The New Sudoku Players' Forum

[ArkieTech]

Code: Select all

 *-----------*
 |4.2|8..|7.9|
 |.91|...|68.|
 |78.|...|.21|
 |---+---+---|
 |...|.1.|..5|
 |...|372|...|
 |2..|.4.|...|
 |---+---+---|
 |65.|...|.92|
 |.27|...|83.|
 |9.8|..6|5.7|
 *-----------*


Play/Print this puzzle online

[Leren]

Either

Code: Select all

*--------------------------------------------------------------------------------*
| 4       36      2        | 8       36      1        | 7       5       9        |
|a35      9       1        |b27     a235     3457     | 6       8      a34       |
| 7       8       356      | 459     3569    3459     | 3-4     2       1        |
|--------------------------+--------------------------+--------------------------|
| 38      3467    3469     | 69      1       89       | 2       467     5        |
| 58      146     4569     | 3       7       2        | 149     146     48       |
| 2       167     69       | 569     4       589      | 139     167     38       |
|--------------------------+--------------------------+--------------------------|
| 6       5      34        |c17      8       37       |c14      9       2        |
| 1       2       7        | 459     59      459      | 8       3       6        |
| 9       34      8        | 12      23      6        | 5       14      7        |
*--------------------------------------------------------------------------------*

als xy-wing: (4=2) r2c159 - (2=7) r2c4 - (7=4) r7c47 => r3c7 <> 4; stte

or

Code: Select all

*--------------------------------------------------------------------------------*
| 4       36      2        | 8       36      1        | 7       5       9        |
| 35      9       1        | 27      235     3457     | 6       8       34       |
| 7       8       356      | 459     3569    459-3    |a34      2       1        |
|--------------------------+--------------------------+--------------------------|
| 38      3467    3469     | 69      1       89       | 2       467     5        |
| 58      146     4569     | 3       7       2        | 149     146     48       |
| 2       167     69       | 569     4       589      | 139     167     38       |
|--------------------------+--------------------------+--------------------------|
| 6       5      c34       | 17      8      d37       |b14      9       2        |
| 1       2       7        | 459     59      459      | 8       3       6        |
| 9       34      8        | 12      23      6        | 5       14      7        |
*--------------------------------------------------------------------------------*

m-wing: (3=4) r3c7 - r7c7 = (4-3) r7c3 = r7c6 => r3c6 <> 3; lclste

Leren

[tlanglet]

ANS(35=2)r2c15-(2=7)r2c4-(7=1)r7c4-(1=4)r7c7-(4=3)r3c7 => r2c9<>3

Ted

[Marty R.]

Code: Select all

+--------------+---------------+------------+
| 4  36   2    | 8   36   1    | 7   5   9  |
| 35 9    1    | 27  235  3457 | 6   8   34 |
| 7  8    356  | 459 3569 3459 | 34  2   1  |
+--------------+---------------+------------+
| 38 3467 3469 | 69  1    89   | 2   467 5  |
| 58 146  4569 | 3   7    2    | 149 146 48 |
| 2  167  69   | 569 4    589  | 139 167 38 |
+--------------+---------------+------------+
| 6  5    34   | 17  8    37   | 14  9   2  |
| 1  2    7    | 459 59   459  | 8   3   6  |
| 9  34   8    | 12  23   6    | 5   14  7  |
+--------------+---------------+------------+


Play this puzzle online at the Daily Sudoku site

Same as leren's first, but with a longer path.

(4=352)r2c195-(2=3)r9c5-(3=7)r7c6-(7=1)r7c4-(1=4)r7c7=>r3c7<>4

[Marty R.]

Code: Select all

+--------------+---------------+------------+
| 4  36   2    | 8   36   1    | 7   5   9  |
| 35 9    1    | 27  235  3457 | 6   8   34 |
| 7  8    356  | 459 3569 3459 | 34  2   1  |
+--------------+---------------+------------+
| 38 3467 3469 | 69  1    89   | 2   467 5  |
| 58 146  4569 | 3   7    2    | 149 146 48 |
| 2  167  69   | 569 4    589  | 139 167 38 |
+--------------+---------------+------------+
| 6  5    34   | 17  8    37   | 14  9   2  |
| 1  2    7    | 459 59   459  | 8   3   6  |
| 9  34   8    | 12  23   6    | 5   14  7  |
+--------------+---------------+------------+


Play this puzzle online at the Daily Sudoku site

Same as leren's first, but with a longer path.

(4=352)r2c195-(2=3)r9c5-(3=7)r7c6-(7=1)r7c4-(1=4)r7c7=>r3c7<>4

Just a quick notation question about the first term. Would (435=2) be any more or less correct than the (4=352) that I used?

