# What is a chess piece mathematically?

Historically, the current "standard" set of chess pieces wasn't the only existing alternative or even the standard one. For instance, the famous Al-Suli's Diamond Problem (which remained open for more than one millennium before getting solved by Grandmaster Yuri Averbakh) was formulated in an ancient Persian variant of chess, called Shatranj, using a fairy chess piece, called Wazir (Persian: counsellor), rather than the conventional queen.

There is a long-standing discussion amongst chess players concerning the best possible configuration of chess pieces which makes the game more exciting and complicated. Also, one might be interested in knowing whether, in a fixed position on the infinitary chessboard, the game value could be changed into an arbitrary ordinal merely by replacing the pieces with new (possibly unconventional) ones rather than changing their positions.

In order to address such questions one first needs to have a reasonable mathematical definition of the notion of a "chess piece" in hand.

Maybe a promising approach inspired by Rook, Knight, and King's graphs is to simply consider a chess piece a graph which satisfies certain properties. Though, due to the different nature of all "reasonable" chess pieces, it seems a little bit hard to find principles which unify all of them into one single "neat" definition. For example, some pieces can move only in one direction, some others can jump out of the barriers, some have a/an finite/infinite range, some can only move among positions of a certain color, etc.

Here the following question arises:

Question. What are examples of mathematical papers (or unpublished notes) which present an abstract mathematical definition of a chess piece? Is such a definition unique or there are several variants?

Update 1. In view of Todd and Terry's comments (here and here), it seems a more generalized question could be of some interest. The problem simply is to formulate an abstract mathematical definition of a "game piece" in general. Are there any references addressing such a problem?

Update 2. As a continuation of this line of thought, Joel has asked the following question as well: When is a game tree the game tree of a board game?

• The chess rules are already abstract and define the structure of chess rigorously. Dec 7, 2017 at 12:05
• @MichaelGreinecker Of course for every variant of chess, we have rigorous well-defined rules but as a game with a long history, "chess" has many variants with different rules and pieces. For example, there are chess games played on a three-dimensional chessboard! (The theory of infinite three-dimensional chess has been studied in some of Joel Hamkins' works. See page 443 of this slides for more information: jdh.hamkins.org/wp-content/uploads/2014/08/… ). I don't see how and why one can consider one of them the standard version and the rest just "non-chess". Dec 7, 2017 at 12:23
• These comments don't really address the question. Dec 7, 2017 at 12:25
• I interpret the question essentially as asking for a definition of what it means to be a chess variant. For example, checkers or draughts is not ordinarily viewed as a chess variant, but Fischer chess, bughouse and horde chess are. My opinion is that the category of chess variants does not have sharp boundaries, and any proposed definition will admit counterexamples. So I don't think there will be a fully satisfactory account. Dec 7, 2017 at 12:46
• If one draws an analogy between chess and fluid dynamics, then the chess pieces are the possible values of Lagrangian coordinates, while the 64 squares of the chessboard are the possible values of Eulerian coordinates. en.wikipedia.org/wiki/… Dec 7, 2017 at 16:39

In terms of mathematical analysis and combinatorial game theory, the essence of any game is captured by its game tree, the tree whose nodes represent the current game state, and to make a move in the game is to move from a node in this tree to a child node. Terminal nodes are labeled as a win for one player or the other, or a draw (and in the case of infinite games, the winner is determined by consulting the set of winning plays, which in a sense defines the game).

In chess, the current game state is not merely a description of what is on the board, for one must also know whose turn it is and also a little about the history of the play, in order to determine whether castling or en passant is allowed or to determine draws by repetition or the 50-move rule.

Once one has the game-tree perspective, the concept of chess pieces tends to fall away, and one might look upon the concept of a chess piece as epi-phenomenal to the actual game, a convenient way to describe the game tree: strategic considerations concern at bottom only the game tree, not pieces.

In the case of chess, for example, the computer chess programs are definitely analyzing and searching the game tree.

You ask for references, and any text in combinatorial game theory will discuss the game tree and prove what I call the fundamental theorem of finite games.

