The New Sudoku Players' Forum

[denis_berthier]

.

Code: Select all

     +-------+-------+-------+
     ! . . . ! . . . ! 3 . . !
     ! 2 . . ! . 8 . ! 1 . . !
     ! . 5 . ! . . 9 ! . . . !
     +-------+-------+-------+
     ! . 8 . ! . 1 . ! 9 . . !
     ! . . 6 ! 7 . . ! . 3 . !
     ! 4 . . ! . . 5 ! . . 2 !
     +-------+-------+-------+
     ! . . 3 ! . 2 . ! . . . !
     ! 1 . . ! . . 4 ! 7 . . !
     ! . 9 . ! 6 . . ! . 8 . !
     +-------+-------+-------+
......3..2...8.1...5...9....8..1.9....67...3.4....5..2..3.2....1....47...9.6...8.
SER = 8.4

Not a new puzzle, but I'm curious about something.

[Edit: changed the title of the thread]

[marek stefanik]

Two elementary steps:

Code: Select all

.-------------------.------------------.----------------------.
| 6789  1467  14789 | 1245  4567  1267 | 3     245679  456789 |
| 2     3467  479   | 345   8     367  | 1     45679   45679  |
| 3678  5     1478  | 1234  3467  9    | 2468  2467    4678   |
:-------------------+------------------+----------------------:
|f357   8    g57–2  | 234   1     236  | 9     4567    4567   |
|f59   a12    6     | 7     49   b28   | 458   3       1458   |
| 4     137   179   | 389   369   5    | 68    167     2      |
:-------------------+------------------+----------------------:
| 5678  467   3     | 1589  2    c178  | 456   14569   14569  |
| 1     26    258   | 3589  359   4    | 7     2569    3569   |
|e57    9     2457  | 6    d357  d137  | 245   8       1345   |
'-------------------'------------------'----------------------'

2r5c2 = (2–8)r5c6 = (8–7)r7c6 = 7r9c56 – (7=5)r9c1 – 5r45c1 = 5r4c3 => –2r4c3

Code: Select all

.------------------.-----------------.---------------------.
| 6789  17   14789 | 1245  4567  26  | 3    245679  456789 |
| 2     37   479   | 345   8    *36  | 1    45679   45679  |
|*3678  5    1478  |*1234 *3467  9   | 246  2467    4678   |
:------------------+-----------------+---------------------:
| 57–3  8    57    | 234   1    *236 | 9    4567    4567   |
| 59    2    6     | 7     49    8   | 45   3       1      |
| 4     137  179   | 39    369   5   | 8    67      2      |
:------------------+-----------------+---------------------:
| 58    4    3     | 589   2     7   | 56   1       569    |
| 1     6    258   | 3589  359   4   | 7    259     359    |
| 57    9    257   | 6     35    1   | 245  8       345    |
'------------------'-----------------'---------------------'

3r3c6\r4c1b2 => –3r4c1, stte

Marek

[RSW]

Also two steps:

Code: Select all

 +-----------------+----------------+--------------------+
 | 6789 1467 14789 | 1245 4567 1267 | 3    245679 456789 |
 | 2    3467 479   | 345  8    367  | 1    45679  45679  |
 | 3678 5    1478  | 1234 3467 9    | 2468 2467   4678   |
 +-----------------+----------------+--------------------+
 |d357  8   e257   | 234  1    236  | 9    4567   4567   |
 |d59  f12   6     | 7    49  g28   | 458  3      1458   |
 | 4    137  179   | 389  369  5    | 68   167    2      |
 +-----------------+----------------+--------------------+
 | 5678 467  3     | 1589 2   a17-8 | 456  14569  14569  |
 | 1    26   258   | 3589 359  4    | 7    2569   3569   |
 |c57   9    2457  | 6   b357 b137  | 245  8      1345   |
 +-----------------+----------------+--------------------+


