The New Sudoku Players' Forum

[denis_berthier]

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Code: Select all

+-------+-------+-------+
! . . . ! . . . ! . . 1 !
! 1 . . ! . . 2 ! . 3 . !
! . 3 . ! 1 4 . ! . 5 . !
+-------+-------+-------+
! . . 6 ! 7 8 . ! . . 2 !
! . 1 8 ! . 2 4 ! 6 7 . !
! 7 2 . ! 6 . 1 ! . 4 . !
+-------+-------+-------+
! . 6 1 ! 2 7 . ! . . . !
! 2 . 4 ! . 1 8 ! . . . !
! 8 7 . ! 4 . 6 ! . . . !
+-------+-------+-------+
........11....2.3..3.14..5...678...2.18.2467.72.6.1.4..6127....2.4.18...87.4.6...

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[DEFISE]

Hello all,

Before basic techniques there is an almost tridagon pattern 359 in b4,b5,b7,b8
I don't think it's a degenerated tridagon because two candidates 3 are missing (in b4 and b7).

But without 3r3c2 clue there is a true tridagon.
So the only guardian 4r4c2 si true. Fortunately the uniqueness of the solution can be proven, which means that the clue is useless, and therefore the proposed puzzle is not minimal. But the level of the resolution is very high: T&E(S2+Xwing,1).

Finally, remembering that 3r3c2 is also true simplifies things considerably, although the puzzle is still difficult (in W8 without UR).

[pjb]

1) type 1 TH => -59 from r4c2
2) replace all 359, 35 39, 59, 3, 5, 9 by xyz
3) replace the xyz's at r4c1, r5c1, r7c1 by x, y, z to give following:

Code: Select all

 4       8xyz    27xyz  | 8xyz   6xyz   7xyz   | 27xyz   26xyz   1     
 1       8xyz    7xyz   | 8xyz   6xyz   2      | 47xyz   xyz     467xyz   
 6       xyz     27xyz  | 1      4      7xyz   | 278xyz  xyz     78xyz   
------------------------+----------------------+---------------------
 x       4       6      | 7      8      xyz    | 1xyz    1xyz    2     
 y       1       8      | xyz    2      4      | 6       7       xyz   
 7       2       xyz    | 6      xyz    1      | 8xyz    4       8xyz   
------------------------+----------------------+---------------------
 z       6       1      | 2      7      xyz    | 4xyz   8        4xyz   
 2       xyz     4      | xyz    1      8      | 7xyz   6xyz     67xyz
 8       7       xyz    | 4      xyz    6      | 12     12       xyz   


4) Solve using basic methods, and xyz at r2c8 resolves to z, and the xyz at r3c8 resolves to x. So z=3, x=5, and therefore y=9

Code: Select all

 4       8       x    | z     6     y   | 7    2     1     
 1       y       7    | 8     x     2   | 4    z     6   
 6       z       2    | 1     4     7   | 8    x     z   
------------------------+----------------------+---------------------
 x       4       6    | 7     8     z   | 1    y     2     
 y       1       8    | x     2     4   | 6    7     z   
 7       2       z    | 6     y     1   | x    4     8   
------------------------+----------------------+---------------------
 z       6       1    | 2     7     x   | y    8     4   
 2       x       4    | y     1     8   | z    6     7
 8       7       y    | 4     z     6   | 2    1     x   

Replacing the x, y, z => solved

Phil

[denis_berthier]

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I proposed this (non-minimal) puzzle as an example of a degenerate-cyclic tridagon with two missing candidates (wrt a non-degenerate form) and a single guardian.
Note: degenerate-cyclic tridagons can be used in exactly the same ways as non-degenerate ones. They are harder to find when there are many guardians.

Without using this pattern, this T&E(1) puzzle (SER = 9.3) is in B13, W16 or gW14, but it falls down to W8 when using it, as follows.

Code: Select all

Resolution state after Singles and whips[1]:
   +-------------------+-------------------+-------------------+
   ! 4569  4589  2579  ! 3589  3569  3579  ! 2479  269   1     !
   ! 1     4589  579   ! 589   569   2     ! 479   3     4679  !
   ! 69    3     279   ! 1     4     79    ! 2789  5     6789  !
   +-------------------+-------------------+-------------------+
   ! 3459  459   6     ! 7     8     359   ! 1359  19    2     !
   ! 359   1     8     ! 359   2     4     ! 6     7     359   !
   ! 7     2     359   ! 6     359   1     ! 3589  4     3589  !
   +-------------------+-------------------+-------------------+
   ! 359   6     1     ! 2     7     359   ! 3459  8     3459  !
   ! 2     59    4     ! 359   1     8     ! 3579  69    35679 !
   ! 8     7     359   ! 4     359   6     ! 12359 129   359   !
   +-------------------+-------------------+-------------------+
152 candidates.

