# [Retros] Math aspects from Yefim Treger and the 5-th example! 08/23/2004

Francois Labelle flab at EECS.Berkeley.EDU
Mon Aug 23 18:54:57 EDT 2004

We've seen an example of positions with different castling rights but the
same "dynamical properties" (the en passant examples with a pin or a check
were considered the same position). I came up with examples where even the
*diagram* looks different, depending on exactly what is meant by "the same
dynamical properties".

Yefim Treger wrote:

> There is a contradiction between definition of Position in chess rules

> and Math (which is: 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.)

I find this "math" definition ambiguous. Two game-trees are the same if
all the moves are the same, but what is a move? Here's a list of
possibilities:

1) In its broadest definition, moves are transitions between chess
positions. The move e2-e4 is represented by the starting position and the
ending position. Since a move includes its starting position two
game-trees are the same if the starting positions are the same. The
"chess" and "math" definitions are the same and there is no paradox.

2) A move is represented by its Long Algebraic notation. The move Na8xb6#
means a knight moved from a8 to b6, captured something, and delivered
mate. Here are two positions with the same dynamical properties using that
definition:

2b3b1/1p1p4/pPkPp3/P1p1Pp2/2P2Pp1/6P1/6R1/4K2B
6b1/1p1p4/pPkPp3/P1p1Pp2/2P2Pp1/6P1/6R1/4K2B
_________________ _________________

| | |

| . . b . . . b . | . . . . . . b . |

| . p . p . . . . | . p . p . . . . |

| p P k P p . . . | p P k P p . . . |

| P . p . P p . . | P . p . P p . . |

| . . P . . P p . | . . P . . P p . |

| . . . . . . P . | . . . . . . P . |

| . . . . . . R . | . . . . . . R . |

| . . . . K . . B | . . . . K . . B |

|_________________|_________________|

White can move his king around, Black can move his g-bishop around. (The
white R and B are there to prevent a draw by A1.3, White can end the game
at any time by moving his rook).

However it is odd to specify that a move is a capture or delivering mate.
So...

3) A move is represented by its starting square and ending square only. My
example is the same except the wK is now on d8.

2bK2b1/1p1p4/pPkPp3/P1p1Pp2/2P2Pp1/6P1/6R1/7B
3K2b1/1p1p4/pPkPp3/P1p1Pp2/2P2Pp1/6P1/6R1/7B
_________________ _________________

| | |

| . . b K . . b . | . . . K . . b . |

| . p . p . . . . | . p . p . . . . |

| p P k P p . . . | p P k P p . . . |

| P . p . P p . . | P . p . P p . . |

| . . P . . P p . | . . P . . P p . |

| . . . . . . P . | . . . . . . P . |

| . . . . . . R . | . . . . . . R . |

| . . . . . . . B | . . . . . . . B |

|_________________|_________________|

Now it's possible to go from the 1st position to the 2nd, so it's possible
to turn this into a 3-repetition of position "paradox" (using the fact
that Kd8xc8 and Kd8-c8 are now considered to be the same move).

4) A move is represented by its Standard Algebraic notation. e2-e4 and
e3-e4 are the same because they're both written "e4". Maybe someone can
make an example based on this.

5) Two game-trees are the same if they're isomorphic as graphs. Then this
position and its 8 symmetrical versions would be the "same".

8/8/8/8/8/2K2k2/1R4r1/8
_________________

| |

| . . . . . . . . |

| . . . . . . . . |

| . . . . . . . . |

| . . . . . . . . |

| . . . . . . . . |

| . . K . . k . . |

| . R . . . . r . |

| . . . . . . . . |

|_________________|

One could make a weird "3-repetition of position" example that goes
through 3 of them. Maybe someone can make an example where the positions
wouldn't even look similar.

Yefim, which of my possible definitions did you have in mind? To define
position equality as game-tree equality, you need to specify exactly what
you store at the nodes or edges of that tree. If you answer "I store the
move", specify exactly what information a move does and does not contain.

Francois