Index and Resource Guide to Abstract Games
Compiled by Michael Keller

In this booklet, we will use a common definition of abstract games as two-player games of perfect information.   Some of the material herein appeared in WGR.    Some major game families (e.g. checkers, reversi) have their own dedicated guides.

Table of Contents
Mancala Games
Alignment Games
  Placement Variants
  Movement Variants
    Delta
Territorial Games
  Go
  Amazons
Other Games
  Fanorona
Bibliography
  Individual Games
  Mancala
  Hoyles
  Other
  Go
  Magazines

Mancala Games
Mancala is a large family of folk games sometimes called "pit and pebble" or "count and capture" games.  They are played on boards consisting of two or more rows of cups, into which some number of stones are placed initially.  Usually there are two larger pits, called storehouses, at the ends of the board, which are used only for storing captured stones (each player uses the storehouse at their right).  A move consists of taking all of the stones from one of the cups on your own side, and sowing them anticlockwise, one stone into each of the succeeding cups, continuing into the opponent's cups if there are enough stones in hand.   Depending on the variant, stones are captured according to a particular set of rules.  A large group of games, of which Wari is the best known, are played on a 2x6 board starting with four stones in each cup.  In Wari, if the last stone sown lands in one of the opponent's cups so as to end up with two or three stones, those stones are captured, along with any one or more consecutive pits prior to the last which also contain 2 or 3 stones.
 
Alignment Games
The objective is to align some number of pieces in a line or some other configuration.  Roughly divided into placement and movement families.

Placement Variants


                                     The three draw positions in Tic-Tac-Toe (X moves first)

Tic-Tac-Toe (Noughts and Crosses) is probably the simplest alignment game, frequently played by children as a pencil-and-paper game.   Players play alternately on a 3x3 board, trying to get three of their pieces in a line (horizontally, vertically, or along either long diagonal).  With best play it is always a draw; every drawn position can be reduced to one of three positions by reflection and rotation (diagram above).    The three-dimensional version on a 3x3x3 board is an easy win for the first player, playing in the middle of the second level.  The 4x4x4 game (marketed many times as Qubic and other names) is much more strategic, though Owen Patashnik proved in 1980 that the first player has a forced win.   There are many other variants, including Ultimate Tic-Tac-Toe, played on a 9x9 grid divided into nine normal 3x3 boards.    A version sometimes called Super Tic-Tac-Toe is played on a 4x4 board, with a win for forming four in a row in any direction, a 2x2 square, or (very unlikely) all four corners (it's probably a draw with best play, but there are several traps).

A popular commercial variant is Connect Four, which is a 7x6 form of what is sometimes called Vertical Tic-Tac-Toe: a piece may be played into any column, but drops to the lowest empty space in that column.   Connect Four was solved in 1988 by James Dow Allen; it is a win for the first player, but only if the first move is in the center column.   John Tromp has computed the number of legal positions at each point in the game (the total is about 4.5 trillion), and the result on other sizes of boards.   There are rumors of early folk versions of the game, including a version called

   

The Game of the Amazons (or simply Amazons) is an abstract game of territory for two players, played on a 10x10 board (a checkered board makes it easier to visualize diagonal moves). Each player has four amazons, which move like chess queens : any number of vacant squares in a straight line -- orthogonally or diagonally.  After it moves, an amazon must fire an arrow in the same manner from its landing square (one or more vacant squares orthogonally or diagonally). The square where the arrow lands is then marked with a poker chip or other counter to indicate it is blocked (in the computer version blocked squares are colored dark red). No amazon or arrow may move into or through a blocked square. The starting position is shown above left. White moves first, moving one amazon and firing an arrow with that amazon (above right).   The players alternate moves, and the player last able to make a move wins.   Draws are impossible, and the game cannot last more than 46 moves.   In practice, in most games, the territory is walled off and the status of every empty square is known earlier than 46 moves, and territory can simply be counted.

