The tridagon rule

Advanced methods and approaches for solving Sudoku puzzles

Re: The tridagon rule

Postby denis_berthier » Sun Sep 11, 2022 7:55 am

.
ORk-FW classification results for mith's 63137 min-expand puzzles in T&E(3):

Puzzles solved in SFin+Trid+Wn+ORkFWn
Code: Select all
-----------------------------------------------------------------------
        n=3                 n=5                 n=7                 n=8    
-----------------------------------------------------------------------
     8 ,196   puzzles solved by SFin+Trid (among 63 137 min-expands)   
-----------------------------------------------------------------------
k=0   8,137              17,532              21,160              22,332
     16,333       9,395  25,728       3,628  29,356       1,172  30,528
-----------------------------------------------------------------------
k=2   3,350               8,713              11,231              12,068   
     19,683      14,758  34,441       6,146  40,587       2,009  42,596   
-----------------------------------------------------------------------
k=3     428               2,255               3,471      
     20,111      16,585  36,696       7,362  44,058      
-----------------------------------------------------------------------
k=4      67                 365                 540      
     20,178      16,883  37,061       7,537  44,598      
-----------------------------------------------------------------------
k=5       2                  28                  99      
     20,180      16,909  37,089       7,608  44,697      
-----------------------------------------------------------------------
k=6       0                   4            
     20,180      16,913  37,093   
-----------------------------------------------------------------------

Lines are separated by dashes, columns are separated by large white spaces.
Each (k, n) cell has three values in it:
- the main one, in the lower right corner, is the total number of puzzles solved by SFin + Trid + Wn + ORkFWn;
- the value above it is the difference with the previous line; it shows what’s gained by increasing k by 1;
- the value on the left of the main number is the difference with the previous cell; it shows what’s gained by increasing n.

Some general conclusions can be drawn from this table:
• for fixed n, as k increases, the difference between two lines decreases quite fast; this shouldn’t be too surprising, as larger k means more chains have to converge to the same candidate;
• for fixed k, as n increases, the difference between two columns decreases quite fast; this shouldn’t be too surprising either, as it already happens with all the “classical” chains (whips…);
• starting from k=2 and n=3, at any point in the table, it is much more fruitful to increase n than to increase k;

I'll give a similar table for ORk-Contrad-Whips, but the calculations are still running.

[Edit]:added case (k=2, n=8)
denis_berthier
2010 Supporter
 
Posts: 3676
Joined: 19 June 2007
Location: Paris

Re: The tridagon rule

Postby denis_berthier » Tue Sep 13, 2022 5:10 am

.
Degenerated trivalue-oddagons: case of 1 decided value

Suppose that, instead of the standard contradictory trivlaue-oddagon pattern, with alll 3 candidates in all 12 cells, one of its cells is a decided value; say r1c1=1.
It is obvious that this pattern is still contradictory. However, it no longer requires T&E(3) to be proven contradictory: this can be done in T&E(2).


The pattern is (modulo isomorphisms):
Code: Select all
+-------------------------------+-------------------------------+-------------------------------+
! 1         123456789 123456789 ! 123       123456789 123456789 ! 123456789 123456789 123456789 !
! 123456789 123       123456789 ! 123456789 123       123456789 ! 123456789 123456789 123456789 !
! 123456789 123456789 123       ! 123456789 123456789 123       ! 123456789 123456789 123456789 !
+-------------------------------+-------------------------------+-------------------------------+
! 123       123456789 123456789 ! 123456789 123456789 123       ! 123456789 123456789 123456789 !
! 123456789 123       123456789 ! 123456789 123       123456789 ! 123456789 123456789 123456789 !
! 123456789 123456789 123       ! 123       123456789 123456789 ! 123456789 123456789 123456789 !
+-------------------------------+-------------------------------+-------------------------------+
! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 !
! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 !
! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 !
+-------------------------------+-------------------------------+-------------------------------+


Here is a way to prove the contradiction in T&E(2), using SudoRules.
Choose T&E(2) in the configuration file.
If you try to apply function "solve-sukaku-grid" to the above resolution state, computations will take too long.
On the other hand, if you try to use function "solve-k-digit-pattern-string" directly, the candidates n2r1c2 and n3r1c1 will not be deleted before starting.

There is a way out of this. Use the following two commands:
Code: Select all
(bind ?*simulated-eliminations* (create$ 211 311))
(solve-k-digit-pattern-string 3 "100100000010010000001001000100001000010010000001100000000000000000000000000000000")


SudoRules outputs a quick and short proof of the contradiction in T&E(2):

Hidden Text: Show
Code: Select all
***********************************************************************************************
***  SudoRules 20.1.s based on CSP-Rules 2.1.s, config = T&E(BRT, 2)
***  Using CLIPS 6.32-r823
***  Running on MacBookPro 16'' M1Max 2021, 64GB LPDDR5, MacOS 12.5
***  Download from: https://github.com/denis-berthier/CSP-Rules-V2.1
***********************************************************************************************
100100000010010000001001000100001000010010000001100000000000000000000000000000000
Simulated elimination of 311
Simulated elimination of 211
naked-single ==> r1c1=1
Resolution state after Singles:
   +-------------------------------+-------------------------------+-------------------------------+
   ! 1         23456789  23456789  ! 23        23456789  23456789  ! 23456789  23456789  23456789  !
   ! 23456789  23        23456789  ! 123456789 123       123456789 ! 123456789 123456789 123456789 !
   ! 23456789  23456789  23        ! 123456789 123456789 123       ! 123456789 123456789 123456789 !
   +-------------------------------+-------------------------------+-------------------------------+
   ! 23        123456789 123456789 ! 123456789 123456789 123       ! 123456789 123456789 123456789 !
   ! 23456789  123       123456789 ! 123456789 123       123456789 ! 123456789 123456789 123456789 !
   ! 23456789  123456789 123       ! 123       123456789 123456789 ! 123456789 123456789 123456789 !
   +-------------------------------+-------------------------------+-------------------------------+
   ! 23456789  123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 !
   ! 23456789  123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 !
   ! 23456789  123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 !
   +-------------------------------+-------------------------------+-------------------------------+

634 candidates, 0 csp-links and 0 links. Density = 0.0%
Starting non trivial part of solution.

*** STARTING T&E IN CONTEXT 0 at depth 1 with 1 csp-variables solved and 634 candidates remaining ***

        STARTING PHASE 1 IN CONTEXT 0 with 1 csp-variables solved and 634 candidates remaining


GENERATING CONTEXT 1 AT DEPTH 1, SON OF CONTEXT 0, FROM HYPOTHESIS n3r6c4.
naked-single ==> r1c4=2

*** STARTING T&E IN CONTEXT 1 at depth 1 with 1 csp-variables solved and 634 candidates remaining ***

        STARTING PHASE 1 IN CONTEXT 1 AT DEPTH 1, with 1 csp-variables solved and 634 candidates remaining


GENERATING CONTEXT 2 AT DEPTH 2, SON OF CONTEXT 1, FROM HYPOTHESIS n2r2c2.
naked-single ==> r3c3=3
naked-single ==> r3c6=1
naked-single ==> r2c5=3
naked-single ==> r4c6=2
naked-single ==> r4c1=3
naked-single ==> r5c2=1
NO POSSIBLE VALUE for csp-variable 155 IN CONTEXT 2. RETRACTING CANDIDATE n2r2c2 FROM CONTEXT 1.

BACK IN CONTEXT 1 with 1 csp-variables solved and 634 candidates remaining.

naked-single ==> r2c2=3
naked-single ==> r3c3=2
naked-single ==> r6c3=1
naked-single ==> r5c2=2
naked-single ==> r5c5=1
NO POSSIBLE VALUE for csp-variable 125 IN CONTEXT 1. RETRACTING CANDIDATE n3r6c4 FROM CONTEXT 0.

BACK IN CONTEXT 0 with 1 csp-variables solved and 633 candidates remaining.


