Vampires in Chess

Thanks for your comment @EudaemonicPhonix .

The starting position is a vampire because it is impossible to reach its mirror image in a standard game of chess. Even if both sides cooperate, after moving the pieces (you can actually only move knights) if you ever reach again the starting array, it will be again White to move. Here is an example:

lichess.org/study/aBmaeSAB/L0k11OKI

Note how I even swapped the knights, the one starting on b1 is now on g1 and vice versa. This did not help much. We didn't manage to reach the starting array with Black to move.

About your first point/question. You are talking about legal moves. As you suggest, the mirror image of the starting position has a valid analogous legal move for every legal move in the starting position. This is true. However, the position from where those moves start is illegal (in the mirror image).

Your reasoning is what allows us to discover that "every vampire comes from a vampire". If you ever reach a position with legal mirror image (a non-vampire), then for every move in it you have a corresponding move in the legal mirror image that leads to a legal position. Consequently, all positions that come after a non-vampire will be non-vampiric.

Let me know if some points need further clarification.

Brilliant! Thank you so much for explaining, @ambrona.

I understand now—"illegal position" means a position unable to be REACHED through a series of legal moves. (A definition that was clear enough in your original posts, but that I glossed over when considering the privileged starting position. I don't usually think of the starting position as a position to "return" to. Sort of passive/aggressive, or maybe catatonic, or perhaps a performance art piece, to play a game in which players attempt to return to the starting position :-)

One additional question: From your second comment, it sounds as if vampirism may be "cured" if, starting from a vampire, you reach a position with a legal mirror image. So in the case of vampire degree zero, is the vampire purged with one move?

Cheers!

Thank you so much for this great article! I'm looking forward to part II!!

@ambrona said in #41:
> Your reasoning is what allows us to discover that "every vampire comes from a vampire". If you ever reach a position with legal mirror image (a non-vampire), then for every move in it you have a corresponding move in the legal mirror image that leads to a legal position. Consequently, all positions that come after a non-vampire will be non-vampiric.

That makes it interesting. This is not just enumeration and detection. Thanks. Some sinus unclogged. Still text. But using this further result to reinforce with the parenthesis, for easy to lose people like me, is a good way to cover the inadequacies of board to text and back to board. We can get there, with enough words.

Repetition is the mother of pedagogy, but with often not mentioned diversity in the repetition, various angles of the same, more corollaries will reduce the fog between the word definition and the defined.

In your next post, when introducing objects with crucial things invisible at first glance, could you use the vertical space to your advantage as you did earlier, in this discussion. Your blog was concise enough, that the extra vertical space would not have hurt, and forced the fast reader to pause on such called-out phrases (not needing to be sentences). I hope this feedback is welcome. I think you have proven your ability to discern how people can think side-ways about the same thing.. Keep that up. Also, a bit of mathematics to states the tuples of variables being talked about (selfish request, reads faster).

Ok I need to parse this carefully. I caught the idea of something being invariant by legal continuation. But text is not my friend, and I know it is the crux of logic flow. But to fully understand I need some space and pauses.. so my visual can follow. Annoying math. subpopulation that likes its mind eye intuition support at every corner... or no logic gets through.

> "every vampire comes from a vampire".
> If
> reach a position with legal mirror image (a non-vampire),
> then
> for every move in it
> you have a corresponding move in the legal mirror image that leads to a legal position.
> Consequently
> all positions that come after a non-vampire will be non-vampiric.

In my words.
a vampire legal continuation (a sequence of positions) can lead to either more vampires or non vampires positions.

a vampire position can only legally retrograde to a vampire position (all vampires come from a vampire).

a non-vampire,
i.e. a position such that mirror operation** is a legal continuation of standard initial position, so both position and mirror are legal, then by definition of the operation same material and flipped placements exist in both "worlds" and same relative empty squares, make for mirror continuations possible and having the same preservation of timing as the initial condition had. (that last part might be hypothesis of interpretation, you would say something else?).

So that all continuations are themselves move mirrored (the thing that change board coordinates, but has other color doing the mirror board change of coordinates).. (This is so visual first. Sorry visual truth for this crawling).

I think my text only first reading of above was a bit enthusiastic. But still. one class of invariant by mobility.
Also a retrograde invariant of sorts. getting somewhere. vampires retrograde to vampires, and non-vampires continue to non-vampires. And using the mirroring as proof generator, for each continuations on each element of the pair of positions, side by side, each successor being mirror of each other. That "corresponding move" does not only say non-vampire but it says mirror game continuations for all continuations.

Does one statement imply the other?