(7)r7c6=(7)r9c56-(7=5)r9c1-(5)r45c1=(5-2)r4c3=(2)r5c2-(2=8)r5c6 => -8r7c6

Singles: 8r5c6 -8r6c4 -8r5c79 8r6c7 -8r3c7 2r5c2 -2r8c2 -2r4c3 6r8c2 -6r7c12 -6r12c2 -6r8c89 1r5c9 -1r6c8 -1r79c9 1r9c6 -1r7c46 -1r1c6 7r7c6 -7r7c12 -7r9c5 -7r12c6 4r7c2 -4r12c2 -4r9c3 -4r7c789 1r7c8

Code: Select all

 +----------------+-------------------+-------------------+
 | 6789 17  14789 | 1245   4567   26  | 3   245679 456789 |
 | 2    37  479   | 345    8     F36  | 1   45679  45679  |
 |#3678 5   1478  |*124-3 *467-3 #9   | 246 2467   4678   |
 +----------------+-------------------+-------------------+
 |#357  8   57    | 234    1     #236 | 9   4567   4567   |
 | 59   2   6     | 7      49     8   | 45  3      1      |
 | 4    137 179   | 39     369    5   | 8   67     2      |
 +----------------+-------------------+-------------------+
 | 58   4   3     | 589    2      7   | 56  1      569    |
 | 1    6   258   | 3589   359    4   | 7   259    359    |
 | 57   9   257   | 6      35     1   | 245 8      345    |
 +----------------+-------------------+-------------------+


Sashimi X-Wing: Digit 3 in columns 1, 6 rows (2), 3, 4, Fin: r2c6 => -3r3c45; stte

[denis_berthier]

.
Indeed, the puzzle is #38 from the Obi Wahn "NoFish" collection in this thread: http://forum.enjoysudoku.com/a-revival-of-broken-wings-t5225.html?hilit=NoFish#p40481:

I'm curious about which oddagons you can find (using no other rules), and their eliminations. Don't worry about the number of steps.

[RSW]

Interesting. The sashimi X-wing (if you can even call it a fish) in my solution didn't appear until after the AIC step.

[P.O.]

Code: Select all

 6789    1467    14789   1245    4567    1267    3       245679  456789           
 2       3467    479     345     8       367     1       45679   45679           
 3678    5       1478    1234    3467    9       2468    2467    4678             
 357     8      a×2±57   234     1       236     9       4567    4567             
 59     e1+2     6       7       49     d2+8     458     3       1458             
 4       137     179     389     369     5       68      167     2               
c56-78  c46-7    3       1589    2      c1+78    456     14569   14569           
 1       26     a2*58    3589    359     4       7       2569    3569             
b5+7     9      a24*57   6       357     137     245     8       1345   


c3n5{r4 r8r9} - r9c1{n5 n7} - r7n7{c1c2 c6} - c6n8{r7 r5} - r5n2{c6 c2} => r4c3 <> 2
singles:( r7c2b7 n4 r7c6b8 n7 r7c8b9 n1 r9c6b8 n1 r6c7b6 n8 r5c9b6 n1 r8c2b7 n6 r5c6b5 n8 r5c2b4 n2 )

Code: Select all

 6789    17      14789   1245    4567    26      3       245679  456789           
 2       ×37     479     345     8      b+36     1       45679   45679           
a±3678   5       1478    12×34   ×3467   9       246     2467    4678             
a+357    8       57      234     1      b2-36    9       4567    4567             
 59      2       6       7       49      8       45      3       1               
 4       137     179     39      369     5       8       67      2               
 58      4       3       589     2       7       56      1       569             
 1       6       258     3589    359     4       7       259     359             
 57      9       257     6       35      1       245     8       345   


c1n3{r3 r4} - c6n3{r4 r2} => r2c2 r3c4 r3c5 <> 3
ste.

