# [Retros] Math aspects of Retro from Yefim Treger 08/21/2004

TregerYefim at aol.com TregerYefim at aol.com
Sat Aug 21 13:42:17 EDT 2004

From Yefim Treger 08/21/2004 Letter to all in retros.at mailing list.
Now, as a mathematician, I am writing a book about Math and Chess where, in
particular, some retro aspects are being used. In connection with this I have
some questions and thoughts... Understanding a legality as key point of retro
topic (and chess itself!) I have come to this "at first look unusual" idea:
there might exist a position which cannot be proved to be legal or illegal.
Let me explain in details.
1. The main idea of my book is analyzing chess game (here: usual game,
recorded, for example, by score sheet) from the math prospective, especially by the
help of sequence theory. Any game is some sequence of positions (it is better
to say so than "it is a sequence of moves" because Position contains specific
details, for example right to castling, ed passant, etc., but there is one to
one correspondence between sequence of moves and positions).
On this way I found some interesting results, but do not object to have
somebody check or control them (I am referring to an ad in USA "Chess Life"
magazine about it; you may use my personal e-mail to discuss book aspects, especially
if you are mathematician and/or fond of math). But that is optional.
2. Legality of position is its ability to form a game - as a sequence (of
positions) from the beginning, the original position. The given position, on the
other hand, as a configuration of pieces (let somebody take chess pieces from
the case and put them on the board, even reasonably) may be legal or illegal.
The main criterion of legality, I think, is a construction of a game from the
original position to the given one. One could use in this way reverse, retro
method, in this case it is enough to reach position, which is definitely legal.
3. Retro solver uses this reverse method, but here some situations are
possible. If he/she solves the given problem, already proved by the composer, then
we know (knew) for sure that the position is (was) legal, only the solver
cannot to find legality.
But let a solver be the solver in the stage of defining legality for the
arbitrary position. An arbitrary position (or simply: any position) is a position
from the side ("from the case" or when composer is creating something and is
vulnerable himself about emerging positions…- below). In this situation two
results are theoretically possible. If he (we, trying to build a game by the
reverse method, backwards, or using "retraction") reaches obviously legal position
then the given position is also legal. Otherwise, the question is open, and
mathematically it is impossible to say about legality (or illegality!).
4. This idea came in my mind when I saw some difficult retro-problems (of
Petrovich, Plaksin and others: AP rule, strange sequences, logic lists of
statements for the reasoning of legality, etc…) I want to specify: a conclusion
that "there might exist position, which cannot be proved to be legal or illegal"
does not mean that such a position is illegal. It means that we cannot
establish its legality. Some ideas about logic and set theory paradoxes come in mind
(...) but I have some arguments in defense of it.
5. At first, I found (and proved), that the movement backward (as retro topic
calls "move retracting") from the given position does not necessarily mean
really the movement to the original position. The point is that all positions in
a tree form levels not by any simple moves, but by type structures (moves,
which change types, approaching us to the structures where pawns are less
advanced, some pieces still are not captured, etc.). For example, it is obvious,
that in position with bared Kings there are retracting moves which do not
actually useful for us in reaching the original (or other, with more pieces)
position. Simply: it is possible cycling, ("recycling"?), which we have to overcome
(but how to do this?)
At second, there are positions which belong to another type, even when we
do not move pawns and/or capture pieces. It is an answer for those who object
to me that in retracting it is enough to create positions with pawn less
advanced locations or appearance of captured peaces - in this situation we take a
risk not to find all levels - look below. I have proved this too (constructed
the position which has moves not by pawns and not by captured after which this
position will never occur again!)
At third. I found that in a movement in the tree by direct method (forwards
to final positions) and reverse method (retracting) the following rule is in
effect: from legal positions by direct movement only legal positions
can be obtained; from illegal - both legal and illegal; using reverse method:
from legal positions both legal and illegal position can be obtained; from
illegal - only illegal ones. It means that we cannot apply both methods
simultaneously (or not attentively) for the given position. Example: I have
constructed illegal position where each move (!) leads to the legal position. Another
example: I have constructed the position where some retracting moves lead to the
legal positions and some others - to illegal ones. In this last case I even
constructed positions of the same type: it means there is a set of illegal
positions, connected by cyclic tread, so the further retracting movement results
in only illegal positions (only due to specific features of positions we can
say that they are illegal; at first sight they look very "usual"). Please recall
what I said little above: how can we overcome cycling or round trip; and even
if we do we take a risk to build a game where some parts consist of illegal
position!…)

from
the original position to the given one (we have to overcome "dead", blokade
positions, which never leads to the final). Mathematically it is better to say:
it is possible that there exists a position relatively which we cannot build
a game even by direct method of choosing all moves (like in Math: there are
irrational numbers but we cannot tell what is a given one is - or something like
this...)
There are some other arguments in my favor, but I suspect that somebody have