Trios Hex Chess

It is notable that all three "standard" Hex Chesses - Wellisch, Glinski, and McCooey - use 91-cell boards of the same shape. Wellisch designed his game specifically for 3 players, each with 15 pieces - covering half the board rounded down (like Shogi) but clearly separated. Glinski designed his game for 2 players, each with 18 pieces, covering 2/5 of the board rounded down which is a little low compared to FIDE or Shogi. Conversely a 3-player version would cover 3/5 which is a little high, with camps rather too close together for comfort. McCooey designed his game primarily for 2 but more easily extrapolated than Glinski's to 3, in both cases each with 16 pieces. 2 players cover less than even Glinski's 2/5, but 3 players cover a respectable figure just over Wellisch's half the board. So, Wellisch and 3-player McCooey seem good-sized 3 player games for that very basic size analysis. What about 2 players?

Glinski's array had an obvious starting point, an empty swathe of seven cells per camp, although I decided to use Brokers as they seem a more consistent Pawn analogue than Migrants. The next question was how many cells to fill and with what. One way, using Curved or Crooked or Bent linepieces, I have already tried in 4 Linepiece Hex Chess, but I wondered what could be done with pieces familiar from the "standard" hex games, and noted that those games had not increased pieces evenly. If the sides to a cell increase from 4 to 6, it might be sensible to have 3 of all analogues to 2-aside pieces rather than just the colourbound one. These trios of pieces are what give this page its name. It then dawned on me that I might be trespassing too closely on ground covered by Graeme Neatham's Fool's Hexagonal Chess, so I checked that page and hove a sigh of relief that I wasn't. His aim was to get a FIDE-like hex game, mine here is to get only a FIDE-like coverage but specifically on a "standard" hex board. I had not been a fool rushing in! It did however give me the idea of going up to 12 front-rank pieces, also half as many again as FIDE Chess.

This raised army size to 23 pieces, covering half the board rounded up, but I had already been considering another idea, adding one or more new piece types. As new compounds of existing pieces did not appeal to me - Unicorn compounds would be a bit too strong to fit in - I thought of having one further simple piece type, another oblique leaper in the style of Modern Kamil - though no compound of both leapers in the style of Wildebeest Chess as I have yet to devise names for such pieces on a hex board. One influence on this was the idea of extrapolating perimeters (see Squirrel). The nth perimeter for square cells comprises the cells reachable in n King steps (any mixture of Wazir and Ferz ones) but not in n-1, and forms a nice square outline. Extrapolating to the steps of the Grandduke for a hex board gives a curious and unintuitive star shape, so I looked to what I term Chinese Perimeters (CPs) defined in terms only of the Wazir (or its royally-restricted version the Chinese General) moves common to both geometries. For square cells this still gives a square outline, albeit turned diagonally, and for hexes a hexagonal one.

Trios Hex versions of Anglojewish Chess and Altorth Chess appear under those variants' respective pages, to save covering their pieces here.

Setup

Aurochs Trios Chess:

Pieces

	The GRANDDUKE moves one step along any orthogonal or hex diagonal and must be kept out of Check. It is the G/McC analogue to the FIDE and Shogi King, under whose page its hex move may be found.
	The DUCHESS moves any distance through empty intermediate cells along any orthogonal or hex diagonal. It is the G/McC analogue to the FIDE Queen, under whose page its hex move may be found, although in the Wellisch interpretation it would be a Marshrider (Rook+Nightrider) analogue.
	The ROOK moves any distance through empty intermediate cells along any orthogonal.
	The UNICORN moves any distance through empty intermediate cells along any hex diagonal. It is the G/McC analogue to the FIDE Bishop, under whose page its hex move may be found, although in the Wellisch interpretation it would be a Nightrider analogue.
	The SENNIGHT makes any root-7 hex leap. It is the G/McC analogue to the FIDE Knight, under whose page its move may be found. It is not used in Wellisch Chess, in whose interpretation it would be an analogue to both Camel and Zebra (see notes).
	The AUROCHS makes any root-13 pure-hex leap. It is not used in "standard" hex games, but in the G/McC interpretation it would be, in some though not all senses, a Camel analogue and in the Wellisch interpretation it would be an analogue to both Giraffe and Antelope. Yet it shares its leap length with the Zebra, to which it is no-one's analogue.
	The BROKER moves one step along the straight-forward orthogonal, except when capturing which it does along either forward hex diagonal (in contrast to the Migrant which captures along the half-forward orthogonals). It is the McCooey analogue to the FIDE Pawn, under whose page its move may be found.

Rules

Brokers ending a move on the far edge must be promoted to some stronger capturable array piece.

Check, Checkmate, and Stalemate are as usual.

Notes

You will notice that in the Pieces section I described the Aurochs as a Camel analogue "in some though not all senses". The senses in which the analogy works are that it is (a) the next shortest-length leaper and (b) the piece that turns only 60° rather than the Camel's 90° between 3:1 coordinates as the Sennight turns only 60° rather than the Knight's 90° between 2:1 ones. In the Wellisch analogy the turn is 120°, with the result that directions "double up".

