Sprouts | Topological Games

The players take turns in joining dots according to simple rules, until one player cannot make a move.

Description

Start by drawing two or more spots on a piece of paper.
Players then take turns to make a move, according to the following rules:

• Draw a curve joining two spots, or a single spot to itself. The curve must not pass through another spot.
• Draw a spot on the new curve.
• No more than three lines can emerge from any spot.

The last player to be able to move wins.
The game is remarkably complicated, and even starting with two spots leads to an interesting game.

Example

In the following sample game with two spots Blue has the first move, and Red wins after 4 moves because Blue has no move:

History

Sprouts was invented by the mathematicians M. S. Paterson and J. H. Conway, and was analysed in Winning Ways, Academic Press, 1982.

Read More

Gale | Topological Games

Players take turns in linking dots on overlapping grids. The first player to draw a continuous chain linking their ends of the board wins.

Description

To create the board first draw a rectangular array of 4 x 5 blue dots. Then draw an overlapping array of 5 x 4 red dots:


The players take turns in linking two adjacent dots of their own colour. No two links may cross. The first player to form a chain of links across the board, from top to bottom (blue) or left to right (red), wins.
The game can be played with larger overlapping arrays of n x n+1 dots. The game cannot be a draw because, to block their opponent, a player must themselves form a continuous chain.

Example

For example, in the following game blue wins by forming a chain from top to bottom:

 

History

Gale, also known as Bridgit, was invented by the mathematician David Gale.
It was described in "The Second Scientific American Book Of Mathematical Puzzles and Diversions", Martin Gardner, The University of Chicago Press, 1961.

Read More

Cram | Topological Games

The players take turns linking pairs of dots on a grid. The first player unable to move loses.

History

The game was originally proposed by Geoffrey Mott-Smith who called it Plugg, and was described by Martin Gardner as the game Cram in "Mathematical Games: Cram, Crosscram and Quadraphage: New Games having Elusive Winning Strategies." Scientific American 230, 106-108, Feb. 1974.
It is related to the game Domineering, which is identical except that one player can only make vertical moves and the other player can only make horizontal moves.

Description

The game is played on a matrix of dots.
The players take turns in linking a pair of adjacent dots with a horizontal or vertical link. No dot can be linked more than once.
In the normal game the first player unable to move loses.
Alternatively, in the misère version, the first player who cannot move wins.

Strategy

Normal Cram has a simple winning strategy on boards with an even side. On an even-by-even grid the second player can win by making a symmetric copy of each of the first player's moves. On an even-by-odd grid the first player can win by making the first move on the centre two squares, and thereafter making a symmetric copy of each of the second player's moves.

Read More

Domineering | Topological Games

The players take turns linking pairs of dots on a grid. The first player unable to move loses.

History

The game was originally proposed by Göran Andersson, and was described by Martin Gardner as the game Crosscram in "Mathematical Games: Cram, Crosscram and Quadraphage: New Games having Elusive Winning Strategies." Scientific American 230, 106-108, Feb. 1974. It can also be played using dominoes to cover up squares on a checkerboard, hence the name Domineering by which it is now more generally known.
It is related to the less-interesting game Cram, in which each player can make either vertical or horizontal moves.
It was described and analysed in the book "On Numbers and Games" by John Horton Conway, Academic Press, 1976.

Description

The game is played on a matrix of dots.
The players take turns in linking a pair of adjacent dots. The first player, Blue, always makes a vertical link, and the other player, Red, always makes a horizontal link. No dot can be linked more than once.
The first player unable to move loses.

Example

In the following game Red is unable to move, and so loses:

Read More