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Brussels Sprouts - A Pencil And Paper Game

Lesson Plans > Mathematics > Problem Solving > Strategy Games
 

Brussels Sprouts - A Pencil And Paper Game

Sprouts is a pencil and paper game designed in 1967 by John H. and by Michael S. Paterson. This game is described in the article Sprouts on this site. Brussels Sprouts is a very simple variation on this game.

In "Brussels Sprouts", the two players begin by drawing a random collection of X's (crosses) on a blank sheet of paper. The number of crosses drawn will effect the length of the game, with a larger number of crosses resulting in a longer game.

After drawing the crosses, the paper should look something like this:
 



The first player now draws a line which connects any two cross-bars of any two crosses. Then the first person draws a "cross-bar" somewhere on the line which was just drawn. A sample first move is shown below:
 


The second player continues by drawing another line connecting two cross-bars, and drawing a new cross-bar. The line drawn cannot intersect the first line.


Play continues in this fashion until no new line can be drawn without crossing any other line. The winner is the last person who can draw a line.

Although this game appears to be quite similar to the game of Sprouts, there is an interesting difference. In Sprouts, the number of available "dots" is always decreasing because each line uses two dots while creating only one. But in Brussels Sprouts, the number of available cross-bars remains constant. Each line uses two cross-bars, but creates two new cross-bars. Thus, when the game ends there are exactly as many cross-bars as when the game began. But none of these cross-bars can be connected to any others.

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