The game is played on a base-7 hexagonal grid of 127 hexagons, each of which is painted one of 3 colors such that no two hexagons which share a side are of the same color, and all digaonals are of one color.
For aesthetic reasons, 2 of the 3 colors should match the colors of the pieces used, and the third should be chosen to be clearly different from them, though it should not clash.
My boards are desert yellow (matching my "White" wooden pieces), brown (matching my "Black" pieces, which are actually brown), and deep green (many hexagonal boards use red rather than green), giving it a somewhat military but earthy feel. I chose green for the color of the center hex for a feel of neutrality.
There is no castling rule, nor is there any need for one since the King and rooks are very mobile and quite well protected.
Ranks and files are composed of hexes in straight lines which share adjacent sides. The 7th (center) rank is the longest, having 13 files.
A diagonally adjacent hex is a hex which can be reached by first moving to one of the 6 adjacent hexes along a rank or file, then moving to one of the adjacent hexes along a 120-degree angle from the initial movement. This is analogous to the square board diagonal, where the turn is 90 degrees rather than 120 degrees.
Rooks can slide any number of hexes in up to 6 directions along the 3 ranks and files composed of hexes in straght lines which share adjacent sides. Rooks capture in the same manner that they move, and they cannot jump pieces. This is the obvious analogue of the FIDE Chess rook.
Bishops can slide any number of hexes in up to 6 directions along the diagonals of the same color hex so long they are not blocked by any pieces along that diagonal.
Since there are 3 colors of hexes and bishops are confined to a given color, not 2 but 3 bishops are required for a reasonable game of HexaChess. Bishops capture in the same way as they move, and they do not "jump," though it appears to some players that they do since pieces on different color hexes do not block the bishop's movement even if the bishop is surrounded on all adjacent hexes of ranks and files. This is completely analogous to the bishop's movement in FIDE Chess.
As in FIDE Chess, the Queen combines the powers of the rook and the bishop. The Queen may slide any number of hexes in either direction along any of the 3 ranks and files as well as any of the 3 diagonals. Thus the powerful Queen has a total of 12 directions of movement.
The King may move to any of the 12 unoccupied hexes which are one Queen's step away from his current position. This is analogous to FIDE Chess, but it should be noted that diagonal moves allow the king to travel rather quickly on the hexagonal board.
Knights have their characteristic jumping move which can be thought of as one step along a rank or file followed by one diagonal step in either of the two directions away from its starting position. Knights capture in the same way as they move, and they cannot be blocked by pieces of either color since they are jumpers. Knights have their choice of up to 12 hexes to which they can move, and they are able to reach any color hex on any given move. In contrast to FIDE chess, I conjecture that Knights are worth a bit more than bishops in HexaChess.
Each pawn may move forward on a file to either of the 2 unoccupied hexes directly in front of it.
As the Queen combines the powers of the Bishop and the Rook, the Vizir (a.k.a. Cardinal, Pegasus, Archbishop, Centaur) combines the powers of the Bishop and the Knight. Thus, this diagonal piece can change colors via its Knight move.
The move of this piece on a square board dates back to at least 1617 (Pietro Carrera, a priest, chess player, and author from Militello, Sicily), and the name used here comes from "Turkish Great Chess" (1797). On a square board, the Vizir is thought to be worth about a pawn less than the Queen, and on a hexagonal board it may be worth slightly less.
As the Queen combines the powers of the Rook and the Bishop, the War Machine (a.k.a. Marshall, Chancellor, Minister, Champion) combines the powers of the Rook and the Knight.
The move of this piece on a square board dates back to at least 1617 (Pietro Carrera, a priest, chess player, and author from Militello, Sicily), and the name used here comes from "Turkish Great Chess" (1797). The War Machine is equal in power to the Queen on a square board, and it is probably even a pawn stronger on a hexagonal board.
The Duke combines the moving and capturing powers of the Knight and the King, except that, unlike the King, threats to the Duke need not be parried, and the game is not over until his King is checkmated. Thus, the Duke makes a formidable guard for the King, and it has offensive potential as well.
I estimate that the Duke is about equal in power to a Rook.
As far as I know, this is a new piece, but I could certainly be ignorant of its earlier existence. In Turkish Great Chess, however, the Giraffe combined the powers of the Knight and the Queen. My guess is that the Giraffe did not survive to the present since it is simply too powerful a piece to make for a truly interesting game of strategy; for example, the Giraffe can easily checkmate a King with no help from supporting pieces, and no piece can threaten a Giraffe without putting itself within capturing range of the Giraffe.
The following set-up is for White, and Black's pieces are the usual reflection of White's pieces.
All 11 pawns begin on the 5th rank, blocking all access of the enemy's sliding pieces as well as restraining your own sliding pieces until they are freed by pawn movements. All 11 pawns begin protected by the pieces which begin on the 3rd rank.
From left to right, the 3rd rank contains the following 9 pieces: Bishop, Knight, Rook, Queen, Bishop, War Machine, Rook, Knight, Bishop.
The King begins on the central square of the 7-hex 1st rank, with the Vizir directly to his right and the Duke directly to his left.
