eE| .  .  .  |
| .  .  .  | .  .  .  | 2C .  2c |
| 2fF.  .  | .  2  .  | .  .  .  |
+----------+----------+----------+
| .  .  .  | 2B .  .  | 2b .  .  |
| .  .  2  | .  .  .  | .  .  .  |
| .  .  .  | .  2a .  | .  2A .  |
+----------+----------+----------+


From the 2 solutions of 2R97B63 :
2r9c5 -> 1 solution : abcdefgh
2r9c8 -> 0 solution : 2A -> 2B, 2D -> 2B2={}

Involved constraints in the exclusion of 2A=2r9c8 : 2R7, 2B3, 2B2
The solutions of those constraints exclude 2r9c8
Eureka notation : (2)[r7c7=r7c4-r123c4=r3c56-r3c9=r2c8]-2r9c8

Some observations on the common exclusions :
2a. Either 2a or 2D -> -2r3c5
2b. Either 2a or 2E -> -2r6c5
2c. => r9c5=2a -> 1 solution
2d. Involved constraints in the exclusion of 2r36c5 : 2R9, 2B3, 2C6
2e. The solutions of those constraints exclude 2r36c5
2f. Eureka notation : (2)[r9c5=r9c8-r2c8=r3c9*-r3c6=r4c6]-2r3c5*.r6c5

3. 2eE <-> 2fF -> 2gG but don't solve the 2s

4a. Either 2d or 2B -> -2r2c4
4b. Either 2g or 2B -> -2r1c4
4c. => r1c2=2gG, r2c8=2d, r9c5=2a -> 1 solution
4d. Involved constraints in the exclusion of 2r12c4 : 2R7, 2B3, 2B1
4e. The solutions of those constraints exclude 2r12c4
4f. Eureka notation : (2)[r7c4=2r7c7-r9c8=r2c8*-r2c1=r1c2]-2r12*c4

5a. Either 2a or 2D -> -2r3c5
5b. Either 2e or 2D -> -2r3c6
5c. => r4c6=2e, r9c5=2a -> 1 solution
5d. Involved constraints in the exclusion of 2r3c56 : 2B2, 2R9, 2B5
5e. The solutions of those constraints exclude 2r3c56
5f. Eureka notation : (2)[r3c9=r2c8-r9c8=r9c5*-r6c5=r4c6]-2r3c5*6

JC

[Sudoku123456789]

What is the technique behind this? Why don't the 2's, for example in B5, don't contribute to this?

Technique : "colouring from each candidate of a well-defined set of constraints". See, for example, here.
Application to the set of candidates of a single digit here.

Thanks JC for this elaborated explanation.

I recently posted another topic about this (here: http://forum.enjoysudoku.com/colouring-trap-t34235.html), but could you tell me your opinion on using the coloring technique? I regard your opinion as highly valuable.

[JC Van Hay]

Single and multiple digits colouring is my favorite (and the most powerfull) way to manually solve any puzzle ...

Here are some simple and complete references for single digit colouring :
http://sudopedia.enjoysudoku.com/
http://sudopedia.enjoysudoku.com/Coloring.html
and especially to go further
http://sudopedia.enjoysudoku.com/X-Colors.html

If you are interested in multiple digits colouring, see for example :
http://forum.enjoysudoku.com/full-tagging-t5624.html and/or http://gpenet.pagesperso-orange.fr/index_fichiers/AWHO.htm
http://sudopedia.enjoysudoku.com/Graded_Equivalence_Marks.html and its unrestricted generalization, Red Green (...) Transport [Note : lost reference for RGT]