Fundamental theorem of finite games. (Zermelo, 1913) In any finite game, one of the players has a winning strategy or both players have drawing strategies.

(Zermelo's actual result was something a little different than this; see the comments below and the interesting paper, Schwalbe and Walker, Zermelo and the early history of game theory.)

This theorem is generalized by the Gale-Stewart theorem (1953), which shows also that every open game is determined, and this is generalized to Borel determinacy and more, and one then enters a realm of sophisticated results in set theory.

Let me mention an example showing how two games can look very different in terms of how they are played, yet at bottom be essentially the same game, with isomorphic game trees.

Consider the game 15, in which players take turns to select distinct numbers from the numbers 1, 2, ..., 9. Once one player takes a number, it is no longer available to the other player. Whichever player can make 15 as the sum of three distinct numbers is the winner.

Please give the game a try!

After a while, the game might begin to be familiar, for we can realize that it is exactly the same game as tic-tac-toe, as can be seen via the following magic square.

$$\begin{array}{ccc} 8 & 1 & 6 \\ 3 & 5 & 7 \\ 4 & 9 & 2 \\ \end{array}$$

At the MoMath museum in New York, they have this game set up with a two-sided display. On one side, for the parents, you see only the numbers in a row. On the other side, for the kids, you see the tic-tac-toe arrangement. How amazed the parents are to be beaten soundly by their kids — all the kids are geniuses!

My point with this is that game of 15 and the game of tic-tac-toe are essentially identical as games, yet in tic-tac-toe there is directly no concept of number or selecting a number, and in 15 there is directly no concept of a corner square or center square. The nature of the number 5 in 15 is game-theoretically similar to the nature of the center square in tic-tac-toe, and this is revealed by the fact that the game trees are isomorphic. Chess pieces are like that.

• (+1) A very good answer as usual! Thanks, Joel! Particularly, I enjoyed reading the last part about the MoMath museum. Dec 7, 2017 at 13:42
• I agree about the primacy of the game tree, and I'd add that the notion of "piece" may be, at least in part, a psychological one: Pieces provide a way to summarize the structure of the game tree in a way that we can understand. It would be interesting to have a game tree with two nontrivially different such summaries --- two seriously different notions of piece. (By "nontrivially", I want to exclude things like the tic-tac-toe example, where the numbers are just a relabeling of the squares.) Dec 7, 2017 at 13:53
• I suppose there are many examples in combinatorial game theory, where various games are seen to be Nim in disguise. Dec 7, 2017 at 14:31
• This seems to miss the point of the question. The question is asking for a definition for a certain subset of games, and saying "it's just a tree" is too broad. There are mathematical questions we can ask about chess variants, that don't make sense for general games. For example, what subsets of pieces guarantee that I can checkmate a lone king? Also see any of the "related" questions. It seems you mention 50 move rule and castling to make the point that it's hopeless to make any definition besides "it's a tree", but in mathematical questions we usually just ignore those rules first. Dec 8, 2017 at 14:55
• @Yakk The game tree of tic-tac-toe is not isomorphic to your game of "would you like your opponent to win?", since every play of tic-tac-toe ends in at most nine moves, but your game has infinite plays. And even if you limit your game to nine moves, then it isn't the same, since in tic-tac-toe, the game tree is branching with nine possible first moves and eight possible second moves and so on, and those plays are not homogeneous, since the game tree below some of them is not isomorphic. Dec 9, 2017 at 11:40

Approaching this from the perspective of a computer programmer rather than a mathematician, my instinct is to try to isolate those properties of a chess piece that are unique to that piece, separating them from rules that apply to all pieces. For example, there's a general rule that you can't do anything that would leave your king in check, and a general rule that you can't move to (or over) a square occupied by one of your own pieces. Of course, we can conceive of pieces that were not subject to these rules, so we have to decide how much we want to generalize. Similarly, when you're dealing with the "aberrations" of castling, en-passent capture, and promotion, you have to decide whether and to what extent you want to create some model which treats these as special cases of something more general.

