The New Sudoku Players' Forum

[denis_berthier]

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Back to another topic of this thread: the generation of puzzles in T&E(1), with extreme B ratings.

Starting from minimal puzzles in some T&E(n), one can first expand them by Singles (not changing their rating/classification) and then expand them by adding a single clue (1-expansion). Some will remain in T&E(d), some will fall into a shallower T&(n-x).
The question here is: what about those that fall into T&E(1) and that have a high B rating, say B ≥ 12 ?
One can look for their minimals, all of which are guaranteed to be in T&E(1) and to have B ≥ 12.

The question is: can we produce many new puzzles with extremely high B ratings that way?
In a previous post, I gave an answer in the following case:
- starting from minimal puzzles in T&E(3), one can get lots of puzzles in T&E(1) with B ≥ 12.
I can now add that, based on 950,685 minimal puzzles (corresponding to 145,614 unique min-expands), one can produce 175,453 minimal puzzles in T&E(1) with B ≥ 12, some exceptional ones of which reach B = 22.
This is about 1.2 minimal puzzles with B ≥ 12 produced for each min-expand in T&E(3).

Here's my new result:
- starting from minimal 6259 puzzles in T&E(2) with BxB ≥ 7 (noted B 7B+ below), assembled from coloin's 4 Mastermind collections, (corresponding to 2,543 unique min-expands), one can produce 10,979 minimal puzzles in T&E(1) with B ≥ 12, some exceptional ones of which reach B = 21.
This is about 4.3 minimal puzzles with B ≥ 12 produced for each min-expand in B7B+ .

The common point between the two starting collections is the presence of a non-degenerate tridagon in all their minimals (except 3 old ones in the B7B+ collection).
In both cases, the process is extremely productive. I'm not sure about why it is still more productive when starting from B7B+ puzzles.
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[denis_berthier]

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P.O.
Basics are irrelevant to the definition of the expansions.
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[P.O.]

Basics are irrelevant to the definition of the expansions.

i know you've already said it, but it's a recurring observation.
what about the order of the values ​​to build the layers?

[denis_berthier]

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I've changed the title of the thread in order to better reflect what's common to all the sub-topics dealt with in it.

[denis_berthier]

what about the order of the values ​​to build the layers?

The layers have been abstractly defined in the 1st post.
Layers and expansion paths are different things.

Suppose you're on the top of a 100 meter high hill (say a convex one to make things simple) - the top is the minimal puzzle.
Every step you make can only make you go down by at least 1m (the 1-expansion), but it may be more (the BRT expansion following it).

You want to reach a bar on the sunny side at the base of the hill (the puzzle on the T&E(3) border).

There are zillions of paths to achieve this goal: one that uses the steepest descent at any time, some that slowly slither down on the sunny side, some that make several complete turns around the hill.
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[P.O.]

exactly, there are 39,916,800 permutations of the 11 values which will define a large number of different layer organizations, what is the rationale for choosing one rather than another to characterize the puzzle?

[denis_berthier]

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The shortest path intrinsically characterises the puzzle; that's what I've defined as the distance to the boundary.
But that doesn't mean the longest paths are not interesting. On the contrary, they show that without any previous knowledge of the effect of adding a clue, one can keep adding clues that don't bring us much closer to the boundary.

This example doesn't show it, but there are cases where the minimal puzzle has lots of guardians and intermediate puzzles progressively have fewer ones.
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[P.O.]

ok why not maximize the number of layers, so for each puzzle the maximum limit is the number of values that keep the puzzle in te3 minus that of the min-expand, which implies no brt expansion and maybe for this puzzle there is such a permutation of the 11 values

[denis_berthier]

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This thread is about (1+BRT) expansions, not about 1-expansions alone. The motivation is to take BRT-equivalence seriously.
In [HCCS2], I've already given stats about the number of clues of minimals in T&E(3) and of their T&E(3)-expands, e.g. for their mean values: 27.11 vs 34.03.
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[denis_berthier]

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One practical good reason for doing slow expansion will appear when you want to search for the minimals of the expanded puzzle(s).
A problem familiar to the puzzle diggers is, the more expansion we do, the more minimals there will be and the longer finding them will take.
Having an expansion path such that each step makes the minimum difference allows some control on this problem.
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[denis_berthier]

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I've finally written all the scripts necessary to extract individual paths from minimal puzzles in T&E(n) to the inner side of the border with T&E(n-1).

