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'For each piece there must exist a 8x8 square which includes at least one piece of the opponent.' ==> which allows a White Rook and a Black Rook to take turns moving away from the main body of pieces, causing the play-area to increase without limit.

I would respectfully disagree that the board is infinite. By the rules, there can be at most 16 pairs of pieces, each in its own 8x8 playing area. If I'm doing the numbers right, that's 1024 squares maximum that can be used, although it is true that these 16 areas can be totally disconnected. 

Well, that seems reasonable, but let's re-think this. Assume the pair of pieces are diagonally separated by 6 empty squares. The opponent piece can move anywhere up to 7 squares away from the friendly one, giving a 15x15 square centered around the friendly piece. That gives 15x15 squares x16 pairs, or 3600 squares max potential board squares. If we don't know who's turn it is, that 15x15 square is duplicated over every piece, friendly and enemy. Doubling 3600 gives 7200, but there's overlap - 64 squares per pair, if I understand this correctly - subtracting 1024 squares from the total, leaving 6176 squares as the maximum potential size of the 'infinite' board. I think the 'actual' maximum size would be 3200, given that we know who moves next. That would occupy about a third of a 100x100 board.

This of course ignores any concept of chess strategy.

'A paradox, a paradox, a most ingenious paradox!'

My statement should be rephrased to read: the smallest rectangular board containing all of the pieces may increase beyond any limit. This can be demonstrated by two pairs of (opposite color) Rooks moving off, one pair in a vertical direction and one pair in a horizontal direction.

Joe Joyce's statement should be rephrased to read: any White move takes place in a playing area consisting of no more than 32 square areas (15x15) centered on the Black pieces. Black's reply must take place in a playing area consisting of no more than 32 square areas (15x15) centered on the White pieces. These 15x15 areas will overlap the previous areas centered on the White pieces.

So the game may be regarded as taking place in (no more than) 32 shifting patches of sunlight scattered across a limitless dark plain. Note that a piece may move thousands of squares in a single move, provided it starts and ends its move within the required distance from an opponent's piece.

David:
    'shifting patches of sunlight scattered across a limitless dark plain'

I never imagined when I first submitted my rules that such a poetic description as David's existed. It is such an appropo description! It seems almost the essense of the game. There's certainly untapped depth to this game. 

Thanks so much to Joe and David's interest and conversation. It adds so much to the mystic of this game for me.

Since each piece 'must' exist within an 8x8 area of an opposing piece, what happens when a piece is stranded? Is the player forced to move that piece back into an appropriate position? And if unable, what then occurs with that piece?

I advocate that a player must immediately correct any inappropriate position, or forfeit the offending piece(s). If multiple pieces are 'stranded', a player would only be able to recover one. If in a checking position, the player could be forced to abandon such to avoid capture of their King. And a King in such a position would be considered in check.

As the number of pieces are reduced during play, the potential size of the playing field would likewise reduce. It still may be quite difficult to promote until well into the end-game.

All in all, a great idea. Definite a brain-squeezer.

Re:Larry,

   There are two solutions:
    
   1. Capture by stranding, where after every move, any opponent pieces that are stranded are considered captured.

   2. Stranded disapear, after every move, any friendly stranded pieced are removed.


Larry, your idea most align with the second choice. I think both are viable, the first being simpler to understand, but the second give the player a 'second chance' of a sort.

The most featured Angel Problems are situated on infinite chessboard: https://en.wikipedia.org/wiki/Angel_problem.

Considering Ji's rule #3 that pieces are stranded if there is no "8x8 square which includes at least one piece of the opponent" then there may be positions where many pieces are lost in one move. For example, if White's pieces (quantity "n-1") are in a legal position because they are within 8 squares of a Black bishop, and if the bishop flees to a legal position by "1" White piece, then White loses all except the "1" piece. I'm not sure if such a position would be reached in a well-played game, but it is interesting that multiple pieces can be lost in a single move. Has anyone played this, or thought it out in more detail?