The New Sudoku Players' Forum

[Mauriès Robert]

Hi all,
I am relaunching a thread on the Oddagon in parallel with those I have read recently, namely :
Here http://forum.enjoysudoku.com/a-revival-of-broken-wings-t5225.html
Here http://forum.enjoysudoku.com/broken-wwing-a-new-use-for-the-turbot-fi-t2666.html
and Here http://forum.enjoysudoku.com/lozenges-8-3-t38439.html#p298073
because I don't find in these threads a clear definition of oddagon apart from the one of RodHagglund who called it Broken Wing.
It seems to me from reading these threads that the Oddagon has a broader definition than that of Broken Wing, even different according to the authors. Especially since it would be necessary to distinguish Oddagons with one, two, or even several numbers.
I understand that what characterizes an Oddagon is a pattern (odd loop, deadly pattern, etc.) leading to a contradiction if some candidates (guardians) are all eliminated. Would that be enough to define them all ?
For example, on this puzzle reported by SpAce, is it an Oddagon or not? For SpAce yes, for Berthier no. Then Oddagon or not Odaggon ?

Code: Select all

+---------------+---------------+-----------------+
|  5   1   27*  | 27*  3   9    | 8    4    6     |
| d27  9   3    |  6   8   4    | 1    5    2-7   |
|  6   4   8    |  1   5   27*  | 239  2379 237   |
+---------------+---------------+-----------------+
|  9   7   6    |  34  2   8    | 5    13   134   |
|  3   2   4    |  5   9   1    | 7    6    8     |
|  1   8   5    |  34  7   6    | 239  239  234   |
+---------------+---------------+-----------------+
|  8   5   1    |  27  4   3    | 6    27   9     |
|  4   6   27*  |  9   1   257* | 23   8   a2357# |
| c27  3   9    |  8   6  b257# | 4    127  1257  |
+---------------+---------------+-----------------+

In 7s, a 5-link oddagon(*) with two guardians(#).
(7)r8c9 == r9c6 - r9c1 = (7)r2c1 => -7 r2c9; stte

SteveC

I would therefore like to know the precise definition of the Oddagon or Oddagons in this thread.
Robert

[Ajò Dimonios]

I think oddagon refers to methods that use the illegal logic of odd length loops which are well described by Allan Barker http://sudoku.allanbarker.com/sweb/gen2/blacklog.htm , which fall under the methods that use the dark logic, meaning by dark logic sets that always have rank = -1.

[Mauriès Robert]

I think oddagon refers to methods that use the illegal logic of odd length loops which are well described by Allan Barker http://sudoku.allanbarker.com/sweb/gen2/blacklog.htm , which fall under the methods that use the dark logic, meaning by dark logic sets that always have rank = -1.

Thank you Paolo, it is an interesting document.
Robert

[StrmCkr]

broken wings were later upgraded to the name oddagon they are one and the same.

they stem from the observation of what happens in a 5 length chain
we can study it in reverse using a 4 length chain and figure out why they work.

skyscraper a 4 link x-chain

Code: Select all

+-----------+-------------+---------+
| .  .    . | -1   .    . | .  .  . |
| /  (1)  / | /    (1)  / | /  /  / |
| .  .    . | -1  .    .  | .  .  . |
+-----------+-------------+---------+
| .  .    . | .    -1   . | .  .  . |
| /  (1)  / | (1)  /    / | /  /  / |
| .  .    . | .    -1   . | .  .  . |
+-----------+-------------+---------+
| .  .    . | .    .    . | .  .  . |
| .  .    . | .    .    . | .  .  . |
| .  .    . | .    .    . | .  .  . |
+-----------+-------------+---------+

raising the question of what happens if box 4 is also a strong link making it a 5 length chain and it completes a loop.

which brings us to turbots the first and easiest oddagon {which now a days is associated with the one above as its shorter and easier}

Code: Select all

+------------+---------------+---------+
| .  .     . |  .   .      . | .  .  . |
| /  (-1)  / | /    (+1)  /  | /  /  / |
| .  .     . |  .   .     .  | .  .  . |
+------------+---------------+---------+
| .  .     . | /    /     /  | .  .  . |
| /  (+1)  / | (1)  /     /  | /  /  / |
| .  .     . | /   (1)    /  | .  .  . |
+------------+---------------+---------+
| .  .     . | .    .     .  | .  .  . |
| .  .     . | .    .     .  | .  .  . |
| .  .     . | .    .     .  | .  .  . |
+------------+---------------+---------+

that's the general idea extrapolated that odd length chains are internally cannibalistic when they are looped back to the start.

the markers for cannibalistic eliminations under broken wings was called guardians. {here it would be R2c5,R5C2)

from the above sky-scrapper noted above we also can conclude that the 4 chain version eliminations are still applicable and indeed reinforces the outcome of the 5 length chain.

definition is a chain of Odd length that is continuous {looped} with at least one part of the chain reused as a node.

as to your question is it an oddagon or not
S

udtyro2 wrote:

Code: Select all

Code: Select all

    +---------------+---------------+-----------------+
    |  5   1   27*  | 27*  3   9    | 8    4    6     |
    | d27  9   3    |  6   8   4    | 1    5    2-7   |
    |  6   4   8    |  1   5   27*  | 239  2379 237   |
    +---------------+---------------+-----------------+
    |  9   7   6    |  34  2   8    | 5    13   134   |
    |  3   2   4    |  5   9   1    | 7    6    8     |
    |  1   8   5    |  34  7   6    | 239  239  234   |
    +---------------+---------------+-----------------+
    |  8   5   1    |  27  4   3    | 6    27   9     |
    |  4   6   27*  |  9   1   257* | 23   8   a2357# |
    | c27  3   9    |  8   6  b257# | 4    127  1257  |
    +---------------+---------------+-----------------+

