The New Sudoku Players' Forum

[Myth Jellies]

(at least I think it is new )

The Multi-Spoking pattern

While playing around with Ruud's idea of tri-coloring and combining it with some characteristics that I have seen while using the molecular method, I came up with the following theoretical base pattern.

[edit - inserting latest description on the spoke pattern in the initial post here for tutorial reasons.]

The spoke pattern consists of a hub, several spokes, and a rim.

The hub is a small piece of the puzzle (often a single cell) where the start of several spokes are all weakly linked to each other.

A spoke is an alternating inference chain (AIC) segment which starts with a strong link out of the hub and ends with a weak link to the rim.

The rim consists of several cells, all in the same house, which form an A^nLS where...

n = number of spokes - 1
A^nLS = N cells which contain candidates for N+n different digit values
A^0LS = LS = Locked Set
A^1LS = ALS = Almost Locked Set
A^2LS = AALS, etc. etc.

The spoke pattern forms a closed AIC network loop which allows you to make the following deductions:

Exactly one of the spoke starting candidates must be true.
All weak links in every spoke may be converted to conjugate links.
The rim cell must contain any digits that are not the weak endpoint of a spoke.

Here is a link to conceptual examples further into the post.

[end edit for tutorial purposes, but note that these early examples came before the spoke pattern was significantly characterized]

Code: Select all

Triple Spoke
 +----------------+-------------------+----------------+
 | .    124  .    | 1A*     .    134  | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+
 | .    2B*  .    | 1a2b3c* .    .    | .    .    .    |
 | .    .    .    | .       .    3C*  | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+
 Legend:  1a & 1A are conjugate colors for digit 1
          2b & 2B are conjugate colors
          3c & 3C are conjugate colors
          * represents any number of extra digits
 r4c4 is the hub cell
 r1c246 are the rim cells

Deduction: Since the colors a, b, and c all see each other, at most only one of them can be true. That means the at least two of the colors A, B, and C must be true. If A & B are true then r1c2 = 4. If A & C are true then r1c6 = 4. If B & C are true then r1c26 equals a locked 14 set. In all cases, r1c26 = 4 and you can remove the 4's from every other candidate in the row. It doesn't stop there, though. We've also just shown that if A, B, and C are true, then we would have two 4's in row 1. Since they all can't be true, exactly one of a, b, or c is true. Thus we can also eliminate all the extra digits in the spoke hub r4c4.

I like this pattern because it can make significant eliminations without requiring any bivalued cells. I have no idea how prevalent it might be (I don't have a single real world example yet) but it might be a key to unlocking some of the harder puzzles. It also seems to have a lot of possibilities for extensibility (can have N spokes in the hub and N digits in the rim cells) and a lot of potential variants to be explored.

From an AIC net perspective (the three endpoint nodes are marked with stars):

Code: Select all


           +----------------------------+
           |                            |
  *(4&1=2or3)r1c26    ------(3c)r4c4 = (3C)r5c6 -----+
        |            /       |                       |
       (2B)r4c2 = (2b)r4c4   |                   (1or3=4)r1c6*
        |            \       |                    |
    *(4=2or1)r1c2     ------(1a)r4c4 = (1A)r1c4 --+
           |                            |
           +----------------------------+

If you use multidigit multicoloring or B&B plots, you can detect potential spoke hubs fairly easily.

I wanted to call this multi-forking; but, since a fork is already a technique on some other forums, I went with spokes (like on a wheel) instead. Probably for the best because it looks more like the spokes on a bicycle than an eating utensil anyway .

[Steve R]

That's a very nice piece of work on compound loops.

You may have sold yourself short on the eliminations, though. I see it as a continuous nice loop, meaning that in addition 1 can be eliminated from the rest of the first row, 2 from the rest of the second column and 3 from the rest of the sixth.

Steve

[Steve K]

I absolutely love this type of wrap around forbidding chain (nice loop).

Here is how one can represent this pattern as a forbidding chain. Take note that I generalize the variables within a forbidding chain to be Boolean variables. This allows forbidding chains to contain forbidding chains.