[DonM]

Code: Select all

+--------------+---------------+------------+
| 4  36   2    | 8   36   1    | 7   5   9  |
| 35 9    1    | 27  235  3457 | 6   8   34 |
| 7  8    356  | 459 3569 3459 | 34  2   1  |
+--------------+---------------+------------+
| 38 3467 3469 | 69  1    89   | 2   467 5  |
| 58 146  4569 | 3   7    2    | 149 146 48 |
| 2  167  69   | 569 4    589  | 139 167 38 |
+--------------+---------------+------------+
| 6  5    34   | 17  8    37   | 14  9   2  |
| 1  2    7    | 459 59   459  | 8   3   6  |
| 9  34   8    | 12  23   6    | 5   14  7  |
+--------------+---------------+------------+


Play this puzzle online at the Daily Sudoku site
Same as leren's first, but with a longer path.

(4=352)r2c195-(2=3)r9c5-(3=7)r7c6-(7=1)r7c4-(1=4)r7c7=>r3c7<>4

Just a quick notation question about the first term. Would (435=2) be any more or less correct than the (4=352) that I used?

You've got it right. The chain is read from left to right: If not (4)r2c9 then you have the locked set (352)r2c195 that would knock out (2)r9c5 etc. This is not so much a question of what is correct math-logic-wise, but, from the reader's point of view, what reflects what is actually going on in the chain.

Edit: In view of the two responses below, I've underlined the above to emphasize what I think is most important point in this notation: Conveying to the reader as clearly as possible what is going on in the chain. During the several years of manual solutions posted on the Eureka forum, this was the most common form of notation I saw used for ALSs (including mine, of course ).

(Also fwiw, although not related to your notation, I never saw a >=3 digit ALS expressed with just the strong link between 2 of the digits. That doesn't show all the digits of the ALS which just makes translation that much more difficult for the reader.)

[ronk]

Same as leren's first, but with a longer path.
(4=352)r2c195-(2=3)r9c5-(3=7)r7c6-(7=1)r7c4-(1=4)r7c7=>r3c7<>4

Just a quick notation question about the first term. Would (435=2) be any more or less correct than the (4=352) that I used?

They are equally correct or equally incorrect, depending on your point-of-view. If one triplet doesn't exist, then the other does. Obviously the <35> is in both.

[daj95376]

Code: Select all

+--------------+---------------+------------+
| 4  36   2    | 8   36   1    | 7   5   9  |
| 35 9    1    | 27  235  3457 | 6   8   34 |
| 7  8    356  | 459 3569 3459 | 34  2   1  |
+--------------+---------------+------------+
| 38 3467 3469 | 69  1    89   | 2   467 5  |
| 58 146  4569 | 3   7    2    | 149 146 48 |
| 2  167  69   | 569 4    589  | 139 167 38 |
+--------------+---------------+------------+
| 6  5    34   | 17  8    37   | 14  9   2  |
| 1  2    7    | 459 59   459  | 8   3   6  |
| 9  34   8    | 12  23   6    | 5   14  7  |
+--------------+---------------+------------+


Same as leren's first, but with a longer path.

(4=352)r2c195-(2=3)r9c5-(3=7)r7c6-(7=1)r7c4-(1=4)r7c7=>r3c7<>4

Just a quick notation question about the first term. Would (435=2) be any more or less correct than the (4=352) that I used?

I'm okay with whichever version you wish to use.

FWIW: I would use (435=352)r2c915 for a loop and (4=2)r2c915 for a chain.

Regards, Danny

[David P Bird]

Just a quick notation question about the first term. Would (435=2) be any more or less correct than the (4=352) that I used?

To clarify where I stand on this:

As others have said, the single and the tuple can be written either way round in the ANS (or AHS) expression. For checking purposes, I personally prefer the digits in the tuple to be in ascending order.

AICs should be capable of being followed either forwards or backwards, so there should be no right or wrong way. In fact, when the inclusion of patterns is first being mastered, it's often useful to check if the logic is still sound when the path direction is reversed. This will show up cases when the external candidates don't see all the necessary internal ones.

Aside: Reversibility is what distinguishes AICs from Forcing Chains. In brief, the argument about their respective acceptability goes like this:
A Forcing Chain assumes a starting node to be true or false and looks for a contradiction.
An AIC observes that two strongly linked nodes must hold at least one truth, and without assuming anything, looks for candidates that can't be true as a result.

From a practical viewpoint AICs are generally more productive. Unlike for Forcing Chains, once AIC segments are found, they can be re-used freely as they don't depend on any assumption (or guess as some would say). The solutions to multiple step puzzles often demonstrate this.

[Marty R.]

