Fairy chess piece

Classification

Fairy chess pieces usually fall into one of three classes, although some are hybrids. Compound pieces combine the movement powers of two or more different pieces. Some chess-problem solving programs, such as WinChloe, recognize hundreds of different fairy pieces.

Leapers

An (m,n)-leaper is a piece that moves by a fixed type of vector between its start square and its arrival square. One of the coordinates of the vector 'start square – arrival square' must have an absolute value equal to m and the other one an absolute value equal to n. A leaper moves in the same way whether or not it captures, the taken unit being on the arrival square. For instance, the knight is the (1,2)-leaper.

The leaper's move cannot be blocked; it "leaps" over any intervening pieces. Leapers are not able to create pins, but are effective forking pieces. The check of a leaper cannot be parried by interposing. All orthodox chessmen except the pawn are either leapers or riders, although the rook does 'hop' over its own king when it castles.

It is convenient to classify all fixed-distance moves as leaps, including (1,0) and (1,1) moves to adjacent squares, because that allows all normal chess moves to be placed in two categories (leapers and riders) without the need to create a third category to describe the king and pawn.

In shatranj, a Persian forerunner to chess, the predecessors of the bishop and queen were leapers: the alfil is a (2,2)-leaper (moving two squares diagonally in any direction), and the ferz a (1,1)-leaper (moving one square diagonally in any direction). The wazir is a (1,0)-leaper (an "orthogonal" one-square leaper). The king of standard chess combines the ferz and wazir. The dabbaba is a (2,0)-leaper. The alibaba combines the dabbaba and alfil, while the squirrel can move to any square 2 units away (combining the knight and alibaba).

The 'level-3' leapers are the threeleaper (0,3), camel (1,3), zebra (2,3), and tripper (3,3). The giraffe is a level-4 leaper (1,4). The table shows the Betza notation for leapers up to (3,3). An amphibian is a combined leaper with a larger range on the board than any of its individual components. The simplest amphibian is the frog, a (1,1)-(0,3)-leaper (HF).

Riders

A rider is a piece that moves an unlimited distance in one direction, provided there are no pieces in the way. There are three riders in orthodox chess: the rook is a (1,0)-rider; the bishop is a (1,1)-rider; and the queen combines both patterns. Sliders are a special case of riders which can only move between geometrically contiguous cells. All of the riders in orthodox chess are examples of sliders. Riders and sliders can create both pins and skewers.

One popular fairy chess rider is the nightrider, which can make an unlimited number of knight moves (that is, (1,2) cells) in any direction (like other riders, it cannot change direction partway through its move). The names of riders are often obtained by taking the name of a leaper which moves a similar cell size and adding the suffix "rider". For example, the zebrarider is a (2,3)-rider.

Hoppers

A hopper is a piece that moves by jumping over another piece (called a hurdle). The hurdle can usually be any piece of any color. Unless it can jump over a piece, a hopper cannot move. Note that hoppers generally capture by taking the piece on the destination square, not by taking the hurdle (as is the case in checkers). The exceptions are locusts which are pieces that capture by hopping over its victim (as in checkers). They are sometimes considered a type of hopper.

There are no hoppers in Western chess. In xiangqi, the cannon captures as a hopper (when not capturing, it is a (1,0)-rider which cannot jump). The most popular hopper in fairy chess is the grasshopper, which moves along the same lines as an orthodox queen, except that it must hop over some other piece and land on the square immediately beyond it.

Marine piece

A marine piece is a combination piece consisting of a rider (for ordinary moves) and a locust (for captures) in the same directions. Marine pieces have names alluding to the sea and its myths, e.g., nereide (marine bishop), triton (marine rook), mermaid (marine queen), or poseidon (marine king).

Games

Some classes of pieces come from a certain game, and will have common characteristics. Examples are the Chinese pieces from xiangqi, a Chinese game similar to chess. The most common Chinese pieces are the leo, pao and vao (derived from the Chinese cannon) and the mao (derived from the horse). Those derived from the cannon are distinguished by moving as a hopper when capturing, but otherwise moving as a rider. Less frequently encountered Chinese pieces include the moa, nao and rao.