I may typo, as I am used to using completely different grid designations for rows and columns. Also, == , a strong link, means at least one of the two endpoints must be true, and --, a weak link, means at most one of the two endpoints can be true. {}'s contain Boolean variables that are expressed as forbidding chains.

r4c4=2 == r4c2=2 -- r1c2=2 == {r4c4=3 == r5c6=3 -- r1c6=3 == {r1c2=1 == r1c2=4 -- r1c6=4 == r1c6=1} -- r1c4=1 == r4c4=1}

Since all the chains contained within the larger wrap around (or nice loop) chain are also wrap around (or nice loop chains), by theory all the weak links in alll the chains are also strong. Thus we conclude:

The following are strong sets: (at least one item of the set is true)
{r4c4=123}
{r14c2=2}
{r15c6=3}
{r1c26=4}
{r1c246=1}

A forbidding matrix gives a much more transparent analysis.

Not sure how to produce tables in this forum, so until I learn, I will not be able to post with the efficiency shown by others here. For that reason, I will forgo the forbidding matrix.

Contained forbidding chains is a simple concept (albeit with complex applications) that I use frequently on extremely difficult puzzles.The use of this idea allows one to generalize almost all sudoku techniques under the umbrella of forbidding chains: Lots of techniques, such as swordfish, triples, ALS are really just contained forbidding chains.

I really have no special names for the techniques that I use, as they are merely the logical consequence of treating the variables within a forbidding chain as Boolean Variables.

[Steve K]

In my last post, I forgot to mention that since there are 5 proven strong sets, there are 5 sets of potential forbiddings. Steve R mentioned 4 of them. The last one is that one can eliminate 4 from the rest of row 1.

[Myth Jellies]

That's a very nice piece of work on compound loops.

You may have sold yourself short on the eliminations, though. I see it as a continuous nice loop, meaning that in addition 1 can be eliminated from the rest of the first row, 2 from the rest of the second column and 3 from the rest of the sixth.

Steve

Good point. For some odd reason, I stopped short of considering those.

It is rather obvious, too. Either 2B is true, or 1A and 3C is true. In the first case r4c2 is 2, in the second r1c2 must be 2. Similar logic will apply for the other numbers as well.

[Myth Jellies]

Time to characterize the various part of the spoke pattern a bit better.

The hub is the branching part where two or more colors or candidates with strong links all see each other. The simplest form of a hub is a simple multi-valued cell with several strong links radiating out from it.

A spoke is an alternating chain that starts with a strong link from the hub and ends with a strong or weak link to the rim. A strong spoke ends with a strong link, and a weak spoke ends with a weak link.

The spoke digit is the value of the candidate on the rim that the spoke connects to. The spoke digit may be different from the starting digit in the hub.

The rim is the house where all the spokes end. A rim cell is the cell where a spoke ends.

A rim cell for a strong spoke can contain any number of candidates.

A rim cell for a weak spoke can contain up to N candidates where N is the number of spokes used in the pattern. One of those candidates is assumed to be the weak spoke digit.

There must be at least one weak spoke.

Set digits in the rim cell of a weak spoke are all the spoke digits of all the strong spokes. Set digits do not actually have to be there in the cell, but they take up an elective digit slot.

You can have up to M elective digits in each weak spoke rim cell where M equals the number of spokes minus 1, minus the number of strong spokes in the pattern. The minus one is because it is assumed that the end of the spoke lies in the rim cell. The elective digits must be the same for all weak spoke rim cells.

If all the requirements for the spoke pattern are met you have a closed AIC net where all binary weak links may be treated as conjugate links, the elective digits are locked into the weak rim cells, and the hub must contain one of the spoke starting candidates.

Those rules are pretty wordy. What you probably need is some visualization aids. Here are some representative spoke patterns. In these patterns, the lower digits are the spoke digits and the higher digits are the elective digits. A '*' represents any number of extra digits. Lowercase/uppercase letters next to a digit indicate candidates that are strongly linked. A '.' can be filled in with any number of digits so long as the strong links indicated still hold. In all cases shown, the rim is row 1.

