Winning Ways for Your Mathematical Plays, from 1982, is a book that marks the beginning of an entire mathematical area, Combinatorial Game Theory, and a new set of numbers, the Surreal Numbers. It was written by Elwyn R. Berlekamp, John H. Conway, and Richard K. Guy. The book contains an impressive number of mathematical techniques and insight and has very hard sections in it (there are more recent books with the goal of introducing the main concepts with a more pedagogical approach). Below, let's call the book just WW (for Winning Ways).
Among the many games explored in the book, some are closer to the idea of abstract games that motivate this blog. This post mentions one of them: Quadraphage.
The rules of Quadraphage (meaning, the square eater) by Richard Epstein in 1973:
- In a NxN empty board, a King is placed on a square
- One player moves the King (the Mover), the other player drops a stone (e.g., a Go stone) on any empty square (the Placer)
- Turns alternate, as usual.
- Goal: if the King reaches any square at the edge, the Mover wins; if the Go stones surround the King, the Placer wins
Since moving first is never a disadvantage, there are three possible outcomes: (a) the Eater always wins, (b) the Mover always wins, (c) the first player to move wins. The book calls a fair position every square for the King to begin, where option (c) occurs.
The book includes the use of other pieces besides the King. It calls Chessgo to this family, and it considers Kinggo (the previous rules) and Dukego (using a Duke, ie, a one-step Rook). Other reasonable options include Knightgo and Ferzgo (using a Ferz, ie, a one-step Queen).
One interesting result from WW is that there are only two possible board sizes where fair positions occur, and that are 33x33 and 34x34 boards (!). On a smaller board the Mover always wins, and for bigger boards the Mover always wins (cf. chapter 19).
Also, the author mentions the game in his 2009's book The Theory of Gambling and Statistical Logic:
This game is an offspring of the medieval Tafl games, and a member of the Fox Games' family.