[Cenoman]

No single, no locked set. In the initial resolution state, there is a POM elimination of 3r8c4

Code: Select all

 +------------------------+-----------------------+---------------------------+
 |  6789   1467   14789   |  1245   4567   1267   |  3      245679   456789   |
 |  2      3467*  479     |  345*   8      367*   |  1      45679    45679    |
 |  3678*  5      1478    |  1234   3467   9      |  2468   2467     4678     |
 +------------------------+-----------------------+---------------------------+
 |  357*   8      257     |  234*   1      236*   |  9      4567     4567     |
 |  59     12     6       |  7      49     28     |  458    3        1458     |
 |  4      137*   179     |  389    369    5      |  68     167      2        |
 +------------------------+-----------------------+---------------------------+
 |  5678   467    3       |  1589   2      178    |  456    14569    14569    |
 |  1      26     258     |  589-3  359    4      |  7      2569     3569     |
 |  57     9      2457    |  6      357    137*   |  245    8        1345     |
 +------------------------+-----------------------+---------------------------+

With the 3s in cells tagged "*", the following BTM (Block Triangular Matrix) can be built:

Code: Select all

 3r2c4 3r2c2       3r2c6
       3r3c1 3r4c1
 3r9c6 3r2c6 3r4c6
 3r4c4             3r4c6 3r4c1
                   3r2c2 3r6c2
------------------------------
-3r8c4

...or for matrices haters, as a kraken using an almost skyscraper and an almost kite:
(3)r2c4
||
(3)r2c2 - [r3c1 = r4c1 - r4c6 = r2c6] = (3)r9c6
||
(3)r2c6 - [r2c2 = r6c2 - r4c1 = r4c6] = (3)r4c4
-----------------------------------------------
=>-3r8c4

An oddagon eliminating the same candidate:

Code: Select all

 +------------------------+-----------------------+---------------------------+
 |  6789   1467   14789   |  1245   4567   1267   |  3      245679   456789   |
 |  2      3467*  479     |  345#   8      367*   |  1      45679    45679    |
 |  3678*  5      1478    |  1234   3467   9      |  2468   2467     4678     |
 +------------------------+-----------------------+---------------------------+
 |  357*   8      257     |  234#   1      236*   |  9      4567     4567     |
 |  59     12     6       |  7      49     28     |  458    3        1458     |
 |  4      137    179     |  389    369    5      |  68     167      2        |
 +------------------------+-----------------------+---------------------------+
 |  5678   467    3       |  1589   2      178    |  456    14569    14569    |
 |  1      26     258     |  589-3  359    4      |  7      2569     3569     |
 |  57     9      2457    |  6      357    137#   |  245    8        1345     |
 +------------------------+-----------------------+---------------------------+

Oddagon (3)r24, c16, b1 having three guardians (#)
(3)r24c4 == r9c6 => -3 r8c4

Not sure it helps to solve the puzzle.

[DEFISE]

My "Simplest First" in (W+S2) with insertion of "Oddagons":

