NoGo

NoGo is a 2005 game by John Moore.

This is a game using the concepts and rules of Go, with some simplifications.

Rules:

• On her turn, the player drops a friendly piece on an empty intersection such that all existing groups of pieces (of either color) have, at least, one liberty (so, no captures or suicides are allowed)
• Wins the last player to move

Black moved last; there is no valid drop left,
so Black wins the match

This is a game that can be analyzed by the tools of Combinatorial Game Theory. The game can be split in a sum of games, when parts of the board no longer interact. For example, using a one-dimensional NoGo, the game o..xxo..xx can be split and simplified by two (in this case equal) subgames: o..x and o..x; this type of property is crucial to computing large game positions.

Moore initially called the game Anti-Atari Go at Sensei's Library. The name No Go meant, back then, another different go variant: After a player has made a move, his opponent may refuse it, and he must make another one. The opponent may not object to this second move (this variant is now known as Forced Takeback Go at Sensei's Library).

There's a ZRF and a Ludii to play the game.

Pencil and paper games: Blocking

Another theme is pencil and paper games is blocking, i.e., as the game advances, less and less squares are available, until the stalemated player loses the game.

A well-known example is 1970's Snort by Simon Norton, aka Cats & Dogs. 

The rules: on an 8x8 grid, players take turns marking an empty square, provided that the square is not orthogonally adjacent to an adversary mark. The first player unable to move, loses the game.

To prevent mirror strategies, in the beginning the first player should play in the central 2x2 area, and the second player must play outside that area.

There's also a faster variant, called Obstruction, that includes diagonal adjacencies as illegal moves.

Slimetrail, designed by Bill Taylor in 1993, is also a pencil and paper game with a blocking theme. To recap, in Slimetrail each player has a home, each one staying in opposite corners. The snail mark starts at the middle of the board. Each player, on his turn, shades the snail mark, and places/marks the snail on an adjacent empty square. Wins the player that moves the snail into his home or stalemates the adversary.

Domineering (aka Stop-Gate, Crosscram) is another example. Designed by John Conway in 1976, the game is also played on a square grid, where each player marks two adjacent empty squares with a domino. One player marks vertical dominoes, while the other player marks horizontal dominoes. The first player to be stalemated loses (i.e., the last player to move wins).

Cram is a variant where both players can mark dominoes horizontally or vertically. It is what's called an impartial game in Combinatorial Game Theory. This means that Cram is just the game of Nim with other clothes (I mean, rules).

Chomp, by Frederik Schuh in 1952, and later by David Gale, is another example. The game is played on a rectangular grid. Each player on his turn selects an empty square, and shades all squares that form a rectangle between itself and the grid's bottom-right corner. The player that moves last loses the game.

Blue lost the match after move 11

Another example comes from JeuMok Plus. There are 24 shared 1x2x4 blocks. The game comes with a rulebook containing five different games (one is a NIM game, another is a Jenga), where the main game rules that players drop blocks in empty squares on a board, and each tile must be adjacent to at least another already on board. The board center cannot be used. The player that moves last wins the game.

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 Placer 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.

Babylon

Babylon is a 2003 game by Bruno Faidutti, published at Interlude (among others).

The game does not have a board, only twelve stackable tiles (three tiles for each one of four colors).

Initially, all tiles are placed in the gaming area. Let's denote a single tile as a stack of size one.

Rules:

• On her turn, the player must place an entire stack on top of another.
• A move is valid if either:
• both stacks are of the same height
• both stacks have the top-most piece with the same color 
• Wins the player making the last move

This is a game that seems very amenable for analysis using Combinatorial Game Theory. But I'm not aware of any article studying it.