The New Sudoku Players' Forum

Website · by **denis_berthier** » Sun Apr 06, 2025 5:37 am

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In [HCCS2, https://www.researchgate.net/publication/381884473_Hierarchical_Classifications_in_Constraint_Satisfaction_Second_Edition], I introduced several definitions that will make the background for this thread.

1) the BRT-expansion of a puzzle P is the (uniquely defined) puzzle obtained by applying it the Basic Resolution Theory (i.e. Singles and elimination by direct contradiction with decided values) - this is commonly called expansion by Singles. (AFAIK, this idea was introduced by mith.)

2) A 1-expand of a consistent puzzle P is any consistent puzzle obtained from P by adding one and only one clue. This is generally applied to single-solution puzzles and the added clue must then be from the solution. If the puzzle has multiple solutions, the added clue must be common to all the solutions.

3) A T&E(d)-expand of a puzzle P at T&E-depth d is any puzzle at T&E-depth d that is an expansion of P and that cannot be further expanded without leaving T&E(d). It is easy to see that any T&E(d)-expand is its own BRT-expand. This is a generalisation of mith's notion of a "max-expand" (it's the same thing if d=3). T&E(d) expands play an important role, as they are on the inner side of the T&E(d) border with shallower T&E-depths. Their 1-expands will be at shallower depths.

4) BRT-equivalence between puzzles (not necessarily minimal): P1 and P2 are BRT-equivalent iff they have the same BRT-expansion.

All the ratings and classifications I've ever defined are compatible with this equivalence relatrion - as should be IMO any reasonable rating system.

As any equivalence relation, BRT-equivalence trivially induces a (non-metrizable) topology on the space of all the consistent puzzles and on the subspace of all the minimals puzzles. For this topology, all my ratings and classifications are continuous. A puzzle is isolated iff it is its own BRT-expand. Puzzles from different solutions grids are isolated from each other.

For each fixed T&E-depth d, (not necessarily minimal) puzzles at this depth can be grouped into layers:
- layer 0 consists of all the minimal puzzles at T&E-depth d, plus all the 1-expands at T&E-depth d of T&E(d+x)-expands, x ≥1, plus all the BRT-expands of all the previous ones (which can only be at T&E-depth d), plus all the puzzles in-between;
- layer p consists of all the 1-expands at T&E-depth d of the BRT-expands in layer p-1, plus all their BRT-expands (which can only be at T&E-depth d), plus all the puzzles in-between.
Layers other than 0 can be considered as layers of successive 1+BRT expansion.

Figure 4.2 of [HCCS2] illustrates this stratification.
Notice that this stratification is not totally unrelated with the number of clues, but it's very different.
In particular, the number of clues is not correlated to anything and has no reason to be used as the basis for a stratification.

The purpose of this thread is to analyse the layered structures of T&E(3), T&E(2) and T&E(1).
One question not solved in [HCCS2] was: how many layers can there be at each T&E-depth.

The puzzle proposed here: http://forum.enjoysudoku.com/sunday-guess-t45628.html gives a first answer: in T&E(3), there are at least 8 layers above the ground layer 0. But this is the result of analysing only a small sample and there may indeed be more layers.

[Edit:] the puzzle proposed here (http://forum.enjoysudoku.com/2nd-sunday-guess-t45646.html) shows there's at least one more layer.
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Website · by **denis_berthier** » Wed Apr 09, 2025 4:58 am

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Basics
in the first posts of this thread, I'll deal mainly with the T&E(3) land and its (10 or more) layers.

1) Starting point
My starting point is mith's latest collection of 4,634,322 minimal puzzles in T&E(3) (http://forum.enjoysudoku.com/t-e-3-puzzles-split-from-hardest-sudokus-thread-t40514-61.html, but I have no link to the collection itself).
Indeed I use the form cleaned by coloin (elimination of non-minimals, of redundant minimals and of redundant solutions): (http://forum.enjoysudoku.com/t-e-3-puzzles-split-from-hardest-sudokus-thread-t40514-104.html);
this list has 4,634,101 different non-isomorphic minimal puzzles in T&E(3) corresponding to 67,539 different non-isomorphic solution grids.

2) Basic organisational principles
In this thread:
a) all the puzzles are in gsf-solution-minlex form with dots as markers for no-clue (as is the case for the coloin's version I use);

b) all the analyses are done solution-grid by solution-grid, which means in practice that:
--- the first step of all my analyses is to define a sample of the 67,539 solution grids; in practice, I have two types of samples: either random or corresponding to continuous slices (e.g. solution grids 1 to 1000 or 1001 to 2000); my current usual sample size is 1000;
--- each sample has a GRIDS sub-directory containing the file "Usols.txt" of its unique solution grids
--- for each unique solution grid #i in the sample, this GRIDS sub-directory has a sub-directory USOL-i where all the results for this grid will appear;
--- the first step of dealing with each sample is creating a "mins.txt" file in each of the USOL-i sub-directories, containing all the minimals of the original collection that have grid #i for their solution;
--- one important feature is also to have clear naming conventions for all the types of puzzles and results at each layer (I may say more about this when needed).