Royal pieces

A royal piece is one which must not be allowed to be captured. If a royal piece is threatened with capture and cannot avoid capture next move, then the game is lost (this is a generalization of the concept of checkmate). In orthodox chess, each side has one royal piece, the king. In fairy chess any other orthodox piece or fairy piece may instead be designated royal, there may be more than one royal piece, or there may be no royal pieces at all (in which case the aim of the game must be something other than to deliver checkmate, such as capturing all of the opponent's pieces). With multiple royal pieces the game can be won by capturing one of them (absolute royalty), or capturing all of them (extinction royalty). The rules can also impose a limit to the number of royals that are allowed to be left under attack at once. The orthodox checking rule corresponds to putting this number at zero. But in Spartan chess you cannot leave both your royals attacked even though they cannot both be captured in one turn, and loss of only one of them would not lose the game.

Parlett's movement notation

In his book The Oxford History of Board Games David Parlett used a notation to describe fairy piece movements. The move is specified by an expression of the form m={expression}, where m stands for "move", and the expression is composed from the following elements:

Distance (numbers, n)

1 – a distance of one (i.e. to adjacent square)

2 – a distance of two

n – any distance in the given direction

Direction (punctuation, X)

* – orthogonally or diagonally (all eight possible directions)

+ – orthogonally (four possible directions)

> – orthogonally forwards

< – orthogonally backwards

<> – orthogonally forwards and backwards

= – orthogonally sideways (used here instead of Parlett's divide symbol.)

>= – orthogonally forwards or sideways

<= – orthogonally backwards or sideways

X – diagonally (four possible directions)

X> – diagonally forwards

X< – diagonally backwards

Grouping

/ – two orthogonal moves separated by a slash denote a hippogonal move (i.e. jumping like knights)

& – repeated movement in the same direction, such as for hippogonal riders (i.e. the nightrider)

. – then, (i.e. an aanca is 1+.nX)

Additions to Parlett's

The following can be added to Parlett's to make it more complete:

Conditions under which the move may occur (lowercase alphanumeric, except n)

(default) – May occur at any point in the game

i – May only be made on the initial move (e.g. pawn's 2 moves forward)

c – May only be made on a capture (e.g. pawn's diagonal capture)

o – May not be used for a capture (e.g. pawn's forward move)

Move type

(default) – Captures by landing on the piece; blocked by intermediate pieces

~ – Leaper (leaps)

^ – Locust (captures by leaping; implies leaper)

Grouping (punctuation)

/ – two orthogonal moves separated by a slash denote a hippogonal move (i.e. jumping like knights); this is in Parlett's, but is repeated here for completeness

, (comma) – separates move options; only one of the comma-delimited options may be chosen per move

() – grouping operator; see nightrider

- – range operator

The format (not including grouping) is: <conditions> <move type> <distance> <direction> <other>

On this basis, the traditional chess moves (excluding castling and en passant capture) are:

King: 1*

Queen: n*

Bishop: nX

Rook: n+

Pawn: o1>, c1X>, oi2>

Knight: ~1/2

Ralph Betza's "funny notation"

Ralph Betza created a classification scheme for fairy chess pieces (including standard chess pieces) in terms of the moves of basic pieces with modifiers.

Capital letters stand for basic leap movements, ranging from single-square orthogonal moves to 3×3 diagonal leaps: Wazir, Ferz, Dabbaba, KNight, Alfil, THreeleaper, Camel, Zebra, and G (3,3)-leaper. C and Z are equivalent to obsolete letters L (Long Knight) and J (Jump) which are no longer commonly used.

A leap is converted into a rider by doubling its letter. For example, WW describes a Rook, FF describes a Bishop, and NN describes a Nightrider. The second letter can instead be a number, which is a limitation on how many times the leap motion can be repeated; for example, W4 describes a Rook limited to 4 spaces of movement.

Combining multiple movement letters into a string means the piece can use any of the available options. For example, WF describes a King, capable of moving one space orthogonally or diagonally.

