Quantum Mechanics and Chess

After the "Physics and Chess" variants (from where Magnetic Chess was invented), Claude and I discussed the introduction of Quantum Mechanics into Chess in rec.games.abstract. Here are some ideas and variants:

Heisenberg Chess

This variant was invented by João Pedro Neto, in January 2000.

1. The FIDE rules apply except in the following:
2. Definitions:
   1. Some moved piece A is seen by B, if that movement includes some square in the capturing range of B.
   2. The distance 'd' between two squares is the maximum between the row and the column differences (egs, d(a1, b3) = 2, d(a1,b2) = 1, d(a1,a5) = 4)
3. By the Heisenberg Principle, every moving piece that is seen by any opponent piece, either change its speed or its position (player's choice):
   1. Change of Speed: The piece start moving (from the square seen by some opponent piece) as if it was a different piece. The new movement is defined by the following order: Q moves as R, R as B, B as N, N as P. (note: for knights, the movement, is defined just by the immediate forward square from the original position. E.g., for Nb1-c3, it's square b2. For Nb1-d2, it's c1). E.g:
      
      . . . . . . . .
      . . . . p b . .
      x . . . x . . .
      n x . x . . . .
      . . . . . . . .
      . x R x B . . .
      x . . . x . . .
      . . . . . x . .
      
      After moving north, Rc3 is seen by the opponent at c5, and changes its direction, start moving as a bishop. It can also take the black bishop.
      
   2. Change of Position: If the piece is seen at square S by N opponent pieces, it changes its position to any empty square at distance N from S (except to those squares that he could move in FIDE chess). E.g.:
      
      . . . . . . . .
      . . . . . b . .
      . . . . . . . .
      n . x . x . . .
      R . . . B . . .
      x . N . x . . .
      x . . . x . . .
      x x x x x . . .
      
      The squares marked by 'x' shows where the Knight can go (it may also capture Na5), if the player decides to change the position, by a Nb5 or Nd5 move. Why is this? Because the immediate forward square for those moves (square c4) is seen by two opponent pieces.

Here some Mutators due to Claude (a mutator is a game modification, check them here):

Mutator Delayed: The principle is that you are allowed do delay the choice among the moves available to you, until some of them interacts with another player's move. When the game is over and all the choices are made, possibly post-mortem, the game should still look like a valid classic game, although possibly a weirdo.   Delayed is the combination of the following mutators:

• Branching: Whenever a player is to make a move, he can submit a set of several moves, as if he didn't know which alternative to choose right now. The game is branching into many potential histories at that point.

Tic-tac-toe example: After I played into the center with 1.b2, you can be very undecided and play a big set with 2.{a1,a2,a3,b1,b3,c1,c2,c3}:

1   . . .  1.b2   . . .  2...   x?x?x?
2   . . .   -->   . o .   -->   x?o x?
3   . . .         . . .         x?x?x?
    a b c

• Collapsing:  A player having left a set of alternatives makes his choice later on, right when another player submits a move which isn't allowed by -- isn't compatible with -- some of the mentioned alternatives. The potential the game history had is collapsing into fewer possibilities.

Example, continuing: If I play 3.a2, my move is not valid in the case your delayed choice is the alternative 2.a2. So I force you to choose. If now you choose indeed to have played 2.c2, my move is alright and the board looks like:

3.a2(2.c2)    . . .
    -->       o o x
              . . .

before the next turn, yours.

• Probing:  A submitted move which turns out to be non-valid is not played. The submitting player must immediately submit another move. The new move may well force again a choice to be made from an opponent.

Example: If you don't choose 2.c2 in the game above but 2.a2 instead, I must play something else, for example b1:

3.a2(2.a2)b1   . o .
    -->        x o .
               . . .

and your position looks worse, because it's as if I probed your state before actually playing.

• Observing:  Only another player can decide when it's time for you to make some choice among a set of alternatives you submitted before. That may be called observing. You cannot observe yourself in order to get your choices to collapse. Of course you can fix your choices in advance in your mind, but you'll need someone else's troublesome, questionable interaction to fix them outside.
  
• Waving and observing:  A player's old set of alternatives must also immediately collapse when another player submits a set of moves among which some move is questionable because of some of the alternatives, as explained above with only a single questionable move.