Note that, as soon as this first step, some interesting observations are possible: the number of minimals per solution-grid varies wildly from one grid to another: from 1 to about 3,000 - considering only the first 5,000 solution grids. I have no idea of the reason - maybe only a result of mith's search algorithms, maybe inherent to the grids, maybe a combination of the two.

c) as this is impossible to manage manually, for each analysis I want to make, I have to write SudoRules scripts that manage all the computations automatically, separately grid by grid.
Currently, I have scripts to:
- create the USOL-i directories and extract the corresponding minimals;
- build all the T&E(3) layers (that's how I found one can have 10 layers, maybe more);
- extract the T&E(3)-expands for each solution grid.

d) tools
I needed several tools to be called by my scripts:
- SudoRules itself for 1-expansions and various calculations (such as the numbers of puzzles in each layer);
- SHC to compute the T&E-depth, BxBB, BxB and B classifications and the BRT-expansions;
- gsf's solver to eliminate redundancies at different steps of the computations, to put puzzles into gsf-solution-minlex form and I'll eventually need it to compute minimals;
- OS operations that are currently only available for Unix but that should have Windows equivalents (contact me if you're interested in a Windows version); this is all the Unix commands you'll need to be able to do on Windows if you want to use my scripts: mkdir (same in Windows), cp, cat file1 >> file2, sed '1d' " input > output (remove first line of a file).

I have written CLIPS interfaces to these tools, so as to do all the computations within SudoRules.
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Website · by **denis_berthier** » Fri Apr 11, 2025 3:59 am

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Here are now the results, T&E(3) layer by T&E(3) layer, of the analysis for the 1,000 first solution grids and corresponding 67368 minimal puzzles in mith's T&E(3) collection (revised by coloin).
T&E(3) layers are named p0, p1, .... starting from the deepest ones. I've added manual comments to layers p0 and p1 in order to describe the layers (i.e. which operation has been done to reach each line from the previous one). The operations on the next layers are like those in p1. I've also kept computation times.

Code: Select all: T&E(3) layer p0 layer p0 total time = 35m 33.56s 67368 mins puzzles (minimals from the collection) 67368 ME puzzles (their min-expands; can only be in T&E(3)) 10586 MEU puzzles (their min-expands after reduction for non-redundancy; can only be in T&E(3)) 67.37 minimals per grid (/ 67368 10586) = 6.3 minimals per min-expand 10.59 min-expands per grid T&E(3) layer p1 layer p1 total time = 4h 4m 29.63s 518402 p1 puzzles (the 1-expands of the MEU) 512822 p1U puzzles (same, after elimination of redundancies) 31349 p1U-d3 puzzles (same, restricted to puzzles in T&E(3)) 13711 p1EU-d3 puzzles (after BRT expansion and elimination of redundancies; can only be in T&E(3)) T&E(3) layer p2 - compute 1-expands time = 7.55s - eliminate-redundancies time = 3h 22m 54.41s - compute T&E-depth time = 1h 4m 22.09s - filter depth 3 puzzles time = 14.22s - compute BRT-expands time = 2m 31.39s - eliminate-redundancies time = 14m 5.27s - layer p2 total time = 4h 44m 14.94s 667058 p1EU-d3-p2 puzzles 653974 p1EU-d3-p2U puzzles 32166 p1EU-d3-p2U-d3 puzzles 9570 p1EU-d3--p2EU-d3 puzzles T&E(3) layer p3 - compute 1-expands time = 6.23s - eliminate-redundancies time = 2h 12m 24.7s - compute T&E-depth time = 40m 33.0s - filter depth 3 puzzles time = 9.46s - compute BRT-expands time = 1m 40.05s - eliminate-redundancies time = 7m 51.56s - layer p3 total time = 3h 2m 45.0s 461153 p1EU-d3--p2EU-d3-p3 puzzles 451438 p1EU-d3--p2EU-d3-p3U puzzles 19182 p1EU-d3--p2EU-d3-p3U-d3 puzzles 4898 p1EU-d3--p3EU-d3 puzzles T&E(3) layer p4 - compute 1-expands time = 4.69s - eliminate-redundancies time = 1h 3m 10.71s - compute T&E-depth time = 19m 19.48s - filter depth 3 puzzles time = 4.82s - compute BRT-expands time = 1m 1.46s - eliminate-redundancies time = 3m 17.08s - layer p4 total time = 1h 26m 58.24s 233079 p1EU-d3--p3EU-d3-p4 puzzles 228400 p1EU-d3--p3EU-d3-p4U puzzles 8374 p1EU-d3--p3EU-d3-p4U-d3 puzzles 2044 p1EU-d3--p4EU-d3 puzzles T&E(3) layer p5 - compute 1-expands time = 3.56s - eliminate-redundancies time = 24m 31.6s - compute T&E-depth time = 8m 0.68s - filter depth 3 puzzles time = 2.0s - compute BRT-expands time = 27.48s - eliminate-redundancies time = 1m 4.55s - layer p5 total time = 34m 9.87s 95812 p1EU-d3--p4EU-d3-p5 puzzles 94125 p1EU-d3--p4EU-d3-p5U puzzles 2862 p1EU-d3--p4EU-d3-p5U-d3 puzzles 676 p1EU-d3--p5EU-d3 puzzles T&E(3) layer p6 - compute 1-expands time = 3.23s - eliminate-redundancies time = 7m 34.03s - compute T&E-depth time = 2m 44.81s - filter depth 3 puzzles time = 0.67s - compute BRT-expands time = 10.62s - eliminate-redundancies time = 16.2s - layer p6 total time = 10m 49.56s 31163 p1EU-d3--p5EU-d3-p6 puzzles 30713 p1EU-d3--p5EU-d3-p6U puzzles 733 p1EU-d3--p5EU-d3-p6U-d3 puzzles 175 p1EU-d3--p6EU-d3 puzzles T&E(3) layer p7 - compute 1-expands time = 3.14s - eliminate-redundancies time = 1m 49.68s - compute T&E-depth time = 45.7s - filter depth 3 puzzles time = 0.18s - compute BRT-expands time = 3.11s - eliminate-redundancies time = 2.78s - layer p7 total time = 2m 44.6s 7925 p1EU-d3--p6EU-d3-p7 puzzles 7843 p1EU-d3--p6EU-d3-p7U puzzles 125 p1EU-d3--p6EU-d3-p7U-d3 puzzles 33 p1EU-d3--p7EU-d3 puzzles T&E(3) layer p8 - compute 1-expands time = 2.94s - eliminate-redundancies time = 18.85s - compute T&E-depth time = 10.33s - filter depth 3 puzzles time = 0.04s - compute BRT-expands time = 0.69s - eliminate-redundancies time = 0.29s - layer p8 total time = 33.14s 1469 p1EU-d3--p7EU-d3-p8 puzzles 1461 p1EU-d3--p7EU-d3-p8U puzzles 12 p1EU-d3--p7EU-d3-p8U-d3 puzzles 4 p1EU-d3--p8EU-d3 puzzles T&E(3) layer p9 - compute 1-expands time = 3.07s - eliminate-redundancies time = 1.78s - compute T&E-depth time = 1.7s - filter depth 3 puzzles time = 0.01s - compute BRT-expands time = 0.0s - eliminate-redundancies time = 0.01s - layer p9 total time = 6.57s 173 p1EU-d3--p8EU-d3-p9 puzzles 173 p1EU-d3--p8EU-d3-p9U puzzles 0 p1EU-d3--p8EU-d3-p9U-d3 puzzles 0 p1EU-d3--p9EU-d3 puzzles

