The New Sudoku Players' Forum

[ArkieTech]

Code: Select all

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


Play/Print this puzzle online

[SpAce]

Code: Select all

.-------------------------.-----------------.-----------------.
|    3456      345  7     | 158  14689  489 | 2      38   369 |
|    2         1    368   | 58   7      89  | 359    4    369 |
|    9         45   468   | 3    468    2   | 57     78   1   |
:-------------------------+-----------------+-----------------:
|  a[37]     a[39]  5     | 178  18     6   | 138-9  2    4   |
| ab[37](4)    6    14-39 | 178  2    b(34) | 1389   5  b(39) |
|    8         2    134   | 9    45     345 | 13     6    7   |
:-------------------------+-----------------+-----------------:
|    456       459  469   | 2    59     1   | 37     37   8   |
|    1         8    2     | 6    3      7   | 4      9    5   |
|    35        7    39    | 4    589    589 | 6      1    2   |
'-------------------------'-----------------'-----------------'

(397)b4p124 = (439)r5c169 => -9 r4c7, -39 r5c3; stte

[Cenoman]

Code: Select all

 +----------------------+----------------------+--------------------+
 |  3456   345   7      |  158   14689   489   |  2      38   369   |
 |  2      1     368    |  58    7       89    |  359    4    369   |
 |  9      45    468    |  3     468     2     |  57     78   1     |
 +----------------------+----------------------+--------------------+
 |  37z   b39z   5      |  178   18      6     |  138-9  2    4     |
 |Aa347y   6     134-9  |  178   2      B34    |  1389   5   B39    |
 |  8      2     134    |  9     45      345   |  13     6    7     |
 +----------------------+----------------------+--------------------+
 |  456    459   469    |  2     59      1     |  37     37   8     |
 |  1      8     2      |  6     3       7     |  4      9    5     |
 |  35     7     39     |  4     589     589   |  6      1    2     |
 +----------------------+----------------------+--------------------+

Death blossom, stem (347)r5c1
(3)r5c1* - (3=9)r4c2
(4)r5c1 - (4=39)r5c69*
(7)r5c1 - (7=39)r4c12*
=> -9 r5c3, r4c7 (-3 r5c3*); ste

Just another presentation of SpAce's solution, so

Code: Select all

 +----------------------+----------------------+--------------------+
 |  3456  e345   7      |  158   14689   489   |  2     d38   369   |
 |  2      1     368    |  58    7       89    |  359    4    369   |
 |  9     c45    468    |  3     468     2     | d57    d78   1     |
 +----------------------+----------------------+--------------------+
 |  7-3  fa39*   5      |  178   18      6     |  1389   2    4     |
 |  47-3   6     1349   |  178   2       34    |  1389   5    39    |
 |  8      2     134    |  9     45      345   |  13     6    7     |
 +----------------------+----------------------+--------------------+
 |  456  ba459*  469    |  2     59      1     |  37     37   8     |
 |  1      8     2      |  6     3       7     |  4      9    5     |
 | a35*    7     39     |  4     589     589   |  6      1    2     |
 +----------------------+----------------------+--------------------+

Almost XY-wing
[(3=5)r9c1-(5*=9)r7c2-(9=3)r4c2] = (4)r7c2 - (4=5)r3c2 - (5=783)b3p278 - r1c2 = (3)r4c2 => -3 r45c1, ste

[Ngisa]

Code: Select all

+---------------------+----------------------+-------------------+
| 3456    345    7    | 158    14689    a489 | 2       38    369 |
| 2       1      368  | 58     7        a89  | 359     4     369 |
| 9       45     468  | 3      468       2   | 57      78    1   |
+---------------------+----------------------+-------------------+
|c37      39     5    | 178    18        6   | 1389    2     4   |
|c347     6      1349 | 178    2         3-4 | 1389    5     39  |
| 8       2      134  | 9      45        345 | 13      6     7   |
+---------------------+----------------------+-------------------+
| 456     459    469  | 2      59        1   | 37      37    8   |
| 1       8      2    | 6      3         7   | 4       9     5   |
|c35      7      39   | 4      589      b589 | 6       1     2   |
+---------------------+----------------------+-------------------+


