The New Sudoku Players' Forum

[P.O.]

Hi Denis, regarding your resolution by templates of this puzzle, your implementation reports: "candidate in no template[1] for digit 7 ==> r7c1≠7"

removing the 7 from r7c1 which is cell 55 means that cell 55 does not appear in any of the possible templates for 7 and since it is eliminated by combinations of 4, combinations of 2 and 3 leave at least one template for 7 which contains cell 55
as only templates for 3,7,8 and 9 have cell 55 it is this combination (3 7 8 9) which eliminates the template(s) for 7 which have cell 55 and allows the elimination r7c1<>7
so by forming this combination and checking all its instances none will be found composed of a template for 3, a template for 8, a template for 9 and a template for 7 having cell 55

it is easy to retrieve the possible templates for a value: among the 46656 templates, these are those which have all the cells of the value considered and no cells of the other values
when initializing the puzzle there are 62 possible templates for 3, 28 for 7, 43 for 8 and 31 for 9

Hidden Text: Show

Code: Select all

3:
62: ((2 13 27 32 44 48 55 69 79) (2 15 27 32 44 46 58 66 79)
     (2 15 27 32 44 48 55 67 79) (2 15 27 32 44 48 58 64 79)
     (2 17 24 32 37 54 58 66 79) (2 17 24 32 45 46 58 66 79)
     (2 17 24 32 45 48 55 67 79) (2 17 24 32 45 48 58 64 79)
     (2 18 24 32 44 46 58 66 79) (2 18 24 32 44 48 55 67 79)
     (2 18 24 32 44 48 58 64 79) (3 13 27 32 44 46 56 69 79)
     (3 15 27 32 44 46 56 67 79) (3 15 27 32 44 46 58 65 79)
     (3 17 24 32 37 54 56 67 79) (3 17 24 32 37 54 58 65 79)
     (3 17 24 32 38 54 55 67 79) (3 17 24 32 38 54 58 64 79)
     (3 17 24 32 45 46 56 67 79) (3 17 24 32 45 46 58 65 79)
     (3 18 24 32 44 46 56 67 79) (3 18 24 32 44 46 58 65 79)
     (4 10 27 32 44 48 56 69 79) (6 10 27 32 44 48 56 67 79)
     (6 10 27 32 44 48 58 65 79) (8 10 24 32 38 54 58 66 79)
     (8 10 24 32 45 48 56 67 79) (8 10 24 32 45 48 58 65 79)
     (4 12 27 32 44 46 56 69 79) (6 12 27 32 44 46 56 67 79)
     (6 12 27 32 44 46 58 65 79) (8 12 24 32 37 54 56 67 79)
     (8 12 24 32 37 54 58 65 79) (8 12 24 32 38 54 55 67 79)
     (8 12 24 32 38 54 58 64 79) (8 12 24 32 45 46 56 67 79)
     (8 12 24 32 45 46 58 65 79) (4 17 19 32 45 48 56 69 79)
     (4 18 19 32 44 48 56 69 79) (6 17 19 32 38 54 58 66 79)
     (6 17 19 32 45 48 56 67 79) (6 17 19 32 45 48 58 65 79)
     (6 18 19 32 44 48 56 67 79) (6 18 19 32 44 48 58 65 79)
     (8 13 19 32 45 48 56 69 79) (8 15 19 32 38 54 58 66 79)
     (8 15 19 32 45 48 56 67 79) (8 15 19 32 45 48 58 65 79)
     (4 17 20 32 45 48 55 69 79) (4 18 20 32 44 48 55 69 79)
     (6 17 20 32 37 54 58 66 79) (6 17 20 32 45 46 58 66 79)
     (6 17 20 32 45 48 55 67 79) (6 17 20 32 45 48 58 64 79)
     (6 18 20 32 44 46 58 66 79) (6 18 20 32 44 48 55 67 79)
     (6 18 20 32 44 48 58 64 79) (8 13 20 32 45 48 55 69 79)
     (8 15 20 32 37 54 58 66 79) (8 15 20 32 45 46 58 66 79)
     (8 15 20 32 45 48 55 67 79) (8 15 20 32 45 48 58 64 79))
7:
28: ((2 18 22 30 41 53 55 70 78) (2 18 22 30 41 53 61 64 78)
     (2 18 22 30 41 53 61 69 73) (2 18 22 33 37 53 59 70 75)
     (2 18 22 33 37 53 61 68 75) (3 18 22 29 41 53 55 70 78)
     (3 18 22 29 41 53 61 64 78) (3 18 22 29 41 53 61 69 73)
     (3 18 22 33 37 53 59 70 74) (3 18 22 33 37 53 61 68 74)
     (3 18 22 33 38 53 59 70 73) (3 18 22 33 38 53 61 68 73)
     (7 10 22 29 41 53 63 66 78) (7 10 22 29 41 53 63 69 75)
     (7 10 22 30 41 53 56 69 81) (7 10 22 30 41 53 63 65 78)
     (7 10 22 30 41 53 63 69 74) (7 10 22 33 38 53 59 66 81)
     (7 10 22 33 38 53 63 68 75) (7 12 22 29 41 53 55 69 81)
     (7 12 22 29 41 53 63 64 78) (7 12 22 29 41 53 63 69 73)
     (7 12 22 33 37 53 56 68 81) (7 12 22 33 37 53 59 65 81)
     (7 12 22 33 37 53 63 68 74) (7 12 22 33 38 53 55 68 81)
     (7 12 22 33 38 53 59 64 81) (7 12 22 33 38 53 63 68 73))
8:
43: ((5 17 21 34 40 47 55 69 81) (5 17 21 34 40 47 63 64 78)
     (5 17 21 34 40 47 63 69 73) (5 17 21 36 40 47 55 70 78)
     (5 17 21 36 40 47 61 64 78) (5 17 21 36 40 47 61 69 73)
     (5 18 21 34 40 47 55 69 80) (5 18 21 34 40 47 55 71 78)
     (5 18 21 35 40 47 55 70 78) (5 18 21 35 40 47 61 64 78)
     (5 18 21 35 40 47 61 69 73) (6 17 21 34 40 47 55 68 81)
     (6 17 21 34 40 47 59 64 81) (6 17 21 34 40 47 63 68 73)
     (6 17 21 36 40 47 59 70 73) (6 17 21 36 40 47 61 68 73)
     (6 18 21 34 40 47 55 68 80) (6 18 21 34 40 47 59 64 80)
     (6 18 21 34 40 47 59 71 73) (6 18 21 35 40 47 59 70 73)
     (6 18 21 35 40 47 61 68 73) (7 14 21 35 40 47 55 69 81)
     (7 14 21 35 40 47 63 64 78) (7 14 21 35 40 47 63 69 73)
     (7 14 21 36 40 47 55 69 80) (7 14 21 36 40 47 55 71 78)
     (8 14 21 34 40 47 55 69 81) (8 14 21 34 40 47 63 64 78)
     (8 14 21 34 40 47 63 69 73) (8 14 21 36 40 47 55 70 78)
     (8 14 21 36 40 47 61 64 78) (8 14 21 36 40 47 61 69 73)
     (7 15 21 35 40 47 55 68 81) (7 15 21 35 40 47 59 64 81)
     (7 15 21 35 40 47 63 68 73) (7 15 21 36 40 47 55 68 80)
     (7 15 21 36 40 47 59 64 80) (7 15 21 36 40 47 59 71 73)
     (8 15 21 34 40 47 55 68 81) (8 15 21 34 40 47 59 64 81)
     (8 15 21 34 40 47 63 68 73) (8 15 21 36 40 47 59 70 73)
     (8 15 21 36 40 47 61 68 73))
9:
31: ((9 10 24 31 43 48 56 71 77) (9 10 24 31 44 48 56 70 77)
     (9 10 24 31 44 48 61 65 77) (9 12 24 31 37 52 56 71 77)
     (9 12 24 31 38 52 55 71 77) (9 12 24 31 43 46 56 71 77)
     (9 12 24 31 44 46 56 70 77) (9 12 24 31 44 46 61 65 77)
     (9 13 19 29 44 51 61 66 77) (9 13 19 30 43 51 56 71 77)
     (9 13 19 30 44 51 56 70 77) (9 13 19 30 44 51 61 65 77)
     (9 13 19 35 38 51 61 66 77) (9 13 19 33 43 48 56 71 77)
     (9 13 19 33 44 48 56 70 77) (9 13 19 33 44 48 61 65 77)
     (9 15 19 31 43 48 56 71 77) (9 15 19 31 44 48 56 70 77)
     (9 15 19 31 44 48 61 65 77) (9 13 20 30 43 51 55 71 77)
     (9 13 20 30 44 51 55 70 77) (9 13 20 30 44 51 61 64 77)
     (9 13 20 35 37 51 61 66 77) (9 13 20 33 44 46 61 66 77)
     (9 13 20 33 43 48 55 71 77) (9 13 20 33 44 48 55 70 77)
     (9 13 20 33 44 48 61 64 77) (9 15 20 31 44 46 61 66 77)
     (9 15 20 31 43 48 55 71 77) (9 15 20 31 44 48 55 70 77)
     (9 15 20 31 44 48 61 64 77))


