Real MATHEMATICS – Game #11

Sprouts

Sprouts is a multiplayer game which was created by M. S. Paterson and brilliant J. H. Conway back in 1967. All you need for playing sprouts are just a piece of paper and a pen/pencil.

  • Game starts with 3 dots on a paper:
    20190429_140440.jpg
  • Players take turns and draw lines from one dot to another (a line can be drawn to the same dot as well). Lines don’t have to be straight and a new dot must be placed on each line:
  • Lines can’t cross one another:
    20190429_140912.jpg
  • A dot is called “dead” if it has 3 lines coming out of it. In other words any dot can be connected to at most 3 lines:

    On right: A, B and E have 3 dots. This means A, B and E are all “dead”.

  • Player who draws the last possible line is the winner.

Brussels’ Sprouts

Brussels’ sprouts is a different kind of sprouts game. It is a multiplayer game just like regular sprouts and all it needs are a paper and a pen/pencil as well. But this time game starts with dots that have thorns. Assume that we will start a Brussels’ with two dots with 3 and 4 thorns on them:

20190429_141609.jpg

Players take turn to draw lines between thorns. When a player draws a line, he/she should mark a new dot that has two thorns on it:

Just like regular sprouts, lines can’t cross in Brussels’ sprouts. And the player who draws the last line wins the game:

Euler and Sprouts

You might wonder how on Earth I get to mention Euler in a game that was created about 200 years after he passed away. I recommend you to check Euler characteristics article.

Euler says:

Let’s imagine that V dots and E lines (which don’t cross one another) are sitting on a plane. If the number of faces on this shape is F (don’t ever forget to count the whole plane as one face), then the equation

V – E + F = 2

will always be satisfied.

Take a finished Brussels’ sprouts game on hand:

20190429_142541.jpg

Find the numbers of the dots, lines and faces:

Apply Euler’s formula:

20190429_142635.jpg

Euler will always be right!

Four Colors

Now take any Brussels’ sprouts sheet and color the faces on it. (Neighboring faces have different colors.)

You will see that four different colors will be enough to color any Brussels’ sheet:

One wonders…

  1. At most how many turns can there be in a regular sprouts game that starts with 3 dots?
  2. Is there a winning strategy for sprouts?
  3. Start a Brussels’ sprouts with 3 dots. If each of them has 3 thorns, at most how many turns can there be?

M. Serkan Kalaycıoğlu

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