GENERATING CONTEXT 3 AT DEPTH 1, SON OF CONTEXT 0, FROM HYPOTHESIS n2r6c4.
naked-single ==> r1c4=3

*** STARTING T&E IN CONTEXT 3 at depth 1 with 1 csp-variables solved and 633 candidates remaining ***

        STARTING PHASE 1 IN CONTEXT 3 AT DEPTH 1, with 1 csp-variables solved and 633 candidates remaining


GENERATING CONTEXT 4 AT DEPTH 2, SON OF CONTEXT 3, FROM HYPOTHESIS n2r2c2.
naked-single ==> r3c3=3
naked-single ==> r6c3=1
naked-single ==> r5c2=3
naked-single ==> r4c1=2
naked-single ==> r5c5=1
NO POSSIBLE VALUE for csp-variable 125 IN CONTEXT 4. RETRACTING CANDIDATE n2r2c2 FROM CONTEXT 3.

BACK IN CONTEXT 3 with 1 csp-variables solved and 633 candidates remaining.

naked-single ==> r2c2=3
naked-single ==> r3c3=2
naked-single ==> r3c6=1
naked-single ==> r4c6=3
naked-single ==> r5c5=1
naked-single ==> r5c2=2
NO POSSIBLE VALUE for csp-variable 141 IN CONTEXT 3. RETRACTING CANDIDATE n2r6c4 FROM CONTEXT 0.

BACK IN CONTEXT 0 with 1 csp-variables solved and 632 candidates remaining.

naked-single ==> r6c4=1

GENERATING CONTEXT 5 AT DEPTH 1, SON OF CONTEXT 0, FROM HYPOTHESIS n3r6c3.
naked-single ==> r3c3=2
naked-single ==> r2c2=3
naked-single ==> r4c1=2
naked-single ==> r5c2=1
naked-single ==> r4c6=3
naked-single ==> r5c5=2
naked-single ==> r2c5=1
NO POSSIBLE VALUE for csp-variable 136 IN CONTEXT 5. RETRACTING CANDIDATE n3r6c3 FROM CONTEXT 0.

BACK IN CONTEXT 0 with 2 csp-variables solved and 612 candidates remaining.

naked-single ==> r6c3=2
naked-single ==> r3c3=3
naked-single ==> r2c2=2
naked-single ==> r4c1=3
naked-single ==> r4c6=2
naked-single ==> r3c6=1
naked-single ==> r2c5=3

PUZZLE 0 HAS NO SOLUTION : NO CANDIDATE FOR RC-CELL r5c5
MOST COMPLEX RULE TRIED = NS
Puzzle 100100000010010000001001000100001000010010000001100000000000000000000000000000000 :
init-time = 0.0s, solve-time = 0.11s, total-time = 0.11s

.
Last edited by denis_berthier on Wed Sep 28, 2022 3:22 am, edited 1 time in total.
denis_berthier
2010 Supporter
 
Posts: 3676
Joined: 19 June 2007
Location: Paris

Re: The tridagon rule

Postby denis_berthier » Tue Sep 13, 2022 5:16 am

.
Degenerated trivalue-oddagons: case of 1 missing candidate

Suppose that, instead of the standard contradictory trivlaue-oddagon pattern, with alll 3 candidates in all 12 cells, one of these candidates is missing; say n3r1c1.
It is obvious that this pattern is still contradictory. However, it no longer requires T&E(3) to be proven contradictory: this can be done in T&E(2).


The pattern is (modulo isomorphisms):
Code: Select all
+-------------------------------+-------------------------------+-------------------------------+
! 12        123456789 123456789 ! 123       123456789 123456789 ! 123456789 123456789 123456789 !
! 123456789 123       123456789 ! 123456789 123       123456789 ! 123456789 123456789 123456789 !
! 123456789 123456789 123       ! 123456789 123456789 123       ! 123456789 123456789 123456789 !
+-------------------------------+-------------------------------+-------------------------------+
! 123       123456789 123456789 ! 123456789 123456789 123       ! 123456789 123456789 123456789 !
! 123456789 123       123456789 ! 123456789 123       123456789 ! 123456789 123456789 123456789 !
! 123456789 123456789 123       ! 123       123456789 123456789 ! 123456789 123456789 123456789 !
+-------------------------------+-------------------------------+-------------------------------+
! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 !
! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 !
! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 !
+-------------------------------+-------------------------------+-------------------------------+