Code: Select all

Double-Spoke (no strong spokes)
 +----------------+-------------------+----------------+
 | .    19   .    | .       29   .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+
 | .    1A*  .    | 1a2b*   .    .    | .    .    .    |
 | .    .    .    | .       2B*  .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+

Note that this is just a simple AIC loop or continuous mixed loop.

Code: Select all

Double-Spoke (one strong spoke)
 +----------------+-------------------+----------------+
 | .    .    .    | 1A*     12   .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+
 | .    .    .    | 1a2b*   .    .    | .    .    .    |
 | .    .    .    | .       2B*  .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+

Yawn, another AIC loop. Note how the strong spoke took away the elective digit option for the weak spoke, though.

Code: Select all

Triple-Spoke (no strong spokes)
 +----------------+-------------------+----------------+
 | .    189  .    | .       289  389  | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+
 | .    1A*  .    | 1a2b3c* .    .    | .    .    .    |
 | .    .    .    | .       2B*  .    | .    .    .    |
 | .    .    .    | .       .    3C*  | .    .    .    |
 +----------------+-------------------+----------------+
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+

Now it's a little more interesting. Note we get two elective digits here. They have to be the same in all rim cells though.

Code: Select all

Triple-Spoke (one strong spoke)
 +----------------+-------------------+----------------+
 | .    .    .    | 1A*     129  .    | 139  .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+
 | .    .    .    | 1a2b3c* .    .    | 3C*  .    .    |
 | .    .    .    | .       2B*  .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+

One of the elective digits goes away.

Code: Select all

Triple-Spoke (two strong spokes)
 +----------------+-------------------+----------------+
 | .    .    1D*  | 2B*     .    123  | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+
 | .    1A*  .    | 1a2b3c* .    .    | .    .    .    |
 | .    .    1d*  | .       .    .    | .    .    .    |
 | .    .    .    | .       .    3C*  | .    .    .    |
 +----------------+-------------------+----------------+
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+

No elective options here.

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Quad-Spoke (no strong spokes)
 +----------------+---------------------+----------------+
 | .    1789 .    | .         2789 3789 | 4789 .    .    |
 | .    .    .    | .         .    .    | .    .    .    |
 | .    .    .    | .         .    .    | .    .    .    |
 +----------------+---------------------+----------------+
 | .    1A*  .    | 1a2b3c4d* .    .    | 4D*  .    .    |
 | .    .    .    | .         2B*  .    | .    .    .    |
 | .    .    .    | .         .    3C*  | .    .    .    |
 +----------------+---------------------+----------------+
 | .    .    .    | .         .    .    | .    .    .    |
 | .    .    .    | .         .    .    | .    .    .    |
 | .    .    .    | .         .    .    | .    .    .    |
 +----------------+---------------------+----------------+

Etc. etc. Three elective digits--you get the idea.


The hub does not have to be a single cell. You can have something like the following...

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Triple-Spoke (no strong spokes)
 +----------------+-------------------+----------------+
 | .    189  .    | .       289  389  | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+
 | .    1A*  .    | 1a*     .    .    | .    .    .    |
 | .    .    .    | .       1b2B .    | .    .    .    |
 | .    .    .    | .       .    1c3C | .    .    .    |
 +----------------+-------------------+----------------+
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+

Note that 1a, 1b, and 1c all exclude each other to form your hub.


You can also have the rim be a box, say box two...

Code: Select all

Triple-Spoke (one strong spoke)
 +----------------+-------------------+----------------+
 | .    .    .    | .       129  .    | .    .    .    |
 | .    .    .    | 1A*     .    .    | .    .    .    |
 | .    .    .    | .       .    139  | .    .    .    |
 +----------------+-------------------+----------------+
 | .    .    .    | 1a2b3c* .    .    | .    .    .    |
 | .    .    .    | .       2B*  .    | .    .    .    |
 | .    .    .    | .       .    3C*  | .    .    .    |
 +----------------+-------------------+----------------+
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+

Finally, you can have almost-spoke patterns. For example...