** trying to get away from text prose

abusing "=" to mean "=" but also by def. or whatever shortens the text torture, such as "element of" or element taken in domain after the equal sign.. = might be like "move" is to chess. "(" ")" also abused.. (my touch typing prison). curly braces are a pain.. specially with my typo rate.. can't think and type anymore..

board = king walks graph (sometimes called kind graph)
diagram = material placements (16 of 1 color, 16 other color) using board coordinates.
turn = (w to move, b to move) = (w, b) = (-1, 1) (whatever).
castle and enpassant undefined but can be switched or flipped. (w caslting, w en passant, etc...).

position = diagram X turn X castling X en passant (getting sloppy).
mirror (position) = Apply( Mirror, position). (applies mirror elemental switch to all components).

mirror(diagram) = ok.. look at pics.. and black still in same AN whereabouts. but having the conformation that white had before mirror, and vice versa.. seems to mean paint the pieces each other colors, and rotate the board from under their feet..

I am also learning chess in all this.. this is not math looking at chess after knowing chess. My big ignorance is about opening data. So when talking about reverse openings. we are talking about black getting both the same board graph material to squares interactions but not necessarily the turn flip? or also the turn flip.

I have been looking at Pawn structures, and there, it is not part of the calling-out specification, which side has the turn.
yet, given only pawn placements are specified (with some abstraction intended and explicitly shared, at least as usual with examples straying from visible specs). I can spot a reversed pawn structure pretty easily. And since the gist of that middle game plan theory based on the extract and abstract pawn slower dynamics, is like in endgame having us to discuss both possibilities of turn, as those have been called out for their global odds symetry (or both sides would not have plan ideas to work with).

I am trying to reconcile various notions of mirrorring..

Finally (we wish!), if above 2 directions of invariance are a valid rephrasing or understanding.
Have the hand-moves and positions that from vampiric positions go on the non-vampire track been characterized. Those might help me in my chess learning approach (that my inability to learn and enjoy depth first data kind of make me prefer, I call it frontal, but it might be iterative deepening after all, without the cheap tricks avoiding width).

As I hinted above, I am chewing on such initial condition signature of the core rules as the games evolve from there. Such as can people tell from a middle game position or near endgame, without knowing the prefix sequence, from which 960 it might have legally be continued from (standard being one of them).

or in one of them, e.g. hand picking with blindfold, say the standard initial position, the early moves imprints through depth: when can retrograde lose focus (?). This might look unmathematical, but not for me. There are notions of spatial configuration information that are not relying only on hand-move sequences information. Such notions could be characterising positions, and surprise notion would not have to only be from the turn-by turn sequences departure from expectation, rather, it would be from the full position information itself, in abstract of its history from initial position.

Your presented invariance which is not just about turn-by-turn in that it propagates whatever turn by turn, for the non-vampiric strains, seems relevant to that kind of chess thinking focussing on the position configuration over which turns were alternated getting there.

But there might be such turns that would be interesting to characterize. I have not completed my converging spiral. I count of imagination not being in short supply as I sense it is the case so far. I really don't like typing, but I crave sharing it seems. Also, I might have babbled enough for one day.

@EudaemonicPhonix said in #42:
> One additional question: From your second comment, it sounds as if vampirism may be "cured" if, starting from a vampire, you reach a position with a legal mirror image. So in the case of vampire degree zero, is the vampire purged with one move?

Exactly, vampirism can be cured! That's a nice way to state it :)
We could also say that the vampiric property is not necessarily inherited. Indeed, most moves in a vampire often lead to a non-vampiric position.

I didn't get your last question. What do you mean by "vampire degree zero"?

The vampiric property can indeed be purged with one move in most positions. It is actually quite challenging to find a vampire where all legal moves preserve the vampiric property, but they exist as far as I can tell.

Hi, @ambrona, Interesting, yes. By "vampire degree zero," I just meant the starting position (the original vampire).

I missed the point of the cured.. but that was my point, also. Can we characterize such cure, in chess examples? or in abstract (well in abstract, they are those that will stay cured.). But I guess I want more wisdom from this. (It might help me get a sense of how wild can vampire and non-vampire position populations be in the legal set. Do they get some phase dependency in their availability.. does that even make sense to ask?

So, most moves lead to non-vampiric.. In line with my interpretation of initial timing "imbalance" for white, is this related to triangulation type of color-side to move to position configuration dependency? That it might be easy to lose this type of quantity as color-side imbalance? perhaps chess is allowing many compensations on the board, beyond chasing or keeping tempi imbalances.

Did we see such examples in the blog or other participants, of curing? Perhaps degree zero, might be some depth bound on vampirism. (I am thinking that my interpretation above is ok). It might also be a synonym to cure, in that post. But I think our curiosities are hooked. Prevalence of vampirism per phase?