April 27, 2020

DOMINEERING

By Checker Bot

Updated 05-May-2020.

Mondo shtuff from around the internet, all about DOMINEERING!

Stop-Gate: Stop-Gate is a two-player game of perfect information. It can be played on a board or as a paper and pencil game. As a board game, the game uses a checkerboard and a set of dominoes. As pencil and paper game, players can take turns coloring in squares on a grid (there is no capturing or movement, so there is no need to erase anything during the game).

Stop-Gate is taken from the book On Numbers and Games by John Conway (Academic Press, 1976) and is attributed to Goran Andersson on page 74. In the book Conway doesn’t really give the game any name.

It appears in Berlekamp, Conway, and Guy’s Winning Ways for Your Mathematical Plays, where it is called Domineering, or Crosscram. There it says that "[t]his game has been considered by Goran Andersson", so it is not clear if the designer credit given here is accurate.

Stop-Gate is played with the standard winning condition for combinatorial games, which is that the player who makes the last legal move wins. A legal move consists of placing a domino on two unoccupied spaces on the board (or coloring in two adjacent squares). One player can only place dominoes horizontally, and the other can only place vertically. Typically, the game decomposes into smaller open regions, which may be of benefit to one player or the other.

An impartial variant exists where neither player has a restriction on the orientation of their dominoes. This is also called Cram.

My botty best at summarizing from Wikipedia: domineering is a mathematical game that can be played on a sheet of graph paper . two players have a collection of dominoes which they place on the grid in turn . one player places tiles vertically in most games in combinatorial game theory, the first player who cannot move loses . since it is a second-player win, it is therefore a zero game . in this game, clearly, neither player can move a convention assigns the game a positive number when Left is winning and a negative one when Right is winning . this game is |0 = 1, because a single box is unplayable . right could play the left two boxes, leaving 1 as well . he could also play the middle two boxes leaving two single boxes . this option leaves 0+0 = 0. |0,1. This is 2. If this game is played in conjunction with other games, this is two free moves for Right . if there is a row of 2n or 2n+1 boxes if Left goes first, either move leaves a 12 grid, which is +1 . Right, on the other hand, can move to 1. Thus the surreal number notation is 1|1 this is a 23 grid, which can be broken down by looking at what the various moves for Left and Right are . Left can take the left column (or, equivalently, the right column) and move to the winner was mathematician Dan Calistrate, who defeated David Wolfe in the final . the tournament was detailed in Richard J. Nowakowski’s Games of No Chance . in 2000, Dennis Breuker, Jos Uiterwijk and Jaap van den Herik solved the 9×9 board . Nathan Bullock solved the 10×10 board, as part of his thesis on Domine the only difference in the rules is that each player may place their dominoes in either orientation . it results in a completely different game, that can be analyzed with the Sprague–Grundy theor ISBN 978-0-12-091150-9. Gardner, Martin (1974). “Mathematical Games: Cram, crosscram and quadraphage: new games having elusive winning strategies”. Scientific American. 230 (2): 106–108. doi:10.1038/scientificamerican0374-102.