Here is a way to prove the contradiction in T&E(2), using SudoRules, in the same way as before.
Choose T&E(2) in the configuration file.
Use the following two commands:
Code: Select all
(bind ?*simulated-eliminations* (create$ 311))
(solve-k-digit-pattern-string 3 "100100000010010000001001000100001000010010000001100000000000000000000000000000000")


SudoRules still outputs a quick and short proof (though not as short as before) of the contradiction in T&E(2):

Hidden Text: Show
Code: Select all
***********************************************************************************************
***********************************************************************************************
***  SudoRules 20.1.s based on CSP-Rules 2.1.s, config = T&E(BRT, 2)
***  Using CLIPS 6.32-r823
***  Running on MacBookPro 16'' M1Max 2021, 64GB LPDDR5, MacOS 12.5
***  Download from: https://github.com/denis-berthier/CSP-Rules-V2.1
***********************************************************************************************
100100000010010000001001000100001000010010000001100000000000000000000000000000000
Simulated elimination of 311
Resolution state after Singles:
   +-------------------------------+-------------------------------+-------------------------------+
   ! 12        123456789 123456789 ! 123       123456789 123456789 ! 123456789 123456789 123456789 !
   ! 123456789 123       123456789 ! 123456789 123       123456789 ! 123456789 123456789 123456789 !
   ! 123456789 123456789 123       ! 123456789 123456789 123       ! 123456789 123456789 123456789 !
   +-------------------------------+-------------------------------+-------------------------------+
   ! 123       123456789 123456789 ! 123456789 123456789 123       ! 123456789 123456789 123456789 !
   ! 123456789 123       123456789 ! 123456789 123       123456789 ! 123456789 123456789 123456789 !
   ! 123456789 123456789 123       ! 123       123456789 123456789 ! 123456789 123456789 123456789 !
   +-------------------------------+-------------------------------+-------------------------------+
   ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 !
   ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 !
   ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 ! 123456789 123456789 123456789 !
   +-------------------------------+-------------------------------+-------------------------------+

656 candidates, 0 csp-links and 0 links. Density = 0.0%
Starting non trivial part of solution.

*** STARTING T&E IN CONTEXT 0 at depth 1 with 0 csp-variables solved and 656 candidates remaining ***

        STARTING PHASE 1 IN CONTEXT 0 with 0 csp-variables solved and 656 candidates remaining


GENERATING CONTEXT 1 AT DEPTH 1, SON OF CONTEXT 0, FROM HYPOTHESIS n3r6c4.

*** STARTING T&E IN CONTEXT 1 at depth 1 with 0 csp-variables solved and 656 candidates remaining ***

        STARTING PHASE 1 IN CONTEXT 1 AT DEPTH 1, with 0 csp-variables solved and 656 candidates remaining


GENERATING CONTEXT 2 AT DEPTH 2, SON OF CONTEXT 1, FROM HYPOTHESIS n1r1c1.
naked-single ==> r1c4=2
NO CONTRADICTION FOUND IN CONTEXT 2.
BACK IN CONTEXT 1 with 0 csp-variables solved and 656 candidates remaining.


GENERATING CONTEXT 3 AT DEPTH 2, SON OF CONTEXT 1, FROM HYPOTHESIS n2r1c1.
naked-single ==> r1c4=1
NO CONTRADICTION FOUND IN CONTEXT 3.
BACK IN CONTEXT 1 with 0 csp-variables solved and 656 candidates remaining.


GENERATING CONTEXT 4 AT DEPTH 2, SON OF CONTEXT 1, FROM HYPOTHESIS n1r1c4.
naked-single ==> r1c1=2
NO CONTRADICTION FOUND IN CONTEXT 4.
BACK IN CONTEXT 1 with 0 csp-variables solved and 656 candidates remaining.


GENERATING CONTEXT 5 AT DEPTH 2, SON OF CONTEXT 1, FROM HYPOTHESIS n2r1c4.
naked-single ==> r1c1=1
NO CONTRADICTION FOUND IN CONTEXT 5.
BACK IN CONTEXT 1 with 0 csp-variables solved and 656 candidates remaining.