Hidden Text: Show

1)
whip[4]: b9n1{r7c8 r9c9}- r5n1{c9 c2}- r5n2{c2 c6}- c6n8{r5 .} => -1r7c6
2)
whip[4]: b7n6{r7c1 r8c2}- c2n2{r8 r5}- r5n1{c2 c9}- c8n1{r6 .} => -6r7c8
3)
whip[5]: c2n3{r6 r2}- c1n3{r3 r4}- c6n3{r4 r9}- r9n1{c6 c9}- r5n1{c9 .} => -1r6c2
4)
whip[5]: c2n3{r6 r2}- c6n3{r2 r9}- b8n1{r9c6 r7c4}- c4n8{r7 r8}- c4n9{r8 .} => -3r6c4
5)
whip[2]: r2n3{c4 c2}- r6n3{c2 .} => -3r3c5
ODDAGON[5]: r2n3{c2 c6},c6n3{r2 r4},r4n3{c6 c1},c1n3{r4 r3},b1n3{r3c1 r2c2}, => -3r8c4
6)
whip[5]: r9c1{n5 n7}- r7n7{c1 c6}- c6n8{r7 r5}- r6c4{n8 n9}- r5n9{c5 .} => -5r5c1
Single(s): 9r5c1, 4r5c5
Alignment: 5r5b6 => -5r4c8 -5r4c9
7)
whip[4]: r6c2{n3 n7}- r6c3{n7 n1}- r3n1{c3 c4}- r3n3{c4 .} => -3r2c2
Single(s): 3r6c2, 3r3c1
Alignment: 3c5b8 => -3r9c6
Naked pairs: 57c1r49 => -7r1c1 -5r7c1 -7r7c1
Hidden pairs: 17r6c38 => -6r6c8
8)
whip[3]: r3c5{n7 n6}- c6n6{r1 r4}- c6n3{r4 .} => -7r2c6
9)
whip[4]: r7n7{c2 c6}- c6n8{r7 r5}- r5n2{c6 c2}- c2n1{r5 .} => -7r1c2
10)
whip[2]: r6n7{c8 c3}- b1n7{r1c3 .} => -7r2c8
11)
whip[4]: r9c6{n1 n7}- c1n7{r9 r4}- r6c3{n7 n1}- b1n1{r1c3 .} => -1r1c6
Single(s): 1r9c6
12)
whip[2]: c2n7{r2 r7}- c6n7{r7 .} => -7r1c3
13)
whip[3]: c6n7{r1 r7}- c2n7{r7 r2}- b1n6{r2c2 .} => -6r1c6
Hidden pairs: 36c6r24 => -2r4c6
14)
whip[4]: c4n1{r1 r3}- c4n2{r3 r4}- c4n3{r4 r2}- c4n4{r2 .} => -5r1c4
15)
whip[4]: r3c5{n6 n7}- c6n7{r1 r7}- c2n7{r7 r2}- b1n6{r2c2 .} => -6r1c5
16)
whip[4]: r1c6{n7 n2}- c4n2{r1 r4}- c4n3{r4 r2}- b2n5{r2c4 .} => -7r1c5
Single(s): 5r1c5
17)
whip[3]: r6n7{c8 c3}- c1n7{r4 r9}- c5n7{r9 .} => -7r3c8
18)
whip[4]: r1c6{n7 n2}- r5n2{c6 c2}- r5n1{c2 c9}- r6c8{n1 .} => -7r1c8
Alignment: 7c8b6 => -7r4c9
ODDAGON[15]: c2n1{r1 r5},r5c2{n1 n2},r5n2{c2 c6},r5c6{n2 n8},c6n8{r5 r7},r7c6{n8 n7},r7n7{c6 c2},b7n7{r7c2 r9c1},c1n7{r9 r4},r4n7{c1 c8},c8n7{r4 r6},r6c8{n7 n1},r6n1{c8 c3},c3n1{r6 r1},b1n1{r1c3 r1c2}, => -7r3c3
Alignment: 7b1r2 => -7r2c9
19)
whip[3]: r1n7{c9 c6}- r7c6{n7 n8}- c1n8{r7 .} => -8r1c9
Alignment: 8r1b1 => -8r3c3
20)
whip[3]: r3c3{n4 n1}- r6c3{n1 n7}- r2n7{c3 .} => -4r2c2
21)
whip[3]: r1n7{c9 c6}- r7n7{c6 c2}- c2n4{r7 .} => -4r1c9
22)
whip[3]: r4c9{n4 n6}- r6c7{n6 n8}- r3n8{c7 .} => -4r3c9
23)
whip[3]: r3n8{c9 c7}- r6c7{n8 n6}- c5n6{r6 .} => -6r3c9
24)
whip[3]: c2n4{r7 r1}- c2n1{r1 r5}- c9n1{r5 .} => -4r7c9
25)
whip[4]: r1n7{c9 c6}- r7n7{c6 c2}- r2n7{c2 c3}- r2n9{c3 .} => -9r1c9
26)
whip[3]: r1c9{n6 n7}- b2n7{r1c6 r3c5}- r3n6{c5 .} => -6r1c8
27)
whip[3]: r1c9{n6 n7}- b2n7{r1c6 r3c5}- r3n6{c5 .} => -6r2c8
28)
whip[3]: r1c9{n6 n7}- b2n7{r1c6 r3c5}- r3n6{c5 .} => -6r2c9
29)
whip[4]: r1c9{n6 n7}- r1c6{n7 n2}- r5n2{c6 c2}- c2n1{r5 .} => -6r1c2
Naked pairs: 14b1p29 => -1r1c3 -4r1c3 -4r2c3
ODDAGON[17]: r1c1{n6 n8},r1n8{c1 c3},r1c3{n8 n9},r1n9{c3 c8},r1c8{n9 n4},r1n4{c8 c2},r1c2{n4 n1},c2n1{r1 r5},r5c2{n1 n2},r5n2{c2 c6},r5c6{n2 n8},c6n8{r5 r7},r7c6{n8 n7},r7n7{c6 c2},c2n7{r7 r2},r2c2{n7 n6},b1n6{r2c2 r1c1}, => -2r1c4
Naked pairs: 14r1c24 => -4r1c8
ODDAGON[17]: r1n6{c1 c9},r1c9{n6 n7},r1n7{c9 c6},r1c6{n7 n2},c6n2{r1 r5},r5n2{c6 c2},b4n2{r5c2 r4c3},r4c3{n2 n5},r4n5{c3 c1},r4c1{n5 n7},c1n7{r4 r9},r9n7{c1 c5},c5n7{r9 r3},r3c5{n7 n6},b2n6{r3c5 r2c6},r2n6{c6 c2},b1n6{r2c2 r1c1}, => -7r2c3

