Sprouts is an interesting paper and pencil game for two players. It was invented in Cambridge in the 1970's.

To begin, mark five large dots on a piece of paper. The two players take turns to draw a line joining two of the dots together, subject to the following restrictions.

  • The line must not cross itself or any other line.
  • No dot may have more than three lines joined to it.
The player then makes a new dot at any position along this line, and play then passes to the opponent. This process is repeated until one player cannot draw a valid line: this player is the loser.

Sprouts is an interesting problem in topology. At the start of the game three lines can be drawn to each of the points, so there are fifteen open 'connections'. On each turn, two connections are removed, and an extra one created, hence the maximum number of turns will be fourteen. The strategy to Sprouts comes from isolating the connections from one another, to reduce the total number of valid moves. Experienced players may use more than five dots to begin with.

Sprouts is a paper and pencil game which appears deceptively simple, but can actually tax the player's two-dimensional topology skills to an immense degree. It was not invented in Cambridge in the 1970's, but was first described (to the best of my knowledge) in the science fiction novel Macroscope written in 1969 by Piers Anthony.

The rules are quite simple:

  • Draw any number of dots at random on a blank piece of paper
  • An open dots is a dot with two or less line segments protruding from it (the dot a player draws on the middle of their line starts with two segments protruding from it)
  • Each player takes turns drawing a single line from one open dot to another without crossing or touching any other line
  • The last player to be able to draw a legal line is the winner
That's it - simple, huh? Actually, that's where the math starts. The game is simple with three dots, which is pretty much the minimum allowable for any type of strategy to be employed, but gets exponentially more complex with each dot added. For each dot, the maximum number of lines can be calculated by
((dots-1) x 3) + 2
Thus with 1 dot you can draw 2 lines; with 2 dots, 5 lines; with 3, 8; with 4, 11 - and so on. The number of line segments grows in a linear way, but the strategy grows more difficult exponentially.

Players can block dots or groups of dots by looping a line around the dots, thus isolating them. Starting with an open field, one loop can subdivide that field into two fields containing smaller amounts of dots. The complexity doubles then and more, as the next player has to deal with two semi-isolated dot fields plus the dot or dots which are on the boundary of the two fields. Each subfield can also be subdibided, quickly adding to the difficulty of the game.

The game is best learned by starting with three dots and moving up a bit at a time. The gameplay quickly becomes engrossing to anyone with interests in mathematics, logic, and reasoning. The organic-looking looped patterns formed by the lines give rise to the name sprouts. I highly recommend it (and Macroscope too.)

Thanks to The Alchemist for clearing up the origin of sprouts. It was indeed invented at Cambridge, but in 1967 by "princeton mathematician" Conway and John Paterson. Piers Anthony must then have had exposure to it while writing Macroscope.
www.sciencenews.org/sn_arc97/4_5_ 97/mathland.htm

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