Example: If I don't submit 3.a2 in the Tic-tac-toe game above but 3.{a1,a2,a3,b1,b3, c1,c2,c3} as you, you are still requested to make a choice, because my a1 for example is not compatible with your a1. Let assumes you choose to have played 2.a2. The boards becomes:

3...(2.a2)   o?o?o?
   -->       x o o?
             o?o?o?

• Waving and probing: After the collapse, the submitting player must immediately replace any move that turned out non-valid. Moves that are already in the set and valid are allowed. In other words, actual non-valid moves can be removed if there is some other valid move in the set.
  
• Nesting or Interference and correlation: Some of your moves can or even may have to be conditional upon previous indeterminate alternatives of yours. In that case you must submit a complete set of moves, that is a set providing any possible choice of yours left undecided with a valid current move (for some of your opponents choices left undecided).
  
• Cascading:  When a move you are submitting isn't valid for several dependent or independent choices to be made by other players, those other players make them all before you can replace your possibly non-valid move with another move. [Note: it may be necessary to describe in which order the choices are made when there is more than one other player involved.] When you have several non-valid moves to replace in a set you've submitted, you replace them all before other players may possibly make new needed choices.

Funnily enough, Delayed can be applied twice and give apparently another, even more abstract game.

But is Delayed really changing anything to any classic game, at least the outcome? It seems to me it isn't for games like Tic-tac-toe where a player's set of valid moves covers the opponent's set of valid moves, so that the classic winner can always submit it and immediately remove any indeterminacy in the opponent's position, thus being able to answer classic moves to a classic position. Is it different for Chess?

And does Delayed have any fixed point at least? How does one find any fixed point of a mutator anyway?

Note that finite, two-person, full-information games of no chance remains so after Delayed mutation. Hence some of Conway's and other's combinatorial theory of games might apply to it, or it might mutates the theory.

Variants that certainly change something:

•   Binary Delayed: You cannot submit more than two alternative moves.
•   Random Delayed: When a delayed choice must be made, sheer randomness is choosing for you.

(maybe Binary and Random are higher Mutators, though the phrasing above would need to be improved to see it?)

There was also other responses to this subject, namely by Stephen Tavener

A couple of quantum-like chess variants I devised a while back...
... in both cases, standard chess rules apart from the exceptions noted.

QUANTUM CHESS

Definition: A piece is "observed" if it is attacked by an enemy piece, or defended by a friendly piece.

Definition: An empty space is "observed" if any piece can move to that space under the normal rules of chess.

Play standard chess, but instead of a standard move, a player may move any of their pieces to any space on the board, so long as the piece is not observed, and the destination space is not observed.

Note: I haven't tried playing this variant.

PRUNED TREE CHESS (call it EIGENSTATE CHESS if you prefer)

Each turn, a player offers N moves. All moves must follow on from one or more of the moves the opponent offered on the previous turn.

Example (N=4)

c4        d4        e4        e3      (White 1)
           |                   |
+----------+         +---------+
|          |         |         |
d5        e5         d5        e5     (Black 1)
           |         |         |
+----------+         |         |
|                    |         |
|          +---------+         |
|          |         |         |
dxe5      Qf3        Qh5      Qf3     (White 2)

... and so it goes.

Note: I tried playing this once by correspondence with N=8; it works, but was a little to slow for postal play, and a pain in the neck setting all the boards up!

Another text by Darry Rubin

How about "Many Worlds Chess"?

In a given player's turn, that player performs all possible moves in parallel.  For each such move, the opposing player performs all possible moves in parallel.  Etc.

If any one of a given turn's possible moves appears real, it is not clear if that is simply a matter of perception or of ultimate reality. Furthermore, it is not clear if the apparent well-ordered time sequence of moves along any given path through the game state space is itself real or merely an illusion.

Many Worlds chess in its present form has the difficulties that, as seen from outside the game, there can be no well-defined win condition.

Please note that Many Worlds chess assumes a flat chess board.  Further work remains to be done to unify this game with another another variant known as Gravitational Chess, which deals with the spatio-temporal structure of the chess board.

written by João Pedro Neto