How to play

Tired of playing tic-tac-toe to survive through a boring lecture?
Here is an interesting alternative you can try.

‘Sprouts’ is a pencil and paper game with interesting mathematical properties. It was invented by mathematicians John Horton Conway and Michael S. Paterson at Cambridge University in the early 1960s.

Rules of the game:

The game is played by two players, starting with a certain number of dots (called spots) drawn on a sheet of paper. Two players make alternate moves, which consists of connecting two dots (spots) with a curve and marking a new dot anywhere on this curve. The segments of curves connecting two dots are called edges. The game play can be summarized by the following rules:

1. No edge should intersect another edge.  (You cannot connect two dots if this current edge (current move) intersects any previously drawn edge)

2. No spot (dot) can have more than three lines attached to it. (Another way of saying it would be – max degree of any spot = 3)

3. A curve may connect a spot to itself. (i.e. – self loops are allowed but be careful with rule number 2. You cannot connect more than 3 lines to a dot)

The player who draws the last curve wins.

Let’s look at some examples to understand the rules.

Before that, some terms I have used:

Spot     =   dot     =   vertex

Curve    =   edge   =   line

Degree of a spot (/dot/vertex) = number of lines (edges) connected to it.

Current curve (/edge/line) refers to the curve drawn by a player as his move. If curves c1, c2 and c3 are drawn during a game in the 1^st, 2^nd and 3^rd move respectively then current curve just after 1^st move is finished and before the 2^nd move is made refers to c1. Likewise current curve after 2^nd move refers to c2 and current curve after 3^rd move refers to c3.

Let’s look at a sample game where the players (assume X & Y) start with 2 dots.

Play a few games of sprouts with someone to get the feel of it. You can start with a small number of initial dots (2-3) and increase gradually. You can also experiment and alter the rules to your taste.

It’s not necessary that one who makes the last move will win the game. You can make your own variations. You can reverse this rule to- one who will make the last move will lose. You can try other variations like – making a new dot on the current curve as optional (my fav).

I didn't like the idea of making a new dot on the current curve after every move. Here is what I do when I play with my friends – (assume X & Y are two players playing the game) If Y made a curve (say c1) in his current move then X has the power to decide if she wants to create a dot on c1. Likewise Y has the power to decide if he wants to make a dot on the current curve after X’s move.

This may seem to complicate things. In optimal play (optimal play means X, in her turn, will make a move such that it enhances her chances of winning. Likewise, in Y’s turn, he will make a move which enhances his chances of winning) before making a move the player has to decide if he/she should mark a dot on the current curve, how the outcome will change with/without the new dot, possible outcome(s) with/without new dot after his/her move … in short- a lot of computations in one’s head.

Complicated?

What is the fun in keeping things simple ;)

More about this variant and other variants in the next post

It will be good to try a few hands at the game before reading further.