- Games are sets of mechanics with unique mathematical properties
This holds for all games, regardless of medium of medium or genre (i.e. board & card games, computer games, puzzle games, sports.) The field of Combinatorial Game Theory is devoted to the mathematical analysis of games toward the goal of solving a given game. The scope of CGT is ever expanding per the capabilities of mathematics and computation to solve problems of increasing complexity.
Games may also include elements (tokens, players, objects, etc.) and these elements are governed by the mechanics.
- Mechanics, simply put, are the specific rules of a given game
By contrast, genre refers to categories of gameplay, which is often conflated with mechanics. But "gameplay" and genre are terms used to connote and categorize games with similar properties and player experience.
Keeping to mathematical definition of structure, m,n,k is a class of games based on connecting k tokens on an m*n game board. All m,n,k-games share this structure and a common set of fundamental mechanics (place tokens, victory condition based on k in a row), and include Tic-tac-toe, Gomoku and Connect Four. The wider genre is "connection games", which includes Hex and TwixT.
Gameplay is a little fuzzier, but all of the connection games listed here can be called "combinatorial" which sometimes refers to a special class of "non-chance, perfect information, sequential, partisan" games.
The concepts of mechanics, gameplay and genre can extended to any type of game, with distinction being that the latter two tend to reflect market-based considerations, specifically relating to players the type of game experience.
- Genre and "gameplay" are distinct from game mechanics in that they are stylistic
- Game mechanics are purely logical/mathematical, which is to say algorithmic