First of all we keep Demonchy's geometrie. But forget, that we play chess with white or black. Now we are both at the same time (grey in a way). The parity (the P in CP-invariance) asks us to make the same move agains both opponents every time, against the right opponent with the white chessmen, against the left one with the black ones. If we move b.e. on the right board the white king's knight from g1 to f3, so we have to move his CP-invariant partner, of course the black king's knight from g8 to f6 on the left board at the same time.
A little experiment of thoughts may explain this state of affairs: Imagine, while you are playing chess with your partner on the top of the board, on the bottom happens the same game with turned colours, where black is the first to move. It is enough to know one of both positions, because one is the redundant reflected image of the other one only with changed colours. Both games content the same information. Therefore we have to look only for one of them. (Which one is all the same, but of cause the FIDE has ruled, how to play Chess.) You see, the 2-Chess-game is dualistic to itself.
Or imagine a chess-cassette, folded up and put upright, with magnetical pieces in the starting position. On the right side, there is the white half of the board, on the left is the black one, the hinges are turned away from you. Every chessman attracts his CP-invariant partner on the other side of the cassette magnetically. If now the king's knight moves (or every other piece), so its CP-partner on the other side moves too, thanks to the magnetic force. You see, I play white and black simultaneously (Do you imagine a Solitaire-Chess?!).
If there are three chess-cassettes of this kind, you may put them upright in an angel of 120 degrees to each other. So you nearly get the threehanded chess-device. But one after another... first we look again on the Demonchy-boards.
Now we have to dicuss the charge of colours (the C in CP-invariance). We distinguish in white and black in the twohanded chessgame. Every player takes one of these two colours. Without the information, who's on the move, it is impossible to declare, how the game will be continued. The conclusion suggests itself: We need 3 colours for 3 players in the threehanded chessgame, and we have to define an order of moves. Simply we take red, yellow and blue for colours and order (you may take others, if you think that Goethe's colour-theory is too conservative). Every player gets both his sets of chessmen, one the reds, another the yellows, the third one the blues. We colour the boards corresponding to the pieces. The first board gets red and yellow squares, the second one yellow and blue squares and the third one blue and red squares. Each outer corner of every niche carries the colour of the player on this place, the inner corners carry the colour of the opponent on this board.
We still miss a usable coordinate-system for the notation of games. On account of the redundance, we rename the rows 8, 7, 6, 5 in 1, 2, 3, 4, because they show (P-invariant as they are) the same position. Both of the red halfs of the chessboards get a roman I, the yellow ones a II, and the blue ones a III. Now every square consists of three coordinates, of course, halfboard-colour (I, II, III), file (a to h) and row (1 to 4), and it exists two times. To intimate the traditinal distinction of white and black, every right board gets a white border, every left one a black border.
Now everything is alright! Red queen - red square (it is the same queen, it is only presented two times!), yellow queen - yellow square, blue queen - blue square, each one on the d-files. The corner-squares a1 carry the colours of the opponents, the corner-squares h1 the player's colour. The CP-invariance keeps maintained, but instead of playing chess in black and white, now we play chess in colours. I gave this game-arrangement the working-title STAR-III-COLOR-CHESS-boards.
Red begins, then it turns against clockwise (in mathematicly positive spin), second yellow, third blue and so on (red, yellow, blue...), until the goal of the game is reached (checkmate one
opponent king). Each third part of a move has to be done with the right hand in a white position and with the left hand in the CP-invariant black position simulteaneously. Once again: It is only one move, it is solely presented two times! (It goes without saying, that citizens with a physical handycap may execute both modes of a move after another.)
Some more introductions:
If a player moves one of his pieces on an opponent halfboard, it disappears on one board and reappears on the other, because it doesn't double itself by crossing the middle (but attention: this is what the triangel-variant is working with). The coordinate-system will help you to find the right square(s). The unconcerned third player now sees this piece from its back, that means it can't get at him, because it always moves away from him! Moves from one opponent to the other opponent are strictly forbidden, of course, instead of breaking the symmetrie. Bishops would change their square-colour by this. (Note: one bishop moves always on the player's colour, the other one on the colours of the opponents.) From your own halfboard the chessmen act against both your opponents. If you have moved it over the middle of the board, the moved piece acts only against the attacked player and in case back to your own board, but not against the opponent on the other halfboard.
If a pawn crosses the middle, it can chose about going to the one or the other opponent. This conclusion is irrevocable, because a pawn never marches back. 2 opponent pawns on the halfboard of the third one may capture each other with a diagonal step from the back. That never happened to them in 2-chess, but in this environment it seems to be absolutely logical. In the chapter about the rules of the game, I will return once again to the mysterious attributes of STAR-III-COLOR-CHESS.
If you like so, the upper described Star-Variant of III-COLOR is an OR-combination of three boards. If you connect the three boards with an AND-combination, then you get the dualistic principle, the Triangle-Variant of III-COLOR. Imagine, you would superpose the positions of your opponents and make the same move against both of them every time. Every move must be legal on both boards, excepted for the pawns. If a pawn takes an other piece, this move may (not has to) be a capturing on one board only, on the other board the same pawn has to be moved as it has captured, too. This rule is a consequence from the pawns' peculiarity, to capture in an other way as they move. Without this generous interpretation of chess-rules, no traditional opening would be possible. If a piece is captured, so its phantom on the other board is captured, too, of course it is only one chessman.
The triangle-coordination-system for notation is ambiguous, because every player is "white and black" at the same time. So we notate only the white moves and take the normal row-coordinates 1 to 8, as we do in 2-Chess.
The names Star and Triangle for the III-COLOR-variants are taken from the electrical technics, better said from the theorie of three-phases-amplitude-current. Here we have a nice example of analogies between two phenomenons, that don't really have other things in common.
Such a game of III-COLOR-CHESS takes one and a half time as a normal chessgame (see also: the III-COLOR-CHESS-Clock). If you play briskly, you may plan 4 to 6 hours for a game. That's a lot of time in this fast times. A useful alternative is playing long-distance III-COLOR-CHESS per e-mail or phone with a third part of a move every day.
There are many other chessboard-games that work very well on this environment: b.e. R. Abbots "Crossover", "Go Moku" (5 in a row), "Oldgerman Checkers" or "Turkish Checkers" (but neither "Reversi" nor "French Checkers" - they are not CP-invariant). Bobby Fisher's "Random-Chess" or "960" and most of normal chessvariants do not make any problem for this game-environment.