The New Sudoku Players' Forum

[P.O.]

in this case basics is very relevant, it gathers all the values ​​that keep the puzzle in te1, that's 36 values, ​​11 more than what you find

[denis_berthier]

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You are not providing any (1+BRT)-expansion path that would have more than my 18 steps.
Anyway, where did I say that I was giving the longest possible expansion path in T&E(1) ?
Currently, expansion paths in T&E(1) are restricted to 20 (1+BRT) steps. No "basics" are used in the calculations.

Note that the hard part in my calculations is not to find the expansions of some given minimal. It's to find the minimals that allow such expansions.
If you want to prove anything, try to start from different minimals (with different expansions).
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[P.O.]

for a puzzle at a t&e(n) depth, the maximum number of values ​​that can be added to this puzzle while maintaining its t&e(n) depth seems to me to be the basis from which to build the organization of its layers, any other approach seems artificial to me

[denis_berthier]

for a puzzle at a t&e(n) depth, the maximum number of values ​​that can be added to this puzzle while maintaining its t&e(n) depth seems to me to be the basis from which to build the organization of its layers, any other approach seems artificial to me

Your opinions are worth the amount of data they rely on: 0. You're welcome to open another thread to explain your results when you have any.

BTW, the puzzle with 65 clues (123456789456789...7981325..234561897815974632967328.5.34.695.7858.247...67.813..5) is reachable in only 6 (1+BRT)-expansion steps.
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[denis_berthier]

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I noticed that (p1+BRT) followed by (p2+BRT) is equivalent to (p1+p2+BRT). With your progressive (1+BRT) expansions, you don't get all the puzzles in T&E(d). Why don't you first do all the pk expansions and only then a unique BRT-expansion?

First, you are right about the equivalence. This is largely used in my scripts (together with p1+p2 = p2+p1) in order to eliminate redundancies at every stage.

You are also right about your assertion: I don't get all the puzzles in T&E(d) - but I claim I get all those of interest.


Now, about the question.

1) This thread relies on the (trivial) observation that two puzzles that have the same BRT-expansion are equivalent for all practical solving purposes. In particular, they have the same T&E-depth, B, BxB, BxBB classifications - which is what matters most in my approach.
As a result, only puzzles that are their own BRT-expands are relevant to a solver. Also 1-expands of such puzzles are relevant, because they allow to fully describe expansion paths. Puzzles in between those two categories may not be reached by my scripts.

2) For a solver, a puzzle being minimal is totally irrelevant. All the puzzles of interest to a solver are reached by my scripts.

3) As of a few years ago, any large scale collection of puzzles had been about minimals. More recently mith introduced the expansion of minimals in T&E(3) by Singles (= BRT-expands) and their max-expands (= T&E(3)-expands). My approach in this thread generalises and systematises these ideas.

4) There's one point that may make puzzles with the same BRT-expand different: they may not have the same sets of minimals. This is of interest only to the puzzle creators. The basic point here is, if you look for the minimals of some puzzle P, you can be sure they'll be at least as hard as P - for any reasonable rating/classification system (which excludes the SER or any system taking uniqueness into account). I've used this extensively to find hundreds of thousands of extremely high B rating (B ≥ 12) puzzles from (T&E(3)+1)-expansions of T&E(3) puzzles.

5) You can also be sure that the minimals of the BRT-expands of P will also be at least as hard as P - thus possibly getting many more minimals than for P.

6) Finding minimals of puzzles is computationally hard. If P2 is an expansion of P1, it's harder for P2 than for P1. How much harder depends on how much the two puzzles differ.

7) As of now, for T&E(3) puzzles, what has been computed is minimals of their BRT-expands. A much wider search might be based on their T&E(3)-expands. However, that would probably lead to untractable computations.
What the progressive (1+BRT)-expansions allow is a better controlled possibility of extending the search for minimals.
In this regard, the most interesting possibilties may not be obtained by using the puzzles appearing in the fastest expansion path out of T&E(3), but on the contrary those on some slowest path. Much remains to be done.
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