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Almost Triple-Spoke (no strong spokes)
 +----------------+-------------------+----------------+
 | .    189  .    | .       289  3589 | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 | .   -.-   .    | .       15   .    | .    .    .    |
 +----------------+-------------------+----------------+
 | .    1A*  .    | 1a2b3c* .    .    | .    .    .    |
 | .    .    .    | .       2B*  .    | .    .    .    |
 | .    .    .    | .       .    3C*  | .    .    .    |
 +----------------+-------------------+----------------+
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+

Spoke Pattern =(or)= (5)r1c6 - (5=1)r3c5 => r3c2 <> 1

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Almost Triple-Spoke (no strong spokes)
 +----------------+-------------------+----------------+
 | .    189  .    | .       289  2389 | .    .    .    |
 | .    .    .    | .      -.-   .    | .    .    .    |
 | .    .    .    | .      -.-   .    | .    .    .    |
 +----------------+-------------------+----------------+
 | .    1A*  .    | 1a2b3c* .    .    | .    .    .    |
 | .    .    .    | .       2B*  .    | .    .    .    |
 | .    .    .    | .       .    3C*  | .    .    .    |
 +----------------+-------------------+----------------+
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+

Spoke Pattern = (2)r1c6 => r23c5 <> 2

Still thinking on how best to represent a spoke pattern in a chain, or succinctly represent a spoke pattern reduction.

[Steve K]

Very interesting pattern.

Here is the way my tangled mass of gray matter sees it:

It seems that all of them are variations of a "pigeonhole" pattern, and all form nice loops. As they get more complex, their representation as a forbidding chain (alternatinig inference chain" here) gets messy. I can write the general idea as a forbidding matrix, using the pigeonhole forbidding matrix. Additionally, this forbidding matrix type will use N strong sets, and prove N new strong sets.

The first Double Spoke Pattern with one strong spoke is much like a Y wing, except that each strong set is a vertex. I call this a Y wing style wrap around chain. As a chain, the idea is easy:
(A == A) -- (B == B) -- (C == C), but A -- C. Therefor:
A == B, C == B, A == C are the conclusions. As a forbidding pigeon hole matrix:

Code: Select all

Double Spoke (One strong spoke) (Y wing style)
 A | A | . |  (A's strong by location)(SL)
 B | . | B |  (B's strong by location)(SL)
 . | A | B |  (AB strong in cell)(SC)
 
First column is weak in a cell, all the other columns are weak by location.


In a typical pigeonhole forbidding matrix, all the rows are strong sets(at least one truth each). All the columns are except the first one are usually weak sets(atmost one truth each). Then by simple counting, one prove the first column strong.
All of these patterns, however, are nicely looped, or wrap around pigeonhole forbidding matrices. In these, all the columns are weak sets. Therefor, an NxN matrix will have at least N truths, but no more than N truths - thus exactly N truths. Furthermore, each row and each column will have exactly one truth each. The important conclusion is that the columns are now proven strong. Thus, anything mutually forbidden by each column member is clearly forbidden.

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Double Spoke (no strong spokes)
 A | A | . | . |     (strong by location) (SL)
 B | . | B | . |     (strong by location) (SL)
 . | A | . | C |     (strong in cell) (SC)
 . | . | B | C |     (strong in cell) (SC)

Again, first column weak in cell, rest of the columns are weak by location.



Here, of course, we can conclude AB are strong in their cell, while AA,BB,CC are all strong in their respective large containers(row,column, boxes). As we continue, a defiinate pattern can be seen using this style:

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Triple Spoke (two strong spokes)
 A | A | . | . | . |    (SL)
 B | . | B | . | . |    (SL)
 C | . | . | C | . |    (SL)
 . | A | . | . | A |    (SL)
 . | . | B | C | A |    (SC)

Again, the first column is weak in a cell, all the other columns are weak by location.

The symmetry is especially compelling to me.

Code: Select all

Triple Spoke (one strong spoke)
 A | A | . | . | . |       (SL)
 B | . | B | . | . |       (SL)
 C | . | . | C | . |       (SL)
 . | A | B | . | D |       (SC)
 . | A | . | C | D |       (SC)

First column weak in cell, rest weak by location

The above idea begs adding a C to the fourth row and a B to the fifth row. I have done a puzzle with exatly that configuration. It is sort of a hidden Sue de Coq cross naked Sue de Coq.

