The tridagon rule

Advanced methods and approaches for solving Sudoku puzzles

Re: The tridagon rule

Postby denis_berthier » Mon Oct 23, 2023 9:08 am

Resilience of the non-degenerate tridagon pattern

Following the last collection of T&E(2) puzzles published by Paquita:

I had a few questions about it, because it had an unusually high proportion of puzzles in B6B.
I went back in time and checked his previous two collections:
They also have unusually high proportions of puzzles in B6B.

But what I had completely overlooked and what there's in common between the three collections is a very high proportion of puzzles with a non-degenerate tridagon (with guardians): 100%, 99.97% and 97.01% resp.

The result is, my questions in the "hardest" thread about the unusual BpB ratings were misguided by my idea that they could mainly result from the search for high SER.
Indeed, I now think they result mainly from the technique of vicinity search.

Starting from puzzles with a non-degenerate tridagon, "many" nearby ones will also have this non-degenerate pattern. Said otherwise, the non-degenerate tridagon pattern is very resilient to "small" changes of givens (even without counting all the trivial isomorphisms).

(For puzzles in T&E(3), this high resilience may also explain why so many puzzles with a non-degenerate tridagon have been found so fast after the first few ones.)

There remains to explain why the presence of a non-degenerate tridagon AND a high SER favours a high BpB, but at least this is easier to imagine than just a high value of the SER (because this explanation alone would contradict our previous vague correlation results).

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Re: The tridagon rule

Postby denis_berthier » Sat May 11, 2024 8:25 am

All the puzzles in mith's max-expand collection ( have a non-degenerate tridagon.

I had previously shown that all the known minimal puzzles in T&E(3) have a non-degenerate tridagon and at that time, I didn't care to look beyond the minimals.

My new calculations allow to conclude that all the known* puzzles in T&E(3) have a non-degenerate tridagon.

(*) and all the implicitly known ones, between the minimals and the max-expands.
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Re: The tridagon rule

Postby denis_berthier » Thu May 16, 2024 2:57 am

What happens to a (non-degenerate) tridagon pattern when clues are deleted or added? The question arises in particular in the context of vicinity search.

1) When a clue is deleted from a puzzle with a tridagon pattern, the puzzle may come to have multiple solutions, but the tridagon is untouched (except that it may come to have more guardians).

2) As for adding a clue, we can be quite precise for puzzles in T&E(3).
Let's start with mith's collection of T&E(3) puzzles. He provides 3 collections in T&E(3): minimals, min-expands and max-expands (which I now prefer to call T&E(3)-expands in order to avoid any ambiguity).
Starting from minimals, clues can be added by applying Singles until one first reaches the min-expands, but still more clues (taken from the solution) can be added until one reaches the T&E(3)-expands, the limit beyond which one leaves T&E(3). Between minimals and max-expands is the whole T&E(3) land and its thickness is variable between 1 (in case the minimal is its own max-expand) and a few puzzles. Nothing really new until now.

But now what happens if one adds one more clue?
First, let's say that a puzzle P' is a 1-expand of a single solution puzzle P if P' has all the clues of P plus one (taken from the solution of P).
Now, starting from all the max-expands in mith's T&E(3) collection (48,071 minus 6 that are there by error), and applying systematically all the possible 1-expansions, one gets 2,257,893 puzzles, thus multiplying the number of puzzles by about 47 (before this number is eventually reduced by isomorphisms).
By the definition of a T&E(3)-expand, we know that none of these puzzles can be in T&E(3) or deeper (this is how I found the small error in the max-expands database). We also know that all the T&E(3)-expand puzzles have a non-degenerate tridagon.
The T&E(3)-expands form the border of T&E(3) from the inner side, and some of their 1-expands form part of the border from the outer side.
Note that these 1-expanded puzzles are not minimals and they may have other minimals than the already known ones, in T&E(3) or not. That's why they are not necessarily on the outer side of the T&E(3) border.

But are they all in T&E(2)? NO, but a high percentage is.
Code: Select all
1,584,186   are in T&E(2), i.e. 70.16 %
598,053   are in T&E(1), i.e. 26.49 %
75,632  are in T&E(0), i.e. 3.35 %

Note that this is not very surprising, as the addition of a clue has unpredictable effects in general.

Back to our original question: what about the tridagon?
For the puzzles in T&E(2), there are
Code: Select all
617,299 with a tridagon, i.e. 38.97 %
1,117,285 with a tridagon or a degenerate cyclic tridagon, i.e. 70.53%

For the puzzles in T&E(1), the percentages are lower, but still noticeable:
Code: Select all
15.20 % with a tridagon, i.e.
36.08 % with a tridagon or a degenerate cyclic tridagon.

Globally, considering the process of 1-expanding all the T&E(3)-expands:
Code: Select all
-in 27.34 % of the cases, one gets a puzzle in T&E(2) that has a tridagon;
-in 49.48 % of the cases, one gets a puzzle in T&E(2) that has a tridagon or a degenerate cyclic tridagon

Said otherwise, about 50% of the 1-expands of all the T&E(3)-expands (before possible reduction by isomorphisms) are potential precursors of T&E(3) puzzles. Which suggests that, in the vicinity search procedure, one shouldn't only check all the minimals of the min-expands, but all the minimals of the 1-expands of all the T&E(3)-expands. Of course, this may quickly become overwhelming work.

I have many more results about such expansions, but I need to do some cleaning before I can sort out the really interesting ones and publish them.
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Re: The tridagon rule

Postby denis_berthier » Sat May 25, 2024 2:39 am

In a previous post (,
I gave the proportion of degenerate-cyclic tridagons (as defined here
in the first small part of the [cbg] collection: 0.22%.

As a background work when my Mac was not busy on some other calculations, I have now finished checking the whole collection of ~6M puzzles.

In the full controlled-bias collection, there is no non-degenerate tridagon, but there are 16,453 puzzles (0.278 % ) with the degenerate-cyclic variant.

The purpose of checking the whole collection was not to improve the precision of the percentage (though it did) but to be able to compute such distributions as the SER or B-ratings and to have enough puzzles for starting further calculations.

Both distributions (SER and B-ratings) are concentrated on much higher values than in the full [cbg] distribution, as shown below.

B distribution (number of puzzles):
Code: Select all
3: 3039
4: 7107
5: 4096
6: 1535
7: 492
8: 138
9: 40
10: 5
11: 0
12: 0
13: 1

SER distribution (number of puzzles) (left column is 10*SER:
Code: Select all
42: 240
43: 0
44: 27
45: 0
46: 0
47: 0
48: 0
49: 0
50: 6
51: 0
52: 0
53: 0
54: 1
55: 0
56: 0
57: 0
58: 0
59: 0
60: 0
61: 0
62: 3
63: 0
64: 0
65: 3
66: 127
67: 96
68: 14
69: 3
70: 16
71: 1814
72: 3564
73: 2015
74: 116
75: 75
76: 220
77: 228
78: 408
79: 102
80: 34
81: 0
82: 253
83: 2641
84: 2013
85: 819
86: 73
87: 28
88: 206
89: 623
90: 599
91: 80
92: 5
93: 1

Even after truncating this sub-collection to the upper B or SER values or both, there remains enough puzzles to be used as seeds in the search for non-degenerate tridagons.

[Edit): Actually, in order to avoid problems with uniqueness, I didn't use the SER but SER with no uniqueness; and I didn't use SudokuExplainer but its FPGX variant, available on GitHub in the programs folder of [cbg]:
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