The New Sudoku Players' Forum

[denis_berthier]

.

Code: Select all

+-------+-------+-------+
! . . . ! . 5 6 ! 7 . 9 !
! . . . ! 7 . . ! . . . !
! . . . ! . 3 . ! 5 . 6 !
+-------+-------+-------+
! . 3 7 ! 5 . . ! . . . !
! 5 1 . ! . . . ! . . . !
! 6 . 9 ! . . . ! . . . !
+-------+-------+-------+
! . . . ! . 9 1 ! . . 4 !
! . . . ! . 7 . ! . 1 2 !
! 9 . . ! . . . ! . 3 . !
+-------+-------+-------+
....567.9...7.........3.5.6.375.....51.......6.9..........91..4....7..129......3.
23 clues

The name should give an idea of what I'm expecting - but a new record is reached.
(Hint: try to add clues without breaking the tridagon.)
.

[Leren]

Not sure what the record is but the TG has RT's for a total of 9 TG related eliminations. My personal record is 11.

[denis_berthier]

Not sure what the record is but the TG has RT's for a total of 9 TG related eliminations. My personal record is 11.

You can count the number of Singles after a tridagon and count records on this also.

But this is not what I'm thinking of. Check my previous Sunday puzzle and my new thread: http://forum.enjoysudoku.com/the-layered-structure-of-t-e-depth-d-t45647.html
.

[Leren]

Here is my first go at maximising the number of clues without disrupting the Trigadon, as per your hint.

1.3.567.9.567.9...79.13.5.6.3756..9.51.9.7.6.6.9.1..75.7..91654..5.7.9129.1..5837

Code: Select all

*-----------------------------------------------*
| 1   *248  3   |*248    5    6    | 7   248 9  |
|*248  5    6   | 7     *248  9    | 13  248 13 |
| 7    9   *248 | 1      3   *248  | 5   248 6  |
|---------------+------------------+------------|
|*248  3    7   | 5      6   *248  | 124 9   18 |
| 5    1   *248 | 9     *248  7    | 234 6   38 |
| 6   *248  9   |*3-248  1    2348 | 24  7   5  |
|---------------+------------------+------------|
| 238  7    28  | 238    9    1    | 6   5   4  |
| 348  468  5   | 3468   7    348  | 9   1   2  |
| 9    246  1   | 246    24   5    | 8   3   7  |
*-----------------------------------------------*

Type 1 Trigadon - Digits 248 in Boxes 1245 => - 248 r6c4;

The number of clues is 48. The post Trigadon RT moves are still there, and this time I used BUG + 1 to finish off.

Leren

[P.O.]

after basics the puzzle has 36 values which makes 13 more than the minimal and is still in te3
adding any other value puts the puzzle in te2
on the other hand for the 4th point here i do not see what should be done

4) It can be expanded by BRT-expansion plus 8 levels of 1+BRT-expansion while still remaining in T&E(3):
(same expanded puzzle)
1+BRT-expand means add a clue and apply BRT-expansion (expansion by Singles).

basics:

Hidden Text: Show

Code: Select all

( n1r9c3   n9r8c7 )

intersections:
((((3 0) (2 7 3) (1 2 3 4 8)) ((3 0) (2 9 3) (1 3 8)))
 (((1 0) (2 7 3) (1 2 3 4 8)) ((1 0) (2 9 3) (1 3 8)))
 (((1 0) (4 5 5) (1 2 4 6 8)) ((1 0) (6 5 5) (1 2 4 8))))

PAIR COL: ((7 7 9) (6 8)) ((9 7 9) (6 8)) 
(((2 7 3) (1 2 3 4 8)) ((4 7 6) (1 2 4 6 8)) ((5 7 6) (2 3 4 6 8)) ((6 7 6) (1 2 3 4 8)) ((7 8 9) (5 6 7 8)) ((9 9 9) (5 7 8)))

TRIPLET ROW: ((2 1 1) (2 4 8)) ((2 5 2) (2 4 8)) ((2 8 3) (2 4 8))
(((2 2 1) (2 4 5 6 8 9)) ((2 3 1) (2 4 5 6 8)) ((2 6 2) (2 4 8 9)) ((2 7 3) (1 2 3 4)) ((2 9 3) (1 3 8)))

( n9r2c6   n9r5c4   n9r4c8   n9r3c2   n6r5c8   n6r4c5   n7r3c1   n1r6c5   n1r3c4
  n1r1c1   n3r1c3 )

intersections:
((((8 0) (4 9 6) (1 8)) ((8 0) (5 9 6) (3 7 8)) ((8 0) (6 9 6) (3 5 7 8)))
 (((4 0) (4 7 6) (1 2 4)) ((4 0) (5 7 6) (2 3 4)) ((4 0) (6 7 6) (2 3 4)))
 (((2 0) (4 7 6) (1 2 4)) ((2 0) (5 7 6) (2 3 4)) ((2 0) (6 7 6) (2 3 4))))


Code: Select all

after basics:
1       248     3       248     5       6       7       248     9               
248     56      56      7       248     9       13      248     13               
7       9       248     1       3       248     5       248     6               
248     3       7       5       6       248     124     9       18               
5       1       248     9       248     23478   234     6       378             
6       248     9       2348    1       23478   234     57      3578             
238     25678   2568    2368    9       1       68      57      4               
348     4568    4568    3468    7       3458    9       1       2               
9       245678  1       2468    248     2458    68      3       57               
144 candidates. 36 values.

