The New Sudoku Players' Forum

[wychwood]

I would welcome tips on the next move in this puzzle please.

Starting posiiton is this:

Code: Select all

 *-----------*
 |.65|1.2|...|
 |...|...|..3|
 |...|.7.|.6.|
 |---+---+---|
 |..9|...|...|
 |6..|.3.|..8|
 |...|...|5..|
 |---+---+---|
 |.7.|.8.|...|
 |4..|...|...|
 |...|5.9|12.|
 *-----------*

Simple solving steps leads me to this point - and then no further. I cannot see the next step at all by any of the methods I know (including X wing, XY wing etc) and Simple Soduko cannot offer a tip at this point either:

*-----------*
|365|192|..4|
|...|...|..3|
|...|.7.|.6.|
|---+---+---|
|..9|...|...|
|6..|.3.|..8|
|..3|...|5..|
|---+---+---|
|.7.|.8.|...|
|4..|...|...|
|836|549|127|
*-----------*


*-----------------------------------------------------------------------------*
| 3 6 5 | 1 9 2 | 78 78 4 |
| 1279 1249 478 | 468 56 4568 | 29 15 3 |
| 129 1249 48 | 348 7 3458 | 29 6 15 |
|-------------------------+-------------------------+-------------------------|
| 1257 12458 9 | 24678 1256 145678 | 3467 1347 126 |
| 6 125 47 | 29 3 15 | 47 19 8 |
| 127 1248 3 | 246789 126 14678 | 5 1479 1269 |
|-------------------------+-------------------------+-------------------------|
| 59 7 12 | 236 8 136 | 346 3459 569 |
| 4 59 12 | 2367 126 1367 | 368 3589 569 |
| 8 3 6 | 5 4 9 | 1 2 7 |
*-----------------------------------------------------------------------------*


Please can someone offer the next step, fully explained in plain Englsh please (I fear that I just do not understand any of the mathematical shorthand that is often used on these message boards):
- please use logic only if at all possible, and thus avoiding the use of 'forcing chains' or similar (which I regard as 'what if' trial and error solutions, not pure logic and reasoning (where the thought flow will start with 'because...');
- please also avoid the use of computer programs and software solvers (Sudopku is after all a human gamne, not a computer game??).

By the way, I was able to solve the next puzzle in the book from which this came by logic and simple solving techniques alone - so not all the puzzles are like this one.

Thanks in advance for all tips received.
Neil

[ravel]

Code: Select all

*---------------------------------------------------------------*
 | 3     6      5    | 1       9     2       | 78    78    4     |
 | 1279  1249   478  | 468    #56    4568    | 29   #15    3     |
 | 129   1249   48   | 348     7     3458    | 29    6     15    |
 |-------------------+-----------------------+-------------------|
 | 1257  12458  9    | 24678  #1256  145678  | 3467  1347  126   |
 | 6     125    47   | 29      3    #15      | 47   -19    8     |
 | 127   1248   3    | 246789  126   14678   | 5     1479  1269  |
 |-------------------+-----------------------+-------------------|
 | 59    7      12   | 236     8     136     | 346   3459  569   |
 | 4     59     12   | 2367    126   1367    | 368   3589  569   |
 | 8     3      6    | 5       4     9       | 1     2     7     |
 *---------------------------------------------------------------*


There is a short xy-chain, that eliminates 1 from r5c8:
Either r2c8=1 or (r2c8=5 => r2c5<>5 => r4c5=5 =>) r5c6=1, so r5c8 cannot be 1.
Brings you here:

Code: Select all

 *---------------------------------------------*
 | 3    6    5    | 1    9  2    | 78  78  4   |
 | 279 #249  478  | 468  5  468  |#29  1   3   |
 | 1   #29   48   | 348  7  348  |#29  6   5   |
 |----------------+--------------+-------------|
 |@57   458  9    | 478  1  478  | 6   3   2   |
 | 6    1   @47   | 2    3  5    |@47  9   8   |
 | 27   248  3    | 9    6  478  | 5  @47  1   |
 |----------------+--------------+-------------|
 |-59   7    2    | 36   8  1    | 34 @45  69  |
 | 4    59   1    | 367  2  367  | 38  58  69  |
 | 8    3    6    | 5    4  9    | 1   2   7   |
 *---------------------------------------------*


There is a unique rectangle type 1 in the # marked cells. Given, that the puzzle has a unique solution, r2c2 cannot be 2 or 9. (Otherwise always at least 2 solutions would be possible, swapping 2 and 9 in the 4 cells)

If you dont want to use it, there is another xy-chain eliminating 5 from r7c1:
Either r4c1=5 or (r4c1=7 => r5c3=4 => r5c7=7 => r7c7=4 =>) r7c8=5.