The most interesting lines of each layer are the first and last ones.
It's interesting to see how the numbers diminish layer after layer, until there is finally no puzzle at T&E-depth 3. In the present case, we have 8 layers. I gave an example in the Puzzles section, from another set of solution grids, where we find 9 layers.

If you look at the "eliminate redundancies" times after the 1-expands lines, you'll notice that they are quite long and they produce no significant reduction. This suggests skipping this step would make computation times shorter - but it would also deprive us of the conclusion that, for each solution grid, 1-expands of its min-expands are not too close to each other and that the min-expands must therefore be some distance apart from each other.

Next post will be about the T&E(3) expands.
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Website · by **denis_berthier** » Sat Apr 12, 2025 3:16 am

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Here are now the results for the T&E(3)-expands for the same sample of solution grids and their repartition in the various layers.
Remember that a layer corresponds to the number of times (1+BRT)-expansion can be done before exiting T&E(3).

Code: Select all: TOTAL = 1068 T&E(3)-expands 64 T&E(3)-expands at layer 0 161 T&E(3)-expands at layer 1 277 T&E(3)-expands at layer 2 213 T&E(3)-expands at layer 3 197 T&E(3)-expands at layer 4 94 T&E(3)-expands at layer 5 44 T&E(3)-expands at layer 6 14 T&E(3)-expands at layer 7 4 T&E(3)-expands at layer 8

If we compare this to the number of min-expands, we get (/ 10586 1068) = 9.91 min-expands per T&E(3)-expand (in the mean for our sample of solution grids)
But things are slightly more complex: any min-expand can have several T&E(3)-expands and any T&E(3)-expand can originate in several min-expands.
And don't forget this is globally for 1000 solutions grids and there are large variations between grids.

The question arises of what happens if one goes further and adds a clue to a T&E(3)-expand. By definition, the T&E(3) border is crossed and the resulting puzzle will no longer be in T&E(3).
It is a result of [HCCS2, section 6.1] that they can be in any of T&E(2, 1 or 0) and a precise distribution was obtained starting directly from a larger sample of T&E(3)-expands:

Code: Select all: 70.16% in T&E(2) 26.49% in T&E(1) 3.35% in T&E(0)

Another question is: how many non-degenerate or degenerate-cyclic tridagons remain after crossing the border? The answer is to be found in the same place and depends on the T&E-depth where the puzzle lands.
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Website · by **denis_berthier** » Sun Apr 13, 2025 5:34 am

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The previous posts were about maximally expanding minimal puzzles in T&E, while remaining in T&E(3).
This post will be about minimally expanding the same minimal puzzles in T&E in a way that leads to leaving T&E(3).

The simplest way of doing this would be to use BRT-expansion, but that keeps the puzzles in T&E(3). Moreover, mith's database is closed under BRT-expansion and minimisation, so that no new minimal of a BRT-expand of his minimals can appear in the process (unless there's some breach of the principle - but that could be only rare, as I've checked it on large random samples).