we easily observe that there are 4 templates for 7 containing cell 55:

Code: Select all

1: (2 18 22 30 41 53 55 70 78)
2: (3 18 22 29 41 53 55 70 78)
3: (7 12 22 29 41 53 55 69 81)
4: (7 12 22 33 38 53 55 68 81)


to form the instances of the combination (3 7 8 9) the simplest is to take a template for 3, look for a template for 7 which is compatible with it, i.e. disjoint, then a template for 8 compatible with the two previous ones and finally a template for 9 in order to form a set of 4 templates pairwise disjoint
doing this i find 17184 instances of the combination (3 7 8 9) whose templates are pairwise disjoint, filtering these combinations to retain only those having the template for 7 with cell 55 i find 1726 and distributing them in the 4 categories formed by the 4 templates for 7 having cell 55, i find:

Code: Select all

 1: 608
 2: 541
 3: 338
 4: 239


here a sample for each category:

Hidden Text: Show

Code: Select all

1:
.73.8...9...9.3.879.87....3..7.3.89..9.87..3.38...9.7.73....9.8..93.87..8...973..
.73.8...9...9.3.87.987....3..7.3.89.9..87..3.38...9.7.73....9.8..93.87..8...973..
.73..8..9...9.3.879.87....3..7.3.89..9.87..3.38...9.7.73..8.9..8.93..7......973.8
.73..8..9...9.3.87.987....3..7.3.89.9..87..3.38...9.7.73..8.9..8.93..7......973.8
.73..8..9...9.3.879.87....3..7.3.89..9.87..3.38...9.7.73....9.8..938.7..8...973..
.73..8..9...9.3.87.987....3..7.3.89.9..87..3.38...9.7.73....9.8..938.7..8...973..
.73....89...983..79.87....3..7.3.89..9.87..3.38...9.7.73....9.8..93.87..8...973..
.73....89...983..7.987....3..7.3.89.9..87..3.38...9.7.73....9.8..93.87..8...973..
.73.8...9..9..3.87..87.9..3..793.8..9..87..3.38....97.79.3....8.3...879.8...973..
.73.8...9...9.3.879.87....3..7.3.89..9.87..3.38...9.7.7..3..9.8.39..87..8...973..
.73.8...9...9.3.879.87....3..7.398.....87.93.389....7.79.3....8.3...879.8...973..
.73.8...9...9.3.87.987....3..7.3.89.9..87..3.38...9.7.7..3..9.8.39..87..8...973..
.73.8...9..9..3.87..87.9..3..793...89..87..3.38....97.79.3..8...3...879.8...973..
.73.8...9...9.3.879.87....3..7.39..8...87.93.389....7.79.3..8...3...879.8...973..
.73..8..9..9..3.87..87.9..3..793.8..9..87..3.38....97.79.38....83....79.....973.8
.73..8..9...9.3.879.87....3..7.3.89..9.87..3.38...9.7.7..38.9..839...7......973.8
.73..8..9...9.3.879.87....3..7.398.....87.93.389....7.79.38....83....79.....973.8
.73..8..9...9.3.87.987....3..7.3.89.9..87..3.38...9.7.7..38.9..839...7......973.8
.73..8..9..9..3.87..87.9..3..793.8..9..87..3.38....97.79.3....8.3..8.79.8...973..
.73..8..9...9.3.879.87....3..7.3.89..9.87..3.38...9.7.7..3..9.8.39.8.7..8...973..