[wychwood]

Thanks Ravel. that seems to make sense to me (althoguh I had not come acrposs these "unique rectangles" before)!

I really overloked that initial XY wing - BUT then it looks more like a forcing chain to me, rather than what I understood was a 'pure' XY wing which I thought only involved 4 cells that can see eacah other?

Anyway, at least these tips take it away from pure guesswork.
Thanks
Neil

[daj95376]

Neil,

Is it possible that you incorrectly copied this puzzle from your book? I can advance you a couple of steps, but then it gets ugly.

Code: Select all

.651.2...........3....7..6...9......6...3...8......5...7..8....4...........5.912.
365192..4........3....7..6...9......6...3...8..3...5...7..8....4........836549127


Code: Select all

# finned Swordfish r358/c269 w/fin cell [r8c8] => [r7c9]<>5
*--------------------------------------------------------------------------------------*
|  3        6        5       |  1        9        2       |  78       78       4       |
|  1279     1249     478     |  468       56      4568    |  29       15       3       |
|  129      1249     48      |  348      7       *3458    |  29       6       *15      |
|----------------------------+----------------------------+----------------------------|
|  1257     12458    9       |  24678    1256     145678  |  3467     1347     126     |
|  6       *125      47      |  29       3       *15      |  47       19       8       |
|  127      1248     3       |  246789   126      14678   |  5        1479     1269    |
|----------------------------+----------------------------+----------------------------|
|  59       7        12      |  236      8        136     |  346      3459     69-5    |
|  4       *59       12      |  2367     126      1367    |  368     #3589    *569     |
|  8        3        6       |  5        4        9       |  1        2        7       |
*--------------------------------------------------------------------------------------*


Code: Select all

# exposes XYZ-Wing at [r8c9]+[r7c9]+[r8c2] => [r8c8]<>9
*--------------------------------------------------------------------------------------*
|  3        6        5       |  1        9        2       |  78       78       4       |
|  1279     1249     478     |  468      56       4568    |  29       15       3       |
|  129      1249     48      |  348      7        3458    |  29       6        15      |
|----------------------------+----------------------------+----------------------------|
|  1257     12458    9       |  24678    1256     145678  |  3467     1347     126     |
|  6        125      47      |  29       3        15      |  47       19       8       |
|  127      1248     3       |  246789   126      14678   |  5        1479     1269    |
|----------------------------+----------------------------+----------------------------|
|  59       7        12      |  236      8        136     |  346      3459    *69      |
|  4       *59       12      |  2367     126      1367    |  368      358-9   *569     |
|  8        3        6       |  5        4        9       |  1        2        7       |
*--------------------------------------------------------------------------------------*


ravel: Your solution involves a Forcing Chain -- because of cell [r4c5] -- and not an XY-Chain. This is from the post you referenced.

Similar to the xy-wing, the xy-chain makes use of the special property of cells with 2 candidates.

[m_b_metcalf]

Is it possible that you incorrectly copied this puzzle from your book?

The code as posted has one, difficult, solution.

Regards,

Mike Metcalf

[ravel]

ravel: Your solution involves a Forcing Chain -- because of cell [r4c5] -- and not an XY-Chain.

Ah yes, thanks. I had a more general definition in mind, which also allows to use bilocation links in the chain. I personally dont mind, if a candidate forces another one because of a bivalue cell or a conjugated pair. But i have to admit, that xy-chains (like the second above) are easier to spot, because you only have to concentrate on bivalue cells.

[re'born]

ravel: Your solution involves a Forcing Chain -- because of cell [r4c5] -- and not an XY-Chain.