As a result, the next simplest case is to consider 1-expansions of the BRT-expands, i.e. the "p1" puzzles in the above notation, or rather the "p1U" ones (i.e. the "p1" with no redundancies). Here again, several options are possible, e.g.
- studying those at T&E-depth 3: I'll skip this because I'm not mainly interested here in producing new T&E(3) puzzles;
- studying those at T&E-depth 2: I'll skip this for now, because the results are not very promising, but I may come back to it later;
- studying those at T&E-depth 1: this will be the topic in the rest of this post. The question here is, can we get "hard" T&E(1) puzzles this way. And the answer is a big YES. "Hard" is here understood as "with very large B rating". And "very large B rating" will in turn be understood as "with B ≥ 12". Note that this is a very reasonable choice: in the whole controlled-bias collection "cbg" of 5,926,343 minimal puzzles, all in T&E(1), there are only 6 with B rating 12, 2 with B rating 13 and none with a larger B rating; B ≥ 12 can therefore naturally be considered as "very high".

Starting from these "p1U" puzzles in the same sample as in my previous posts, several actions can be done:
- start: 512,822 "p1U" puzzles (non-minimal)
- filter those in T&E(1) ===> 126,985 "p1U-d1" puzzles (non-minimal)
- filter those with B ≥ 12 ===> 1,638 "p1U-B12" puzzles (non-minimal) - note that this is already much more that in the whole cbg collection.
- find all their minimals ===> 16,153 "p1U-B12-mins" minimal puzzles (which can a priori be at any T&E-depth ≥ 1)
- filter those in T&E(1) ===> 7,226 "p1U-B12-mins-d1" minimal puzzles (which can a priori only have B ≥ 12)
Now, we have many more minimal puzzles with B≥12 than among the non minimal "p1U-B12". This was of course to be expected at the end of minimisation. But the question is, how much harder can they be than B12? And that's were we find interesting results: they can be as hard as B19 - i.e.. extremely rare T&E(1) puzzles - so rare that no probability for them can be computed.

In particular, we find 86 minimal puzzles in B18 and 35 minimal puzzles in B19 (shown below):

Code: Select all: 1...5678....78...3...1.3.65......83.....7.....6783.51..9.56.....42.1....8........ #125 1...5678....78...3...1.3.65......837..........6783.51..9.56.....42.1....8........ #126 1...5678....78.1.3.....3.65......83.....7.....6783.51..9.56.....42.1....8........ #127 1...5678....78.1.3.....3.65......837..........6783.51..9.56.....42.1....8........ #128 12...67.9.56......7.9.2..........95.....6...1....123.85...7....6.2.91..7.9...5.1. #656 12...67.9.56...1..7.9.2..........95.....6...1....123.85...7....6.2.91..7.9...5... #657 ..34....94.6.8.....8.1...6......19.4...3..81......8.363........5.296.....97...... #1912 ..34....94.6.89....8.1...6......19.4...3..81......8.363......9.5.2.6.....97...... #1913 ..34.6..94.6.8.....8.1..4.......19.4...3..81......8.363........5.296.....97...... #1914 ..34.6..94.6.89....8.1..4.......19.4...3..81......8.363......9.5.2.6.....97...... #1915 1.34.....4.6.89....8.1...6......19.4...3..81......8.363......9.5.2.6.....97...... #1916 1.34.6...4.6.89....8.1..4.......19.4...3..81......8.363......9.5.2.6.....97...... #1917 1.3.......567.....78...356...16.7....6..18..7...5........8..29....3....58.5...3.1 #2704 1.3...7...56......78...356...16.7....6..18....7.5........8..29....3....58.5...3.1 #2707 1.3...7...56......78...356...16.7....6..18..7...5........8..29....3....58.5...3.1 #2710 1.3.......5678....78...356...16.7....6..1...7..85........8..29....3....58.5...3.1 #2713 1.3.......5678....78...356...16.7....6..18..7...5...........29....3....58.5...3.1 #2714 1.3...7...56.8....78...356...16.7....6..1.....785........8..29....3....58.5...3.1 #2719 1.3...7...56.8....78...356...16.7....6..1...7..85........8..29....3....58.5...3.1 #2720 1.3...7...56.8....78...356...16.7....6..18....7.5...........29....3....58.5...3.1 #2721 1.3...7...56.8....78...356...16.7....6..18..7...5...........29....3....58.5...3.1 #2723 ...4...8.....8912....1.24.52.4....9..3.9.....96.5.1.....82.591.............8..254 #3043 ...4...89....8912....1.24.52.4.......3.9.....96.5.1.....82.591.............8..254 #3044 .2.45.7.9...7.9..3....3254..........5...7493.9..5.32...7.....9.64.......8.1.....5 #6228 .2.45.7.9...7.9..3....3254........5.5...7493.9....32...7.....9.64.......8.1.....5 #6230 .2.45.7.9.5.7.9..3....32.4..........5...7493.9..5.32...7.....9.64.......8.1.....5 #6237 .2.45.7.9.5.7.9..3....32.4........5.5...7493.9....32...7.....9.64.......8.1.....5 #6240 ...45...9.5...9..37.9.32...2.....6..3.7....15....7.4.25.23.7..4..4......9....5... #7215 ...45...945...9..37.9.32...2.....6..3.7....15....7.4.25.23.7..4.........9...45... #7216 ..345...9.5...9..37.9..2...2.....6..3.7....15....7.4.25.23.7..4..4......9....5... #7217 ..345...945...9..37.9..2...2.....6..3.7....15....7.4.25.23.7..4.........9...45... #7218 .2.45...9.5...9..37.9.32.........6..3.7....15....7.4.25.23.7..4..4......9....5... #7219 .2.45...945...9..37.9.32.........6..3.7....15....7.4.25.23.7..4.........9...45... #7220 .2345...9.5...9..37.9..2.........6..3.7....15....7.4.25.23.7..4..4......9....5... #7221 .2345...945...9..37.9..2.........6..3.7....15....7.4.25.23.7..4.........9...45... #7222