2:
.37.8...9..9..3.87..87.9..3.7.93.8..9..87..3.38....97.79.3....8..3..879.8...973..
.37.8...9...9.3.879.87....3.79.3.8.....87.93.38...9.7.79.3....8..3..879.8...973..
.37.8...9...9.3.879.87....3.7..398.....87.93.389....7.79.3....8..3..879.8...973..
.37.8...9..9..3.87..87.9..3.7.93...89..87..3.38....97.79.3..8....3..879.8...973..
.37.8...9...9.3.879.87....3.79.3...8...87.93.38...9.7.79.3..8....3..879.8...973..
.37.8...9...9.3.879.87....3.7..39..8...87.93.389....7.79.3..8....3..879.8...973..
.37..8..9..9..3.87..87.9..3.7.93.8..9..87..3.38....97.79.38....8.3...79.....973.8
.37..8..9...9.3.879.87....3.79.3.8.....87.93.38...9.7.79.38....8.3...79.....973.8
.37..8..9...9.3.879.87....3.7..398.....87.93.389....7.79.38....8.3...79.....973.8
.37..8..9..9..3.87..87.9..3.7.93.8..9..87..3.38....97.79.3....8..3.8.79.8...973..
.37..8..9...9.3.879.87....3.79.3.8.....87.93.38...9.7.79.3....8..3.8.79.8...973..
.37..8..9...9.3.879.87....3.7..398.....87.93.389....7.79.3....8..3.8.79.8...973..
.37..8..9..9..3.87..87.9..3.7.93...89..87..3.38....97.79.3..8....3.8.79.8...973..
.37..8..9...9.3.879.87....3.79.3...8...87.93.38...9.7.79.3..8....3.8.79.8...973..
.37..8..9...9.3.879.87....3.7..39..8...87.93.389....7.79.3..8....3.8.79.8...973..
.37...8.9..9.83..7..87.9..3.7.93..8.9..87..3.38....97.79.3....8..3..879.8...973..
.37...8.9...983..79.87....3.79.3..8....87.93.38...9.7.79.3....8..3..879.8...973..
.37...8.9...983..79.87....3.7..39.8....87.93.389....7.79.3....8..3..879.8...973..
.37....89..9.83..7..87.9..3.7.93.8..9..87..3.38....97.79.3....8..3..879.8...973..
.37....89...983..79.87....3.79.3.8.....87.93.38...9.7.79.3....8..3..879.8...973..

3:
.3..8.7.9..79.3.8.9.87....3.79.3.8.....87.93.38...9.7.79.3....88.3..7.9.....983.7
.3..8.7.9..79.3.8.9.87....3.7..398.....87.93.389....7.79.3....88.3..7.9.....983.7
.3..8.7.9..79.3.8.9.87....3.79.3...8...87.93.38...9.7.79.3..8..8.3..7.9.....983.7
.3..8.7.9..79.3.8.9.87....3.7..39..8...87.93.389....7.79.3..8..8.3..7.9.....983.7
.3..8.7.9..79.3..89.87....3.79.3..8....87.93.38...9.7.79.3..8..8.3..7.9.....983.7
.3..8.7.9..79.3..89.87....3.7..39.8....87.93.389....7.79.3..8..8.3..7.9.....983.7
.3...87.9..79.3.8.9.87....3.79.3.8.....87.93.38...9.7.79.3....8..3.87.9.8...9.3.7
.3...87.9..79.3.8.9.87....3.7..398.....87.93.389....7.79.3....8..3.87.9.8...9.3.7
.3...87.9..79.3.8.9.87....3.79.3...8...87.93.38...9.7.79.38......3..789.8...9.3.7
.3...87.9..79.3.8.9.87....3.7..39..8...87.93.389....7.79.38......3..789.8...9.3.7
.3...87.9..79.3.8.9.87....3.79.3...8...87.93.38...9.7.79.3..8....3.87.9.8...9.3.7
.3...87.9..79.3.8.9.87....3.7..39..8...87.93.389....7.79.3..8....3.87.9.8...9.3.7
.3...87.9..79.3..89.87....3.79.3.8.....87.93.38...9.7.79.38....8.3..7.9.....9.387
.3...87.9..79.3..89.87....3.7..398.....87.93.389....7.79.38....8.3..7.9.....9.387
.3...87.9..79.3..89.87....3.79.3..8....87.93.38...9.7.79.38......3..789.8...9.3.7
.3...87.9..79.3..89.87....3.7..39.8....87.93.389....7.79.38......3..789.8...9.3.7
.3...87.9..79.3..89.87....3.79.3..8....87.93.38...9.7.79.3..8....3.87.9.8...9.3.7
.3...87.9..79.3..89.87....3.7..39.8....87.93.389....7.79.3..8....3.87.9.8...9.3.7
.3....789..7983...9.87....3.79.3.8.....87.93.38...9.7.79.3....88.3..7.9.....983.7
.3....789..7983...9.87....3.7..398.....87.93.389....7.79.3....88.3..7.9.....983.7

4:
.3..8.7.9..79.3.8.9.87....3..9.378...7.8..93.38...9.7.79.3....88.3.7..9.....983.7
.3..8.7.9..79.3.8.9.87....3..9.378...7.8..93.38...9.7.79.3....8..3.78.9.8...9.3.7
.3..8.7.9..79.3.8.9.87....3..9.37..8.7.8..93.38...9.7.79.3..8..8.3.7..9.....983.7
.3..8.7.9..79.3.8.9.87....3..9.37..8.7.8..93.38...9.7.79.3..8....3.78.9.8...9.3.7
.3..8.7.9..79.3..89.87....3..9.37.8..7.8..93.38...9.7.79.3..8..8.3.7..9.....983.7
.3..8.7.9..79.3..89.87....3..9.37.8..7.8..93.38...9.7.79.3..8....3.78.9.8...9.3.7
.3...87.9..79.3.8.9.87....3..9.37..8.7.8..93.38...9.7.79.38......3.7.89.8...9.3.7
.3...87.9..79.3..89.87....3..9.378...7.8..93.38...9.7.79.38....8.3.7..9.....9.387
.3...87.9..79.3..89.87....3..9.37.8..7.8..93.38...9.7.79.38......3.7.89.8...9.3.7
.3....789..7983...9.87....3..9.378...7.8..93.38...9.7.79.3....88.3.7..9.....983.7
.3....789..7983...9.87....3..9.378...7.8..93.38...9.7.79.3....8..3.78.9.8...9.3.7
.3....789..7983...9.87....3..9.37..8.7.8..93.38...9.7.79.3..8..8.3.7..9.....983.7
.3....789..7983...9.87....3..9.37..8.7.8..93.38...9.7.79.3..8....3.78.9.8...9.3.7
..3.8.7.9..79.3.8..987....3....3789.97.8...3.38...9.7.73....9.88.937........983.7
..3.8.7.9..79.3.8..987....3....3789.97.8...3.38...9.7.73....9.8..9378...8...9.3.7
..3..87.9..79.3.8..987....3....37.9897.8...3.38...9.7.73..8.9....937.8..8...9.3.7
..3..87.9..79.3..8.987....3....3789.97.8...3.38...9.7.73..8.9..8.937........9.387
..3..87.9..79.3..8.987....3....3789.97.8...3.38...9.7.73..8.9....937..8.8...9.3.7
..3...789..7983....987....3....3789.97.8...3.38...9.7.73....9.88.937........983.7
..3...789..7983....987....3....3789.97.8...3.38...9.7.73....9.8..9378...8...9.3.7