If the optional pieces are not used, then no pieces begin on the 1st rank, and the King is placed in the War Machine's position; this variant might be called Standard HexaChess, though I prefer Grand HexaChess since it is useful --not to mention more fun and interesting-- to have more mating power given the King's mobility and the large board. The relative proportions of jumpers and sliders is also more balanced in Grand HexaChess.
With Standard HexaChess, optionally, the board could be altered such that the 1st two ranks and the last two ranks are removed from play; this would give the King less mobility, for example. Finally, as with FIDE Chess it is also possible to play Pre-HexaChess, where pieces are placed on the board alternately by each player, or Screen HexaChess, where players secretly deploy their forces however they choose on their side of the seventh rank before movement begins.
Since inventing this game, I discovered that several hexagonal chess variants predate mine. However, it is my opinion that my rules make for a more interesting game which is closer in spirit to FIDE Chess.
In particular, the movement and capturing rules for the pawns in addition to the initial positions of the pieces seem to be a little strange in most other variants.
Comparing the movement of pieces in hexagonal variants to the movement of pieces in FIDE Chess, it is clear that the pieces should have 50% more mobility each since there are 12 directions on hexagonal grids vs. 8 directions on a square grid. This is one of the reasons that I allow pawns to move in 2 directions rather than the one direction allowed, for example, in Glinski's and McCooey's variants. Another good reason for this is that there are insufficient pawns to cover every file; thus it is useful to be able to shift pawns around somewhat.
My pawn captures, like McCooey's, are to adjacent diagonal hexes, but because of McCooey's rotated board when his pawns capture away from the center file they advance two steps closer to promotion. Glinski may have thought of this, and this is why his pawns capture to one of the two adjacent hexes which are one rook's step forward into one of the two adjacent files. However, the result is that when Glinksi's pawns capture toward the center file they make absolutely no progress toward promotion. Moreover, it is odd for that his pawns do not capture along a true diagonal. Also, the total number of hexes influenced by Glinksi's and McCooey's pawns is only 3, the same number influenced on a square grid; my pawns influence 4 hexes, reflecting the need for increased mobility on the hexagonal grid. I do not allow pawns to capture diagonally straight toward the enemy's back rank since this would not preserve the pawn's distance to promotion; this form of capture is allowed, however, in some other hexagonal chess variants which use pawn movements similar to mine.
As for the initial set-up of pieces, my set-up is unusually close to FIDE Chess since all pawns begin protected, and all bishops and rooks are blocked by their own pawns; this is not true for most other hexagonal chess variants.
Also, I use just enough pawns to totally block enemy attacks and block the movements of the pieces behind the pawns; I've seen variants which use twice as many pawns to achieve this purpose, but this seems unnecessary to me. It is not a simple task for White to grab the center hex in my variant since Black can amply defend it if he chooses; though it is not clear to me that the center hex has special significance, this property of fairness shows that White's first-move advantage is not too great. Along these lines, it is important in my set-up that pawns may not be advanced 2 hexes on their first move.
I do not consider this a disadvantage since my guess is that the double pawn advance and en passant captures evolved in square chess to speed the opening and give the pawns a little more mobility since they begin so far apart; this is not necessary in HexaChess. Similarly, I have no castling rule since there is no need for one with the King's and Rook's mobility in HexaChess. Finally, my pawns begin equidistant from the enemy's pawns; again, this is not true of several variants including McCooey's.
Since the movements of most of my pieces are identical to those of McCooey's and Glinkski's variants, I expect that McCooey's endgame database results also hold approximately for my variant. The only differences are that my board is one ring of hexes larger, my Duke may not be in his database, and my pawns move and capture differently than his (though he may not have included pawns in his endgame analyses anyway). McCooey also has results which include the Vizir and the War Machine, though I do not yet know what he has found.
Thus I predict that, as in McCooey's and Glinksi's variants:
King & 2 Knights can checkmate a lone King, though this is very
King & Rook beats King & Knight, with excellent play.
King & Rook beats King & Bishop
King & 2 Bishops cannot checkmate a lone King, except for some very rare positions.
King, Knight & Bishop cannot checkmate a lone King, except for some very rare positions.
King & Queen draws against King & Rook.
King & Rook can checkmate a lone King.
I have also come up with rules for a 3-player hexagonal chess variant which uses a base-9 hexagonal grid, though I have not yet played this game. Players pieces begin on alternating back ranks of the board in the order specified above for White in 2-player Standard HexaChess, and each player's 11 pawns begin on his 3rd rank. Pieces move as in 2-player HexaChess, but pawns only promote upon reaching one of the enemy's back ranks. Play proceeds clockwise, but there are some unique rules to limit the "petty diplomacy" problem of such a 3-player perfect information strategy game. Below, I will state these rules in general for an n-player chess variant since they can easily be adapted to improve other such games. For 3-Player HexaChess, n=3 and k=2 in what follows.
Finally, I have come up with a sensible board design for a 4-player hexagonal chess variant, though it's difficult to describe the shape in words. It uses a subset of the hexes of a base-10 hexagonal grid. The pieces move as in my 2- and 3-player variants, but now two players are allied against the other 2 players, and the game is won by the team who first checkmates one of the players of the opposing team.