STTE

[DEFISE]

.
I'm curious about which oddagons you can find (using no other rules), and their eliminations. Don't worry about the number of steps.

Sorry I had not read carefully.

[denis_berthier]

There are no Singles and no whips[1] at the start.
So, this is the starting point:

Code: Select all

Resolution state after Singles:
   +----------------------+----------------------+----------------------+
   ! 6789   1467   14789  ! 1245   4567   1267   ! 3      245679 456789 !
   ! 2      3467   479    ! 345    8      367    ! 1      45679  45679  !
   ! 3678   5      1478   ! 1234   3467   9      ! 2468   2467   4678   !
   +----------------------+----------------------+----------------------+
   ! 357    8      257    ! 234    1      236    ! 9      4567   4567   !
   ! 59     12     6      ! 7      49     28     ! 458    3      1458   !
   ! 4      137    179    ! 389    369    5      ! 68     167    2      !
   +----------------------+----------------------+----------------------+
   ! 5678   467    3      ! 1589   2      178    ! 456    14569  14569  !
   ! 1      26     258    ! 3589   359    4      ! 7      2569   3569   !
   ! 57     9      2457   ! 6      357    137    ! 245    8      1345   !
   +----------------------+----------------------+----------------------+
206 candidates, 1209 csp-links and 1209 links. Density = 5.73%

From here, there are 3 oddagons that can solve the puzzle (with Singles + a whip[1] after the first two).


The first has already been found by Cenoman (it's available right at the start):
oddagon[5]: r2n3{c2 c6},c6n3{r2 r4},r4n3{c6 c1},c1n3{r4 r3},b1n3{r3c1 r2c2} ==> r8c4≠3
with z-candidates = n3r2c4 n3r9c6 n3r4c4

and the state is now:

Code: Select all

   +----------------------+----------------------+----------------------+
   ! 6789   1467   14789  ! 1245   4567   1267   ! 3      245679 456789 !
   ! 2      3467   479    ! 345    8      367    ! 1      45679  45679  !
   ! 3678   5      1478   ! 1234   3467   9      ! 2468   2467   4678   !
   +----------------------+----------------------+----------------------+
   ! 357    8      257    ! 234    1      236    ! 9      4567   4567   !
   ! 59     12     6      ! 7      49     28     ! 458    3      1458   !
   ! 4      137    179    ! 389    369    5      ! 68     167    2      !
   +----------------------+----------------------+----------------------+
   ! 5678   467    3      ! 1589   2      178    ! 456    14569  14569  !
   ! 1      26     258    ! 589    359    4      ! 7      2569   3569   !
   ! 57     9      2457   ! 6      357    137    ! 245    8      1345   !
   +----------------------+----------------------+----------------------+

The second is an oddagon[9].
I let you time to try again: these oddagons are worth seeing.

[denis_berthier]

My "Simplest First" in (W+S2) with insertion of "Oddagons":

Hi François,
It depends on where you put the oddagons[15 and 17] in the hierarchy (I put them just before generic chains of same length), but I'm surprised you need so long ones in addition to whips[5].
The puzzle is solvable in W5 or W5+O5.

(But what's interesting is a solution using only Oddagons - in addition to Singles and whips[1])

[DEFISE]

Hi Denis,

I saw the 3 oddagons you gave at
a-revival-of-broken-wings-t5225-90.html#p291137

oddagon[5]: b2n3{r2c6 r3c5},r3n3{c5 c1},c1n3{r3 r4},r4n3{c1 c6},c6n3{r4 r2} ==> r8c4 ≠ 3
oddagon[9]: r5n1{c2 c9},b6n1{r5c9 r6c8},c8n1{r6 r7},r7n1{c8 c6},r7c6{n1 n8},c6n8{r7 r5},r5c6{n8 n2},r5n2{c6 c2},r5c2{n2 n1} ==> r7c6 ≠ 1
oddagon[9]: r5n1{c2 c9},b6n1{r5c9 r6c8},c8n1{r6 r7},r7c8{n1 n6},b9n6{r7c8 r8c9},r8n6{c9 c2},r8c2{n6 n2},c2n2{r8 r5},r5c2{n2 n1} ==> r7c8 ≠ 6

Indeed the oddagon[5] can be found above all basics.
But my program cannot find the 2nd and the 3rd since it uses a too restrictive definition of oddagon.
So it seems difficult to me to solve this puzzle with only oddagons.

[denis_berthier]

I saw the 3 oddagons you gave at
a-revival-of-broken-wings-t5225-90.html#p291137

oddagon[5]: b2n3{r2c6 r3c5},r3n3{c5 c1},c1n3{r3 r4},r4n3{c1 c6},c6n3{r4 r2} ==> r8c4 ≠ 3
oddagon[9]: r5n1{c2 c9},b6n1{r5c9 r6c8},c8n1{r6 r7},r7n1{c8 c6},r7c6{n1 n8},c6n8{r7 r5},r5c6{n8 n2},r5n2{c6 c2},r5c2{n2 n1} ==> r7c6 ≠ 1
oddagon[9]: r5n1{c2 c9},b6n1{r5c9 r6c8},c8n1{r6 r7},r7c8{n1 n6},b9n6{r7c8 r8c9},r8n6{c9 c2},r8c2{n6 n2},c2n2{r8 r5},r5c2{n2 n1} ==> r7c8 ≠ 6

Indeed the oddagon[5] can be found above all basics.
But my program cannot find the 2nd and the 3rd since it uses a too restrictive definition of oddagon.
So it seems difficult to me to solve this puzzle with only oddagons.

OMG, I had totally forgotten that I had already posted these oddagons. As such, they are not enough to solve the puzzle.
However, and that's the new thing I've found, each of the last two allows more eliminations.

The funny part remains to be found.

[denis_berthier]

.

As no one has found what's noticeable about the weird two oddagons, here they are:
(Remember that only Singles, whips[1] and oddagons are used - otherwise the puzzle can be solved in W5).