GENERATING CONTEXT 6 AT DEPTH 2, SON OF CONTEXT 1, FROM HYPOTHESIS n1r2c2.
naked-single ==> r1c1=2
naked-single ==> r1c4=1
naked-single ==> r3c3=3
naked-single ==> r3c6=2
naked-single ==> r2c5=3
naked-single ==> r4c6=1
naked-single ==> r4c1=3
naked-single ==> r5c2=2
NO POSSIBLE VALUE for csp-variable 155 IN CONTEXT 6. RETRACTING CANDIDATE n1r2c2 FROM CONTEXT 1.

BACK IN CONTEXT 1 with 0 csp-variables solved and 656 candidates remaining.


GENERATING CONTEXT 7 AT DEPTH 2, SON OF CONTEXT 1, FROM HYPOTHESIS n2r2c2.
naked-single ==> r1c1=1
naked-single ==> r1c4=2
naked-single ==> r3c3=3
naked-single ==> r3c6=1
naked-single ==> r2c5=3
naked-single ==> r4c6=2
naked-single ==> r4c1=3
naked-single ==> r5c2=1
NO POSSIBLE VALUE for csp-variable 155 IN CONTEXT 7. RETRACTING CANDIDATE n2r2c2 FROM CONTEXT 1.

BACK IN CONTEXT 1 with 0 csp-variables solved and 656 candidates remaining.

naked-single ==> r2c2=3

GENERATING CONTEXT 8 AT DEPTH 2, SON OF CONTEXT 1, FROM HYPOTHESIS n1r2c5.
naked-single ==> r5c5=2
naked-single ==> r4c6=1
naked-single ==> r5c2=1
naked-single ==> r6c3=2
naked-single ==> r3c3=1
naked-single ==> r1c1=2
NO POSSIBLE VALUE for csp-variable 114 IN CONTEXT 8. RETRACTING CANDIDATE n1r2c5 FROM CONTEXT 1.

BACK IN CONTEXT 1 with 0 csp-variables solved and 656 candidates remaining.

naked-single ==> r2c5=2
naked-single ==> r5c5=1
naked-single ==> r5c2=2
naked-single ==> r6c3=1
naked-single ==> r4c1=3
naked-single ==> r3c3=2
naked-single ==> r1c1=1
NO POSSIBLE VALUE for csp-variable 114 IN CONTEXT 1. RETRACTING CANDIDATE n3r6c4 FROM CONTEXT 0.

BACK IN CONTEXT 0 with 0 csp-variables solved and 655 candidates remaining.


GENERATING CONTEXT 9 AT DEPTH 1, SON OF CONTEXT 0, FROM HYPOTHESIS n2r6c4.

*** STARTING T&E IN CONTEXT 9 at depth 1 with 0 csp-variables solved and 655 candidates remaining ***

        STARTING PHASE 1 IN CONTEXT 9 AT DEPTH 1, with 0 csp-variables solved and 655 candidates remaining


GENERATING CONTEXT 10 AT DEPTH 2, SON OF CONTEXT 9, FROM HYPOTHESIS n1r1c1.
naked-single ==> r1c4=3
NO CONTRADICTION FOUND IN CONTEXT 10.
BACK IN CONTEXT 9 with 0 csp-variables solved and 655 candidates remaining.


GENERATING CONTEXT 11 AT DEPTH 2, SON OF CONTEXT 9, FROM HYPOTHESIS n2r1c1.
NO CONTRADICTION FOUND IN CONTEXT 11.
BACK IN CONTEXT 9 with 0 csp-variables solved and 655 candidates remaining.


GENERATING CONTEXT 12 AT DEPTH 2, SON OF CONTEXT 9, FROM HYPOTHESIS n1r1c4.
naked-single ==> r1c1=2
NO CONTRADICTION FOUND IN CONTEXT 12.
BACK IN CONTEXT 9 with 0 csp-variables solved and 655 candidates remaining.


GENERATING CONTEXT 13 AT DEPTH 2, SON OF CONTEXT 9, FROM HYPOTHESIS n3r1c4.
NO CONTRADICTION FOUND IN CONTEXT 13.
BACK IN CONTEXT 9 with 0 csp-variables solved and 655 candidates remaining.


