[Retros] 16 combinations for castles yefimT 08/24/2004

[TregerYefim at aol.com]

Hello, Yefim 08/24/2004.
I am glad that somebody asked math question, I will try to answer (and will 
not speak very "hard" for many of us to understand).
I want to defense my definition of Position: two positions are the same if 
sets of all games emerging from them are the same (not paying attention to 
numbers of moves; also good is graph definition: if trees are the same then 
Positions are equal.
1. Two positions may be connected only by one move, no more moves!  It means, 
in particular, that:
    a) if they are connected directly (White->Black or Black->White) then 
only one move is possible as a transfer between them (or they are not directly 
connected, or even illegal, that is different story). For example: move 1.e2-e4 
played in all games is unambiguous Not because it is written in full Algebraic 
notation but due to it defines two concrete position. One ("starting" as 
Francois writes) is the Original position (it is called "initial" in Fide rules, 
with 4 castles, if after knight-rook dance any castle is lost then we have 
other positions).
     b) Move e2-e4 may be made in 16 different cases, positions distinguished 
only by combinations of right of castles (all pieces are at the same squares 
as in Original position). Why 16? Let's agree denote each castle as letter 
A,B,C,D where A- stands for White 0-0-0; B stands for White 0-0; C -Black 0-0-0; 
D-Black 0-0. Then word ABCD with only 2 values, 1 and 0 for each letters (1 - 
castle is present; 2- absence), gives us full information about position. But 
this is binary system, so we have 16 combinations. Thus, 1111 stands for 
Original position, 0000 stands for position after full dance of knights and rooks. 
For example word 1011 describes position after 1.Nf3 Nf6 2.Rg1 Ng8 2.Rh1 Nf6 
3.Ng1 Ng8 where we lost right to short side White castle. See the difference 
between e2-e4 after that (in 1011 "type") and e2-e4 in 1111 type (which, by the 
way, may be not by number 1, for example: 1.Nf3 Nf6 2Ng1 Ng8 3.e2-e4)?
  c) So, Math definition of move implies that it defines both starting 
position and ending position. There is a mapping between two different direct 
connected positions and the move between them. That is why I (and we all) talking 
mostly about position. Move means (has to mean) Two positions but often we do 
not need the ending position.
  d) There is essential feature of position: set of all direct moves. But 
this is Not sufficient for the definition: as we have seen; all 16 positions 
("types" of initial position, that is why I am saying the Original Position, 
implying only type 1111) have the same set of direct moves, but these are different 
positions. They take Different places in a tree of all positions.
  e) We may attribute binary system to any position as additional information 
for it (for all combinations with castles the same concerns en passant); for 
example after 1. e2-e4 we may begin knight-rook dance; so we have here 16 
types (I prefer to use word subtypes, but about my book later…). In this situation 
we may have so called "defective" types (subtypes) when some castle is later 
proved to be impossible but we at first attributed to it some subtype. That 
was my the first example with Black pawn e2 preventing White from castle 
(reminder: and we cannot get rid of it what sometimes it is difficult to prove…) So, 
upon my system at first we had denoted position as subtype 0100 but it is 
0000? Contradiction? No, if we understand that this position has defective types, 
some of them coincide. Of course, almost always one may find contradiction 
(here 0100 is not 0000) but we have just solved it! Arbiter Gijssen (I saw him in 
Moscow, but spoke only about time control) mathematically is not right about 
rule about castle (if King and/or rook had not moved then castle always 
possible) but he is right about a convention about this, because after this we do 
not have to think about castle-repetition problem (as I said sometimes it is 
difficult to prove something…). On other hand, we have a right to offer something 
positive in this matter and really maybe some rules have to be emulated (If I 
see him next time maybe I tell him about this matter to know his opinion…) 
    f) Answer to question about moves with captures and/or delivering mate. 
Francois writes:
"The move Na8xb6# means a knight moved from a8 to b6, captured something, and 
delivered mate…"As I said, it is not complete (math) move; it always defines 
two positions, so we know captured piece and check or mate position (here 
game-sequence properties are hidden, for example, mate positions do not have 
moves, that is another story, below).   
   2. There is one and only tree of all positions, but we do not know many of 
its properties. There are positions with unknown properties (at least so 
far), including unknown rights to castles or something else (for example, how they 
connected, if we knew all of them all composer problems would be solved 
instantly !). But one position takes only one place in this tree and one place is 
taken only by one position (so called the one to one correspondence) and again: 
one move is possible between them. So it is direct, oriented graph and 
according to graph theory all graphs may be (as Frankois writes) isomorphic. That is 
true, but I told about games. Tree is not a game (s)! Tree itself shows how 
positions are connected and game show path in this tree from the Original 
position to final one (or to any position before agreement for a draw or resigning; 
even 3 times repetition and 50 moves are human  rules and theoretically 
possible play eternally, especially after introducing current time controls…)  
   a) Positions without pawns may be reflected in the mirror, explaining idea 
of isomorphic graphs (in such case all positions emerging from it will be 
"mirror" ones) but I have never understood that they are the same because games 
still are different (and if they were I have not given for discussing this 
isomorphic idea…)
   b) We may paint all White positions (in a tree) to the Black ones and vise 
versa and receive isomorphic graphs (Black begins first). We may set up 
chessboard incorrectly but have the same graphs. Yes, it is possible to think of 
idea of isomorphic games, but about that next time.
  3. What is more important that sets of all possible games for the equal 
positions are equal. What is a game from the Math point of view? (for this e-mail 
I am finishing). It is a sequence (of positions). Usually a game implies a 
sequence from the Original Position and may end by any legal position but we can 
count a wide definition of a game as sequence from any starting (here: the 
first) legal position to any other (here: the last one) legal position. Further, 
if the first or last one is the final one (mate, stalemate) then that game 
consists of only such position. So, game always has at least one legal position 
and a set of all games is never empty (as Michael writes; it would be empty 
only if some positions were illegal…). There might be still a question about 
equality of sets (of games) but as I said, next time!…
    Bye, Yefim Treger. P.S. Computer program are still working incorrectly, 
showing "Draw" after 1.Nf3 Nf6 2.Rg1 Rg8 3.Rh1 Rh8 4.Ng1 Ng8 5. Nf3 Nf6…

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