By Tamsen Hunter
Special Topics
2003

• Invented in 1967 by John H. Conway, and
Michael S. Paterson
• Paterson and Conway were Princeton
Professors working at the University of
Cambridge
• Conway once wrote, “The day after sprouts sprouted, it seemed that everyone was playing it.”

1. Play with a pen and a piece of paper
2. Starting positions are the dots
3. Draw a loop/line (make sure not to cross any other loop/line)
4. