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MatPlus.Net Forum General Largest n for "Something happens in n"

### Largest n for "Something happens in n"

As I wrote there, n is potentially infinite (e.g. the Dawson lunar flight shown in the link),
and I think the n for the next longer boards here were actually printed in Fabel? Grasemann?,
but those not even scratched the value 2^59+2^57+58 *actually* printed in the new SCHWALBE
in an article. (Stipulation was "Aim in n", a load of fairy conditions applied, and
the problem implemented a waiting queue algorithm scaling with 2^(N^2), with N board size.)

So, can you beat ~10^18 *printed* somewhere under a problem? (Probably math-heavy.)

Any halting Turing Machine can be interpreted as a series-auto-stalemate on a finite(!) board with an ad hoc fairy condition.
The Busy Beaver function beats the Ackermann function and there are published very small machines (2-state 6-symbol) that require more than 10↑↑10↑↑10↑↑3 steps to halt.

Edit: Sorry, you asked about scaling with the board size, not with the rules complexity.
Edit2: Though we probably can shrink the board size arbitrarily by complicating the rules.

No need to be sorry, as half-a-mathematician I find your info still interesting ;-)
(But naturally, any editor would refuse to print Mate in 10↑↑10↑↑10↑↑3 (not
10↑↑10↑↑10↑↑3, naturally, but the exact number) - the printing costs :-)

Digital publication is also an avenue. Just store it in a super computer. Technically scrollable.

Hauke is kidding, the number of bits required to store this number by far exceeds the number of particles in the observable universe.
This number, by the way, is the lower bound for the number of "pieces" left on "board" when the solution ends, the number of moves is slightly larger.
This particular machine was found by Pavel Kropitz in 2023, here's the detailed analisys of why it stops and how long it takes (some math and domain specific jargon):
https://www.sligocki.com/2023/05/20/bb-2-6-p3.html

(= 2+1 )

Longest possible game?

Because it's not a retro, the 50-move rule and the 3-fold repetition are not enabled, so the longest possible game has infinity moves.

The question reminded me that I had the idea to add a problem on the bottom of all the pages of my electronic book published on Vaclav Kotesovec's website (http://problem64.beda.cz/silo/arno_tungler_2009.pdf). All pages would have a 50x7 diagram on the bottom with a white pawn on the sixth row of each of the 50 columns. On page 1 the diagram would have wKa7 sKa1 and sPa6 on the very first column. And on the last page the very last column would have in addition to the wP on row 6 a wS on row 7 and bR on row 1... All together, on 526 pages, this would build a 26300x7 diagram with the stipulation r+26299 (reflex-check in 26299 moves) with the solution 1.b7=S! Kb1 2.c7=S Kc1 ... 26298. [Column 26299]7=S Rx[Column 26300]6 26299. Sxa6 Rxa6+. If you are interested, please check on page 224 of the above book the very simple reason why with this stipulation you will have on any Nx7 board a correct r+[N-1] with consecutive [N-2] knight-promotions. As I here mirrored the position and removed one row for economy, you can also see a sample diagram with N=20 on https://i.ibb.co.com/PmF7xjs/fen.png?=K18N/pPPPPPPPPPPPPPPPPPPP/20/20/20/20/k18rb

Draw by triple rep is always enabled in compositions, irrespective of retro status. It is a question what is the longest game possible under Codex rules. I think it s about 10^90.

Codex says about 3-rep 'is considered', not 'is'. So for the question of 'longest game possible?' it's not a terminating condition.

(10) Posted by Andrew Buchanan [Thursday, Jan 18, 2024 14:39]

I don’t think the weasel word “considered” should be taken too seriously. The third occurrence will stop the game. Boring were it not to work like that, and those compositions which operate in the space rely on that termination condition

My dislike of the 50-move rule and 3-rep rule in chess composition is well-known. They're artificial rules that have nothing to do with *chess* an sich but only with *playing* chess. and chess composition isn't *playing*.
And wording in rules is very important. If a rule should be enforcing, it shouldn't contain wishy-washy phrases like 'is considered'.

@Joost,

I don't understand why the 50-move rule (or any arbitrary number of moves without progress, for that matter) and the 3-repetitions rule (or an arbitrary number of repetitions, for that matter) should be considered any different than a fairy aim (e.g., MAFF, double-checkmate, capture, capture by en passant, castling, etc), but I am happy to learn more about alternative perspectives on this matter.

If you are simply arguing the FIDE Codex should not consider this an orthodox chess problem, we are in complete agreement (that said, I do dislike your use of the word "artificial" -- there is nothing artificial about fairy stipulations, fairy goals, or fairy aims -- they are simply unorthodox).

I would urge the FIDE Codex to provide a clear definition which distinguishes orthodox problem elements from fairy problem elements.
There are difficult questions which require resolution here (even for me).

For example, consider an A=>B problem which starts from the initial game array of Upside Down Chess (I published such a problem on Julia's Fairies in 2023).
If there is no need to define a fairy condition (the rules are the same), and there is no fairy stipulation (no fairy aim, no fairy goal), it begs the question: should we consider this a fairy problem (does an alteration to the initial game array constitute a fairy condition) or is this an orthodox problem (which happens to employ a starting diagram, A, which can not be reached by legal means)? Can it be considered either (an overlap in the Venn Diagram of Orthodox/Fairy)?
Should we consider any A=>B problem with an illegal diagram (A) as a fairy? If so, does that also hold for all orthodox problems (#n, h#n, s#n, etc) with an illegal diagram?