GENERATING CONTEXT 14 AT DEPTH 2, SON OF CONTEXT 9, FROM HYPOTHESIS n1r2c2.
naked-single ==> r1c1=2
naked-single ==> r3c3=3
naked-single ==> r6c3=1
naked-single ==> r4c1=3
naked-single ==> r4c6=1
naked-single ==> r3c6=2
naked-single ==> r2c5=3
NO POSSIBLE VALUE for csp-variable 155 IN CONTEXT 14. RETRACTING CANDIDATE n1r2c2 FROM CONTEXT 9.

BACK IN CONTEXT 9 with 0 csp-variables solved and 655 candidates remaining.


GENERATING CONTEXT 15 AT DEPTH 2, SON OF CONTEXT 9, FROM HYPOTHESIS n2r2c2.
naked-single ==> r1c1=1
naked-single ==> r1c4=3
naked-single ==> r2c5=1
naked-single ==> r3c6=2
naked-single ==> r5c5=3
naked-single ==> r4c6=1
naked-single ==> r5c2=1
naked-single ==> r6c3=3
NO POSSIBLE VALUE for csp-variable 133 IN CONTEXT 15. RETRACTING CANDIDATE n2r2c2 FROM CONTEXT 9.

BACK IN CONTEXT 9 with 0 csp-variables solved and 655 candidates remaining.

naked-single ==> r2c2=3

GENERATING CONTEXT 16 AT DEPTH 2, SON OF CONTEXT 9, FROM HYPOTHESIS n1r2c5.
naked-single ==> r5c5=3
naked-single ==> r4c6=1
naked-single ==> r1c4=3
naked-single ==> r3c6=2
naked-single ==> r3c3=1
naked-single ==> r1c1=2
naked-single ==> r4c1=3
NO POSSIBLE VALUE for csp-variable 163 IN CONTEXT 16. RETRACTING CANDIDATE n1r2c5 FROM CONTEXT 9.

BACK IN CONTEXT 9 with 0 csp-variables solved and 655 candidates remaining.

naked-single ==> r2c5=2

GENERATING CONTEXT 17 AT DEPTH 2, SON OF CONTEXT 9, FROM HYPOTHESIS n1r3c3.
naked-single ==> r6c3=3
naked-single ==> r3c6=3
naked-single ==> r1c4=1
naked-single ==> r4c6=1
naked-single ==> r4c1=2
NO POSSIBLE VALUE for csp-variable 111 IN CONTEXT 17. RETRACTING CANDIDATE n1r3c3 FROM CONTEXT 9.

BACK IN CONTEXT 9 with 0 csp-variables solved and 655 candidates remaining.

naked-single ==> r3c3=2
naked-single ==> r1c1=1
naked-single ==> r1c4=3
naked-single ==> r3c6=1
naked-single ==> r4c6=3
naked-single ==> r5c5=1
naked-single ==> r5c2=2
NO POSSIBLE VALUE for csp-variable 141 IN CONTEXT 9. RETRACTING CANDIDATE n2r6c4 FROM CONTEXT 0.

BACK IN CONTEXT 0 with 0 csp-variables solved and 654 candidates remaining.

naked-single ==> r6c4=1

GENERATING CONTEXT 18 AT DEPTH 1, SON OF CONTEXT 0, FROM HYPOTHESIS n3r6c3.

*** STARTING T&E IN CONTEXT 18 at depth 1 with 1 csp-variables solved and 633 candidates remaining ***

        STARTING PHASE 1 IN CONTEXT 18 AT DEPTH 1, with 1 csp-variables solved and 633 candidates remaining


GENERATING CONTEXT 19 AT DEPTH 2, SON OF CONTEXT 18, FROM HYPOTHESIS n1r1c1.
naked-single ==> r4c1=2
naked-single ==> r4c6=3
naked-single ==> r5c5=2
naked-single ==> r5c2=1
naked-single ==> r3c3=2
naked-single ==> r2c2=3
naked-single ==> r2c5=1
NO POSSIBLE VALUE for csp-variable 136 IN CONTEXT 19. RETRACTING CANDIDATE n1r1c1 FROM CONTEXT 18.

