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Does the same rule apply as in your 5x5x5 game, that checkmating one king wins the game even if the other is safe?
Checkmating one King wins the game. The idea is to make wins more likely and draws less likely. Note that a King fork is considered 'mate' if the forking piece is immune from capture. This is because you only have one move and need to protect BOTH Kings.
That is a sensible use of two Kings. In fact a King fork would be checkmate even if both Kings CAN escape check (as long as the checking piece cannot be captured), as whichever one is moved leaves the other in check.
One more question. Am I right in understanding that the triagonal move is not used in this game?
The King and Queen may use the triagonal (3D diagonal) in this variant.
This game is fun! You can take all of the opponents pieces-including the king-and then the applet cannot move any black pieces, and you cannot move since it is blacks turn. Still, it must be hard to make an applet so I'm not going to say it's bad (and its seriously isn't). Also, I moved my rook infront of the king, checking it, and the King moved diagonally into another check instead of forward and taking my rook :). It's fun to mess around with the computer!
Where can I find the applet you are referring to?
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