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I don't think that 'conceptually' really means much.

It refers to how human beings normally conceive of the rules of a game, particularly for the purpose of playing it themselves with physical equipment. What you have proposed could go on behind the scenes in a computer program, but it would not reflect how people would actually play the game with a physical set. Instead of thinking that a piece changed its type with each move to a square with a different arrangement of arrows, human players with a physical set would normally think of each piece as a singular thing capable of moving in the directions indicated by the arrows on its square.

As far as I am concerned game rules are an abstract mathematical concept: a set of game states acting as nodes of a directed graph that indicate the legal transistions between one game state and another.

I can understand a programmer thinking of game rules in those terms, but most people are not programmers and will not understand game rules in that way. Furthermore, keeping the code close to the conceptual understanding of the rules can make it easier for other programmers to understand. Consider the code in the interactive diagram for this game. It works as a way of allowing a computer program to play the game, but it's about as obfuscated as code can get, and looking at this code is not going to give anyone much idea about how to play. But my own code for Game Courier would give someone who examined it a much better idea of how the game is played.

Any verbal description that unambiguously defines this graph is as good as any other, and concepts like turn, piece, move, cell, type are not more than personal preferences of the person or machine learning the game, to help him remember and apply the rules.

Machines do not have to work with concepts, and they do not have personal preferences. If it works, a machine can be programmed to play a game in a manner very different from how humans would normally conceive of the rules. This might be due to the limitations of a programming language or for the sake of efficiency. But when the language allows it, and it can be done at no cost to efficiency, it is best to reflect a normal human understanding of the game in the code. Such an understanding will normally be common to many people thanks to being grounded in the common experience of playing with physical pieces on physical boards. This makes it normal for people to distinguish between the pieces and the terrain, whereas a computer program, not thinking about the game conceptually at all, could easily merge the pieces and terrain together in a way that humans would not normally do.