so whatever the template(s) for 7 with cell 55 eliminated by the combinations of 2 and 3 the one(s) which remain are not eliminated by the combinations of 4
and my conclusion is: at this stage of the resolution by combination of templates, with a combinatorial argument it is not possible to eliminate 7 from r7c1, so the question i ask myself now is: how does the logic you use work to know which template to eliminate?

[Hajime]

.I've found nothing special about the 9 puzzles in T4. If anyone wants to analyse them further, here they are:

Code: Select all

......7..4....9...6.8..7.......1537.3.5.2..91...9.35.2...3.19.7.3.5..12..1...2.53;893;16198 #4956
..34.678...718..36....374.129..6....5........73....6.837..4.........3.......718.3;933;22275 #5067
..34.678...718..36....374.129..6....37....6.85........73..4.........3.......718.3;934;22276 #5074
.2345.78.4.7.8.2...8..........5.4.7.....6....7..91....3.5...4.78....532...2..3.58;1245;23134 #6120
.2..5.......1....6.8.73.....1........98...62...4...8.1.61..49..8.2...46..4...8.12;1247;29659 #6124
.2..5.......1....6...73.....1........98...62...4...8.1.61..49.88.2...46.94...8.12;1247;28794 #6125
12..56......1.92.6...23.1........5.1.4.......87...2.....2.9.36....3..9.2..962..15;1449;62715 #6696
1..45....4.71.9.......271.42.4...........16.8......3...4279..1.57..1..9.9.15.....;1451;62714 #6700
12.4.........8...6....731..2.45..96....9...1.91....5...4....62.59....4.16........;2317;298786 #9524

.

In SiSeSuSo I implemented Nested Forcing Nets. At level 2 it solves every Sudoku before 2022.
When Tridagon were discovered in 2022 Nested Forcing Nets at level 3 will solve them.
I thing NFN(3) is equal to T&E(3), but I did not read all the theories of Denis.
The first 4 Sudoku's in above list are all solved with NFN(3).

[denis_berthier]

how does the logic you use work to know which template to eliminate?

This is the only part of your post I can answer: I have no means to follow the details of the thousands (sometimes even millions) of eliminations of templates[1, 2, 3, 4]. As I said in the first post, I don't consider templates as a human solving technique.
Moreover, even for this small part, I can't answer in the way you expect.

The overall logic is the same as for any rules in SudoRules: the rules are the program. The part of SudoRules for templates doesn't have a single line of procedural code.
The rules are applied in the simplest-first way (wrt the priorities mentioned in the first posts: BRT > T1 > T2 > T3 >T4). In case several rule instantiations have the same priority (e.g. all the T2 or all the T3), the one that is applied is selected arbitrarily by the inference engine (which has no practical consequence on the final result, as each of the Ti's has the confluence property). Most of the choices done during resolution are therefore arbitrary.
Think of it this way: if you want to put a seed in each of the 81 cells of a grid, you can put them linearly from 1 to 81 - or you can put them in a (pseudo) random order; SudoRules works the second way and there is no way to control this.

One more point useful to better understand how the templates (and indeed all of CSP-Rules) work is, when a pattern P2 depends on the effective presence of another pattern P1 (e.g. a template[1] depending on a candidate or a template[3] depending on a template[2]), if P1 is eliminated by some rule, then P2 is also automatically eliminated (said otherwise, logical dependencies are automatically dealt with by the inference engine). This is extremely powerful but there's no easy way to replace this by what you call "logic", i.e. I guess an algorithm or combinations of "if then else" statements.

If you have any doubts about my rules or their implementation, consider that they have now solved more than 100,000 puzzles with none leading to a (false) "this puzzle has no solution" message. It means that no rule can make any invalid elimination (such rules are detected after trying to solve only a handful of puzzles). In theory, it leaves the possibility that they don't make all the valid eliminations, but then they would lead to larger depths than yours, not smaller ones.
.

[Hajime]

I think NFN(3) is equal to T&E(3)....

I now see that T&E(3) is quite different than T(3) or T(4).

[denis_berthier]

I think NFN(3) is equal to T&E(3)....

I now see that T&E(3) is quite different than T(3) or T(4).

The T&E(n) and Templates techniques are totally different in their principles and in their classification results.

What we know as of now:
T&E(0) = T0 = BRT, by definition (BRT is 29.17% of the puzzles in unbiased stats);
T&E(1) has puzzles in T1, T2, T3 (based on 21,375 cbg-000 puzzles); I haven't yet seen any in T4 but it may come; (T&E(1) is almost all the rest of the puzzles in unbiased stats);
T&E(2) has puzzles in T3 and T4; but my sample is still very small (153);
T&E(3) has puzzles in T3 and T4 (based on 40,000 puzzles in T&E(3)); those in T3 are rare;
No puzzle requiring T5 is known.

[P.O.]