Standard chess pieces except Pawns (which are particularly complex) and Knights (which are a basic leap movement) have their own letters available as shorthand; K = WF, Q = WWFF, B = FF, R = WW.

Lowercase letters in front of the capital letters modify the component. Often used modifiers are: forward, backward, right, left, sideways, vertical, move only, capture only, z crooked (moving in a zigzag line like the Boyscout), grasshopper, jumping (i.e., it must jump, cannot move without a hurdle), non-jumping like the Chinese Elephant, o cylindrical (moving off one side of the board loops to the other), pao (travels through captured piece), then (for pieces that start moving in one direction and then continue in another, like the Gryphon and Aanca), and q circular movement (like the Rose).

In addition, Betza has also suggested adding brackets to his notation: q[WF]q[FW] would be a circular king, which can move from e4 to f5 (first the ferz move) then g5, h4, h3, g2, f2, e3, and back to e4, effectively passing a turn, and could also start from e4 to f4 (first the wazir move) then g5, g6, f7, e7, d6, d5, and back to e4.

Example: The standard chess pawn can be described in Ralph Betza's funny notation as mfWcfF (ignoring the initial double move).

There is no standard order of the components and modifiers. In fact, Betza often plays with the order to create somehow pronounceable piece names and artistic word play.

Addition to Betza's

Betza does not use the small letter i. It is used here for initial in the description of the different types of pawns. The letter a is used here to describe for again, indicating the piece can make the move on which it is prefixed multiple times, possibly with new modifiers mentioned behind the a. Directional specifications for such a continuation step should be interpreted relative to the first step (e.g. aW is a two-step orthogonal move that can change direction; afW is a two-step orthogonal move that must continue the same direction).

Notable examples

The following table shows game pieces of unorthodox chess, from fairy chess problems and chess variants (including historical and regional ones), and the six orthodox chessmen.

A – B – C – D – E – F – G – H – I – J – K – L – M – N – O – P – Q – R – S – T – U – V – W – X, Y, Z

Relative value of pieces

While a large amount of information can be found concerning the relative value of variant chess pieces, there are few resources where it is in a concise format for more than just a few piece types. One challenge of producing such a summary is that piece values are dependent upon the size of boards they are played on, and the combination of other pieces on the board.

On an 8×8 board, the standard chess pieces (pawn, knight, bishop, rook, and queen) are usually given values of 1, 3, 3, 5, and 9 respectively. When the basic pieces wazir (W), ferz (F), and mann (WF = K), are played with a similar mix of pieces, they are typically valued at around 1.2, 1.5, and 3.2 points respectively. Three popular compound pieces, the archbishop (BN), chancellor (RN), and amazon (QN) have been estimated to have point values around 8, 9, and 11.5 respectively. (Due to the powerful ability of the three later pieces, it is uncommon for more than one to be played on a standard 8x8 board).

Apart from these, reliable estimates are not be well established for many other pieces. Even when the same game format is assumed (board size and combination of other pieces), there is often little agreement on the specific value of many other pieces. Compound pieces are sometimes approximated as the sum of their component pieces, or estimated to be slightly higher due to synergistic effects (such as it is for the archbishop and chancellor).

For purely jumping pieces (including moves of a single square), one formula has been developed, sometimes good-naturedly called "Muller's Short Range Leaper Law" which estimates a piece's value as:

v a l u e = α ( 30.0 ( N ) + 0.625 ( N ) 2 ) , centipawns where: α = factor based on scale for other pieces (1.0: classical, i.e. knight = 300; 1.1: Kaufman, i.e. knight = 325) N = squares attacked

Although regarded as reliable, this formula is limited in that it only applies to leapers jumping not more than two squares. Furthermore, it doesn't take into account some factors which may influence a piece's value, such as the distance of jumps (shorter jumps are usually worth more), and direction (pieces favoring forward captures are typically worth more than symmetrically capturing pieces).

For sliding pieces, and sliding-leaping compounds, there seems to be no corresponding formula to estimate piece values. It is generally presumed that with bigger boards, sliding pieces (i.e. bishop and rook) generally become more valuable relative to short-range jumpers, if for no other reason that they can travel about the board more quickly.