hidden-pairs-in-a-row: r9{n1 n2}{c7 c8} ==> r9c8≠9, r9c7≠9, r9c7≠5, r9c7≠3

Code: Select all

Degen-Cycl-Trid-OR1-relation for digits 3, 5 and 9 in blocks:
        b4, with cells (marked #): r4c2, r5c1, r6c3
        b5, with cells (marked #): r4c6, r5c4, r6c5
        b7, with cells (marked #): r8c2, r7c1, r9c3
        b8, with cells (marked #): r8c4, r7c6, r9c5
with 1 guardians (in cells marked @): n4r4c2

   +-------------------+-------------------+-------------------+
   ! 4569  4589  2579  ! 3589  3569  3579  ! 2479  269   1     !
   ! 1     4589  579   ! 589   569   2     ! 479   3     4679  !
   ! 69    3     279   ! 1     4     79    ! 2789  5     6789  !
   +-------------------+-------------------+-------------------+
   ! 3459  459#@ 6     ! 7     8     359#  ! 1359  19    2     !
   ! 359#  1     8     ! 359#  2     4     ! 6     7     359   !
   ! 7     2     359#  ! 6     359#  1     ! 3589  4     3589  !
   +-------------------+-------------------+-------------------+
   ! 359#  6     1     ! 2     7     359#  ! 3459  8     3459  !
   ! 2     59#   4     ! 359#  1     8     ! 3579  69    35679 !
   ! 8     7     359#  ! 4     359#  6     ! 12    12    359   !
   +-------------------+-------------------+-------------------+

Degen-Cycl-Trid-ORk-relation with only one candidate => r4c2=4

The end is in W8, with nothing noticeable.

@Defise: there are many minimal puzzles for the one proposed here. The origin of this puzzle will become clear in a forthcoming post in the tridagon thread.
@pjb: good idea to use eleven's replacement technique
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[DEFISE]

Hi Denis,
I did not read the whole thread "the tridagon rule" and so I did not know the "cyclic degenerated tridagon".
Same thing for the documentation on the TH if it exists.

[pjb]

Hi DEFISE

If you are interested in a short description of Thor's Hammer technique with many working examples, have a look at https://www.philsfolly.net.au/Sudoku/tridagon_help.htm.

Phil

[denis_berthier]

Hi Denis,
I did not read the whole thread "the tridagon rule" and so I did not know the "cyclic degenerated tridagon".
Same thing for the documentation on the TH if it exists.

Hi François
I introduced the degenerate-cyclic tridagon here: http://forum.enjoysudoku.com/the-tridagon-rule-t39859-118.html
There were two purposes:
1) to define an interesting case of degeneracy that would avoid too much degeneracy and trivial cases, in the same way as, for me, two Pairs don't make a Quad;
2) to define a potential precursor of the non-degenerate case that would help understand how the pattern could appear via vicinity search.

The posts after the above-mentioned one give results about the presence of the pattern in all the known large collections.
The forthcoming second edition of [HCCS] will give more results about it.

As for documentation about the non-degenerate case, see chapter 14 of [AUM] or [UMNR], available in pdf format on ResearchGate: https://www.researchgate.net/profile/Denis-Berthier/research
.

[DEFISE]

@Phil : thank you very much.

@Denis :
You wrote about eleven’s puzzles : "10861 out of 26,370 puzzles (41 %) have a degenerate cyclic anti-tridagon."
Very interesting indeeed !

One question: no one noticed this property, not even eleven himself ?

[denis_berthier]

@Denis :
You wrote about eleven’s puzzles : "10861 out of 26,370 puzzles (41 %) have a degenerate cyclic anti-tridagon."
Very interesting indeeed !
One question: no one noticed this property, not even eleven himself ?

Neither the degenerate-cyclic tridagon nor the non-degenerate tridagon were defined when eleven built his collection. He had no reason to look for them.

If you want a much bigger surprise, the tridagon pattern has been in [ph2010] since 2012, in 4 puzzles by dobrichev and it went unnoticed for 12 years, until a few weeks ago when mith started a systematic search in the old collections. I had started a search also, but first restricted to SER ≥ 11.6 and then to part of SER ≥ 11.2. They happen to have SER 11.1.
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