No puzzle requiring T5 is known.

there may not be known puzzles in T5 but there are known puzzles in 5-Template and 6-Template

i have no doubt that your implementation is correct in the sense that it does not eliminate candidates that are part of the solution otherwise you will quickly get contradictory grids, but how can it apply, using only combinations of 4, the rule that cell 55 is not part of any template for 7 and therefore 7 can be eliminated from cell 55 when there are thousands of valid size 4 combinations integrating cell 55 in a template for 7 as i have shown: it is factual, the puzzle is not solved by using only all combinations of 4

your engine does not just form combinations of templates, note the disappearance of some of them and draw the consequences on the evolution of the grid, elimination or assertion of candidates, which is the mechanics of resolution by template

he does more and something else than that and even some kind of anticipation mechanism using dependency relations cannot explain the contradiction of an assertion, this candidate can be eliminated from this cell, by facts, the production of thousands of combinations which invalidates at this stage of the resolution process the use of the elimination rule used by the resolution by templates

on the other hand if your implementation classifies as T4, that is to say using at most combinations of 4, puzzles for which i use at least combinations of 5 or 6 will you ever need more than T4 to solve any single-solution puzzle?

have you tried to solve a multi-solutions puzzle? with a template procedure you get all the solutions and your engine will have to resort to all the sizes of templates up to 9

[denis_berthier]

your engine does not just form combinations of templates, note the disappearance of some of them and draw the consequences on the evolution of the grid, elimination or assertion of candidates, which is the mechanics of resolution by template

On the contrary, it's exactly what it does and it's the only thing it does. But it does it to the full extent allowed by the logic of templates.

he does more and something else than that and even some kind of anticipation mechanism using dependency relations cannot explain the contradiction of an assertion,

Dependency relations are not an anticipation mechanism; they only allow to automatically keep track of the real resolution state at any stage of the computation. Whenever a candidate or a template is deleted, all the other candidates and templates whose logical existence depended on it are automatically deleted before anything else can happen. I don't think you can easily do this in a procedural program. In Clips, this is done by using only one word in the conditions of the rules: "logical". It tells the system to keep track of all the dependencies of each fact. Of course, this has a cost in terms of computation time and memory, but it's so powerful that it's worth the cost.
In a procedural language, the most direct and dumbest analog would be to re-start all the template computations after each assertion/deletion of a candidate.

have you tried to solve a multi-solutions puzzle? with a template procedure you get all the solutions and your engine will have to resort to all the sizes of templates up to 9

I haven't tried, but it will not give me all the solutions - no more than SudoRules does with other rules. It will give me only sssetions and eliminations that are common to all the solutions and reachable by the template technique.
.

[P.O.]

One more point useful to better understand how the templates (and indeed all of CSP-Rules) work is, when a pattern P2 depends on the effective presence of another pattern P1 (e.g. a template[1] depending on a candidate or a template[3] depending on a template[2]), if P1 is eliminated by some rule, then P2 is also automatically eliminated (said otherwise, logical dependencies are automatically dealt with by the inference engine). This is extremely powerful but there's no easy way to replace this by what you call "logic", i.e. I guess an algorithm or combinations of "if then else" statements

"a template[3] depending on a template[2]" this expression is slightly ambiguous it seems to suggest a consideration of T3 in T2

in my implementation templates are eliminated when combinations of size n are produced from combinations of size n-1, templates that cannot enter any combination disappear
when a candidate is asserted i recalculate the set of possible templates, when a candidate is eliminated without this resulting in an assertion it is not necessary and the template will be eliminated during recombinations

that said i still can't understand how the existence of thousands of valid combinations (3 7 8 9) where the template for 7 is one of the 4 having cell 55 does not prevent your engine to trigger the rule: candidate in no template[1] for digit 7 ==> r7c1≠7

i did the same check for the rule "candidate in no template[1] for digit 3 ==> r2c4≠3" and i get equivalent results
r2c4 is cell 13, at initialization there are 4 possible templates for 3 with this cell, there are 1247 combinations (1 3 4 9), the templates having cell 13 count for 30 of them
here are the results:

Hidden Text: Show

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1: (2 13 27 32 44 48 55 69 79)
13.4....94.93...1.....19.43.1.93.4..9.4...13...3.419..39.1.4....4...3.91..1.9.3.4
13.4....94.93...1.....19.43.1.93...49.4...13...3.419..39.1.4........3491.41.9.3..
13.4....94.93...1.....19.43..193.4..9.4...13...3.419..39.1.4....4...3.91.1..9.3.4
13.4....94.93...1.....19.43...9314...14...93.9.3.4.1..39.1.4....4...3.91..1.9.3.4
13.4....94.93...1.....19.43...931..4.14...93.9.3.4.1..39.1.4........3491.41.9.3..

2: (3 13 27 32 44 46 56 69 79)
1.34....94.93.1........9143...93.41..94.1..3.3.1.4.9..93.1.4....4...3.91.1..9.3.4
1.34....94.93...1.....19.43.1.93.4...94...13.3...419..93.1.4....4...3.91..1.9.3.4
1.34....94.93...1.....19.43.1.93...4.94...13.3...419..93.1.4........3491.41.9.3..
1.34....94.93...1.....19.43..193.4...94...13.3...419..93.1.4....4...3.91.1..9.3.4
1.34....94..3.9.1..9..1..43...9314...14...93.3.9.4.1..93.1.4....4...3.91..1.9.3.4
1.34....94..3.9.1..9..1..43...931..4.14...93.3.9.4.1..93.1.4........3491.41.9.3..
1.34....94.93...1.....19.43...9314...94...13.3.1.4.9..93.1.4....4...3.91.1..9.3.4

3: (8 13 19 32 45 48 56 69 79)
1..4...394.93...1.3...19.4..1.93.4...94...1.3..3.419..93.1.4....4...3.91..1.9.3.4
1..4...394..3.9.1.39..1..4..1.93.4....4...1939.3.41....3.1.49...49..3..1..1.9.3.4
1..4...394.93...1.3...19.4..1.93...4.94...1.3..3.419..93.1.4........3491.41.9.3..
1..4...394.93...1.3...19.4..1.93...4..4...1939.3.41....3.1.49...9...34.1.41.9.3..
1..4...394..3.9.1.39..1..4..1.93...4..4...1939.3.41....3.1.49....9..34.1.41.9.3..
1..4...394.93...1.3...19.4...193.4...94...1.3..3.419..93.1.4....4...3.91.1..9.3.4
1..4...394..3.9.1.39..1..4...193.4....4...1939.3.41....3.1.49...49..3..1.1..9.3.4
1..4...394..3.9.1.39..1..4....9314...14....939.3.4.1...3.1.49...49..3..1..1.9.3.4
1..4...394.93...1.3...19.4....931..4.14....939.3.4.1...3.1.49...9...34.1.41.9.3..
1..4...394..3.9.1.39..1..4....931..4.14....939.3.4.1...3.1.49....9..34.1.41.9.3..