In 7s, a 5-link oddagon(*) with two guardians(#).
(7)r8c9 == r9c6 - r9c1 = (7)r2c1 => -7 r2c9; stte

SteveC

the problem is there is an underling fish that is relevant to the elimination and depending on whom answered didn't look past the shorter fish cycle to find the oddagon as they tend to overlap significantly as seen above.

like this one:
Finned Mutant Jellyfish: 7 r8c1b12 r1c69b7 efr2c1 => r2c9<>7

[Mauriès Robert]

Hi StrmCkr,
Thank you for all these explanations which comfort me in my understanding of the oddagon.
In the example of SteveC that I recall at the beginning of the thread, indeed the elimination of 7r2c9 can be done directly without using the oddagon pattern, but it is indeed an oddagon that Steve used.
Denis Berthier to whom I had asked the question (see here) considers that no because (this is my opinion) in his conception of the oddagon the guardians are z-candidates (see here in answer to eleven), which is not the case in this example. This leads me to say that Denis Berthier's conception of the oddagon is more restrictive. But perhaps Denis Berthier will express himself differently about this example and his conception of the oddagon.
Robert

[StrmCkr]

The kicker for this one is the
# guaridans delete r8c6 for digit 7, changing the chain to a full closed loop fullfilling the elimination
Which was there desite adding the extra col to the mix which made it more difficult to pinpoint how they were using it as an oddagon to begin with. The r8c7 elimination is also valid by the oddagon itself but its not mentioned.

[SpAce]

Hi StrmCkr,

broken wings were later upgraded to the name oddagon they are one and the same.

I don't quite agree with that. I think "oddagon" is a generalized version of "Broken Wings". But, as I said here, I make no claims about the official definitions.

[StrmCkr]

it could be, but i haven't seen the name broken wings used in a long time most of them have been replaced with oddagon which generalize it to all chains instead of limiting it to a singular digit chain.

broken-wings name was for the no-fish patterns where it was prominently used at the time, however years latter these are frequently associated with oddagon name
so I safely imply my comment.

I'm more surprised you didn't chime on in the turbot reference and counter what i said they used to be.

[Cenoman]

the problem is there is an underlying fish that is relevant to the elimination

I don't catch your point. Do you mean that the oddagon pattern should not be invoked when there is fish, how complex it might be ?

To me, the oddagon is another presentation of the same elimination and in the example above, it is far much simple and legible than the finned mutant jellyfish !

[StrmCkr]

It only appears simpler as it removed and shorted the notation to not disply all of cells of the chain for example start with r3c6
Which contains a self elimination to imply that r8c9 =7 (which breaks aic construction rules) as they arent bi directional chains.

There is smaller fish that resolve this elimination directly or indirectly
i picked one that used the same cells as close as possible.
For me id choose fish first, once there isnt fish left then i resort to nofish(brokenwing/oddagons) to explain if applicable.

Again my preference i find base/cover easier to understand then digesting the full notation of that chain

Try writing an aic for the jelly fish as it has less cells to use technically by 1.

Im not saying they arent applicable, my point was mearly some people trying to digest the full chain might see the fish and stop there and go oh we dont need an oddagon to explain this elimination.

[Pupp]

deleted

[denis_berthier]

.
Starting from the given resolution state:

Code: Select all

   +----------------+----------------+----------------+
   ! 5    1    27   ! 27   3    9    ! 8    4    6    !
   ! 27   9    3    ! 6    8    4    ! 1    5    27   !
   ! 6    4    8    ! 1    5    27   ! 239  2379 237  !
   +----------------+----------------+----------------+
   ! 9    7    6    ! 34   2    8    ! 5    13   134  !
   ! 3    2    4    ! 5    9    1    ! 7    6    8    !
   ! 1    8    5    ! 34   7    6    ! 239  239  234  !
   +----------------+----------------+----------------+
   ! 8    5    1    ! 27   4    3    ! 6    27   9    !
   ! 4    6    27   ! 9    1    257  ! 23   8    2357 !
   ! 27   3    9    ! 8    6    257  ! 4    127  1257 !
   +----------------+----------------+----------------+

there are several paths (as always).

The most elementary one doesn't involve any oddagon:

Code: Select all

finned-x-wing-in-columns: n7{c3 c4}{r1 r8} ==> r8c6 ≠ 7
finned-x-wing-in-columns: n7{c6 c8}{r3 r9} ==> r9c9 ≠ 7
finned-x-wing-in-rows: n7{r2 r8}{c9 c1} ==> r9c1 ≠ 7
stte

If you're specifically looking for oddagons and activate only them, there are 3 of them (may be more along other resolution paths):

Code: Select all

oddagon[5]: c4n2{r1 r7},r7n2{c4 c8},c8n2{r7 r3},r3n2{c8 c6},b2n2{r3c6 r1c4} ==> r3c8 ≠ 2
oddagon[5]: c4n7{r1 r7},r7n7{c4 c8},c8n7{r7 r3},r3n7{c8 c6},b2n7{r3c6 r1c4} ==> r3c8 ≠ 7
whip[1]: c8n7{r9 .} ==> r8c9 ≠ 7, r9c9 ≠ 7
oddagon[5]: c3n7{r1 r8},r8n7{c3 c6},c6n7{r8 r3},b2n7{r3c6 r1c4},r1n7{c4 c3} ==> r3c6 ≠ 7
stte

Both cases are perfectly defined patterns. I have no idea what "dark logic" could be.
Note that n7r2c9 can't be eliminated by an oddagon - but it can be eliminated by a z-chain[3]:

Code: Select all

z-chain[3]: b1n7{r2c1 r1c3} - r8n7{c3 c6} - r3n7{c6 .} ==> r2c9 ≠ 7
stte