Code: Select all

+--------------+---------------+------------+
| 4  36   2    | 8   36   1    | 7   5   9  |
| 35 9    1    | 27  235  3457 | 6   8   34 |
| 7  8    356  | 459 3569 3459 | 34  2   1  |
+--------------+---------------+------------+
| 38 3467 3469 | 69  1    89   | 2   467 5  |
| 58 146  4569 | 3   7    2    | 149 146 48 |
| 2  167  69   | 569 4    589  | 139 167 38 |
+--------------+---------------+------------+
| 6  5    34   | 17  8    37   | 14  9   2  |
| 1  2    7    | 459 59   459  | 8   3   6  |
| 9  34   8    | 12  23   6    | 5   14  7  |
+--------------+---------------+------------+


Play this puzzle online at the Daily Sudoku site

Same as leren's first, but with a longer path.

(4=352)r2c195-(2=3)r9c5-(3=7)r7c6-(7=1)r7c4-(1=4)r7c7=>r3c7<>4

Just a quick notation question about the first term. Would (435=2) be any more or less correct than the (4=352) that I used?

As others have said, the single and the tuple can be written either way round in the ANS (or AHS) expression. For checking purposes, I personally prefer the digits in the tuple to be in ascending order.

Thanks to Don, Ron, Danny and David for the feedback.

(Thanks David for introducing me to a new word--tuple )

As to number sequence, I'm still relatively new at this still have problems with ALS, ANS, AHS, whichever is correct.
My intention with (435=2) is to show that 4 would be the first number in the sequence, equivalent to (4=2).
My intention with (4=352)r2c195 is to show 2 is the last number and the number that is located in the last cell referenced, r2c5.

[eleven]

A Forcing Chain assumes a starting node to be true or false and looks for a contradiction.
An AIC observes that two strongly linked nodes must hold at least one truth, and without assuming anything, looks for candidates that can't be true as a result.

From a practical viewpoint AICs are generally more productive. Unlike for Forcing Chains, once AIC segments are found, they can be re-used freely as they don't depend on any assumption (or guess as some would say). The solutions to multiple step puzzles often demonstrate this.

I don't want to discuss notation issues, i accept anything i can understand.

But this is not true.
Different to a contradiction chain a forcing chain (which consists of 2 or more chains with alternative starting points) does not look for a contradiction, but for a common output (implication) like this cell must or cannot be a number. There is no logical difference between a forcing chain starting with 2 possibilities like r1c1 = 2 or 3 and an AIC using the link (2=3)r1c1. You can split an AIC in each strong inference and make a forcing chain from there in both directions.
The advantage of forcing chains is, that there is no problem with starting with 3 or more possibilities, while you have troubles to write an AIC for that.
The only real advantage i see in AIC's is, when you have a loop. Since all possibilities for these cells are explicitly notated, it is easier to determine the candidates, which can be eliminated.

[daj95376]

When Jeff wrote his thread on Forcing Chains, he included numerous definitions and scenarios. Among his definitions are:

Implication Stream - a sequence of nodes and links where strong or weak inferences are made from one node to the other(s) unidirectionally from left to right. (Refer definitions for "node", "link", "strong inference" and "weak inferences" below)

Forcing Chain - a chain that has 2 or more implication streams that start from one node and end in another node where the outcomes of inferences merge from the 2 implication streams. In a forcing chain, a node can only infer the next successive node downstream.

Forcing Net - same as a forcing chain, except that in a forcing net, a node could infer 2 or more nodes downstream. Such inference is regarded as a multiple inference. (Refer definition of "multiple inference" below)

These definitions seem to match eleven's comments above. However, Jeff also includes:

Error Net or SIN (Single Implication Network) - a network with one implication stream that starts with a candidate selected in one node and propagates with or without multiple inferences until a contradiction is revealed. Due to this contradiction, such as 'empty cell', 'one digit appears 2 times in a unit' and 'no place for a digit in a unit', it can be concluded that the candidate selected at the start is invalid. This is also the principle of a "backtest" (Refer definition of a backtest below).

This provides the contradiction that DPB mentioned, but only when a single implication stream is involved. I have a SIN module in my solver, and its output is often difficult to translate into chain notation. However, it's common that I can translate the output into a Kraken Cell/Column/Row.

I've been in several discussions on AIC vs. Forcing Chains. Myth Jellies tried to explain their difference in another forum, but I was unable to understand his position. Finally, aran convinced me that any candidate that forces both ends of an AIC to be false must itself be false ... and can be eliminated. Bottom Line: a Forcing Chain can be constructed from any AIC strong inference and produce identical eliminations (mentioned by eleven), but the underlying principles are different.