1.3.567.9...7.9...79.13.5.6.3756..9.51.9...6.6.9.1........91..4....7.9129.1....3.

after T&E(2,Singles)
1     248   3     248   5     6     7     248   9             
248   56    56    7     248   9     13    248   13             
7     9     248   1     3     248   5     248   6             
248   3     7     5     6     248   124   9     18             
5     1     248   9     248   37    234   6     378           
6     248   9     2348  1     347   234   57    3578           
23    2567  256   368   9     1     68    57    4             
348   468   4568  36    7     35    9     1     2             
9     567   1     248   248   2458  68    3     57             
126 candidates. 36 values.


[eleven]

Code: Select all

*-----------------------------------------------*
| 1   *248  3   |*248    5    6    | 7   248 9  |
|*248  5    6   | 7     *248  9    | 13  248 13 |
| 7    9   *248 | 1      3   *248  | 5   248 6  |
|---------------+------------------+------------|
|*248  3    7   | 5      6   *248  | 124 9   18 |
| 5    1   *248 | 9     *248  7    | 234 6   38 |
| 6   *248  9   |*3-248  1    2348 | 24  7   5  |
|---------------+------------------+------------|
| 238  7    28  | 238    9    1    | 6   5   4  |
| 348  468  5   | 3468   7    348  | 9   1   2  |
| 9    246  1   | 246    24   5    | 8   3   7  |
*-----------------------------------------------*

Type 1 Trigadon - Digits 248 in Boxes 1245 => - 248 r6c4;

In this 48 clue puzzle Sukaku needs a 9.0 dynamic contradiction forcing chain with 3 outside candidates to eliminate a single digit from the tridagon pattern (hardest step, followed by three 8.8, two 8.3, one 7.7, one 5.7 step to get a number) - so dumb.
Also Denis' original 11.6 puzzle is not hard to solve with tridagon,UR and remote triple.

[denis_berthier]

Here is my first go at maximising the number of clues without disrupting the Trigadon, as per your hint.
1.3.567.9.567.9...79.13.5.6.3756..9.51.9.7.6.6.9.1..75.7..91654..5.7.9129.1..5837

Code: Select all

*-----------------------------------------------*
| 1   *248  3   |*248    5    6    | 7   248 9  |
|*248  5    6   | 7     *248  9    | 13  248 13 |
| 7    9   *248 | 1      3   *248  | 5   248 6  |
|---------------+------------------+------------|
|*248  3    7   | 5      6   *248  | 124 9   18 |
| 5    1   *248 | 9     *248  7    | 234 6   38 |
| 6   *248  9   |*3-248  1    2348 | 24  7   5  |
|---------------+------------------+------------|
| 238  7    28  | 238    9    1    | 6   5   4  |
| 348  468  5   | 3468   7    348  | 9   1   2  |
| 9    246  1   | 246    24   5    | 8   3   7  |
*-----------------------------------------------*

Type 1 Trigadon - Digits 248 in Boxes 1245 => - 248 r6c4;
The number of clues is 48.

OK. My idea was to stop then process before leaving T&E(3), but the expanded puzzle you've found is interesting also. It shows one can go very far in adding clues and preserving the tridagon, while keeping the puzzle non trivial.
.

[denis_berthier]

In this 48 clue puzzle Sukaku needs a 9.0 dynamic contradiction forcing chain with 3 outside candidates to eliminate a single digit from the tridagon pattern (hardest step, followed by three 8.8, two 8.3, one 7.7, one 5.7 step to get a number) - so dumb.

Actually, the puzzle can be solved in W5 - no forcing chain needed here.

[denis_berthier]

for the 4th point here i do not see what should be done

The idea is, if you start from the min-expand puzzle (which has already more clues than the given minimal) and you add clues one by one, each time you add a new clue, you may systematically apply BRT-expansion to the result: this can't change any rating/classification of the 1-expanded puzzle.
So, what really matters is not the total number of added clues, but the total number of 1+BRT expansion steps.
.

[Leren]

Let's try this clue adding process with Loki.

57....9..........8.1.........168..4......28.9..2.9416.....2.....6.9.82.4...41.6..

So far I've got to here with the Trigadon still intact and the puzzle solvable without the Trigadon but I wouldn't say trivial.

576...9..493.....8218........168..42...1.28.9..2.9416.....26.9..6.9.82.4...41.68.