4: (8 13 20 32 45 48 55 69 79)   
1..4...394.93...1..3..19.4..1.93.4..9.4...1.3..3.419..39.1.4....4...3.91..1.9.3.4
1..4...394.93...1..3..19.4..1.93...49.4...1.3..3.419..39.1.4........3491.41.9.3..
1..4...394.93...1..3..19.4..1.93...4..4...1939.3.41...3..1.49...9...34.1.41.9.3..
1..4...394.93...1..3..19.4...193...4..4...1939.3.41...31...49...9...34.1.4.19.3..
1..4...394.93...1..3..19.4...193.4..9.4...1.3..3.419..39.1.4....4...3.91.1..9.3.4
1..4...394.93...1..3..19.4....9314...14...9.39.3.4.1..39.1.4....4...3.91..1.9.3.4
1..4...394.93...1..3..19.4....931..4.14...9.39.3.4.1..39.1.4........3491.41.9.3..
1..4...394.93...1..3..19.4....931..4.14....939.3.4.1..3..1.49...9...34.1.41.9.3..


doesn't that question you?
this suggests to me that your engine is not just using the information contained in the combinations but is adding information from another source to make its decisions.

[denis_berthier]

"a template[3] depending on a template[2]" this expression is slightly ambiguous it seems to suggest a consideration of T3 in T2

There's nothing ambiguous in what I wrote - but you're reversing the meaning.

here are the results:
[...]
doesn't that question you?

Not in the least. SudoRules doesn't have to follow the same resolution path as your solver.

this suggests to me that your engine is not just using the information contained in the combinations but is adding information from another source to make its decisions.

Yes, I included a crystal ball. Don't worry, I'll publish the code soon; you'll be able to search it for it.

[P.O.]

it is not a question of resolution path but of technique, at a given level all the combinations that have been formed from the previous one are available and it is their analysis that leads to the composition of the combinations of the next level and it is during this process that the templates that cannot integrate a combination are eliminated

i have shown that combinations of size 4 containing candidates 7 and 3 in cells 55 and 13 respectively exist and have therefore been formed from the previous ones, consequently level 4 keeps these candidates in these cells

only the composition of the combinations of the following level can make the templates considered disappear and if this is the case allow the elimination of the candidates

[denis_berthier]

.
I'm not going to debug your program for you. If I've chosen Clips to implement SudoRules, it's precisely to avoid having to deal with implementation details that are dealt with automatically in an inference engine.
There are several ways templates can be eliminated. Eliminations work both upwards and downwards.
.

[P.O.]

that's the problem with Clips, it's a black box to you, you don't know what it's doing.

[denis_berthier]

that's the problem with Clips, it's a black box to you, you don't know what it's doing.

I know perfectly what it does at the logical level, which is the only thing that matters. Implementation details are irrelevant.
.

[denis_berthier]

.
For people who need to debug their programs, here are 3 lists of 100 puzzles each at respective template-depths 1, 2 or 3.
The puzzles are from the controlled-bias collection.
Good debugging practice: start from T1 and then proceed upwards until you find the first discrepancies.

T1:

