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 What are examples of mathematical papers (or unpublished notes) which present an abstract mathematical definition of a chess piece?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. – Michael Greinecker Dec 7 '17 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". – Morteza Azad Dec 7 '17 at 12:23
• These comments don't really address the question. – Nik Weaver Dec 7 '17 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. – Joel David Hamkins Dec 7 '17 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/… – Terry Tao Dec 7 '17 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. – Morteza Azad Dec 7 '17 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.) – Andreas Blass Dec 7 '17 at 13:53
• I suppose there are many examples in combinatorial game theory, where various games are seen to be Nim in disguise. – Joel David Hamkins Dec 7 '17 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. – Tim Carson Dec 8 '17 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. – Joel David Hamkins Dec 9 '17 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. – Gerry Myerson Dec 9 '17 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. – Michael Kay Dec 10 '17 at 0:14

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 castle 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 '18 at 2:25
• @Bill That sounds about right. – J.G. Mar 15 '18 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.