The key to this can be found on the cubic 3d board. Consider the direction with coprime coordinates x+y:x:y, where y is the smallest. The Square of Leap Length (SOLL) for the piece with those coordinates is (x+y)²+x²+y² = (x²+y²+2xy)+x²+y² = 2x²+2y²+2xy = 2(x²+y²+xy). The SOLL of the coprime x:y leaper's G/McC analogue is x²+y²+xy. It can quickly be seen that the hex piece's SOLL is the sum of either three odd numbers or one odd and two even, and is therefore odd itself, and that the cubic piece's SOLL, being twice that odd number, divides by 4 with remainder 2. The cubic piece's leap can also be seen as two Bishop moves: x:x:0 followed by y:0:y, in other words x Ferz steps followed by a 60° turn and then y Ferz steps, corresponding to the hex piece in Wazir steps. As well as G/McC analogue to the x:y leaper, the hex piece is Wellisch analogue to the x+y:x and x+y:y leapers through seeing the cubic piece's move as x+y:0:x+y followed by 0:x:-x and also x+y:x+y:0 followed by 0:-y:y. The pattern is as follows, with the SOLLs is brackets:

x y Cubic piece       Hex piece   is G/McC      is Wellisch
                                  analogue to   analogue to
- - ----------------- ----------- ------------- ---------------------------
1 0 1:1:0 Ferz    (2) Wazir   (1) 1:0 Wazir (1) 1:0 Wazir (1), 1:1 Ferz (2)
1 1 2:1:1 Sexton  (6) Viceroy (3) 1:1 Ferz  (2) 2:1 Knight (5) twice over
2 1 3:2:1 Fortnight   Sennight    2:1 Knight    3:1 Camel (10),
                 (14)         (7)           (5)              3:2 Zebra (13)
3 1 4:3:1 Arbez       Aurochs     3:1 Camel     4:1 Giraffe (17),
                 (26)        (13)          (10)           4:3 Antelope (25)
3 2 5:3:2 Sustainer   Student     3:2 Zebra     5:2 Satyr (29),
                 (38)        (19)          (13)              5:3 Gimel (34)
4 1 5:4:1 Votary      Overscore   4:1 Giraffe   5:1 Zemel (26),
                 (42)        (21)          (17)             5:4 Rector (41)
4 3 7:4:3 Ognimalf    Goose  (37) 4:3 Antelope  7:3 Samel (58), 7:4 Ox (65)
                 (74)                      (25)               
5 1 6:5:1 Endower     Newlywed    5:1 Zemel     6:1 Flamingo (37),
                 (62)        (31)          (26)             6:5 Parson (61)
5 2 7:5:2 Bandicoot   Barnowl     5:2 Satyr     7:2 Stork (53),
                 (78)        (39)          (29)              7:5 Famel (74)
5 3 8:5:3 Hemel       Heptagram   5:3 Gimel     8:3 Huckster (73),
                 (98)        (49)          (34)           8:5 Agronome (89)
6 1 7:6:1 Pipistrelle Prizemouse  6:1 Flamingo  7:1 Namel (50),
                 (86)        (43)          (37)             7:6 Curate (85)
7 1 8:7:1 Beggar      Bettong     7:1 Namel     8:1 Bittern (65),
                (114)        (57)          (50)            8:7 Deacon (113)

Even here the G/McC analogy can be seen breaking down, as the SOLL length order does not match between square and hex pieces. However a more serious breakdown is the third sense of the Camel in which the Aurochs analogy fails. This is the Camel as being the Knight's colourbound dual - the piece that is to Ferz moves what the Knight is to Wazir ones. Well the piece that is to the Viceroy (Ferz analogue) move what the Sennight (Knight analogue) move is to the Wazir move is the piece with SOLL 3x7, the Overscore. This is the piece that is bound to one is three cells, as the Camel is to one in two. The Overscore however was rather too unwieldy for so small a board, so I settled for the Aurochs. The Aurochs is unbound (which makes for easier placement in the array) and actually has its own bound hex dual in the form of the Barnowl, whose SOLL is 3x13.

Finally, using the Aurochs fills the 4th CP, as using the Sennight does the 3rd. The Overscore can fill a CP, the 5th, only together with the Student. Here is a list of leapers required to fill each perimeter, assuming the existence of the Rook and either Bishop or Unicorn.

Peri- Standard,                    Chinese,         Chinese,
meter square cells                 square cells     hex cells
----- ---------------------------- ---------------- ------------------
1st   none needed                  none needed      none needed
2nd   Knight                       none needed      none needed
3rd   Camel, Zebra                 Knight           Sennight
4th   Giraffe, Charolais, Antelope Camel            Aurochs
5th   Zemel, Satyr, Gimel, Rector  Zebra, Giraffe   Student, Overscore
6th   Flamingo, Crane, Chamois,    Zemel, Charolais Srene, Newlywed
                 Zherolais, Parson
7th   Namel, Stork, Samel,         Flamingo,        Goose, Barnowl,
                 Ox, Famel, Curate Satyr, Antelope          Prizemouse

Note that for any two their compound can be substituted - Bison for Camel and Zebra, Gamewarden for Zebra and Giraffe, et cetera. In AltOrth Hex Chess these are analogues only for the compounds, with asymmetric pieces representing the components.