One question one might ask is, do all these exceptional puzzles have a tridagon? For the whole list of 7,226 minimals with B ≥ 12, the answer is no:
- only 5,282 have a non-degenerate tridagon
- only 6,113 have a non-degenerate or degenerate-cyclic tridagon
For an example that has none of these, see http://forum.enjoysudoku.com/t-e-1-nightmare-t45664.html
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Website · by **denis_berthier** » Mon Apr 14, 2025 3:09 pm

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Here are now the results of the search for high B-rating puzzles for the second sample of 1000 solution grids. The results are still better than in the first 1000.
Same notations as before.

Code: Select all: start with 496,242 "p1U" puzzles 123,029 "p1U-d1" puzzles 2,042 "p1U-B12" puzzles 26,485 "p1U-B12-mins" puzzles 11,625 "p1U-B12-mins-d1" puzzles (which can only have B ≥ 12)

What's new is, we find minimal puzzles up to B = 22.

525 in B16
155 in B17
101 in B18

24 in B19:

Code: Select all: 12...6...4.....1...89.......6..41......3.7..2.3726.4.......427.6...7..3.....236.1 #2115 12...6...4.....1...89.......6..41......3.7.62.372..4.......427.6...7..3.....236.1 #2116 ...45.......7.9..3.89..2....7.86.39.........7.983..2...37...9.8..2...6...6..7.... #2843 ...45...9...7.9..3.89..2....7.86.3..........7.983..2...37...9.8..2...6...6..7.... #2845 1......894.6...1.3.8..3.......3.46..8......529.....3....184.....98.63....4.9.1... #3943 1..4...894.6...1.3.8..3.......3.46..8......529.....3....18......98.63....4.9.1... #3944 .2.4.67..4.67.912...9......2.....47...7..4.9.9.4...6.1..28.5.......6....6.13.7... #4034 ...4.67.94.67.912..........2.....4.........929.4.7.6.1..28.5.1.....6...76.13..... #4063 ...4.67.94.67.912..........2.....47........929.4.7.6.1..28.5.1.....6....6.13.7... #4064 1..4...894...8.12.......5...48....9567..9.....3..4.......51..48....2.9.1...9..25. #6655 1..4...894...8.12.......5..248.....567..9.....3..4.......51..48....2.9.1...9..25. #6657 1..45..894...8.12...........48....9567..9.....3..4.......51..48....2.9.1...9..25. #6665 1..45..894...8.12..........248.....567..9.....3..4.......51..48....2.9.1...9..25. #6667 1..4...894...8.12.......5....8....9567..9.4...3..4.......51..48....2.9.1...9..25. #6675 1..4...894...8.12.......5..2.8.....567..9.4...3..4.......51..48....2.9.1...9..25. #6677 1..45..894...8.12............8....9567..9.4...3..4.......51..48....2.9.1...9..25. #6685 1..45..894...8.12..........2.8.....567..9.4...3..4.......51..48....2.9.1...9..25. #6687 1..4...894...8.12.......5....8....9567..9.....3..4.......51..48....2.9.1.1.9..25. #6693 1..45..894...8.12............8....9567..9.....3..4.......51..48....2.9.1.1.9..25. #6699 .23.5.7.94........7.9.325.42.5...3..........793....6.8....94......27.....4.3.5... #8379 .23.5.7.94........7.9.325..2.5.....1........793....6.8....94......27.4...4.3.5... #8398 .234..7.945.......7.9.325..2.5.....1........793....6.8....9.......27.4...4.3.5... #8399 .2345.7.94........7.9.325..2.5.....1........793....6.8....9.......27.4...4.3.5... #8400 .23.5.7.94........7.9.325.42.5...3.......3..79.....6.8....94......27.....4.3.5... #8402

9 in B20:

Code: Select all: .23......45....1....9....6....87.....7..238.....6.13....8...61...731.2.8..2.6...7 #4899 .23......456...1....9....6....87.....7..238.....6.13....8...61....31.2.8..2.6...7 #4900 .23..6...45....1....9....6....87.....7..238.....6.13....8...61....31.2.8..2.....7 #4901 .....6......78.....89.32....41...8....85...1.59...143..15...9.8.3.....41..4...35. #5803 .....6......78.....89.32....41...8.5..85...1..9...143..15...9.8.3.....41..4...35. #5804 .....6.8....78.....89.32....41........85...1.59...143..15...9.8.3.....41..4...35. #5805 .....6.8....78.....89.32....41.....5..85...1..9...143..15...9.8.3.....41..4...35. #5806 12.45.........9.2.7...3...4.6..1...7.7.64..92..4.......1....9..6.7.....1.92....46 #11228 12.45.........9.2.7...3...4.6..1...7.7164..92..4............9..6.7.....1.92....46 #11229