(4=89)r12c6 - (89=5)r9c6 - (5=374)r945 => - 4r5c6; stte

Clement

[Sudtyro2]

Code: Select all

+-----------------+-----------------+-----------------+
| 3456  345  7    | 158  14689 489  |   2     38  369 |
| 2     1    368  | 58   7     89   |   359   4   369 |
| 9     45   468  | 3    468   2    |   57    78  1   |
+-----------------+-----------------+-----------------+
| 37   a39   5    | 178  18    6    |  138-9  2   4   |
| 347   6   b1349 | 178  2     34   |  1389   5  d39  |
| 8     2   c134  | 9    45    345  | d13     6   7   |
+-----------------+-----------------+-----------------+
| 456   459  469  | 2    59    1    |  37     37  8   |
| 1     8    2    | 6    3     7    |  4      9   5   |
| 35    7    39   | 4    589   589  |  6      1   2   |
+-----------------+-----------------+-----------------+

9r4c2 = (9-1)r5c3 = r6c3 - (1=39)b6p67 => -9 r4c7; stte

SteveC

[SteveG48]

Code: Select all

 *--------------------------------------------------------------------*
 | 3456   345    7      | 158    14689  489    | 2      38     369    |
 | 2      1      368    | 58     7      89     | 359    4      369    |
 | 9      45     468    | 3      468    2      | 57     78     1      |
 *----------------------+----------------------+----------------------|
 |a37     9-3    5      | 178    18     6      | 1389   2      4      |
 |b347    6     b149-3  | 178    2     b34     | 1389   5     b39     |
 | 8      2      134    | 9      45     345    | 13     6      7      |
 *----------------------+----------------------+----------------------|
 | 456    459    469    | 2      59     1      | 37     37     8      |
 | 1      8      2      | 6      3      7      | 4      9      5      |
 | 35     7      39     | 4      589    589    | 6      1      2      |
 *--------------------------------------------------------------------*


(3=7)r4c1 - (7=3491)r5c1369 - (1=473)b4p149 => -3 r4c2,r5c3 ; stte

Hmm. Looks like another variant on SpAce's solution.

[Sudtyro2]

(4=89)r12c6 - (89=5)r9c6 - (5=374)r945 => - 4r5c6; stte

Hi Clement,
I like your AIC path, but the second node (a single cell) is actually an AALS and should be written as (8|9=5)r9c6. What might be even better is to combine your first two nodes into a single ALS. That would give a nice 2-node AIC solution...
(4=895)r129c6 - (5=374)r459c1 => -4r5c6

SteveC

[SpAce]

Hi Steve,

(4=89)r12c6 - (89=5)r9c6 - (5=374)r945 => - 4r5c6; stte

Hi Clement,
I like your AIC path, but the second node (a single cell) is actually an AALS and should be written as (8|9=5)r9c6. What might be even better is to combine your first two nodes into a single ALS.

I agree on both accounts, but it's a waste of breath considering how many times it has been brought up already. That being said, some prominent sudoku masters wouldn't even see anything wrong with it, because it can be argued that the '|' is implied between multiple candidates in a single cell (there's no real ambiguity). It's true, but I still don't like it because it's harder to read for someone used to the explicit form. It also makes it easy to make mistakes when it's actually needed.

That would give a nice 2-node AIC solution...
(4=895)r129c6 - (5=374)r459c1 => -4r5c6

That leads to an interesting terminology question that has been bothering me for a long time. People seem to use "node" for two different meanings (I have too), which is a bit confusing. I don't know if it's actually defined anywhere, but personally I think any boolean argument separated by a link is a node. Therefore I see four nodes in that chain. After all, it's just a shorthand of this:

(4)r1c6 = (895)r129c6 - (5)r9c1 = (374)r945c1 => -4 r5c6

My solution actually has two nodes because it only has one link.