Code: Select all

Resolution state after Singles:
   +----------------------+----------------------+----------------------+
   ! 6789   1467   14789  ! 1245   4567   1267   ! 3      245679 456789 !
   ! 2      3467   479    ! 345    8      367    ! 1      45679  45679  !
   ! 3678   5      1478   ! 1234   3467   9      ! 2468   2467   4678   !
   +----------------------+----------------------+----------------------+
   ! 357    8      257    ! 234    1      236    ! 9      4567   4567   !
   ! 59     12     6      ! 7      49     28     ! 458    3      1458   !
   ! 4      137    179    ! 389    369    5      ! 68     167    2      !
   +----------------------+----------------------+----------------------+
   ! 5678   467    3      ! 1589   2      178    ! 456    14569  14569  !
   ! 1      26     258    ! 3589   359    4      ! 7      2569   3569   !
   ! 57     9      2457   ! 6      357    137    ! 245    8      1345   !
   +----------------------+----------------------+----------------------+
206 candidates, 1209 csp-links and 1209 links. Density = 5.73%

The first oddagon, available at the start, has already been mentioned and it has nothing noticeable:

Code: Select all

oddagon[5]: r2n3{c2 c6},c6n3{r2 r4},r4n3{c6 c1},c1n3{r4 r3},b1n3{r3c1 r2c2} ==> r8c4≠3
   with z-candidates = n3r2c4 n3r9c6 n3r4c4

The state is now:

Code: Select all

   +----------------------+----------------------+----------------------+
   ! 6789   1467   14789  ! 1245   4567   1267   ! 3      245679 456789 !
   ! 2      3467   479    ! 345    8      367    ! 1      45679  45679  !
   ! 3678   5      1478   ! 1234   3467   9      ! 2468   2467   4678   !
   +----------------------+----------------------+----------------------+
   ! 357    8      257    ! 234    1      236    ! 9      4567   4567   !
   ! 59     12     6      ! 7      49     28     ! 458    3      1458   !
   ! 4      137    179    ! 389    369    5      ! 68     167    2      !
   +----------------------+----------------------+----------------------+
   ! 5678   467    3      ! 1589   2      178    ! 456    14569  14569  !
   ! 1      26     258    ! 589    359    4      ! 7      2569   3569   !
   ! 57     9      2457   ! 6      357    137    ! 245    8      1345   !
   +----------------------+----------------------+----------------------+

This is where the next two oddagons appear:

oddagon[9]: r5c2{n1 n2},r5n2{c2 c6},r5c6{n2 n8},c6n8{r5 r7},r7c6{n8 n1},r7n1{c6 c8},c8n1{r7 r6},b6n1{r6c8 r5c9},r5n1{c9 c2} ==> r7c6≠1, r9c6≠7, r9c5≠7, r7c6≠8, r7c2≠7, r7c1≠7, r2c6≠7, r1c6≠7
with z-candidates = n7r7c6
singles ==> r7c6=7, r5c6=8, r5c2=2, r8c2=6, r7c2=4, r5c9=1, r7c8=1, r9c6=1, r6c7=8
whip[1]: b8n3{r9c5 .} ==> r3c5≠3, r6c5≠3

oddagon[5]: c1n3{r3 r4},r4n3{c1 c4},b5n3{r4c4 r6c4},c4n3{r6 r3},r3n3{c4 c1} ==> r6c4≠3, r4c6≠6, r4c6≠2, r4c4≠3, r4c1≠3, r2c6≠3
with z-candidates = n3r4c6
stte

What's noticeable is, each of them eliminates a target that belongs to its defining chain. They are autophage oddagons.

I don't know if such cases have appeared before.

[marek stefanik]

Hi Denis,

These cases are quite common if you're willing to disregard simpler chains.

In the first one, the oddagon contains this bivalue chain:
c8n1{r7 r6},r5n1{c9 c2},r5n2{c2 c6},c6n8{r5 r7} ==> r7c6≠1

The second oddagon contains this finned x-wing:
n3{r3 r4}{c1 c4} ==> r6c4≠3

As for the other eliminations, I don't understand how you even got them, the oddagons don't explain them.

What happened to the guardians n1r7c49 in the first one (r7n1{c6 c8}) and n3r24c4 in the second one (c4n3{r6 r3})?
Eliminating your only z-candidate for the latter doesn't break the template.

Marek