5 in B21:

Code: Select all: 12.........6....2378..3.......56..9...59.1..6....23.15.9..1....5..69..32...3....1 #3306 12.........6....2378..3.......56..9...59.1..6....23.1539..1....5..69..32........1 #3307 12........56....2378..3.......56..9....9.1..6....23.15.9..1....5..69..32...3....1 #3308 12........56....2378..3.......56..9....9.1..6....23.1539..1....5..69..32........1 #3309 123........6....2378..........56..9...59.1..6....23.15.9..1....5..69..32...3....1 #3310

2 in B22:

Code: Select all: .23......45.7.9...7....25.4.....3.75.9....6.1.4.....92...2.4.....2.95...9..37.... #7017 .23......45.7.9...7.9..25.4...9.3.75......6.1.4.....92...2.4.....2.95...9..37.... #7018

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Website · by **denis_berthier** » Wed Apr 16, 2025 5:17 am

.
Here's some update to my previous results.
Still about the same collection.

I've now analysed the first 5 continuous samples of 1,000 solutions grids.

1) Variations among solution grids and samples
As I said previously, there are very large variations between different solution grids, e.g. in the numbers of minimal puzzles for them in the collection. These numbers can vary from 1 to several thousands. This is one interesting result of splitting the global collection of minimals.
Are those variations smoothed out by glueing together 1000 solutions grids into a sample? Yes, of course, but not to the point of not remaining obvious. For the 5 samples, the total numbers of minimals are respectively:
67368
56409
74198
62512
67929
I've also started to study a slice of solution grids in the middle of the file (sample #30, i.e. solution grids 29001 to 30000) for which the number is
53771

2) The T&E(3) layers
As of now, the largest number of (1+BRT)-layers I've found above the p0 layer (consisting of minimals and their BRT-expands) remains 9.

3) The p1U puzzles and their minimals at T&E-depth 1
In [HCCS2,section 6.1 https://www.researchgate.net/publication/381884473_Hierarchical_Classifications_in_Constraint_Satisfaction_Second_Edition,
I noticed that the 1-expands of T&E(3)-expands (i.e. of mith's max-expands) can be in any of T&E(0, 1 or 2) and that those in T&E(1) can have very large B ratings and I found a few thousand puzzles with B ≥ 12.

In the previous posts of this thread, instead of starting from T&E(3)-expand puzzles, I started from puzzles closer to the T&E(3)-minimals, namely the "p1U" puzzles, i.e. the 1-expands of the min-expands (with no redundancies). The result is very large numbers of non-isomorphic minimal puzzles at T&E-depth 1 with extremely large B ratings.
After analysing the 5 samples, I now have the followng numbers of puzzles with B ≥ 12, for each sample:
7226
11625
22339
9704
running
Here also, one can see the large variations between samples.

I also have several thousands of puzzles with B ≥ 15.
The maximum B rating reached remains 22.

(Notes:
- the largest known B rating is 30, for a puzzle due to Mauricio; only 2 puzzles with B=30 are known, the second one is due to 1to9only;
- before the above results, there was no collection of puzzles with large B ratings, say B ≥ 12.
)
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by **eleven** » Wed Apr 16, 2025 8:54 am

A solution to the B22 above:
.23......45.7.9...7....25.4.....3.75.9....6.1.4.....92...2.4.....2.95...9..37....

Hidden Text: Show

Tridagon 168 (*): 5r7c3 = 3r2c5
3r5c1 = r5c8 - r3c8 = r3c5 == 5r7c3 - r56c3 = 35r56c1
=> -28r5c1

Hidden Text: Show

If 3r2c5 and not 5r7c3, there is a remote triple 168r27c3,r7c5 from the tridagon, and 3r3c8, -3r7c8
This is not possible, because the digit in r7c8 (1,6, or 8) goes to r2c3 (RT), and can't be in box 3.
So 5 must be in r7c3 because of the tridagon.

Hidden Text: Show

kite 1: 1r1c1 = r78c1 - r9c2 = r9c6 => -1r1c6
8 in row 2:
8r2c3 - r5c3 = r5c6
8r2c5
8r2c9 - r9c9 = kite 8 [8r1c1 = r78c1 - r9c2 = 8r9c6]
=> -8r1c6

Hidden Text: Show

remote pair 18: r3c4 - r2c2 - r9c2 - r9c6 => -18r8c4, stte

by **jco** » Wed Apr 16, 2025 12:38 pm

eleven wrote:A solution to the B22 above:
.23......45.7.9...7....25.4.....3.75.9....6.1.4.....92...2.4.....2.95...9..37....
(...)

Nice solve! Thank you for sharing it.

Website · by **denis_berthier** » Wed Apr 16, 2025 2:30 pm

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Some of the high B-rating puzzles still have a non-degenerate tridagon, with 1 or more guardians. It works as for the T&E(3) puzzles, i.e. it makes them simpler but what remains may be easy or extremely hard. One could say exactly the same of any rating and any pattern not considered in the rating: adding the pattern has totally unpredictable effects.
Some puzzles don't have such a tridagon, but they have some pattern close to it: I've given an example in the Puzzles section.

Considering the way the puzzles were obtained, having a tridagon or a very close pattern shouldn't be a surprise.

