continuous chess

[Uri Blass]

In chess you can say that every piece can be on a single square that you can describe by 2 numbers.

number of rank 1<=x<=8
number of file 1<=y<=8

x,y must be integer numbers and I thought that it is possible to have generalization that allow x,y to be also non integer for example
(1.1,4.32)

Queen can move exactly to the same geometrical directions when the difference in continuous chess is that the queen
can move also from (2,2) to (1.9,1.9) or to (2,4.65)

In order to make things simple let say that except the fairy queen we have no more pieces that can move non integer number of squares.

What is the value of the fairy queen.

Certainly the value is more than the normal value of the queen
because it can also choose to move only integer number of squares.

Note that there is going to be a problem to analyze a game with it with computers because unlike normal chess there is an infinite number of moves that the queen can move so the question is if there is a theoretical way to solve this game mathematically(from the computer point of view the number of possibilities is finite because there is a limited memory but even if we allow only 10 digits it is practically not a good idea to try brute force).

Uri

[Uri Blass]

I think that one of the advantages of the fairy queen that can move non integer number of squares is that non fairy pieces cannot capture it if it is in non integer square and it lead me to think about another type of fairy queen that move like a normal queen but it is impossible to capture it.

This new fairy queen is a very strong piece because it can capture everything with no risk(something that the continuous fairy queen cannot do because the continuous fairy queen cannot hide at the time that she capture a piece).

To make this new fairy queen less strong you can decide that the new fairy queen can be captured only after it captured a piece before moving again.

[zullil]

In chess you can say that every piece can be on a single square that you can describe by 2 numbers.

number of rank 1<=x<=8
number of file 1<=y<=8

x,y must be integer numbers and I thought that it is possible to have generalization that allow x,y to be also non integer for example
(1.1,4.32)

Queen can move exactly to the same geometrical directions when the difference in continuous chess is that the queen
can move also from (2,2) to (1.9,1.9) or to (2,4.65)

In order to make things simple let say that except the fairy queen we have no more pieces that can move non integer number of squares.

What is the value of the fairy queen.

Certainly the value is more than the normal value of the queen
because it can also choose to move only integer number of squares.

Note that there is going to be a problem to analyze a game with it with computers because unlike normal chess there is an infinite number of moves that the queen can move so the question is if there is a theoretical way to solve this game mathematically(from the computer point of view the number of possibilities is finite because there is a limited memory but even if we allow only 10 digits it is practically not a good idea to try brute force).

Uri

A short game of continuous chess:

Code: Select all

1. d4        d5
2. Qd√2

Faced with this irrational opening, Black immediately resigned.

[Rein Halbersma]

]

A short game of continuous chess:

Code: Select all

1. d4        d5
2. Qd√2

Faced with this irrational opening, Black immediately resigned.

Black could have meditated a bit about white's move and reply with

Code: Select all

2...Bhπ

to transcend all that irrationality.

[hgm]

I think most non-integer positions a Continuous Queen can occupy are actually equivalent. For a Continuous Rook, if it is between two squares, it doesn't really matter where between the squares it is, if all other pieces are 'discrete', i.e. have 'quantized' positions. So you could map the game to a 15x15 board, where all discrete pieces move in double steps.

Actually the game would need additional specification on whether the discrete pieces are blocked or can jump over a continuous piece that is halfway between the squares between which they are want to move. (E.g. if there is a Rook on f2.5, could I move a Pawn from f2 to f3?) Normally riders cannot be blocked on squares they cannot visit.

If there are more continuous pieces it also becomes important of whether pieces positioned between the same rays are exactly aligned with each other. And it should always be possible to put a second piece between the same rays on either side of the other piece, no matter how close he already is to the integer ray. This suggests you cannot map the game to a finite board. You could, however, if you change the rules for moving such that continuous pieces are allowed to push other continuous pieces off their ray. E.g. if both sides had one Continuous Rook, you could put two intermediate ranks/files between the normal ones used by the other pieces, and if white moves to f2.3, rounded to f2+, and black wanted to move to b2.2 to be fetween second rank and that Continuous Rook, you could put the white CR on b2+, and push the black CR to f3-. Then you would still have a finite game-state space, albeit with somewhat more complex moves.