Code: Select all

Triple Spoke (no strong spokes)
 A | A | . | . | . | . |     (SL)
 B | . | B | . | . | . |     (SL)
 C | . | . | C | . | . |     (SL)
 . | A | . | . | D | E |     (SC)
 . | . | B | . | D | E |     (SC)
 . | . | . | C | D | E |     (SC)

First column weak in cell, rest weak by location

There might be a typo in Myth Jellies' Almost Triple-Spoke (no strong spokes) example. Assuming that 15 should be 25:

Code: Select all

 A | A | . | . | . | . | . |    (SL)
 B | . | B | . | . | . | . |    (SL)
 C | . | . | C | . | . | . |    (SL)
 . | A | . | . | D | E | . |    (SC)
 . | . | B | . | D | E | . |    (SC)
 . | . | . | C | D | E | F |    (SC)
 . | . | B | . | . | . | F |    (SC)

First column weak in cell, rest weak by location (row, column, box)

Interestingly enough, even after adding the ALS type configuration, there are still clearly N sets proven strong by N native strong sets.

In my opinion, then the pattern relies upon the following conditions:
Weakness in one cell from which emanates strength by location.
The strength in location converges back into strength in cells.
The strong cells use weakness in location to fold back into the original cell
weakness.

[Steve K]

I could not find the maximum depth 5 puzzle that I mentioned above. I created one that is probably similar:

Code: Select all

|.8.|.95|...|
|9.5|...|..7|
|...|.6.|...|
-------------
|82.|1..|...|
|3..|6..|.9.|
|..1|2..|3.8|
-------------
|1..|32.|...|
|26.|..7|.8.|
|...|...|...|

In this puzzle, after performing a locked candidate elimination on the 6's in row 1, one can immediately apply a triple spoke pattern starting from cell r9c8 and using strong cells r1c7, r1c9. One caveat though - the strength in location is grouped in box 3.

Happily, then the puzzle solves with "unique possibilities" - or singletons in cells, rows, columns, boxes - to the end. Just as happily, the puzzle is resistant to solution using most other methods that I am aware of.

[Steve K]

Sorry about the multiple posts:

My preferred technique is forbidding chains, and they are my preferred language to represent a technique. To make a technique such as this one easy to write, it may require allowing subsitution within a forbidding chain - called alternating inference chains here.

Suppose we have the multi spoke pattern, which essentially means:

n candidates {a1, ... ,an} share weakness in a cell - call this cell X
Each of these N candidates have a strength in location emanating from cell X.
These strengths, either directly or eventually, converge back into an Almost Locked Set configuration involving these N candidates, indirectly of directly, plus perhaps a few volunteer candidates. In every event, though, the ALS becomes a naked tuple upon the introduction of the implied strength from each candidate in {a1,...,an}

Let i,j be integers {1, ..,n} and i<>j.

Here is the forbidding chain: (AIC) (This is a mistake!! - corrected in later posts)

(X=ai) == (cell? = ai) -- {cells containing ALS = t} == {cells containing ALS = r} -- ( cell ?? = aj) == (X=aj)

Since X=aj -- X=ai, we have a nice loop on all ai. Since each weak link is strong, we have proven strong sets = number of native strong sets in the loop.
Of course, in most cases t=ai and r=aj, but that is not really a requirement. (albeit for t<>ai, for example, some chains exist within the weak link)

I am curious how others view this type of chain substitution. IMHO, it can be a powerful language tool in presenting a technique.

[Steve K]

I suppose that in the puzzle I posted, one must use an Almost Almost Locked Set configuration. The forbidding chain representation required for such a configuration is much more problematic, IMHO. I have a few drafts on this idea, but none of them is wholly satisfactory. I would be much obliged if someone can help me in such a case.

Although I can easily prove strength in the hub, showing transparently using only AIC (or forbidding chains) that the AALS becomes a naked tuple containing the volunteer candidate seems to elude me.

Naturally, though, a wrap around (or nicely looped) pigeonhole forbidding matrix easily provides transparency to all the eliminations.

