Sudoku Players' Forum (clone)

by **denis_berthier** » Sun Apr 06, 2025 2:37 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.

Note: as no puzzle is know at T&E-depth > 3, layer 0 for T&E-depth 3 is very simple:
- minimal puzzles at T&E-depth 3;
- their BRT-expands;
- any puzzle between the above two.

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 expansion phases from the minimals. 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 expansion phase.
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[Edit]: in order to avoid any confusion, I replaced where necessary the word "layer" by "expansion phase".
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by **denis_berthier** » Wed Apr 09, 2025 1:58 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) expansion phases.

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 expansion phase (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) expansion phases (that's how I found one can have 10 expansion phases, 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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[Edit]: in order to avoid any confusion, I replaced where necessary the word "layer" by "expansion phase".
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by **denis_berthier** » Fri Apr 11, 2025 12:59 am

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Here are now the results, expansion phase by expansion phase within T&E(3), 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) expansion phases are named p0, p1, .... starting from the deepest ones. I've added manual comments to expansion phases p0 and p1 in order to describe the expansion phases (i.e. which operation has been done to reach each line from the previous one). The operations on the next expansion phases are like those in p1. I've also kept computation times.

Code: Select all: T&E(3) expansion phase p0 expansion phase 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) expansion phase p1 expansion phase 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) expansion phase 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 - expansion phase 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) expansion phase 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 - expansion phase 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) expansion phase 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 - expansion phase 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) expansion phase 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 - expansion phase 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) expansion phase 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 - expansion phase 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) expansion phase 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 - expansion phase 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) expansion phase 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 - expansion phase 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) expansion phase 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 - expansion phase 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 expansion phase are the first and last ones.
It's interesting to see how the numbers diminish expansion phase after expansion phase, until there is finally no puzzle at T&E-depth 3. In the present case, we have 8 expansion phases. I gave an example in the Puzzles section, from another set of solution grids, where we find 9 expansion phases.

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.

[Edit): in order to avoid any ambiguities, I replaced the word "layer" by "expansion phase".
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by **denis_berthier** » Sat Apr 12, 2025 12:16 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 expansion phases.
Remember that an expansion phase 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 638 terminal puzzles at expansion-phase 0 922 terminal puzzles at expansion-phase 1 835 terminal puzzles at expansion-phase 2 565 terminal puzzles at expansion-phase 3 353 terminal puzzles at expansion-phase 4 156 terminal puzzles at expansion-phase 5 62 terminal puzzles at expansion-phase 6 18 terminal puzzles at expansion-phase 7 4 terminal puzzles at expansion-phase 8 0 terminal puzzles at expansion-phase 9

Note that, as a given puzzle can be the result of several expansion paths, it may appear as terminal for several expansion phases. As a result, the number of T&E(3)-expands is less than the sum of the above:
1059 unique T&E(3)-expands

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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[Edit]: in order to avoid any confusion, I replaced where necessary the word "layer" by "expansion phase".
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by **denis_berthier** » Sun Apr 13, 2025 2:34 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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by **denis_berthier** » Mon Apr 14, 2025 12:09 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

.

by **denis_berthier** » Wed Apr 16, 2025 2:17 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) expansion phases
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
[Edit] ~~running~~18109 [/Edit]
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.
)
.

by **eleven** » Wed Apr 16, 2025 5:54 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 9:38 am

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.

by **denis_berthier** » Wed Apr 16, 2025 11:30 am

.
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
.

by **denis_berthier** » Tue Apr 22, 2025 5:24 am

.
I've now analysed 9 samples of 1000 solution grids each (about 7.5% of the collection of solution grids), corresponding to 576697 minimal puzzles.

The maximum number of T&E(3) layers I found remains 9.

I've also studied the T&E-expands ("max-expands" in mith's vocabulary).
The above samples together give rise to 9614 T&E(3)-expands.
The question I asked was: how do they occupy the various layers? Said otherwise, at which layer do T&E(3)-expands occur?

