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The red hexes with fat borders shown on the board are the castles of the players. The object of the game is to move a piece into the opponents' castle, thereby capturing it.
There are three types of pieces: Pawns, rooks and bishops.
A pawn (P) can move to a neighbouring hex either up-left or up-right (up being away from the home castle). If a pawn enters one of the two red hexes in the penultimate row that do not have any hexes above them (neither up-left nor up-right), they may move one hex sideways. Unlike in normal chess, pawns capture the same way that they move, nor may they move two hexes in their first move (and hence there are no en-passant captures). There is no promotion of pawns.
A few examples of pawn moves are shown below. White moves up the board and black down the board.
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A rook (R) can move any number of hexes in a straight line in any of the six primary directions (left, right, up-left, up-right, down-left or down-right).
An example of rook moves is shown below.
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A bishop (B) can move to any of the six nearest hexes in the secondary directions (along the edges separating two neighbouring hexes). Hence, they always stay on the same colour. Bishops are not blocked by pieces in the hexes they move alongside. They are thus somewhat of a cross between bishops and knights from normal chess.
A few examples of bishop moves are shown below.
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The initial set-up is shown below.
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A player that has no pieces left that can legally occupy the opponents' castle (i.e, only blue and green bishops) loses.
A position may not be repeated. A move that would repeat a previous position is hence illegal.
The no-repeat rule above means that no game can end in a stalemate. However, I don't think it will happen very often that games are lost by the inability to make a new position. After all, pawn moves and captures are irreversible.
I have concentrated on making the game simple, which is one reason for the small number of different pieces and the relatively simple movement rules. The only exception is the special case for pawns in the red hexes of the penultimate lines, which I introduced to avoid pawns being stuck there.
Making the object of the game the capture of a special hex rather than a special piece was prompted by the shape of the board. Initially I did have a king piece starting in the castle, but I wanted to have a bishop of each colour and the castle seemed like a logical place to put it. And I don't think a king will add anything of value to the game.
The red bishop is much more valuable than the green or the blue. Not only is it capable of entering the opponents castle, there are also has three more red hexes than blue or green.
If playing with pieces from a single normal chess set, you will be one bishop short. In view of the higher value of the red bishop, I suggest you use the queen piece for that.