A chess piece is characterized primarily by the moves that are possible from a given square, which in turn can be characterized as the set of squares that would be reachable in the absence of obstacles, less the squares that are obstructed. The reachable squares are a function of the piece and the starting square, while the obstructed squares are a function of the state of the board (independent of which piece you are moving). You can generalize the concept of a "move" to a "transition in the state of the board" that includes other pieces moving (castling) or disappearing (captures), or pieces being transformed into other pieces (promotion). And you can generalize "the state of the board" to include not just the current positions of pieces, but also the history of the game, or a "distilled history" that contains only as much information as is needed to determine legality of moves (whether the King has castled, whether pawns are subject to en-passent capture).

In summary, the way to model a chess piece mathematically depends on how much you want to generalize from the rules of chess as they are, to the rules of all conceivable chess-like games. As always, the right amount of generality rather depends on what you want to do with the model.

• "[There's] a general rule that you can't move to (or over) a square occupied by one of your own pieces." Except, of course, that a knight can move over a square occupied by one of your own (or your opponent's) pieces. Dec 9, 2017 at 3:20
• Well, if you define the rule in terms of lines between the centres of squares, the knight doesn't move over the centre of another square so you don't have to treat is as an exception. Dec 10, 2017 at 0:14
• Then castling is an exception, as you move the rook over the center of the square occupied by the king. Aug 22, 2023 at 0:13

The easy part is to say, "a pawn is one instance of type-pawn chess pieces". Hereafter I'll assume "a chess piece" is a piece type rather than an instance, which moves us onto the hard part.

The first temptation is to say it's a graph showing which $$A$$-to-$$B$$ moves are possible; for example, you might think, "oh, the rook graph has edges between those vertices denoting squares connected by a rank or file". But in practice whether the piece can move a certain way depends on other details, such as what piece instances (including their colours) are in the way and the game's history, which has implications for castling, en passant, whether the pawn can move two squares etc.

So a second stab at it is to say a piece is a function from past-and-present-state descriptions of the game to such legal-move graphs. For example, if there were a hypothetical piece that can go from $$A$$ to $$B$$ regardless of history and both piece colours' current placements, so long as $$A,\,B$$ are in the right relation, this function would be constant, always returning the same graph. But all the pieces you'll ever consider are non-constant functions.

If we now consider capture and promotion, however, we realise that the move to $$B$$ also lets a piece do something when it gets there; and if we consider castling, we realise the move also lets the King and Rook do certain things to each other. So an even better attempt at defining pieces, and hopefully the last one we need, is a function from past-and-present-state descriptions to the graph of legal changes in the game state. (Of course, states are so numerous this would be a graph with a huge number of indices.) Actually, I'll reword that once more: it's a directed graph of all legal history-to-history transitions. Each directed edge leads to a state in which the player's elapsed turn count increases by $$1$$.

• So each piece is a subgraph of the graph of all possible state-to-state transitions. I.e., the subgraph composed of all the transitions where the piece moves?
– Bill
Mar 15, 2018 at 2:25
• @Bill That sounds about right.
– J.G.
Mar 15, 2018 at 6:25

A chess board represents an abstract space and a chess piece is a member of a set of finite elements where to each element is assigned a rule on how to change its coordinates in the abstract space.

These rules also depend on the positions of other pieces. For example, a rule for a bishop is that it cannot proceed past another piece in its way. But no such rule exists for a knight.

One possible way to represent a chess piece mathematically is to abstract away things like starting position or colour and only focus on what it can do. Thus it can be represented it as a triple (M, R, E) like this:

M is the set of vectors, representing all possible moves that the piece can take on an infinitely large board with no other pieces (or restrictions) on it.

R is the set of logical formulae, which given the history of the moves, the current position of the piece and a move m from M tells you if m can be done given the current state of the board.

E is a function from M, the piece's current position and the current board state to a new board state. This will tell you what (other) effects the move will have. For example, taking a piece or moving your rook if you are castling.

Using these 3 you can represent any chess piece and also compose them together to find out all possible moves that can be taken in a given board state. You could also represent other board game's pieces in a similar way, but you might also need to generalise one of the components depending on the rules of the game.

