3-player Honeycomb Chess

2d Hex variants can often be extrapolated to 3 players. My own hex variants for 2 or 3 players on the same board (and page) include first Anglojewish Chess, from which the name of the Finch below derives, and more recently Hexgi and AltOrth Hex Chess. As this depends on players being in 3 of the 6 directions away from the centre, applying it to 3 dimensions means rotating the board. This replaces the change-of-rank rule for Bishops and Knights with a change-of-level rule, so there is still some restriction on these. This change now reflects the 2d change-of-file rule, which as a bonus restores the ratio of capturing to noncapturing Pawn moves to that of FIDE Chess.

Since posting this variant I have gone on to use a smaller board with this geometry (but for only 2 players) for Lengthleaper Hex Chess, and larger ones (with 4 players) for the second and third variants in my Armies of Faith series.

Setup

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In the basic variant corner columns contain the pieces RQKQR, like a 5-file version of the back ranks of John Groeneman's Half Chess, with Queens on their own colour. The two columns adjacent to each corner have the same player's BNFNB, where F is a Finch physically represented by an inverted Rook. The three remaining columns adjoining each of those pairs are filled with Pawns. The third army can be represented by both armies of a third set distinguishable from the two identical sets.

This diagram applies to levels A and E:

This diagram applies to levels B and D:

This diagram applies to level C:

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In 3-player Wildhoney Chess corner columns become (from top to bottom) RQKGR, where G is a Gnu. The adjacent columns become BCNCB, where C is a Camel. For this version it may be best to use three easily distinguishable sets, one for each army, with e.g. White pieces representing KQBBRP and Black ones representing -GCCNP.

This diagram applies to levels A and E:

This diagram applies to level B:

This diagram applies to level C:

This diagram applies to level D:

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Pawnless subvariants of both are also possible for the same reason as Half Chess, the shortness of the second-rank pieces' moves on the board. These would not however save on physical pieces, as they would still require the back ranks of three FIDE sets.

Pieces

	The PAWN always moves to an adjacent column. Red Pawns increment their column by 1 or 11, Yellow Pawns increment by 10 or decrement by 1, and Blue Pawns decrement by 10 or 11. Vertically, Pawns move to an adjacent level when capturing and otherwise remain on the same level, corresponding to how 2d Pawns move relative to files. Pawns have 2 noncapturing and 4 capturing moves on inner levels and 2 of each on Rook levels, twice the FIDE numbers of moves. Each player starts with 15 Pawns.
	The ROOK moves any distance through empty intermediate cells along Hex orthogonals within levels, cycling through the level's three colours, or along columns between levels, alternating between the column's two opposite colours. These add up to 8 directions, twice the FIDE number, falling to 7 from a Rook level, 6 from a rectangular face, 5 from the edge of a Rook level, 4 from a corner column, and 3 from a vertex. Each player starts with 2 Rooks.
	The BISHOP moves any distance through empty intermediate cells along square diagonals, always changing level and column at the same rate, as it does file and rank in 2d, and cycling through all six colours in the obvious spectral order. There are up to 12 such directions, 3 times the FIDE number, falling to 8 from a rectangular face, 6 from a Rook level, 4 from an edge, and 2 from a vertex. It cannot start and end a move on the same column or level, and can reach the entire board. Each player starts with 4 Bishops.
	The QUEEN combines the Rook and Bishop moves. This gives up to 20 directions, falling to 14 from a rectangular face, 13 from a Rook level, 9 from the edge of a Rook level, 8 from a corner column, and 5 from a vertex. Each player starts with 2 Queens in the basic variant and 1 in the Wildhoney one.
	The KING moves in the same directions as the Queen but only by one step per move. Each player starts with 1 King, and must hang on to it!
	The KNIGHT moves either 2 columns (in the same direction) and 1 level or 1 column and 2 levels. From a centre cell (3c5, 3c6, 4c6) it has 24 destinations, but this number drops rapidly as it approaches a face. It cannot be blocked. It cannot start and end a move on the same column or level, although unlike in FIDE Chess it can lose the move in 5 moves. It always moves to a cell of a different, but not opposite, colour. Each player starts with 4 Knights in the basic version and 2 in the Wildhoney version.

The FINCH is a curved Rook, which makes a 60° turn left or right between each step of its move, but not mixing the two in one move. It is blocked by intervening pieces on the circuit, though not on the cell being encircled. For example the Red Finches can make their first move along the circuit 1c2-1c3-2c4-3c4-3c3-2c2 once the 1c3 and 3c3 Pawns are clear of it, even if the 2c3 Pawn is unmoved. The Finch alternates between cells of the two colours other than that of the cell being encircled. As there are no orthogonals at 60° to columns, Finches moving between levels can only do so one step straight up or down. Each player starts with 2 Finches in the basic variant.

The CAMEL moves either 3 columns (in the same direction) and 1 level or 1 column and 3 levels. On a large enough hex-prism board it too would have 24 destinations but on this one the best that it can manage is 18, from the middle of a Queen level (3b5, 3b6, 4b6, 3d5, 3d6, 4d6). It cannot be blocked. It cannot start and end a move on the same column or level, or an even number of levels apart. It always moves to a cell of an adjacent or opposite colour. Both the last two characteristics reflect its inability to lose the move, although like the Bishop it does lose its 2d and cubic colourbinding. Each player starts with 4 Camels in the Wildhoney version.

The GNU combines the Knight and Camel moves. On a large enough board it would have 48 destinations but on this one the best that it can manage is 36 (24 Knight + 12 Camel) from the centre cells. It cannot be blocked. It can triangulate in a Camel move and two Knight ones. Each player starts with 1 Gnu in the Wildhoney version.

Rules

Pawns that have no further move are promoted to another capturable piece from the relevant variant's array.

A player is checkmated when their King is threatened by the player about to move. The first player checkmated loses and their pieces are recruited by the checkmating player. Whichever remaining player checkmates the other wins.

Notes

In these variants the hex diagonal and hybrid diagonal are not used. As with the 2-player version, double-strength subvariants are also possible in which hex diagonals are treated the same as orthogonals, hybrid diagonals the same as standard diagonals, and obliques extrapolated from there. This version has the same effect on most pieces as it does for two players except that Pawns gain the hex-diagonal forward noncapturing move, as well as capturing moves along the two forward hybrid diagonals above and below it, to become Pawnouts. Note that the double-strength counterpart of the Finch is termed a COHEN as of February 2008.