Finally, since I am completely new here, if all this is just a repeat of what others have already done, I am sorry. Moreover, if it all seems just like garbage, I also apologize.

[Steve R]

MJ, I hope you won’t mind a couple of suggestions.

Does defining the rim as a house represent the clearest approach? Take your first example.

Code: Select all

Triple Spoke
 +----------------+-------------------+----------------+
 | .    124  .    | 1A*     .    134  | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+
 | .    2B*  .    | 1a2b3c* .    .    | .    .    .    |
 | .    .    .    | .       .    3C*  | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+
 Legend:  1a & 1A are conjugate colors for digit 1
          2b & 2B are conjugate colors
          3c & 3C are conjugate colors
          * represents any number of extra digits
 r4c4 is the hub cell
 r1c246 are the rim cells


If, instead, the rim, R, is taken as {r1c2, r1c6} we have a continuous nice loop:
R -1- r1c4 =1= H
R -2- r4c2 =2= H
R -3- r5c6 =3= H
R, like the hub, then forms part of the loop. It may of course lie in the intersection of a line and a box, when eliminations may be made in both houses.

Secondly I wonder whether the distinction between strong and weak spokes is altogether necessary:

Code: Select all

Triple-Spoke (one strong spoke)
 +----------------+-------------------+----------------+
 | .    .    .    | 1A*     129  .    | 139  .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+
 | .    .    .    | 1a2b3c* .    .    | 3C*  .    .    |
 | .    .    .    | .       2B*  .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 | .    .    .    | .       .    .    | .    .    .    |
 +----------------+-------------------+----------------+


Here (my suggested) R = {r1c5, r1c7}. We have:
R -1- r1c4 =1= H
R -2- r4c2 =2= H
R -3- r5c6 =3= H
Although the picture is different, the underlying theory is, I think, the same.

The general rule would be that the rim has n degrees of freedom and n + 1 (distinct weak) spokes entering it. The “elective digits” are just the candidates for the rim which are not the labels of any of these weak links. As you rightly observe, there may be none; in such cases eliminations are confined to the hub and the labels attached to the links in the continuous loop.

Steve

[Steve K]

Here is the forbidding chain: (AIC)

(X=ai) == (cell? = ai) -- {cells containing ALS = t} == {cells containing ALS = r} -- ( cell ?? = aj) == (X=aj)

But I was not thinking clearly:

Additonally, let P be a group of cells forming an ALS with (N-1) degrees of freedom.

Double spoke configuration: (N=2)

(X=ai) == (cell? = ai) -- { P= t} == {P = r} -- ( cell ?? = aj) == (X=aj)

N spoke configuration derived by induction:

(X=ai) == (cell? = ai) -- {P= t} =={N-1 Spoke}

[ronk]

The Multi-Spoking pattern

Very interesting, but I note that most of the diagrams are for continuous loops.

I've never seen statistics on this, but there are unquestionably many more deductions made with discontinuous loops rather than continuous ones. For the ALS xz-rule, for example, there are many more singly-weakly-linked sets than doubly-linked ones (sometimes called ALS xz mutual exclusion rule).

That observation coupled with the lack of actual examples, causes me to be skeptical of the usefulness of this technique. Does anyone have any real-world examples

[Steve K]

Perhaps better:

I suppose there is may be no reason to begin the induction at N=3?

N=1 spoke:

(X=ai) == (cell? = ai) -- {{ P= t} == false}}

Then the induction for all N>1.

(X=ai) == (cell? = ai) -- {P= t} == {(N-1) Spoke}

[Steve K]

Does anyone have any real-world examples

I have used this idea in the real world, but not often. More often, I have found a double hub. The following link is to a very tough puzzle that is advanced slightly with the technique. A double spoke pattern exists using 2 strong cells, plus 2 strong links out of the hub.

http://sudoku.com.au/1V8-6-2006-sudoku.aspx

Of course, there is also the puzzle I created above. (Does that disqualify it from the real world?)

Since the technique had no name at the time, it will be hard to search my archived puzzle proofs for futher examples in a timely fashion.