Ah yes, thanks. I had a more general definition in mind, which also allows to use bilocation links in the chain. I personally dont mind, if a candidate forces another one because of a bivalue cell or a conjugated pair. But i have to admit, that xy-chains (like the second above) are easier to spot, because you only have to concentrate on bivalue cells.

Another way to make your exclusion is with the ALS xz-rule:

A={r5c4, r246c5}, B={r5c8}, x=9, z=1.

However, I like your generalization of XY-chains a lot, and will keep an eye out for more examples of this type.

Another solution, again designed to find ravel's exclusion uses 3D coloring (which is definitely not trial and error), and so I hope wychwood will find this solution acceptable.

In the standard notation:
[r5c6]-1-[r7c6]=1=[r7c3]=2=[r7c4]-2-[r5c4]-9-[r5c8]-1-[r5c6]
and so r5c6<>1.
To let you see that there was nothing up my sleeves, here is the coloring I did:

Code: Select all

.---------------------.---------------------.---------------------.
| 3      6      5     | 1      9      2     | 78     78     4     |
| 1279   1249   478   | 468    56     4568  | 29     15     3     |
| 129    1249   48    | 348    7      3458  | 29     6      15    |
:---------------------+---------------------+---------------------:
| 1257   12458  9     | 24678  1256   145678| 3467   1347   126   |
| 6      125    47    | 2C9c   3      15-   | 47     1c9C   8     |
| 127    1248   3     | 246789 126    14678 | 5      1479   1269  |
:---------------------+---------------------+---------------------:
| 59     7      1a2A  | 2a36   8      1A36  | 346    3459   569   |
| 4      59     12    | 2367   126    1367  | 368    3589   569   |
| 8      3      6     | 5      4      9     | 1      2      7     |
'---------------------'---------------------'---------------------'


Since 'a' and 'C' are weakly linked (r57c4), any placement that would make 'A' and 'c' simultaneously false, must be false. In particular, a 1 in r5c6 would kill the 'A' in r7c6 and the 'c' in r5c8. Therefore, r5c6<>1.

[Para]

Hi

Code: Select all

*---------------------------------------------------------------*
 | 3     6      5    | 1       9     2       | 78    78    4     |
 | 1279  1249   478  | 468    #56    4568    | 29   #15    3     |
 | 129   1249   48   | 348     7     3458    | 29    6     15    |
 |-------------------+-----------------------+-------------------|
 | 1257  12458  9    | 24678  #1256  145678  | 3467  1347  126   |
 | 6     125    47   | 29      3    #15      | 47   -19    8     |
 | 127   1248   3    | 246789  126   14678   | 5     1479  1269  |
 |-------------------+-----------------------+-------------------|
 | 59    7      12   | 236     8     136     | 346   3459  569   |
 | 4     59     12   | 2367    126   1367    | 368   3589  569   |
 | 8     3      6    | 5       4     9       | 1     2     7     |
 *---------------------------------------------------------------*


This is actually a move i look for often when there are distant pairs in a grid.
R3C8 and R5C6 both are {15}. C5 has only 2 options left for digit 5 and they are both seen by R3C8 or R5C6. So R3C8 and R5C6 can't both be 5. So any cell that can see both R3C8 and R5C6 can't contain a 1. Thus we can eliminate 1 from R5C8. This logic isn't really a forcing chain, is it? It is basically the same logic used for xy-wings but then 2 values in 1 cell are used instead of 2 values in a house.

greetings

Para

[udosuk]

Another way to make your exclusion is with the ALS xz-rule:

A={r5c4, r246c5}, B={r5c8}, x=9, z=1.

rep'nA, you should mention that this ALS results in the elimination r5c6<>1, which is not exactly the same as ravel's exclusion (though they achieve the same results several eliminations later)... Also, ALS A is one of those complex ones that's not within a house which some might not like...