And, if we consider this orthodox, what about the alternation of square colors in Upside Down Chess (does that render it a fairy condition)?
If I publish an orthodox #2 problem, but I alter the colors of the squares on the board, should it be considered a fairy problem (assuming I never admit the rotated diagram was a blunder)?

These matters should be clearly defined, and the FIDE Codex is the only proper authority to resolve these questions.
I happen to think the best way to go about this is to consider the variety of ways a problem *might* be considered unorthodox:
1) a problem might employ unorthodox units (grasshoppers, Nightriders, etc),
2) a problem might employ unorthodox rules (which constrain, expand, or alter normal movement of orthodox units),
3) a problem might employ unorthodox stipulation (based upon an unorthodox aim, based upon an unorthodox goal, or based upon a false stipulation which conceals unorthodox rules),
4) a problem might employ an unorthodox board (larger, smaller, holes, dynamically shifting, some variation in the connection of squares -- like Grid Chess, or Cylinder forms),
5) a problem might employ an alternation to the initial game array,
6) a problem might employ a diagram which can not be reached by legal means (from the orthodox initial game array),
7) etc.

Like you -- like anyone who has pondered these issues -- I have my own opinions what is the most elegant way to resolve these matters (while we can all agree most of the time, the lack of definitional clarity will always allow somebody to disagree).
But if you allow yourself to look for loopholes, I must admit, it is not challenging to discover special cases which are not easily resolved.
Our definitions require careful wording, as they define categorical divisions (which can impact problem placement, judgements, album points, and titles).

It is not my intent to divert this interesting conversation into yet another discussion about the poor definition for the elements of a chess problem (a long lasting soap box issue for me) -- my intent here is only to remind readers that this lack of definition is precisely the issue driving this thread (the failure of the Codex to provide unambiguous definitions is almost always the underlying issue driving the disagreements voiced on this forum).

I look forward to the day when we all resolve to pressure the Codex to address our broader issues (which would inform everyone of the perspective necessary to resolve these lesser concerns).

Careful, unambiguous definitions for the broader terms (a clear line between fairy and orthodox) would save us from countless disagreements (saving an uncountable number of words, for those whose opinions are informed by word count efficiency).

> I do dislike your use of the word "artificial"

Both rules were implemented to avoid neverending games. In that aspect they're artificial concepts, they're not part of the 'core' rules that define chess, but of the additional rules that define how to practically *play* chess under certain conditions. The exact number is chosen fairly randomly, especially in the 50-move rule.
About following FIDE rules: A lot of rules are irrelevant for a lot of chess compositions. Take e.g. rule 1.4. That's only valid for mate compositions: selfmate, stalemate, any other stipulation go directly against this rule. For non-mate compositions, the DP rule is also not general enough. A h= composition with wK wS vs bK for instance is a perfectly valid composition, although 1.5 states that the game should be over because there's no mating potential. So what? The goal isn't mate anyway, so that rule should be 'impossible to reach the goal', not 'impossible to deliver mate'. Anti-circe isn't described in the FIDE rules, so why should the 3-rep rule be used in Anticirce compositions? Etc, etc.

On a 1 dimensional board of height x+3, you have a (help-)stalemate in x(+0.5) with the position wK1, bK3, assuming Black is to move. This scales linear.

Exponential scaling can likely be achieved with some fairy shenanigans.

@Joost,

QUOTE

Both rules were implemented to avoid neverending games. In that aspect they're artificial concepts, they're not part of the 'core' rules that define chess, but of the additional rules that define how to practically *play* chess under certain conditions. The exact number is chosen fairly randomly, especially in the 50-move rule.

When you frame your usage of the word "artificial" this way, I find it impossible to disagree; arbitrary elements are indisputably external to the "core" essence of chess.
In fact, I would take your point a step further: there is something artificial about even the arbitrary initial game array in chess.
Though the starting game position is very well chosen (for the default variant), there is no denying that our arbitrary initial arrangement is non-essential (it exists outside the core essence of chess).

This is why I personally prefer Circe variants with rebirth rules that do not depend upon the initial game array.
While I do not deny that beautiful problems arise from variants which depend upon this arbitrary rule (same is true with other arbitrary rules -- including 3-reps, and 50-move rule), I prefer to avoid rebirth rules which constantly call attention back to a non-essential element (the arbitrary initial game array).

Furthermore, I often think the arbitrary initial arrangement is a hinderance to proofgame composers (we really should be exploring more opportunities in A=>B problems, rather than depending so much upon the artificial arrangement for a starting position), but at least proofgames (which require half the diagrams of an A=>B problem) show some escape from this arbitrary arrangement (whereas some Circe/Acirce forms constantly draw us back to what may be considered artificial).

I suppose we might even consider the arbitrary board size to be artificial -- an infinite board is the more natural state (more to the point, an infinite anchor ring would seem the most natural state) -- but at least this artificial board size element brings some benefit (corners and edges can be very beneficial, just as the elimination of zero-movements is certainly beneficial).

I am less troubled by artificial constraints imposed upon a stipulation (e.g., 3-Move-Repitition, the 50-move rule, and/or Dead Reckoning rules), because, like the arbitrary board configuration, these artificial constraints confer some practical benefit.

I don't mind artificial sweeteners, as long as they prove their sweetness.
I still say our artificial authority (the FIDE Codex Committee) has a responsibility to classify these non-essential elements as fairy -- providing nutritional information would be very sweet (this could potentially render artificial the vast majority of debates found in this forum).