For the B22 puzzle, I have three solutions:
- one in B22 (by definition);
- one using only the tridagon, in W7+Trid-OR2-W7;
- one using the tridagon + eleven's EL14c30 3-digit impossible pattern (used for only 1 elimination), in W5+(Trid+EL14c30)-OR3-W5 - below:

Code: Select all: Resolution state after Singles and whips[1]: +-------------------+-------------------+-------------------+ ! 168 2 3 ! 14568 14568 168 ! 1789 168 6789 ! ! 4 5 168 ! 7 1368 9 ! 138 2 368 ! ! 7 168 9 ! 168 1368 2 ! 5 1368 4 ! +-------------------+-------------------+-------------------+ ! 1268 168 168 ! 9 12468 3 ! 48 7 5 ! ! 2358 9 578 ! 458 2458 78 ! 6 348 1 ! ! 13568 4 15678 ! 1568 1568 1678 ! 38 9 2 ! +-------------------+-------------------+-------------------+ ! 1568 13678 1568 ! 2 168 4 ! 13789 1368 36789 ! ! 168 13678 2 ! 168 9 5 ! 13478 13468 3678 ! ! 9 168 4 ! 3 7 168 ! 2 5 68 ! +-------------------+-------------------+-------------------+ 177 candidates

hidden-pairs-in-a-column: c2{n3 n7}{r7 r8} ==> r8c2≠8, r8c2≠6, r8c2≠1, r7c2≠8, r7c2≠6, r7c2≠1
hidden-pairs-in-a-row: r1{n7 n9}{c7 c9} ==> r1c9≠8, r1c9≠6, r1c7≠8, r1c7≠1
hidden-pairs-in-a-row: r1{n4 n5}{c4 c5} ==> r1c5≠8, r1c5≠6, r1c5≠1, r1c4≠8, r1c4≠6, r1c4≠1

Code: Select all: Trid-OR2-relation for digits 1, 6 and 8 in blocks: b1, with cells (marked #): r1c1, r2c3, r3c2 b2, with cells (marked #): r1c6, r2c5, r3c4 b7, with cells (marked #): r8c1, r7c3, r9c2 b8, with cells (marked #): r8c4, r7c5, r9c6 with 2 guardians (in cells marked @): n3r2c5 n5r7c3 +----------------------+----------------------+----------------------+ ! 168# 2 3 ! 45 45 168# ! 79 168 79 ! ! 4 5 168# ! 7 1368#@ 9 ! 138 2 368 ! ! 7 168# 9 ! 168# 1368 2 ! 5 1368 4 ! +----------------------+----------------------+----------------------+ ! 1268 168 168 ! 9 12468 3 ! 48 7 5 ! ! 2358 9 578 ! 458 2458 78 ! 6 348 1 ! ! 13568 4 15678 ! 1568 1568 1678 ! 38 9 2 ! +----------------------+----------------------+----------------------+ ! 1568 37 1568#@ ! 2 168# 4 ! 13789 1368 36789 ! ! 168# 37 2 ! 168# 9 5 ! 13478 13468 3678 ! ! 9 168# 4 ! 3 7 168# ! 2 5 68 ! +----------------------+----------------------+----------------------+ EL14c30-OR3-relation for digits: 1, 6 and 8 in cells (marked #): (r2c3 r3c5 r3c4 r3c2 r1c8 r1c6 r1c1 r8c4 r8c1 r9c6 r9c2 r7c8 r7c5 r7c3) with 3 guardians (in cells marked @) : n3r3c5 n3r7c8 n5r7c3 +----------------------+----------------------+----------------------+ ! 168# 2 3 ! 45 45 168# ! 79 168# 79 ! ! 4 5 168# ! 7 1368 9 ! 138 2 368 ! ! 7 168# 9 ! 168# 1368#@ 2 ! 5 1368 4 ! +----------------------+----------------------+----------------------+ ! 1268 168 168 ! 9 12468 3 ! 48 7 5 ! ! 2358 9 578 ! 458 2458 78 ! 6 348 1 ! ! 13568 4 15678 ! 1568 1568 1678 ! 38 9 2 ! +----------------------+----------------------+----------------------+ ! 1568 37 1568#@ ! 2 168# 4 ! 13789 1368#@ 36789 ! ! 168# 37 2 ! 168# 9 5 ! 13478 13468 3678 ! ! 9 168# 4 ! 3 7 168# ! 2 5 68 ! +----------------------+----------------------+----------------------+

whip[5]: c7n9{r7 r1} - c7n7{r1 r8} - r8c2{n7 n3} - r8c9{n3 n6} - r9c9{n6 .} ==> r7c7≠8
Trid-OR2-whip[5]: c1n3{r5 r6} - c1n5{r6 r7} - OR2{{n5r7c3 | n3r2c5}} - b3n3{r2c7 r3c8} - r5n3{c8 .} ==> r5c1≠2
singles ==> r4c1=2, r5c5=2
At least one candidate of a previous EL14c30s-OR4-relation between candidates n2r4c5 n4r4c5 n5r7c1 n3r2c5 has just been eliminated.
There remains an EL14c30s-OR3-relation between candidates: n4r4c5 n5r7c1 n3r2c5