[DonM]

When Jeff wrote his thread on Forcing Chains, he included numerous definitions and scenarios. Among his definitions are:

Implication Stream - a sequence of nodes and links where strong or weak inferences are made from one node to the other(s) unidirectionally from left to right. (Refer definitions for "node", "link", "strong inference" and "weak inferences" below)

Forcing Chain - a chain that has 2 or more implication streams that start from one node and end in another node where the outcomes of inferences merge from the 2 implication streams. In a forcing chain, a node can only infer the next successive node downstream.

Forcing Net - same as a forcing chain, except that in a forcing net, a node could infer 2 or more nodes downstream. Such inference is regarded as a multiple inference. (Refer definition of "multiple inference" below)

These definitions seem to match eleven's comments above. However, Jeff also includes:

Error Net or SIN (Single Implication Network) - a network with one implication stream that starts with a candidate selected in one node and propagates with or without multiple inferences until a contradiction is revealed. Due to this contradiction, such as 'empty cell', 'one digit appears 2 times in a unit' and 'no place for a digit in a unit', it can be concluded that the candidate selected at the start is invalid. This is also the principle of a "backtest" (Refer definition of a backtest below).

This provides the contradiction that DPB mentioned, but only when a single implication stream is involved. I have a SIN module in my solver, and its output is often difficult to translate into chain notation. However, it's common that I can translate the output into a Kraken Cell/Column/Row.

I've been in several discussions on AIC vs. Forcing Chains. Myth Jellies tried to explain their difference in another forum, but I was unable to understand his position. Finally, aran convinced me that any candidate that forces both ends of an AIC to be false must itself be false ... and can be eliminated. Bottom Line: a Forcing Chain can be constructed from any AIC strong inference and produce identical eliminations (mentioned by eleven), but the underlying principles are different.

IMO, these terms can have different meanings/implications depending on whether you are programming a solver or manually solving a puzzle. As a generic term, discontinuous AICs are forcing chains and if one is programming a computer solver, the difference between the two would probably be negligible.

However, as a manual solving method, a forcing chain is quite different from an AIC. With the former, in the simplest form you plug in one value in a starting bivalue cell and sees how what it results in in another bivalue cell and then plug in the other value and see if the results in the other cell are the same (for instance, lead to the same exclusion). When manual solvers first started using the 'forcing chain method', they were often solving simple xy-chains.

When it comes to more difficult puzzles with few, if any, bivalue cells, the weakness in the forcing chain method for manual solvers becomes more apparent, but again, not so much if one is using or programming a computer solver.

SINs were interesting from a theoretical point-of-view, but in practice the concept, not the least of which when it came to notation, was unwieldy for manual solvers and was never picked up as a practical manual method (see Carcul's: the-notation-used-in-nice-loops-and-sins-t3628.html). In fact, the only time I recall ever seeing a SIN solution, manually derived or otherwise, used in a puzzle was from your solver.

The way I see it, the introduction of AICs pretty much brought to an end the discussion of things like forcing chains and SINS. AICs allowed for a starting point and overall propagation that was less assumptive, allowed for the use of more complex constructs and/or nets and blended characteristics that had previously been assigned to the terms 'forcing chains' and SINs.

[David P Bird]

In <Sudopedia> it states "A Forcing Chain is therefore also known as an implication chain" which I the understanding I've been using.

Myth Jellies' line of argument was that solvers don't need to consider which of the linked candidates was true or false to be able to construct an AIC, they just have to follow the linking rules. In comparison an "implication chain" starts by assuming a candidate is true and follows the consequences.

At the time I raised the issue that the "Sudoku Thought Police" could never possibly know what mental processes solvers used before they notated their solutions as AICs, so it could only be a matter of conscience.

If there are two implication chains that start from strongly linked candidates, then together more often than not they are equivalent to an AIC.

If two implication streams starting from the same candidate produce a contradiction, then that candidate is false. Again they can usually be translated into AICs: strongly or weakly link the candidates producing the contradiction and work the two legs back to the original starting candidate.

AFAIAC where the contradiction involves more than two chains the result is generally a net (and always when the chains all have different lengths).

[eleven]

However, as a manual solving method, a forcing chain is quite different from an AIC.

Excuse me, but this is just rubbish. Logically they are exactly the same (when you restrict forcing chains to 2 alternative starting points), as well as for a manual solver. What is the difference, when you say "one of x and y must be true, if x then .. and if y then ..." or (x=y) with an AIC left and right?

When it comes to more difficult puzzles with few, if any, bivalue cells, the weakness in the forcing chain method for manual solvers becomes more apparent

Do you really know, what a forcing chain is ?

SINs were interesting from a theoretical point-of-view, but in practice the concept, not the least of which when it came to notation, was unwieldy for manual solvers and was never picked up as a practical manual method

If you ever have tried to solve hard puzzles manually, you will know that when trying to expand one direction of an AIC, it will happen, that this number runs into a contradiction. Embarrassing ?