In a previous post, I gave an example in T&E(3), and in another post I mentioned a puzzle in T&E(1) with a still longer path (18 steps) towards T&E(0). Here are now the details of this latter example.
Before that, I need to recall that it's the path that allows the maximum number of BRT-independent additions of clues, not necessarily the path with the largest BRT-distance to T&E(0). This longest path has 18 independent steps (but the BRT-distance of the minimal puzzle to the T&E(0) border is only 6.)

The last line below is the minimal puzzle. One must read upwards in order to see the successive additions of clues (alternating +p and +BRT). I've added manually the B (=W) ratings of the puzzles. It's noticeable that the minimal puzzle has a very low W rating (3) but can nevertheless be expanded so many times within T&E(0).
(A puzzle resulting of a BRT-expansion is not rated, as it can only have the same rating as the puzzle below it in the list.)

Code: Select all

1234..789456789.2.798132564.345..8.7815.746.2.67..845.34..9..7.589.47...67...394. 54c +BRT -> --p18EU
1234..789456789.2.798132564.345..8.7815..46.2.6...845.34..9..7.589.47...6....394. 51c +p18 W1
1234..789456789.2.798132564.345..8.78.5..46.2.6...845.34..9..7.589.47...6....394. 50c +BRT -> --p17EU
1234..789456789.2.79813256..345..8.78.5..46.2.6...845.34..9..7.589.47...6....39.. 48c +p17  W1
1234..789456789.2.7981325...345..8.78.5..46.2.6...845.34..9..7.589.47...6....39.. 47c +BRT -> --p16EU
1234..789456789.2.7981325...345..8.78.5..46.2.6...845.34..9..7.589.47...6....39.. 47c +p16  W1
1234..789456789.2.7981325...345..8.78.5..46.2.6...845.3...9..7.589.47...6....39.. 46c +BRT -> --p15EU
1234..789456789.2.7981325...345..8.78.5..46.2.6...845.3...9..7.589.47...6....39.. 46c +p15  W1
1234..789456789.2.7981325...345..8.78.5..46.2.6...8.5.3...9..7.589.47...6....39.. 45c +BRT -> --p14EU
1234..789456789.2.7981325...345..8.78.5..46.2.6...8.5.3...9..7.589.47...6....39.. 45c +p14  W1 <=======
1234..789456789.2.7981325...345..8.78.5..46.2.6...8.5.3...9..7.589.47...6....3... 44c +BRT -> --p13EU
1234..789456789.2.7981325...345..8.78.5..46.2.6...8.5.3...9..7.589.47...6....3... 44c +p13  W2
1234..789456789.2.7981325...345..8.78.5..46.2.6...8.5.3...9..7.58..47...6....3... 43c +BRT -> --p12EU
1234..789456789.2.7981325...345..8.78.5..46.2.6...8.5.3...9..7.58..47...6....3... 43c +p12  W2
1234..789456789.2.7981325...345..8.78.5..46.2.6.....5.3...9..7.58..47...6....3... 42c +BRT -> --p11EU
1234..789456789.2.7981325...345..8.78.5..46.2.6.....5.3...9..7.58..47...6....3... 42c +p11  W2
1234..789456789.2.7981325...345..8.78.5...6.2.6.....5.3...9..7.58..47...6....3... 41c +BRT -> --p10EU
1234..789456789.2.7981325...345..8.78.5...6.2.6.....5.3...9..7.58..47...6....3... 41c +p10  W2
1234..789456789.2.7981325...345..8..8.5...6.2.6.....5.3...9..7.58..47...6....3... 40c +BRT -> --p9EU
1234..789456789.2.7981325...345..8..8.5...6.2.6.....5.3...9..7.58..47...6....3... 40c +p9  W2 <=======
1234..789456789...7981325...345..8..8.5...6.2.6.....5.3...9..7.58..47...6....3... 39c +BRT -> --p8EU
