Quadraphage on Winning Ways

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.

Ko-an

Ko-an is a 1994 game by David Welch and Paul Whitehorn, published at Image Games.

Each player has two types of pieces: five squared pieces and six octagonal pieces. The board is composed of a grid of yellow squares and green octagons.

This is the initial setup:

• Pieces only move forwards, not backwards nor sideways, to an empty space. There's a further move restriction: a piece cannot move between squares [in a better designed board, like the one below, it's easy to see why: squares do not touch on a 4.8.8 grid].
• Pieces can also move forwards to a space occupied by an enemy piece, which is then captured (captures are by replacement).
  
• However, captures also depend on piece type: octagonal pieces can only capture pieces on octagons, and squared pieces can only capture pieces on squares.
  
• Capturing is not mandatory.
  
• Wins the player that moves a friendly piece to the last row or stalemates the adversary (which includes capturing all the enemy army).

Here's a board (by r0cka) to play the game:

From a 1994 review by Richard Breese:

[...] Considering its simplicity, the game can develop in a surprising variety of ways. For example, pieces of both players may bunch up on one side of the board, several 'skirmishes' may occur all over the board, or a large stand off may arise across the center of the board. This variety adds to the attractiveness of the play.

Of the two types of playing pieces the octagonal piece is stronger and this factor probably explains why in most games it is a octagonal shaped piece which makes the winning break through. This strength arises from the situation where an octagonal shaped piece confronts a squared shaped piece. As the square spaces do not join each other it is necessary for all pieces to move onto the octagonal shaped spaces - the spaces on which the octagonal shaped pieces can capture. Consequently an octagonal shaped piece is much more likely to be able to capture a square shaped piece than vice versa.

Some pics from the original game package:

Pusher, Leverage, and Physics

Pusher is a 1993 game by Werner Falkhof, and published by ASS Altenburger Spielkarten.

The game works for two or three people, and has a dexterity element to it.

The rules,

Since there is Physics involved (players push balls in non-deterministic ways) this is a game that looks abstract, but isn't. Something between a mini-pool and Subbuteo, which is very hard to design well.

Indeed, the last part of a review by Ben Baldanza:

Pusher is another great-looking game that disappoints in play. I give Theta credit for trying to integrate a dexterity mechanic into a placement and seemingly strategic context, but the result is more chaos than strategy.

 

 

A game that better approaches the problem of mixing Abstract Strategy and Physics is Leverage:

 

 

No, I didn't forget Jenga: not obscure enough :-)

[addendum] Regarding Jenga, there is a 2016 game by designer Masoccer, called Tumiki Maze that is very interesting:

• There are five shapes of wood tiles (ten pieces of each type?) and two pawns for each player.
• First place the board on the table, and then put the pawns appropriately on it.
• On his turn, the player places two pieces on the structure, and then move one of his pawns that must be placed higher.
• The player unable to do it, or the player that collapses the structure, loses the game.