The New Sudoku Players' Forum

[Leren]

Hi Marek,

First of all, let me thank you for your patience and detailed analyses.

At this stage I'll refer you to a purported proof by eleven and backed up by totuan, that if a cell is solved in one of the four Trigadon boxes, then the other 3 TG cells in the rectangle of cells in the other three Trigadon boxes must all be different.

You will find their posts here.

Could you please review those them and say whether or not they are complete and correct. You seem to have found a counter example in the Trigadon boxes (but not in the puzzle as a whole).

Also eleven did not include Puzzle 6 in his list of qualifying puzzles. Perhaps he can explain why my proposed RT a does not qualify.

Leren

[marek stefanik]

The proofs are correct, but they require an almost TH, which requires a 3+1 split of the diagonals or equivalently one rectangle and one 8-loop:

Code: Select all

---------------------------
a123 .   .   |b123 .   .
 .  AX   .   | .  B123 .
 .   .  e123 | .   .  f123
---------------------------
h123 .   .   | .   .  g123
 .  D123 .   | .  C123 .
 .   .  d123 |c123 .   .
---------------------------

3 diagonals going down-right, 1 diagonal going down-left
rectangle marked as ABCD, 8-loop as abcdefgh

The cells you marked in puzzle 6 have all diagonals going down-right and there are three rectangles (which you marked), so TH-related techniques cannot be used.

Code: Select all

*--------------------------------------------------------------------------------*
| 2478    2678    4578     | 3789    378     378      | 456     459     1        |
| 478     678     4578     | 789     1       2        | 456     3       59       |
| 139     39      13       | 4       5       6        | 7       8       2        |
|--------------------------+--------------------------+--------------------------|
| 14     b378     14       |b378     2       5        | 9       6       378      |
| 29      29     c378      | 6      c378     378      | 13458   1457    3578     |
|a378     5       6        | 1       4      a9        | 2       47      378      |
|--------------------------+--------------------------+--------------------------|
|a378     1       9        | 25      6      a378      | 358     257     4        |
| 5      b378     2        |b378     9       4        | 138     17      6        |
| 6       4      c378      | 25     c378     1        | 358     2579    35789    |
*--------------------------------------------------------------------------------*

Marek

[Leren]

Hi Marek, OK I think I've got it sorted now.

Code: Select all

*--------------------------------------------------------------------------------*
| 2478    2678    4578     | 3789    378     378      | 456     459     1        |
| 478     678     4578     | 789     1       2        | 456     3       59       |
| 139     39      13       | 4       5       6        | 7       8       2        |
|--------------------------+--------------------------+--------------------------|
| 14     A378     14       |B378     2       5        | 9       6       378      |
| 29      29      378      | 6       3478    378      | 13458   1457    3578     |
| 378     5       6        | 1       4-378   9        | 2       47      378      |
|--------------------------+--------------------------+--------------------------|
| 378     1       9        | 25      6       378      | 358     257     4        |
| 5      C378     2        |D378     9       4        | 138     17      6        |
| 6       4       378      | 25      378     1        | 358     2579    35789    |
*--------------------------------------------------------------------------------*

So this is puzzle 6 with the rectangle cells ABCD. Box 5 has the TH digits in an up-right pattern. Since the rectangle cells can't solve to a non-TH digit RT's can't be found.

Code: Select all

*--------------------------------------------------------------------------------*
| 2369    34679   35689    | 3679    3689    369      | 2459    245789  1        |
| 169     4679    5689     | 1679    1689    2        | 459     3       4579     |
| 1239    379     389      | 1379    4       5        | 29      2789    6        |
|--------------------------+--------------------------+--------------------------|
|b369     1       4        | 5       7      b369      | 8       269     239      |
| 5      c369     2        | 8      c369     1        | 7       469     349      |
| 7       8      A369      |B369     2       4        | 1       569     359      |
|--------------------------+--------------------------+--------------------------|
|b369     5       1        | 24      369     8        | 23469   2479    2479     |
| 4       2      C369      |D1-369   5       7        | 369     19      8        |
| 8      c369     7        | 24      1369    369      | 234569  12459   2459     |
*--------------------------------------------------------------------------------*

Compare this with Puzzle 5, where one of the rectangle cells, D has just been solved to a non-TH digit 1, which ensures 3 sets of RT's - Cells ABC in the rectangle and the 6 cells marked b and c in Boxes 5, 6 and 8.