Rereading the questions, I would simply answer that with the basic rule of chess one can consider the pieces as instances (or initializations on a game board) take from mobility classes that directly decsribe all the different mobility rules or a complete description of the board, possibly including a useful coordinate system to simplify the notation to write the different classes of mobility. Is there a paper that does that, I do not know of any. Should that be a reason not to share the above paragraph, i do not think so.

One can consider the king graph, and its representation embedded in R2 (a finite subset of R2 of course. The grid of small square being actually a grid of points, with the links following the king mobility class (figure any way to create the link set, without referring to king mobility class, this is just to avoid typing in math.which i can't. Or if R2 is not to ones liking, just go crazy with finite description of all the grid points.

In anycase, one can construct in finite human time, such mathematical objects. And what would the question want more that such set description. The word class is just used above the math. here. It the question was about the piece in a particular game, i would say, an initial condition on a game of the set of legal target grid points i called mobility class of the name of the piece you want. In exaustive forumlation you would have to list all the source grid points, and all the target grid points, on an empty board.

If the conceptual bug is about the empty board description, then I need to go to the full formalism of a game or even position. Take the cartesian product of all pieces having been initialized in standard initial position.

legal moves can be reduced from all possible moves in that big multidimensional set, so that there are no collisions, and all the ruleset non-legal things one may list. Using R2, is just to make such descriptions more compact, and generic. One may even use intersection of 8x8 subset in R2 of bigger boards (even infinite) to reduce the typing even more. instead of making all the rim cases. without blasphemy i would suggest to picture a real chess board from above. as a 2D grid of points. assign placement of pieces to those point.. and notice that the ambient R2 space does not distract from the mathematics.

An answer above made similar point about whether to interpret the question as about the piece as instance or as mobility definition on the board. As to the game tree question, I also share it. It puzzles me, when considering where all the games are evolving. A game tree seems to assume only on game description, not all the games. I mean a unique representation of all the games. Transposition seem to be getting in the way. I would consider unique positions and moves as positions transitions. Each game can be view as a sequence of decisions among all the legal moves on such position defined representation, however the whole game world might not be "bijective" to a game tree (2 nodes on such tree, could be the same position).

sorry for bad English. I am in process of editing.

• I can delete the last paragraph. as this is me asking question on top of initial third question. upon uproar to have it deleted. I am also looking for ways, to open my answer for those with better english that might understand the main points I am trying to express. and if someones can point to existing papers doing the same. please share.
– dbdb
Sep 2, 2023 at 1:30
• I think I forgot to mention that the big multidimension set of empty board with one piece on each defined by a mobility set of pairs of grid points to choose at each move, is meant to be superimposed on the same 2D board in order to apply the legal chess rules for obtaining a legal position successors. both position and moves are described exploded and then superimposed back.. collisions being not legal for example.. go big then reduce. nothing really magical about it.
– dbdb
Sep 2, 2023 at 1:47
• more erratum.. edit the above is dangerous for more details pouring.. so 8x8 intersection a mobility set of pairs of conveniently morphed Z2 subset embedded on R2. each piece on board can have its own mobility set applied to any of its placements on board. using the possibly shiffted in parts or dilated in parts infinte Z2 to describe the mobility in compact form to delegate the problem to implementation of the 8x8 intersection to instance computations, is still math. right?
– dbdb
Sep 2, 2023 at 3:56
• the more i explain the more i have to explain.. since no paper out there found in the answers, why not accept, ideas as answer. if i could open this as a wiki, perhaps, people with same ideas but clearer expression, and math. touch typing confort. could fix this. what matters is that the symetries of the board are kept in the morphing of Z2 within R2. but some translation properties have to be kept. so that one can intersect properly a full infinite discrete space compact mobility figure math statement with 8x8 (also in same morphism).
– dbdb
Sep 2, 2023 at 3:58
• one more precision.. i give up after that. asking for some imagination on reader beyond that. discussion welcome. not just opinions though. not all translations with vectors in R2. only those compatible with the king walk directions. there are 4 signed directions. (or is that orientation, i said signed did i not).
– dbdb
Sep 2, 2023 at 4:28