What I added was

1. Three solved cells at the time the Loki Trigadon is found.

2. A naked pair (89) which was in the Loki hit Box 9.

3. Make the whole of Box 1 solved.

1. Works obviously. I think 2. works if the NP digits are not the Trigadon digits and both don't interact with any cell that has a Trigadon digit.

The Box 1 thing was a bit of a guess, but I reasoned that it can't see any of the Trigadon boxes, so it was worth a try.

Also, when I got to there, there were 14 other solved cells with the Trigadon still intact, but I didn't include them and do a without Trigadon solve test.

Anyway, 2 out of 2 is a good start.

Leren

<edit>

I added the 14 other solved cells and the modified Loki looks like 57684.92.49326...8218..94.69.168..426..1.28.9..2.9416.....26.9..6.9.82.4.2941.68. 49 clues, TG still intact, non TG solution still not trivial.

I've tried this with another random Trigadon, did all the stuff above, and kept adding solved cells as clues at the Trigadon find stage until the extra solved cells stopped. 51 clues, TG still intact, non TG solution still not trivial.

3 out of 3 is looking pretty good.

[denis_berthier]

.
Hi Leren,
I haven't tried systematically the maximum expansion of puzzles as in your 3 above cases, but I think this would work in many cases.

What I've done until now is expansion up to the T&E(3) border (what mith calls the max-expands). We already knew that this preserves the tridagons for all the known T&E(3) puzzles. What seems to be new is the very large numbers, both of clues and of (1+BRT)-expansions, that can be added between the minimals and the T&E(3) border.
And, as shown by your 3 examples for clues, also beyond the T&E(3) border.
.

[Leren]

Hi denis,

I've tried the same thing on a number of other Trigadon puzzles and it works most of the time.

I've even coined a new term the Remote Box, being the Box that none of the four Trigadon boxes can see. eg If the Trigadon Boxes are 1,2,4,5 the Remote Box is 9, and so on.

So the trick is to solve the puzzle, then fill in the Remote Box solution as clues. If the Trigadon doesn't collapse, you can also fill in other solved cells as clues.

Do the extra solved cell thing recursively until they stop appearing. You can also fill in the solution to certain naked pairs as I've described above.

It's not uncommon to get up to about 50 clues with the Trigadon still intact. Failures seem to occur when the solution becomes almost completely trivial.

The final step is to ignore the Trigadon solution and see how the puzzle solves. For the Loki expansion Hodoku solves with about 13 chains and a score of about 3000. Not hard but still respectable.

I don't know what all this proves but it's interesting to see puzzles with about 50 clues, still having a non trivial solution.

Leren

[denis_berthier]

So the trick is to solve the puzzle, then fill in the Remote Box solution as clues. If the Trigadon doesn't collapse, you can also fill in other solved cells as clues.

You mean if it doesn't collapse after applying Singles? Because it can't collapse just by filling in the remote block.

I don't know what all this proves but it's interesting to see puzzles with about 50 clues, still having a non trivial solution.

I think there are several points of interest here:
1) not only minimal puzzles are worth considering (anyway, being minimal is irrelevant to players);
2) tridagons are extremely resistant to additions of clues;
3) ... and totally resistant to deletion of clues; which explains why [-p +q] search can hardly get rid of them;
4) tridagons are not isolated from the rest of the puzzle, but they show some independence from it;
5) having a tridagon in a puzzle doesn't allow to conclude much about its difficulty, because the rest of the puzzle may short-circuit it;
6) the large number of (1+BRT) layers in T&E(3) (http://forum.enjoysudoku.com/the-layered-structure-of-t-e-depth-d-t45647.html) is probably due to the presence of a tridagon in each known T&E(3) puzzle, but it shows more: one can add clues in a way that not only preserves the tridagon but keeps the puzzle in T&E(3).
.

[Leren]

Understand all your points except why the TG can't ever collapse by just filling the Remote Box. My intuition said that it would work some of the time, but why is it obvious it will work all of the time ?

Anyway, if it does, the way forward looks fairly easy. Trying another puzzle I did learn a couple of points.

For this puzzle some naked pairs looked like they made no difference, but in fact if you look carefully they all make the TG slightly degenerate. I think that should be avoided.

I did get an error with a UR move that involved the TG boxes. May be a UR bug in my code or something that should be avoided. Turning off URs fixed the problem.

So starting with this ..1...93..763.......2......4..89..131..5.48.9....3154..14.83.9....9.51.4......38. I got to this 851...93.9763.84.1342..9..84.589..131..5.48.9....3154..14.83.9....9.51.4......38.

Leren

[denis_berthier]

Understand all your points except why the TG can't ever collapse by just filling the Remote Box. My intuition said that it would work some of the time, but why is it obvious it will work all of the time ?

No candidate in the remote box is linked to the 4x3x3 tridagon candidates. Choosing one as a clue can't therefore directly eliminate any of these candidates.
But, of course, if you follow this by applying Singles, you may indirectly eliminate some of them.

As for degeneracy, some degenerated forms of the tlridagon remain interesting: I've already mentioned the "degenerate-cyclic" form. It still allows the same eliminations.
.