Code: Select all

 *--------------------------------------------------*
 | 3    6    5    | 1    9    2    | 78   78   4    |
 | 279  249  478  | 468  5    468  | 29   1    3    |
 | 1    29   48   | 348  7    348  | 29   6    5    |
 |----------------+----------------+----------------|
 |@57   458  9    | 478  1    478  | 6    3    2    |
 | 6    1   @47   | 2    3    5    |-47   9    8    |
 | 27   248  3    | 9    6    478  | 5    47   1    |
 |----------------+----------------+----------------|
 |#59   7    2    |#36   8    1    |#34   45  #69   |
 | 4    59   1    | 367  2    367  | 38   58   69   |
 | 8    3    6    | 5    4    9    | 1    2    7    |
 *--------------------------------------------------*

Here, an alternative approach to ravel's xy-chain is another ALS-xz:
A={r4c1,r5c3}, B={r7c1479}, x=5, z=4 => r5c7<>4

[Para]

Code: Select all

.---------------------.---------------------.---------------------.
| 3      6      5     | 1      9      2     | 78     78     4     |
| 1279   1249   478   | 468    A56    4568  | 29     15     3     |
| 129    1249   48    | 348    7      3458  | 29     6      15    |
:---------------------+---------------------+---------------------:
| 1257   12458  9     | 24678  A1256  145678| 3467   1347   126   |
| 6      125    47    | B29    3      -15   | 47     B19    8     |
| 127    1248   3     | 246789 A126   14678 | 5      1479   1269  |
:---------------------+---------------------+---------------------:
| 59     7      12    | 236    8      136   | 346    3459   569   |
| 4      59     12    | 2367   126    1367  | 368    3589   569   |
| 8      3      6     | 5      4      9     | 1      2      7     |
'---------------------'---------------------'---------------------'

I think you can show that first ALS-xz move easier if you use

A = {R246C5}, B = {R5C48}, x = 2, z = 1

exactly the same cells, except both ALS's are in 1 house.

Para

[wychwood]

Hi all you respondents to this, and thanks for your thoughts and replies.

OK, to clarify/comment on these a little:
1. there was no error in the original puzzle as coded.
2. I regret that the shorthand formats and mathematical expressions used by, for example, udosuk with all the brackets and things just baffles me - me no understand them.
3. I have no ides what rep'nA is talking about with "ALS xz-rules" etc. These are yet more things that completely bamboozle me (although you guys may well find them helpful and intelligible between yourselves).
4. I liked the logic approach that para used and I could follow it because it was written in English (as requested in my initial submission).
5. I am intrigued by Mike's response and would be interested in seeing how he starts off his solution from where I had got to. But helpful to know that it is a single solution puzzle after all - that did concern me initially.

Finally, just to say that I am a part-timer on this, and do not have the time to read message boards, pages and pages of complicated solving tips etc.

But thanks again for some of these responses. which make some logical sense to me occasionally.

Cheers
Neil

[udosuk]

2. I regret that the shorthand formats and mathematical expressions used by, for example, udosuk with all the brackets and things just baffles me - me no understand them.
3. I have no ides what rep'nA is talking about with "ALS xz-rules" etc. These are yet more things that completely bamboozle me (although you guys may well find them helpful and intelligible between yourselves).

Neil, although you didn't understand those expressions, I must thank you for posting this puzzle and allow the rest of us who have learnt those advanced techniques an opportunity to study it...

ALS stands for "Almost Locked Subsets" and is a powerful but very logical technique for advanced-level puzzles like this one... I personally find it more elegant than xy-chains etc because if you understand it, the logic is as clear and simple to follow as xy-wings... (In fact an xy-wing can be interpreted as a very simple version of ALS...) Whereas xy-chains, especially long ones, have a chaotic feeling about them (again just personal taste)...

I don't think you have time to read the forum pages to study these moves, and if you ever do, I definitely recommend you to try to learn ALS, especially if like me, you prefer logical moves which are more elegant than forcing chains...

[m_b_metcalf]

5. I am intrigued by Mike's response and would be interested in seeing how he starts off his solution from where I had got to. But helpful to know that it is a single solution puzzle after all - that did concern me initially.

Well, I just fed your original puzzle into a brute-force solver that counts the exact number of solutions. The result was 1. I then fed it into a logic solver that told me it was difficult. These are just tools in my toolbox that I've written. I didn't look at your partial solution. [Edit: now I have and it's correct as far as it goes.]