BACK IN CONTEXT 18 with 1 csp-variables solved and 633 candidates remaining.

naked-single ==> r1c1=2
naked-single ==> r4c1=1
naked-single ==> r5c2=2
naked-single ==> r5c5=3
naked-single ==> r4c6=2
naked-single ==> r3c3=1
naked-single ==> r3c6=3
NO POSSIBLE VALUE for csp-variable 114 IN CONTEXT 18. RETRACTING CANDIDATE n3r6c3 FROM CONTEXT 0.

BACK IN CONTEXT 0 with 1 csp-variables solved and 632 candidates remaining.

naked-single ==> r6c3=2

GENERATING CONTEXT 20 AT DEPTH 1, SON OF CONTEXT 0, FROM HYPOTHESIS n3r5c5.
naked-single ==> r4c6=2
naked-single ==> r5c2=1
naked-single ==> r4c1=3

*** STARTING T&E IN CONTEXT 20 at depth 1 with 2 csp-variables solved and 612 candidates remaining ***

        STARTING PHASE 1 IN CONTEXT 20 AT DEPTH 1, with 2 csp-variables solved and 612 candidates remaining


GENERATING CONTEXT 21 AT DEPTH 2, SON OF CONTEXT 20, FROM HYPOTHESIS n1r1c1.
naked-single ==> r3c3=3
naked-single ==> r2c2=2
naked-single ==> r2c5=1
NO POSSIBLE VALUE for csp-variable 136 IN CONTEXT 21. RETRACTING CANDIDATE n1r1c1 FROM CONTEXT 20.

BACK IN CONTEXT 20 with 2 csp-variables solved and 612 candidates remaining.

naked-single ==> r1c1=2
naked-single ==> r2c2=3
naked-single ==> r3c3=1
naked-single ==> r3c6=3
NO POSSIBLE VALUE for csp-variable 114 IN CONTEXT 20. RETRACTING CANDIDATE n3r5c5 FROM CONTEXT 0.

BACK IN CONTEXT 0 with 2 csp-variables solved and 611 candidates remaining.

naked-single ==> r5c5=2
naked-single ==> r4c6=3
naked-single ==> r4c1=1
naked-single ==> r1c1=2
naked-single ==> r1c4=3
naked-single ==> r2c5=1
naked-single ==> r2c2=3

PUZZLE 0 HAS NO SOLUTION : NO CANDIDATE FOR RC-CELL r5c2
MOST COMPLEX RULE TRIED = NS
Puzzle 100100000010010000001001000100001000010010000001100000000000000000000000000000000 :
init-time = 0.0s, solve-time = 0.25s, total-time = 0.25s
s

.
Last edited by denis_berthier on Wed Sep 28, 2022 3:23 am, edited 1 time in total.
denis_berthier
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Re: The tridagon rule

Postby denis_berthier » Fri Sep 16, 2022 4:19 pm

.
ORk-CW classification results for mith's 63137 min-expand puzzles in T&E(3):

Here are results for ORk-Contrad-Whips similar to those for ORk-Forcing-Whips in the first post of this page.

Puzzles solved in SFin+Trid+Wn+ORkCWn
Code: Select all
-----------------------------------------------------------------------
        n=3                 n=5                 n=7                 n=8    
-----------------------------------------------------------------------
     8 ,196   puzzles solved by SFin+Trid (among 63 137 min-expands)   
-----------------------------------------------------------------------
k=0   8,137              17,532              21,160              22,332
     16,333       9,395  25,728       3,628  29,356       1,172  30,528
-----------------------------------------------------------------------
k=2   1,700               6,276               8,863               9,944   
     18,033      13,971  32,004       6,215  38,219       2,253  40,472   
-----------------------------------------------------------------------
k=3     286               1,379               2,413      
     18,319      15,064  33,383       7,249  40,632      
-----------------------------------------------------------------------
k=4      49                 319                 478      
     18,368      15,534  33,702       7,408  41,110      
-----------------------------------------------------------------------
k=5       6                  25                 111      
     18,374      15,353  33,727       7,494  41,221      
-----------------------------------------------------------------------

Same conventions as above:
Lines are separated by dashes, columns are separated by large white spaces.
Each (k, n) cell has three values in it:
- the main one, in the lower right corner, is the total number of puzzles solved by SFin + Trid + Wn + ORkFWn;
- the value above it is the difference with the previous line; it shows what’s gained by increasing k by 1;
- the value on the left of the main number is the difference with the previous cell; it shows what’s gained by increasing n.