1.34..789456789...7981325...345..8..8.5...6.2.6.....5.3...9..7.58..47...6....3... 38c +p8  W3
..34..789456789...7981325...345..8..8.5...6.2.6.....5.3...9..7.58..47...6....3... 37c +BRT -> --p7EU
..34..789456789...7981325...345..8..8.5...6.2.6.....5.3...9..7.58..47...6....3... 37c +p7  W3
..34..789456789...79813.5...345..8..8.5...6.2.6.....5.3...9..7.58..47...6....3... 36c +BRT -> --p6EU
..34..789456789...79813.5...345..8..8.5...6.2.6.....5.3...9..7.58..47...6....3... 36c +p6  W3
..3...789456789...79813.5...345..8..8.5...6.2.6.....5.3...9..7.58..47...6....3... 35c +BRT -> --p5EU
..3...789456789...79813.5...345..8..8.5...6.2.6.....5.3...9..7.58..47...6....3... 35c +p5  W3
..3...789.56789...79813.5...345..8..8.5...6.2.6.....5.3...9..7.58..47...6....3... 34c +BRT -> --p4EU
..3...789.56789...7.813.5...345..8..8.5...6.2.6.....5.3...9..7.58..47...6....3... 33c +p4  W3
..3...789.5678....7.813.5...345..8..8.5...6.2.6.....5.3...9..7.58..47...6....3... 32c +BRT -> --p3EU
..3...789.5678....7.813.5...345..8..8.5...6.2.6.....5.3...9..7.58..47...6....3... 32c +p3  W3
..3...789.567.....7.813.5...345..8..8.5...6.2.6.....5.3...9..7.58..47...6....3... 31c +BRT -> --p2EU
..3...789.567.....7.813.5...345..8..8.5...6.2.6.....5.3...9..7.58..47...6....3... 31c +p2  W3
..3...789.567.....7.81..5...345..8..8.5...6.2.6.....5.3...9..7.58..47...6....3... 30c +BRT -> --p1EU
..3...789.567.....7.81..5...345..8..8.5...6.2.6.....5.3...9..7.58..47...6....3... 30c +p1  W3
..3...789.56......7.81..5...345..8..8.5...6.2.6.....5.3...9..7.58..47...6....3... 29c +BRT -> MEU
..3...789.56......7..1..5...345..8..8.....6.2.......5.....9..7.58..47...6....3... 25c mins  W3 <=======

Needless to say that computations to obtain such results must explore unimaginably many paths within T&E(1) and take a very long time.
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[P.O.]

just an observation

Code: Select all

..3...789.56......7..1..5...345..8..8.....6.2.......5.....9..7.58..47...6....3... minimal      25c
..3...789.56......7.81..5...345..8..8.5...6.2.6.....5.3...9..7.58..47...6....3... min-expand   29c

after min-expand you add 25 values ​​to get this puzzle which is solved with basics:
1234..789456789.2.798132564.345..8.7815.746.2.67..845.34..9..7.589.47...67...394.  54c

but after min-expand it is possible to add 36 values ​​to obtain this puzzle which is not solved with basics:
123456789456789...7981325..234561897815974632967328.5.34.695.7858.247...67.813..5  65c


and this puzzle is the one obtained after applying basics to the minimal
but of course basics does not gather all the values ​​that keep a puzzle in its t&e depth as this puzzle shows

the question is: what is the set of values ​​that we consider to maintain this puzzle in te1?

[denis_berthier]

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As I've already said several times, basics are irrelevant to the analyses in this thread.
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[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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