Code: Select all: p0 p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 6,21 17,19 24,55 21,87 16,37 8,08 3,99 1,32 0,35 0,05 0,00

Thus:
- 6,21% of the T&E(3)-expands are in layer p0, i.e. after a simple BRT-expansion;
- the largest percentages occur in layers p2 (24.55%) and p3 (21.87%);
- there are still 0.05% of T&E(3)-expands in layer p9. That's a very small percentage, but if you multiply by the total number of T&E(3-expands, it makes 5 T&E(3) puzzles on the p9 border.

Globally, this shows one can have a significant number of (1+BRT)-expansions before leaving T&E(3). My interpretation is, the non-degenerate tridagon pattern is extremely stable under the random addition of clues and under BRT-expansion. Moreover, it is not so easy to short-circuit its internal T&E3) complexity by using other parts of the puzzles (as must be the case for T&E(2 or 1) puzzles that have the pattern).

My plans are now to analyse different collections at different T&E-depths, with and without tridagons, and to see how many layers one can obtain while remaining at the same T&E-depth. Don't be too impatient about it; that will require some time.

The scripts I'm writing will be published in SudoRules after I've finished testing them.
.

by **denis_berthier** » Wed Apr 23, 2025 12:38 am

.
Once the layered structure of T&E(n) is defined, it's easy to define the distance of a puzzle in T&E(n) to the T&E(n-1) boundary: it's the minimum number of (1+BRT)-expansion steps (after an initial BRT-expansion) before reaching the boundary, i.e. one of its T&E(3)-expands.
Note: there may be many expansions paths to the boundary; distance considers only the shortest one(s).

For the moment, let's stick to T&E(3).
I mentioned it previously: any minimal puzzle in T&E(3) may have several (1+BRT) expansions until it reaches one of its T&E(3)-expands and any T&E(3)-expand may come from several minimal puzzles.
As a result, the following distributions are generally different (although the max value for distances can only be ≤ the number of layers):
- the distribution of layers to which the T&E(3)-expands belong (given in my previous post for 9,000 solution grids);
- the distribution of distances of the minimal puzzles to the inner side of the T&E(3) boundary with T&E(<3).

Below are the two distributions for the sample of the first 1000 solution grids.

Code: Select all: Distribution (in %) of layers "on the boundary": p0 p1 p2 p3 p4 p5 p6 p7 p8 5.99 15.07 25.94 19.94 18.45 8.80 4.12 1.31 0.37

Code: Select all: Distribution (in %) of distances of minimals to the boundary: 0 1 2 3 4 5 6 7 8 0.65 4.31 13.21 11.51 21.70 18.59 17.34 8.85 3.84

.

by **denis_berthier** » Wed Apr 30, 2025 7:41 am

.
I've now analysed 13,000 solution grids (about 19%) of the above-mentioned T&E(3) collection.
The max number of layers within T&E(3) is now 10 and the updated distribution of layers (in %) on the T&E(3) side of the T&E(3) to T&E(<3) boundary is:

Code: Select all: p0 p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 6,17 17,35 24,61 22,48 16,08 7,87 3,86 1,20 0,34 0,04 0,01

.

by **coloin** » Thu May 01, 2025 8:05 am

Perhaps you could post the minimal puzzle with 10 layers !! ?

Perhaps an example of a puzzle expansion is useful to explain ..?