El Juego de las Amazonas was invented in 1988 by Walter Zamkauskas of Argentina, and first published (in Spanish) in issue number 4 of the puzzle magazine El Acertijo in December of 1992. In 1993, I informally introduced it to the postal gaming club The Knights of the Square Table, where it gained immediate popularity. An authorized translation was published in January 1994. The first international match was a friendly team match, played by fax between Argentina and the United States in 1994-1995; the six games were split 3-3.  In early 1994, I wrote what I believe was the first computer version, in Fortran with a simple keyboard interface.  It had a weak computer opponent.   The Windows version, from which the screenshot diagrams above are taken, is a slightly later development of the same program, now written in Visual Basic.

Amazons became an event in the Mind Sports Olympiad in 1997, and in the 5th Computer Olympiad in 2000.   An introduction by Paul Yearout appeared in Abstract Games 16 (2003, pp. 12-13).  A video introduction to Amazons with Dr. Elwyn Berlekamp can be found on YouTube.    A number of very strong programs have been written to analyze and play Amazons, and mathematical papers have been published by, among others, Hensgens (2001), Müller and Tegos (2002), Lieberum (2005), Kloetzer et al. (2008), and Song and Müller (2014).   However, the game has been solved only on very small boards.

A number of variants have been proposed, among them Phil Cohen's Non-Parthian Amazons, in which amazons may not fire directly behind them after moving (among other effects, this nullifies the last square in each territory), and Fergal O'Hanlon's Centaurs, in which amazons may fire from any square along their movement path (including the starting square).

El Juego de las Amazonas (The Game of the Amazons) is a trademark of Ediciones de Mente.

Other Games


Fanorona
Dating back centuries, and known as the national game of Madagascar, Fanorona has two unusual capturing methods called approach and withdrawal (the latter inspired one of the pieces in Robert Abbott's chess variant Ultima).   It is played on the intersections of a 9x5 board (see the empty board and the initial arrangement of pieces above), which

(e.g. a1b2 is legal, but not b1a2).  A move may capture opposing pieces along its line of movement in one of two ways:
(a) Approach capture: if the piece moves next to an opposing piece along its line of movement, it takes that piece and any adjacent opposing pieces along the line of movement.  For example, in the initial position, d2e3 captures f4 and g5.
(b) Withdrawal capture: if the piece moves directly away from an opposing piece or line of pieces along its line of movement, it captures all of them. For example, if White (from the initial position) plays e2e3, capturing e4 and e5, Black may reply f4e5, capturing g3, h2, and i1.

A move may not capture by both approach and withdrawal; if both are possible, the moving player must choose which capture to make. If White (from the initial position) plays d3e3, he may capture f3 (by approach) or c3 (by withdrawal), but not both.   A player must make a capturing move on each turn if possible, but may choose any available capture or series of captures; a non-capturing move (paika) is only allowed if no captures are available.  After making a capture, if the moving piece may move along a different line and make another capture, it may do so on the same turn, and may continue making further captures as long as it is able. These additional captures are not compulsory; a player may stop at any time after making at least one capture.  Each capture must involve a change of direction, and may not reverse direction along the same line just traversed.  A piece making a series of captures may not return to its starting point or any other point it occupied during the turn. Captured pieces are removed after each move in the series, so a piece making a series of captures may later move to a point which was originally occupied by an opposing piece.   The object is to capture all of the opponent's pieces; a draw may be agreed if neither player can force any further captures.

Notation: White opens the game with d3e3:c3.  Black may play d4d3:d2,dl; d3c3:e3; c3d2:el; d2e3:cl (or may stop anywhere during the series of captures).  Since a capture always captures a complete string, there's no need to write all of the pieces taken.  Just indicate whether the capture is by withdrawal (-) or approach (+), and the number of pieces taken. For example, the capture just mentioned is d4d3(+2)c3(-1)d2(+1)e3(-1).


                     

    

Most recently edited on November 5, 2024.
This article is copyright ©2024 by Michael Keller.  All rights reserved.