EL14c30s-OR3-whip[4]: r5n3{c1 c8} - r5n4{c8 c4} - OR3{{n4r4c5 n5r7c1 | n3r2c5}} - b3n3{r2c7 .} ==> r5c1≠5
Trid-OR2-whip[3]: OR2{{n3r2c5 | n5r7c3}} - c1n5{r7 r6} - r6n3{c1 .} ==> r2c7≠3
z-chain[4]: r4n6{c3 c5} - b5n4{r4c5 r5c4} - r5n5{c4 c3} - c3n7{r5 .} ==> r6c3≠6
z-chain[4]: r4n1{c3 c5} - b5n4{r4c5 r5c4} - r5n5{c4 c3} - c3n7{r5 .} ==> r6c3≠1
Trid-OR2-whip[4]: r5n3{c1 c8} - r3n3{c8 c5} - OR2{{n3r2c5 | n5r7c3}} - c1n5{r7 .} ==> r6c1≠3
singles ==> r5c1=3, r6c7=3
Trid-OR2-whip[5]: OR2{{n3r2c5 | n5r7c3}} - r5n5{c3 c4} - b5n4{r5c4 r4c5} - r4c7{n4 n8} - r2c7{n8 .} ==> r2c5≠1
z-chain[2]: r9n1{c2 c6} - b2n1{r1c6 .} ==> r3c2≠1
whip[5]: r2c7{n8 n1} - b1n1{r2c3 r1c1} - r1n8{c1 c6} - r3n8{c4 c2} - r9n8{c2 .} ==> r2c9≠8
whip[1]: c9n8{r9 .} ==> r7c8≠8, r8c7≠8, r8c8≠8
whip[5]: c5n5{r6 r1} - c5n4{r1 r4} - r4c7{n4 n8} - r2n8{c7 c3} - b4n8{r4c3 .} ==> r6c5≠8
whip[5]: r3c2{n8 n6} - r3c4{n6 n1} - r1c6{n1 n6} - r9n6{c6 c9} - r2n6{c9 .} ==> r3c5≠8
finned-swordfish-in-columns: n8{c5 c7 c3}{r7 r2 r4} ==> r4c2≠8
z-chain[5]: r3n1{c5 c8} - r2c7{n1 n8} - r1n8{c8 c1} - c2n8{r3 r9} - r9n1{c2 .} ==> r1c6≠1
whip[1]: b2n1{r3c5 .} ==> r3c8≠1
finned-x-wing-in-columns: n1{c6 c2}{r9 r6} ==> r6c1≠1
whip[1]: r6n1{c6 .} ==> r4c5≠1
biv-chain[5]: r3n1{c4 c5} - r3n3{c5 c8} - r2c9{n3 n6} - r9c9{n6 n8} - c2n8{r9 r3} ==> r3c4≠8
biv-chain[3]: r3n8{c2 c8} - r2c7{n8 n1} - b1n1{r2c3 r1c1} ==> r1c1≠8
Trid-OR2-whip[4]: OR2{{n5r7c3 | n3r2c5}} - b2n8{r2c5 r1c6} - r5c6{n8 n7} - c3n7{r5 .} ==> r6c3≠5
z-chain[5]: r4n4{c5 c7} - c7n8{r4 r2} - b3n1{r2c7 r1c8} - r1c1{n1 n6} - r6n6{c1 .} ==> r4c5≠6
whip[1]: b5n6{r6c6 .} ==> r6c1≠6
naked-pairs-in-a-row: r4{c5 c7}{n4 n8} ==> r4c3≠8
z-chain[3]: c1n6{r8 r1} - b1n1{r1c1 r2c3} - r4c3{n1 .} ==> r7c3≠6
Trid-OR2-whip[5]: c3n6{r2 r4} - c3n1{r4 r7} - OR2{{n5r7c3 | n3r2c5}} - b3n3{r2c9 r3c8} - r3n8{c8 .} ==> r2c3≠8
hidden-single-in-a-block ==> r3c2=8
naked-pairs-in-a-block: b3{r2c9 r3c8}{n3 n6} ==> r1c8≠6
naked-pairs-in-a-column: c3{r2 r4}{n1 n6} ==> r7c3≠1
x-wing-in-rows: n8{r2 r4}{c5 c7} ==> r7c5≠8
t-whip[2]: r1n6{c6 c1} - b7n6{r8c1 .} ==> r9c6≠6
biv-chain[3]: r7c5{n6 n1} - r9c6{n1 n8} - r9c9{n8 n6} ==> r7c8≠6, r7c9≠6
finned-x-wing-in-rows: n6{r7 r1}{c1 c5} ==> r3c5≠6, r2c5≠6
swordfish-in-rows: n6{r2 r4 r9}{c9 c3 c2} ==> r8c9≠6
biv-chain[3]: r9c6{n1 n8} - r1c6{n8 n6} - r3c4{n6 n1} ==> r8c4≠1
finned-x-wing-in-rows: n1{r1 r8}{c1 c8} ==> r7c8≠1
stte
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The layered structure of T&E-depth d

The layered structure of T&E-depth d

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Re: The layered structure of T&E-depth d

Re: The layered structure of T&E-depth d

Re: The layered structure of T&E-depth d

Re: The layered structure of T&E-depth d

Re: The layered structure of T&E-depth d

Re: The layered structure of T&E-depth d

Re: The layered structure of T&E-depth d

Re: The layered structure of T&E-depth d