The points of confusion for me were: 1. What was the rectangle that was being referred to ? and 2. The fact that one of the rectangle cells must solve to a non-TH digit. This was marked as X in the proofs I mentioned but they don't explicitly say that X was a non-TH digit, which confused me at least.

It might be helpful for non-experienced TH-ers to have a list of the Rising (Up-Right) and Falling (Down-Right) TH digit patterns. Up-right and down-right don't work well with the forum's search function, so I suggest that Rising and Falling be used.

Code: Select all

    | .  .  * |    | *  .  . |    | .  *  . |      | *  .  . |    | .  .  * |    | .  *  . |
    | .  *  . |    | .  .  * |    | *  .  . |      | .  *  . |    | *  .  . |    | .  .  * |
    | *  .  . |    | .  *  . |    | .  .  * |      | .  .  * |    | .  *  . |    | *  .  . |
 
                 Rising patterns                                Falling patterns

Leren

[denis_berthier]

It might be helpful for non-experienced TH-ers to have a list of the Rising (Up-Right) and Falling (Down-Right) TH digit patterns. Up-right and down-right don't work well with the forum's search function, so I suggest that Rising and Falling be used.

Code: Select all

    | .  .  * |    | *  .  . |    | .  *  . |      | *  .  . |    | .  .  * |    | .  *  . |
    | .  *  . |    | .  .  * |    | *  .  . |      | .  *  . |    | *  .  . |    | .  .  * |
    | *  .  . |    | .  *  . |    | .  .  * |      | .  .  * |    | .  *  . |    | *  .  . |
 
                 Rising patterns                                Falling patterns

Good. And to be complete, you only need to add that the tridagon pattern is when you have an odd number of rising (or falling patterns) in the 4 blocks.

[Leren]

Code: Select all

  . . * | * . . 
  . * . | . * .
  * . . | . . *
  ------+------
  * . . | * . .
  . * . | . * .   
  . . * | . . *

   TH Pattern

As Denis said, and Marek said earler in the thread, for a TH Pattern you must have a 3/1 split of rising and falling patterns in the 4 TH boxes. If a particular TH pattern doesn't look exactly like the diagram, you can morph the puzzle with row and column permutations so that 3 falling boxes are the same as in the picture. The 4th box will turn out to be whatever is left over. If it's any rising pattern you are golden. If it's also falling you have a 4/0 split and you have lucked out.

I have to conclude this post with a diagram of Mith's Loki puzzle.

For TH tragics this must be a thing of pure beauty. No morphing and the rising/falling patterns in the boxes are the easy to read ones, exactly as in the exemplar with the rectangle marked in cells ABCD.

Code: Select all

*--------------------------------------------------------------------------------*
| 5       7       3468     | 238     346     13       | 9       123     136      |
| 23469   2349    3469     | 2357    34567   13579    | 3457    12357   8        |
| 234689  1       34689    | 23578   34567   3579     | 3457    2357    3567     |
|--------------------------+--------------------------+--------------------------|
| 379     359     1        | 6       8      *357      |*357     4       2        |
| 3467    345     34567    | 1     A*357     2        | 8     B*357     9        |
| 378     358     2        |*357     9       4        | 1       6      *357      |
|--------------------------+--------------------------+--------------------------|
| 134789  34589   345789   |*357     2       6        |*357     89      1357     |
| 137     6       357      | 9     D*357     8        | 2     C*1-357   4        |
| 23789   23589   35789    | 4       1      A357      | 6       89     *357      |
*--------------------------------------------------------------------------------*

Leren

[denis_berthier]

As Denis said, and Marek said earler in the thread,...

Yes, but if you're assigning credit for the pattern, it should go to mith.

[Leren]

Hi Denis, looks like we cross posted. I actually got the diagram from Phil's site and probably should have posted it with the other pictures. Leren