Regards,

Mike Metcalf

[udosuk]

Neil, here is an attempt to explain to you the logic of the ALS-xz moves in plain English:

Code: Select all

 *-----------------------------------------------------------------------------*
 | 3       6       5       | 1       9       2       | 78      78      4       |
 | 1279    1249    478     | 468    @56      4568    | 29      15      3       |
 | 129     1249    48      | 348     7       3458    | 29      6       15      |
 |-------------------------+-------------------------+-------------------------|
 | 1257    12458   9       | 24678  @1256    145678  | 3467    1347    126     |
 | 6       125     47      |#29      3      *15      | 47     #19      8       |
 | 127     1248    3       | 246789 @126     14678   | 5       1479    1269    |
 |-------------------------+-------------------------+-------------------------|
 | 59      7       12      | 236     8       136     | 346     3459    569     |
 | 4       59      12      | 2367    126     1367    | 368     3589    569     |
 | 8       3       6       | 5       4       9       | 1       2       7       |
 *-----------------------------------------------------------------------------*

1. There can only be one 2 in block 5.

2. Especially, the 3 cells r5c4, r4c5 & r6c5 can have at most one 2 among them.

3. Therefore either r5c4 isn't 2, or r4c5+r6c5 don't have 2, or all three of them aren't 2.

4a. r5c4 isn't 2 would make r5c4=9 and r5c8=1.
4b. r4c5+r6c5 don't have 2 would make r2c5+r4c5+r6c5 a naked triple {156} with the 1 locked in r4c5+r6c5.
4c. all three of them aren't 2 would make both effects above (r5c8=1 and 1 locked in r4c5+r6c5).

5. So no matter what, at least one of r5c8, r4c5 & r6c5 must be 1.

6. Hence, r5c6, sharing a row/block with these 3 cells, cannot possibly be 1, and we can eliminate 1 from this cell.

Code: Select all

 *--------------------------------------------------*
 | 3    6    5    | 1    9    2    | 78   78   4    |
 | 279  249  478  | 468  5    468  | 29   1    3    |
 | 1    29   48   | 348  7    348  | 29   6    5    |
 |----------------+----------------+----------------|
 |@57   458  9    | 478  1    478  | 6    3    2    |
 | 6    1   @47   | 2    3    5    |*47   9    8    |
 | 27   248  3    | 9    6    478  | 5    47   1    |
 |----------------+----------------+----------------|
 |#59   7    2    |#36   8    1    |#34   45  #69   |
 | 4    59   1    | 367  2    367  | 38   58   69   |
 | 8    3    6    | 5    4    9    | 1    2    7    |
 *--------------------------------------------------*

1. There can only be one 5 on column 1.

2. Especially, the 2 cells r4c1 & r7c1 must have exactly one 5 between them.

3. Therefore either r4c1 isn't 5, or r7c1 isn't 5.

4a. r4c1 isn't 5 would make r4c1=7 and r5c3=4.
4b. r7c1 isn't 5 would make r7c1=9, r7c9=6, r7c4=3 and r7c7=4.

5. So no matter what, at least one of r5c3 & r7c7 must be 4.

6. Hence, r5c7, sharing a row/column with these 2 cells, cannot possibly be 4, and we can eliminate 4 from this cell.


Of course, if you have learnt the formal rules of ALS, all these logical reasonings become standard patterns and you could easily spot the eliminations when you find the corresponding cells...

[daj95376]

I've been reviewing ravel's chain from different angles. Observe the (excerpt) pattern below. If [r2c8]=5 and [r5c6]=5, then all of the <5>s in [b2] are eliminated. However, [r5c8]=1 forces these two assignments. Therefore, we can conclude that [r5c8]<>1. I wonder how often this scenario exists when puzzles get difficult.

Code: Select all

*-----------------------------------*
|  .  .  5  |  .  .  .  |  .  .  .  |
|  .  .  .  |  .  5  5  |  . 15  .  |
|  .  .  .  |  .  .  5  |  .  .  5  |
|-----------+-----------+-----------|
|  5  5  .  |  .  5  5  |  .  .  .  |
|  .  5  .  |  .  . 15  |  . 19  .  |
|  .  .  .  |  .  .  .  |  5  .  .  |
|-----------+-----------+-----------|
|  5  .  .  |  .  .  .  |  .  5  .  |
|  .  5  .  |  .  .  .  |  .  5  5  |
|  .  .  .  |  5  .  .  |  .  .  .  |
*-----------------------------------*