Hidden Text: Show

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1...56..9.5...9.2....1..4....5.9..7..3.2...5.96..3....3...4.....9......864...5... 668321  #20
.23....8.......12.7.91.........97.46...3....29..6......6.9..5.457.....98.....4... 55363  #22
...45...........23..91.34...1...753.....1.8..6.5.......723...48.6......7.3.5.8... 962725  #31
.2....78...6.89..3...12..6.2.......1......24.8.129.....6......454..6......83..... 278879  #48
.2..56...4..7....3......46.....9753...8....72...23..4.59.3.4...6....5...81..7.... 175024  #72
..3456..........2.7..1..4...4.9.5.7.........8..1...542..5.6..9.67.3.8.....2...6.4 169879  #81
.23.5..8.45.7...2.7...2..6.....9...73.856....5..8......4.2.....8.....6.1..7....32 39575  #84
..3.56...4...8.1...8....4.52......7.....475.6...6...3.6.....8...759.26........... 171198  #87
...45.........91.3.8...3..5.6.....71...2473..9..6...4.3.....6.....8....2.42.3.... 175202  #88
1...5..89...78.1....91...652.8........756.....1.8......7......2.......3..35..46.8 72229  #93
1......894.6..........2.4....1......34.9..2.6..5...87......76......94.1..6281.3.. 78668  #95
..34...8..5......37.....5...14..5........8...9.......1.61..2..7.4.8...96.....73.. 13978  #99
1..4......5...9.23.......64.1....97.5.........9.247...3..8......61..2.3.9...65..7 32666  #103
1......8.....89.23.....3..4..46....85..91.24....27.....92.....16......9.8.......7 563947  #104
1.........5.7....3.891......3.....9.5.7.9..31.4...1.57...678...69....8....5...... 482636  #134
..34..78.....89...7...2.......6..9..64......8.....5.31.1...2...57.3....6.6....35. 520640  #135
1....6.8..5......3.89.2.5.42..8...4....3..8.....6...7237.....915.19.......4.....7 33798  #152
1234...8.4.6.....3...1..5...38..5..6..4....9.6.7...35.3..6.2.7........32.7.8..... 353558  #165
.....6......7...2....13.46521.8...9.......842........75.2..86...64..5...93..4.... 366711  #187
.2.4..7.945........891.........9..3.........296..7.5..5...28..4.3.9....8.42..59.. 8676  #189
.2..56...4.6.891..7.............5....95.14..28.42.....348...6....76...3.......5.8 415168  #200
...4...8..5..8...37....25.62.4.....8.......6..98.1...7....7..9287...3...9..5.8... 721299  #204
...4...89....8...37.91.....2.4.7.6...6.9....5.3.......34..........2.3.6.6..59..1. 158048  #205
12...6.89..678...........4..1..9....3....82..8..2...355.......7...52...1.3....4.2 141427  #206
1.34.......6.8..2.7.......6...8..6....4.2..7..976.13...4...7.68...3.4....7....... 535815  #228
..3....8.4......2...91.2..6.1..6..98.....42.18....73.........5....9.3..7..2.7..3. 483346  #236
..3..6....5.7.....7.9...5..2...9.8.....8..97.....2.43..6.9.8...8.2.1........73.1. 178003  #239
..3.5..89.5.7...2.7..1........67.9....8...6.7..4..3.1.3......9...5.....29.2..4..5 311975  #268
12...6..94.....1.......25.6.3...5.7.5.86...9.....7..1...59....86...48.....25..... 1.03336e+06  #269
1...5.78.........3.891.....2..9....4.673...9..9....3.1.........6.5..3..2...617.3. 156312  #276
...4.....4..7.9.2..8..3...6..45.1698.6.2.8...........73.5...8...42....7........14 472137  #277
....56..9....8.12....1...4.23..6..1..9...3.7...89..23.31...4...86.....52....7.... 18681  #289
1.3....8..5..891........5.62...6.8......4....9.1.2836..4..1....69...4.3...2...4.5 561803  #292
.2...67...5.........9132..6.....5.1.5.1.4.9.289.....6.....9.27.6....8.9.....7.... 66327  #293
.2....7..4.......3.89.......358...9..9..2.4......6.......6..8.5.4..7.96...75.82.. 70372  #303
.2...6.......891.....13...62....1.74..1.7.89...8...2....7....5..1.3.796..9..2.... 114475  #306
.2..5.7.94.6....2.......54...589..7....2.....891..............5..89..43.917....6. 17592  #308
1....67...567....3.8.1......35.74..8.....32..9...1...4..8...4...1.3.....6.....9.1 376482  #315
.2.....8...6..91.....1.2.4..3.615...5.....2...48......3..871.6.......3......6..72 335306  #320
........9..67...2..8....546.3.8.......4.7.9..91..43.7.......8.....36..126..52.... 184122  #325
..3...7.9.5.7....3.89....4.2.7.15......2....1.9.8..........8....625.13.7....2.6.8 121415  #328
1.........56....2.7.9...5.......58....43...6.....743..3..591.7.6..8....29...2...1 30858  #335
....5..8...678..237....2.4.24.....95.....5...93....8..3......52..7.1..........4.7 580780  #343
.2.4..78...67....3..9...5...4....8......2..7.697...2...1...5....7596.3....281.... 173660  #344
.2.456.8.....8...37.......6...5.7.9..98.2....6...1....3.58.19..81......5..4...... 185528  #345
..3..67..4..7.9.23......54..4..97.5.....2.4.85.......2.7...3...8.......491.2...3. 196656  #374
.2..5.789.........7.9..2..6.48.176..5.1.........32......4..82....5.....8....41.5. 211457  #375
.....6.8..5.7..1.3.8..325.........78......3.29..2.741........946.2..4....9...32.. 401958  #378
.2..5.7.9.5..8....7..........1..8..5375.9.6.....61.......8....761...3.5.93....2.. 234111  #379
12....7.9....8.......1......6...4.3..9....6518..3.........4.968...9.......8.7.2.4 112203  #386
.2..56.8.4.67....37...3..4.......91..95..............4....1...76.7.98.......6.83. 2302  #405
1...56.......8..237..1......6....3.....5..2.85.8..1..........1.842.....5.3..4..6. 152628  #409
12..56....5..............46...8...1..3...1.......752.8..1.678.26...489....52..... 141665  #413
1......8.4.6..9....89.......7.36...46.52.4......59.....4.6..8.1.6..2......78..2.5 376236  #419
...4.678...6.89..3...1.......5.18.6....5.....8.19....5......618.......979....3.5. 149989  #423
.......8....7.91.3.8.132..6..59........62....6.4...2.....81..9....5.76....1.6..57 157004  #431
.2.4..7....6....2.7.9......27...83....82......14.7...8.3....6.....3...958...15... 160110  #437
.2....7.9.5.78......9...54....6.5........82..6....3.1.3......5..7...4..1.14....3. 306415  #441
12..5.7......89.2.7..1....6.9..61...3.7.2..68...3...........2...75.946..9.2.....5 4028  #443
1.3.......5......3789...5.4.....7....6.52.9..59.6..4.......5..8.....327.9..871... 55533  #450
1.3.5..8...6......7.9..2......6....8....14.72.......51.6.5.......5..13469.2..7... 134329  #454
.2..56..9.5.7.9.2..8.....6..........3..54.....4.2.7.......2463...2...84..7.36.9.. 184560  #470
1.34.....4.6....23.8....5......7..36...6.........1..4...48.......1....9.9..5.74.8 1859  #485
12.4....9.56....2....1.256...7.9....3..2....56.....24...4.7.....7.9....1.....5.7. 658190  #490
1.3....8.4...8...........6.....43.......2.4...6.5..2...71....5...5..73.88.23...4. 157304  #491
.2..5....4.67.......9...56......5.4......783.6.7.4.......9...1...2.139.6....6..58 169701  #500
