HONEYCOMB CHESS
Charles Gilman
This variant uses a board of hexprism cells. Each of the 8 ranks, numbered 1 to 8, is a 15cell hex board laid vertically on one edge. This gives 15 files, running horizontally at right angles to the ranks, and lettered a bc def ghij klmno. Files a, e, g, j, and m are coloured red on odd ranks and green on even ones. Files b, f, h, k, and n are yellow and purple. Files c, d, i, l, and o are blue and orange. Colouring serves as a guide to moves. Unfortunately the diagram cannot show these colours behind pieces, but they can be extrapolated from odd/even empty ranks. Two FIDE sets are used, minus a King and Pawn aside.
The variant's origin is perhaps surprising: to devise a 3d game which, like Quadlevel, preserves the 2d rule that diagonal and oblique moves always change rank, but without the artificiality and loss of strength. My first thought was to add the samerank diagonal to the Rook and the Unicorn move to the Bishop, but this still seemed artificial. Then I hit on the idea of making the ranks hex boards to increase sideways orthogonals, forward diagonals, and backward diagonals from four to six each. As orthogonals are the only FIDE directions on a hex board, the rule about changing rank FOLLOWS AUTOMATICALLY. Note that I interpret orthogonal directions as passing through cell boundaries at right angles, not necessarily being at right angles to each other.
The Rook moves along either files or any orthogonal within ranks. As on square and cubiccell boards it cannot change file and rank in the same move. Along a file it alternates between the file's two opposite colours. Within a rank it cycles through the rank's three colours.
The Bishop moves on squareboard diagonals, both obvious ones such as k1l2m3n4o5 and those recognisable by rotation such as a1c2f3j4o5. As hex boards simply DO NOT HAVE this kind of direction the bar on a samerank move FOLLOWS AUTOMATICALLY. Bishops cycle through all six colours in the obvious spectral order and can reach the entire board, not by any extra kind of move but because THIS IS HOW A STANDARD, UNENHANCED BISHOP BEHAVES ON THIS KIND OF BOARD! Note how the sample diagonals have one shared cell, o5, and one mutually orthogonally adjacent pair, j4 and n4.
The Queen combines the Rook and Bishop moves, and the King moves one cell in any of the same twenty directions  forward, backward, six sideways, six forward diagonal, six backward diagonal. As in most of my 3d and/or 4player variants, there is no castling.
The Knight makes a 2:1 leap, where the 2 and the 1 are AT RIGHT ANGLES. As on square and cubic boards, this is the same as any root5 leap. Come to think of it the two are synonymous on a hex 2d board, which has none of either! The Knight always moves to a cell of a different, but not opposite, colour and  again WITH NO EXTRA KIND OF MOVE  can lose the move in 5 steps. Again the bar on a samerank move is AUTOMATIC.
The Pawn, when not capturing, moves directly forward one cell, or two from its array position. It captures by moving diagonally forward one cell, giving up to six capturing destinations. As Pawns have no samerank move even on a cubic board, it is unsurprising that they have none here. En passant and promotion are exactly as in FIDE Chess.

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For further hexprism variants it is worth noting the board's effect on other pieces AS DEFINED FOR A SQUARECELL BOARD. The Wazir becomes  and the Gnu and Goldgeneral remain  able to triangulate. The latter's destinations (14) still outnumber by one those of the Silvergeneral (13), which as it always moves an odd number of ranks still cannot lose the move. For the same reason nor can the Ferz and Camel, which like the Bishop become unbound, but the Zebra can. The Generals could appear in a 3d Shogi close to the 2d original (array suggestions are welcome), and the oblique pieces in a 3d Wildebeest Chess (Wildebeest being a synonym for Gnu) which I term WILDHONEY CHESS, first rank R NC BKB CQGN RNBCR. All in all this geometry is something of a Holy Grail for 3d games with simplydefined pieces.
A piece making use of a Hex geometry is the FINCH, first introduced in my 2d variant Anglojewish Chess and piece article Mighty Like a Rose. Within a rank it takes a curved path of up to four steps, turning 60° at each intermediate cell, and alternating between cells of the two colours OTHER than that of the cell that it encircles. Between ranks it moves only as a Wazir as there is no 60° turn to make. This piece, together with the existing Rook, suggests to me a subvariant inspired by Keats' poem (and perhaps named) LA BELLE DAME SANS MERCI in which a Knight feels unable to tear himself away from a desolate lateyear scene where "...no birds sing". The Pawns start on the third rank and lose their double move, Finches replace the Knights on the first rank, and the second rank is filled with Knights (perhaps represented by most of a third set of smaller build). Knights can move only if their army includes three Rooks, three Finches, or one of each. Only Queens can capture Rooks and Finches, and only Bishops can capture Queens. This may affect the choice of promotees.
If the Finch proves insufficiently powerful in this subvariant it could be replaced by the GOLDFINCH, a Goldgeneral similarly enhanced to continue its withinrank moves on a curved path. This name fits the theme equally well as a bird name. I decided against using Flamingoes as they only work well on a board of large minimum dimension and the name is a little exotic for the poem's setting.
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Two types of radial not yet mentioned are those such as a1e1m1 and those such as a1e2m3, as using them detracts from a simple extrapolation of squareboard pieces. One approach is a DOUBLESTRENGTH HONEYCOMB CHESS adding them to orthogonals and diagonals respectively, to preserve rankchange restrictions. Rooks become DUCHESSES, Bishops PONTIFFS, and the King and Queen a WORLDRULER and WORLDRIDER. Pawns become PAWNBREAKERS, not to be confused with the PAWNBROKERS common (in every sense!) on cubic boards but impossible on this one. Knights gain rankchanging root7 and root13 leaps (but NOT samerank ones, which also exist) to become BREEDERS (i.e. those who breed horses). A really radical approach is DELAYEDSTRENGTH HONEYCOMB CHESS, starting with FIDE pieces promotable to their, or in the case of Pawns other pieces', doublestrength versions.
Finally there could be doublestrength forms of offshoots. The Wazir becomes a DUKE and the Ferz a LANDLORD. The Camel's logical extension the SHEIKH is harder to describe as it has (among others) some but not all root28 leaps. Goldgenerals and Silvergenerals gain 12 extra destination cells each to become DUKEGENERALS and LANDLGENERALS. Shogistyle linepiece promotion involves adding the Worldruler's move and the WORLD prefix. Finches become the JEWISH MOTHERS of AJ Chess, strong within ranks but weak between them, and Goldfinches DUKEMOTHERS, with many forwardonerank moves as well.
Written by Charles Gilman.