Here is a minimal puzzle from the WheresMyHammer puzzle ... TE3

Code: Select all: +---+---+---+ |89.|..7|..5| |5.7|4..|.96| |...|...|...| +---+---+---+ |...|.4.|...| |4..|.75|6..| |65.|8.9|4..| +---+---+---+ |.8.|...|.61| |76.|...|.32| |...|.8.|9..| +---+---+---+ 28 clues minimal --> with 3 singles.. +---+---+---+ |89.|..7|..5| |5.7|4.8|.96| |...|...|...| +---+---+---+ |.7.|.4.|...| |4..|.75|6..| |65.|8.9|4..| +---+---+---+ |.8.|...|.61| |76.|...|832| |...|.8.|9..| +---+---+---+ 31 clues ---> 3 more singles after pointing on row9 +---+---+---+ |89.|..7|.45| |5.7|4.8|.96| |.4.|...|...| +---+---+---+ |.7.|.4.|...| |4..|.75|6..| |65.|8.9|4..| +---+---+---+ |.8.|...|.61| |76.|...|832| |...|.8.|9.4| +---+---+---+ 34 clues --> 2 more singles after claiming and hidden pair box 2 +---+---+---+ |89.|.67|.45| |5.7|4.8|.96| |.46|...|...| +---+---+---+ |.7.|.4.|...| |4..|.75|6..| |65.|8.9|4..| +---+---+---+ |.8.|...|.61| |76.|...|832| |...|.8.|9.4| +---+---+---+ 36 clues TE3 limit reached... dratted uniqeness r8c45 inserts a 4@r8c6 +---+---+---+ |89.|.67|.45| |5.7|4.8|.96| |.46|...|...| +---+---+---+ |.7.|.4.|...| |4..|.75|6..| |65.|8.9|4..| +---+---+---+ |.84|...|.61| |76.|..4|832| |...|.8.|9.4| +---+---+---+ 38 clues not TE3 anymore

so how many layers / depth has this original minimal puzzle ?

EDIT
The 36 clue "max-expand" has 175 minimal puzzles
on applying my expansion [gsf -E] this gives 3 expands....

Code: Select all: 89..67.455.74.8.96.46.......7..4....4...756..65.8.94.........6.76....832....8.9.4 # 34 clues 89..67.455.74.8.96.46.......7..4....4...756..65.8.94...8.....6.76....832....8.9.4 # 35 clues 89..67.455.74.8.96.46.......7..4....4...756..65.8.94...8.....6176....832....8.9.4 # 36 clues 89..67.455.74.8.96.46.......7..4....4...756..65.8.94...8.....6176....832....8.9.4 # 36 clues max expand

by **P.O.** » Thu May 01, 2025 11:51 am

coloin wrote:Perhaps an example of a puzzle expansion is useful to explain ..?

i would also like to see the details of what Denis does when he talks about (1+BRT)-expansion
same analysis for this puzzle:
89...7..55.74...96.............4....4...756..65.8.94...8.....6176.....32....8.9..
basics:

Hidden Text: Show

Code: Select all: after basics: 8 9 123 123 6 7 123 4 5 5 123 7 4 123 8 123 9 6 123 4 6 59 59 123 1237 1278 378 1239 7 12389 1236 4 1236 1235 1258 389 4 123 12389 123 7 5 6 128 389 6 5 123 8 123 9 4 127 37 239 8 23459 23579 2359 234 57 6 1 7 6 1459 159 159 14 8 3 2 123 123 1235 123567 8 1236 9 57 4 151 candidates. 36 values. 89..67.455.74.8.96.46.......7..4....4...756..65.8.94...8.....6176....832....8.9.4 after T&E(2,Singles): 8 9 123 123 6 7 123 4 5 5 123 7 4 123 8 123 9 6 123 4 6 59 59 123 1237 278 78 1239 7 89 16 4 1236 25 1258 389 4 123 289 123 7 5 6 128 389 6 5 123 8 123 9 4 127 37 29 8 45 2357 2359 234 57 6 1 7 6 459 159 159 14 8 3 2 123 123 1235 67 8 26 9 57 4 128 candidates. 36 values.

the format is (cell value)
becomes te1
((25 7) (26 2) (28 9) (31 1) (33 6) (34 2) (35 5) (39 2) (55 2) (58 7) (59 9) (61 5) (66 9) (75 5)
(76 6) (78 2) (80 7))
becomes te2
((3 1) (4 2) (7 3) (11 2) (14 3) (16 1) (19 3) (22 9) (23 5) (24 1) (27 8) (30 8) (36 3) (38 1)
(40 3) (44 8) (45 9) (48 3) (50 2) (53 1) (54 7) (57 4) (60 3) (67 5) (68 1) (69 4) (73 1) (74 3))

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(1+BRT) expansion paths within T&E(n) and beyond

(1+BRT) expansion paths within T&E(n) and beyond

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

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

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