123......4.......3.8....56.2.....9...9..18.4.....9..185....4.72...3.........7.4.1 996877  #501
.23.....9....8.1.3...1.2.6.........76.......5..52.4.1...156.9......413..89....... 688128  #502
1...56......7...2.78.....6.2..97.34.37.2..9..96.8..2.....64....63...8.71.....7... 410356  #504
....5..8...67.....7....25...316...4........9....218....95.....2..4...9...12.4.63. 400707  #510
1....6.8......9..37..1.2..4..4.......6.....32.9.....5.3..9......4..7.2.5...5.4.7. 21778  #523
1....6.89...7.......9.32564234...6.5.97........1.....2...5.8.....5..4..7...67.... 3199  #530
...4...8......91..789.3.....34...6......6..9.6.1........2.9.876...32.......678.42 80968  #533
1...56....5...9..3...1..5..2.....4....4.7.2.88.7....9....9..6..5.2.4.91......7.3. 599191  #540
.2.4..7.94.6............5....5.4..7..7...521.9.8..........186.25...2.8.....5.3... 13036  #543
...4.6..9...78.1....91.........7..4.6179.8.5.....2......48...955.......8..2...47. 45190  #573
1...5.7.9...7....3.8...2...2....7...59.........18.4...3....8.5..4.2..3.1..5..1.42 335343  #574
..3....8.4.6...1......3...42.89..6...97.....86...7.3.....2.....84.....75.7.84...6 410444  #577
.2......94..7..........256..3...5..6.1.248..5..5..78...9.6.1...5.2..3..1.......3. 91280  #578
..34...89.5.7.....7....2...23..1..45.6......7.1.64.2..............5.83...7.3..4.6 123244  #579
..3..67...5678...3...1......4.5..9....8....1.....9.2.53..6.4..1......63.....1..5. 137929  #589
.2.....8.4.6.8.1.3.....25..24...7......64.....6..1...739.52...6.72....1....9..3.. 199286  #598
..3......4.6.8....7.....564..5.....73..9...4.....413..........1.1.59...8.926...3. 5465  #602
1........4...891...8...2.....56..3.7...2..4..9.75...183.....6.......48....8..5..2 36011  #608
....5..894........7...3..6..4.8......61..7..5.3..4.6....2..8....7.....9.69...4.72 453604  #614
....567.......91.378......4..759...6......8.2....24....1..6....6..9472..9.4..5... 193226  #627
...4...89...78....7.9...........7.....5.1.2..91...5..6.6.....95.9.6.43..8.......2 315816  #639
....5.789..67..1.....1.....248.......9.51.6......2.97.5...932.........9..72.....5 328215  #648
1.........5.....237.91...64.6.345..7...9....5......4......9.....35....76.42.6.... 2368  #650
...4.6....5..8...37.9.3...426....9......7..4...19......1...3..7.3...1..69..6..3.. 181216  #667
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............78..23..9...5..2....493...5...4...7.63......7.6.8..5.12...9..9....... 244509  #131
.2...67..45......37..1..5......1.6....123...8.7.9.....36......5..8..2....4.67.... 243685  #136
..3...7894........78..23........8.....1.3...2...94..1534...5.....2.9.6...6......8 17394  #137
..3...7......8..2..8.1.3........5.915..9..4......48.7.31.26..4.....9..1..42...6.. 163106  #139
..3..6.894..7......89.2..........91.........887194.3...1....6.25.7.....1.4..7..3. 113462  #142
.2.......45......3..91..5......9.6..8..3.4.1......2.3....6.5..1.4..37.9.6....8..5 62054  #143
..34..78.4....91.37...2..........6.5...34.......2.5..1.9...8....45.......1.57.9.. 7236  #144
1......89.5...9..3...........4...6..5.7..1....9...2.4.3..9..2..6.5217....1.5....6 37839  #145
.2......94....9......1235...3..78...6..5.1.3...5...4.73.2....4...8...3..94...72.6 407488  #146
....5.789..67..1...8.12..6.2.5.9.....173......9........4.8...1...897.43.....3.... 47565  #148
.2.....8..5.789.237...........6..84167.3...5...1..5............5...32.76...5..3.. 11842  #149
....56.8....78.1.3........42..........12..3..97.6....1..2......5..3..9..8.4.726.. 31350  #151
.23..6...4..78..2...912.....35...6...4...7....9..4.3.....2..91..189....6.....5... 224663  #153
1.........567...2.7..1.35.4.3.5.8..1..1........429..........9..6.8........2.754.8 75503  #155
1.3....8..5.7......89.......3......5..1..263.9.....21......5....74318.......9.34. 507546  #157
.....678.....89...7.9..3...2....86..5.....3.....31.2.7.....2...615......87....4.5 186926  #160
.....6...4..789........35.42..5....65.1.948...7....2...9.....1.........28...3...5 176939  #161
.2.4..78......9..3.....35..2.8...9....7....4...481..7.........167..94....925..... 315393  #163
123...7....6.891...............47..5.6..9.83.9.1...6..31.5......47..2...8......7. 3288  #164
....5.7.......9...789.2...42..6...38......27..7..3....3..5..41..17....9286...1... 83098  #168
.23.........7.9...7....3.6...56..93....2..6...9....2..5.234.....3....4.1..45...7. 21168  #170
.2.4567.....7.......9...5.4..5......63.......97.8.4.5......5..7........28.4.1..9. 2192  #171
.2.4.6....56...12.7.9.2...........9......5..7..72.165....8...1..6.3.48....2....3. 5250  #172
.234.......67.9...7...2..........8.6.9..4.27....2....5...3.1..2.34...91...2.6.3.7 296993  #173
.23....8...6...1..78...35.4..8..7....9...2.516..3.........4..9..3...5....726..... 86220  #174
.2..5.7.94.67.9..3..9....6...456.....9...83.6....74...3....52..8....1.......2.... 695782  #176
.2.4....9..67891..........5..4..8.9.....215..9.......2....64...6.2.1..7.8.1....5. 16712  #179
.....6.8....7....37.91.....2.4.7...8...89..1.......3.45.........6.3..5.2.375.86.. 367655  #180
....5..8.4....91.378..3..652....4..8.4.9..67.6..8...1....39......1.4.2...7....... 609002  #185
12...6......7.......91.2.65.156..34.3....4...86............8.17.4.3..8..9........ 181914  #186
..34..7.9....89...7....2....1...4......26..........25.5..6.8.7.6.1..35....2...91. 88371  #188
1..4.6..945.7....3.8...2.....5...6..3............61.52....2..4...2.935.78........ 471104  #190
12.45.78.4....9..3...............5..34.82........1..74....7....6....1.5.8.13..9.. 160421  #193
.2.45.7...5...91..............5..6....82...4..4..1...23....8.97.6.9..2..9...2..1. 447791  #198
1...5.7..4.6...12..8...2..5.......5.8.16.....9...248......6..9......3.7...25.134. 70262  #203
1....678..5.....2...9.3.5....43.8......6.1.9..6..4..1.3..5.....59.......6.19..85. 53619  #208
......7.94.6.8.1......32....1.8........6....2...5..3.4.42.1.6.7.9...7...8...6..9. 33670  #213
.....67...5.7..123.8......6.1.89.....6.3.1..293......4.......9.5.12..6.......32.7 151740  #215
..3...7...567..123.....25...1..6.......2.7...8.....9..5...2..9.6.8.9..3.94.6...5. 185004  #217
.2........567.91..7.....5....5.4..9.3....7.1...85.3.64....2....6..8.4....92..5... 349539  #223
1...5..8..5.7.9..3..91....62..6.........934.5.3.....6.....6..74..291..3..7....... 41229  #224
....56..94.....1.3.89....4.21.648.....4.1.6.......3..2.....4....3197.......8.13.. 20679  #225

[Edit: correcte the ill-copied T3 list]
.

[P.O.]

Hi Denis, T2 and T3 are the same.