To avoid ambiguity, I suggest that we stick to this definition of "node" and use something else for the other meaning. Personally I've been calling it "term". Thus, your chain above would have four nodes and two terms, while the longer form I wrote has four of both. Accordingly, Clement's original has six nodes and three terms.

[Sudtyro2]

That would give a nice 2-node AIC solution...
(4=895)r129c6 - (5=374)r459c1 => -4r5c6

That leads to an interesting terminology question that has been bothering me for a long time. People seem to use "node" for two different meanings (I have too), which is a bit confusing. I don't know if it's actually defined anywhere ...

Sudopedia defines "node" here, which "includes complex structures such as ALS." My suggestion above was to simply link two clearly separate ALS. For now, I'll stick with Sudopedia's definition.

SteveC

[SpAce]

Sudopedia defines "node" here, which "includes complex structures such as ALS." My suggestion above was to simply link two clearly separate ALS.

Oh, you found a definition? Great! Except that it doesn't really resolve this issue at all:

A node is an item in a chain or loop. It can be a cell, a candidate or a complex structure like an Almost Locked Set.

The term vertex is sometimes used as an alias.

The nodes are connected with links or edges.

None of that disagrees with my adopted definition. Too bad it doesn't really disagree with yours either, though I think especially the last line is more in line with my interpretation (but not conclusively). Maybe it could also be argued that the first line is more in line with yours, but it's not conclusive either. In your definition a node can have an internal link and contain two booleans. In my definition all linked booleans are their own nodes. If your definition is correct, then what should we call the internal booleans? Either way we need a word for that.

My usage is based on that I see chains in terms of boolean logic instead of physical grid elements. That's why I care more about how many linked booleans a chain has than how the writer chose to group them or how they're manifested in the grid. That's especially true about chains with ALSs because they can be written in so many ways. For example, all of these are logically equivalent (and not even the only possibilities to write it):

Code: Select all

(4=895)r129c6 - (5=374)r459c1 => -4r5c6
(4)r1c6 = (895)r129c6 - (5)r9c1 = (374)r945c1 => -4 r5c6
(489)r129c6 = (895)r129c6 - (537)r945c1 = (374)r945c1 => -4 r5c6
(489=895)r129c6 - (537=374)r945c1 => -4 r5c6
(4=5)r129c6 - (5=4)r945c1 => -4 r5c6


If your definition of "node" is used, then how many nodes does each of those chains have? I think the second chain has four and all the rest have two. Would you agree? Either way, they all have exactly four linked booleans which are much easier to count accurately. What should we call them if not "nodes"?

It's also significant that the same logic can't be (easily) written with fewer linked booleans, which makes it different from my solution that could be written with just two (like I did) instead of the normal four of an ALS-XZ:

Code: Select all

(397)b4p124 = (439)r5c169 => -9 r4c7, -39 r5c3; stte
(397=4)b4p124 - (4=39)r5c69 => -9 r4c7, -39 r5c3; stte

Those two are actually different chains. The first one can only be written one way, but the latter has just as many options as the other example above (being a normal ALS-XZ). That's one reason why I like to write one-linkers when possible, because it's actually a significant difference between ALS patterns. I like to call the first form ALS-Z, because it skips the X part. All ALS-Z can be written as ALS-XZ, but not vice versa.

For now, I'll stick with Sudopedia's definition.

I could say that just as well. I don't though, because it's an ambiguous definition. It supports both interpretations pretty much equally, making it worthless. Thus I suggest we continue this discussion because I think it would be helpful to get an agreement about chaining terminology. Right now "node" doesn't have an unambiguous meaning (to me at least), which is very irritating because it's used a lot. Sudopedia doesn't seem to help much, so it's up to us to choose.

Clearly we need two different words for the two meanings. So far I've been using "node" for all linked booleans (also within parenthesized ALS) and "term" for the bigger chain chunks (including parenthesized ALS); i.e. number of nodes >= number of terms. Is the common usage the other way around, or something else?