And also same general conclusions:
• for fixed n, as k increases, the difference between two lines decreases quite fast; this shouldn’t be too surprising, as larger k means more chains have to converge to the same candidate;
• for fixed k, as n increases, the difference between two columns decreases quite fast; this shouldn’t be too surprising either, as it already happens with all the “classical” chains (whips…);
• starting from k=2 and n=3, at any point in the table, it is much more fruitful to increase n than to increase k;

Plus a new one:
For any fixed k and n, ORk-Forcing-Whips are more powerful than ORk-Contrad-Whips
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Re: The tridagon rule

Postby denis_berthier » Wed Oct 26, 2022 7:14 am

.
ORk-W classification results for mith's 63137 min-expand puzzles in T&E(3):

I've almost completed calculations similar to the above ones, for the classification of the 63137 min-expand database:
1) using ORk-whips instead of ORk-forcing-whips or ORk-contrad-whips;
2) using all the ORk-chains together.

The global results and general conclusions are similar to the above ones, with some additional conclusions:
- for any k and n, there is a large overlap between what can be solved with ORk-whips[n] and what can be solved with ORk-forcing-whips[n];
- for any k and n, ORk-whips[n] (which include ORk-contrad-whips as a special case) have a greater resolution power than ORk-forcing-whips[n];
- most of the puzzles that have a solution with ORk-forcing-whips[n] also have one with ORk-whips[n] but the converse is not true: for instance, only 96 puzzles can be solved in SFin+Trid+W5+OR5FW5 but not in SFin+Trid+W5+OR5W5; whereas 1894 can be solved in SFin+Trid+W5+OR5W5 but not in SFin+Trid+W5+ OR5FW5; and the difference is still larger for n=7;
- the difference between ORk-forcing-whips and ORk-whips is not well compensated by increasing the FW lengths: for instance, only 55 puzzles can be solved in SFin+Trid+W5+OR5FW5 but still not in SFin+Trid+W7+OR5W7; whereas 683 can be solved in SFin+Trid+W5+OR5W5 but still not in SFin+Trid+W7+OR5FW7.

It is so complicated to format a table as those I've posted before that I don't try to do it for the ORk-whips. You will find all the details in the forthcoming new version of CSP-Rules manual.

I consider these results as important when one wants to choose which kinds of rules to use.

Notice also that:
- as much as 13% of the puzzles can be solved using only SFin+Trid (Subsets + Finned Fish + the elementary tridagon elimination rule with only 1 guardian); this may give the wrong idea that puzzles with the anti-tridagon pattern can easily be reduced to easy puzzles;
- BUT, even with whips[≤8] and all the ORk-chains[≤8], a noticeable proportion of the puzzles (21%) remains unsolved.
This is to be compared with the result that +99,97% of all the Sudoku puzzles (unbiased statistics - see [PBCS]) can be solved by whips[≤8].

This should be food for thought for people who might have had the idea that finding it was all there is to the anti-tridagon pattern.

As an aside remark, the above results also show that there are several different rating systems for puzzles in T&E(3). Even in case we choose the system based on all the types of ORk-chains, k remains as a parameter.
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Re: The tridagon rule

Postby denis_berthier » Thu Dec 01, 2022 4:06 am

.
I don't plan to analyse mith's extended database of 158,276 min-expand puzzles (http://forum.enjoysudoku.com/t-e-3-puzzles-split-from-hardest-sudokus-thread-t40514.html) in as much detail as I've analysed the previous 83,177 one (mainly in this thread).
But here is an important result that remains valid after its publication:

All the known 9x9 sudoku puzzles in T&E(3) are indeed at most in T&E(W2, 2).
With the large number of minimal puzzles involved (847,778), this seems to entrench a new frontier of complexity.

(More detail here: http://forum.enjoysudoku.com/t-e-3-puzzles-split-from-hardest-sudokus-thread-t40514-18.html)
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