How to play (2 player game):
Draw two (+) signs or crosses on a piece of piece of paper. One move entails connecting the arm of one cross (tine) to an available tine on the same cross or on a different cross with a smooth curve, then drawing a short line to bisect this connecting line—creating two more tines to replace those you used up. The curves cannot touch or cross other curves on the plane or come too close to the edge of the paper. The objective is to be the last player to make a valid move so that your opponent is unable to complete his/her turn, alternating between players each turn.
Although it is not immediately apparent in a two player game, the number of moves in such a round of “Brussels Sprouts” is fixed at 8; therefore, the player to move second should always be the winner. Although this epiphany renders strategy obsolete and spoils any element of fun, the game is nonetheless intriguing—and not just in the realm of gambling, a usage that is not recommended under conventional standards of fair play. In fact, the trick of predicting outcomes in “Brussels Sprouts”–which applies to both the traditional two-player game and to games with varying numbers of crosses; requires a logical mindset and a knowledge of mathematical concepts.
At first glance, it appears as though the game should be able to continue on infinitely due to the fact that, each time a player uses up two tines, these expended tines are replaced by two fresh tines. However, when looked at in terms of space partitioning and regions, one can see that seemingly “open” tines become systematically inaccessible, whether or not one consciously attempts to employ strategy. Assuming that crosses are used as opposed to figures with five or more tines, the number of moves in one game turns out to be exactly 5n – 2, where n is the starting number of crosses. Naturally, an odd number of turns corresponds to the victory of the player who moves first, whereas an even number of turns corresponds to the victory of the player who moves second.
Furthermore, the game of “Brussels Sprouts” is intrinsically linked to the Euler characteristic. The formula for the Euler characteristic of a planar graph, 2 = i – e + r, where “i” is the number of intersections, “e” is the number of edges, and “r” is the number of regions; can also be used to predict the duration of the game.
http://www.madras.fife.sch.uk/maths/games/brusselsprouts.html
http://www.math.cornell.edu/~mec/2006-2007/Games/sprouts.html
–See Page 129
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