I wish everyone would participate in this discussion because it would help the most if we had a consensus about it. I'm perfectly fine with switching to another usage of the terms if need be. It's quite possible that I'm the one that has been using them the wrong way. The only significant part for me is that we need two different words for two different concepts, and we should preferably agree to what they are.

[Sudtyro2]

I wish everyone would participate in this discussion because it would help the most if we had a consensus about it. I'm perfectly fine with switching to another usage of the terms if need be. It's quite possible that I'm the one that has been using them the wrong way. The only significant part for me is that we need two different words for two different concepts, and we should preferably agree to what they are.

9r4c2 = (9-1)r5c3 = r6c3 - (1=39)b6p67 => -9 r4c7; stte

Regarding my own posted solution (shown above) for this puzzle, I think the conventional interpretation of, say, the fourth node in the chain would most likely be the ALS (1=39)b6p67 rather than 1r6c3. However, I also fully agree that additional reader comments about this issue would be very much appreciated.

SteveC

[SpAce]

9r4c2 = (9-1)r5c3 = r6c3 - (1=39)b6p67 => -9 r4c7; stte

Regarding my own posted solution (shown above) for this puzzle, I think the conventional interpretation of, say, the fourth node in the chain would most likely be the ALS (1=39)b6p67 rather than 1r6c3.

After reading some old stuff, I think you might be right about the conventional interpretation. Even so, I think we still a better definition of "node". ALSs and other patterns confined to certain cells and possibly including an internal strong link are pretty clear-cut. What if you have a weak link within parentheses? Is that a node too? (I would say yes, and based on your counting above, I guess you would too.) What about patterns where the strong link is external, such as finned fishes or AHS? Are the fish body and the fin(s) two separate nodes? (Again, I would say yes.) What about parenthesized location links, like in my 3D notation? (That would be a maybe for me.)

I guess it would be simplest to say that everything within any type of brackets (and any element that doesn't need brackets) is a single node, whether it includes any type of internal link or not (or possibly many, like a nested AIC).

More importantly, if "node" is reserved for that, then we still need another name for the individual boolean elements. Your chain above has six of them. What do you call them? I'm still more interested in that part because there's no ambiguity about identifying and counting those pieces, and they're the ones that actually form the chain. For example, what do you call 1r5c3 in your chain? It's the third what?

However, I also fully agree that additional reader comments about this issue would be very much appreciated.

Indeed.

[SteveG48]

Interesting discussion.

I generally use "term" and "node" pretty much interchangeably, but when I think about it I'm inclined to think of a node as a unique cell or set of cells, and a term as a major element in a logic chain. Suppose, then, I write a simple chain:

(1=2)r1c1 - (2=3)r1c5

Here we have two nodes, r1c1 and r1c5. We also have two terms, (1=2)r1c1 and (2=3)r1c5. However, if we expand it and write:

1r1c1 = 2r1c1 - 2r1c5 = 3r1c5,

We now have 4 terms, but we still have 2 nodes.

[SpAce]

Thanks for the comments, Steve!

I generally use "term" and "node" pretty much interchangeably, but when I think about it I'm inclined to think of a node as a unique cell or set of cells, and a term as a major element in a logic chain. Suppose, then, I write a simple chain:

(1=2)r1c1 - (2=3)r1c5

Here we have two nodes, r1c1 and r1c5. We also have two terms, (1=2)r1c1 and (2=3)r1c5. However, if we expand it and write:

1r1c1 = 2r1c1 - 2r1c5 = 3r1c5,

We now have 4 terms, but we still have 2 nodes.

So... you're complicating things even further!! That's ok. I'm actually willing to buy your definitions. They make more sense than anything I suggested before.

HOWEVER, now we need a third distinct name for the individual boolean elements!! No one has still said anything about that. For me it's the most important question, and it remains totally unanswered. Well, not totally, since we now know that they're neither nodes nor terms (if the definitions above are accepted). So what are they?

[SteveG48]

So what do you mean by an individual Boolean element? Any one of the four terms in my expanded example? If so, I'd call them simplified terms. A simplified term would represent the truth value of a single candidate in a single node.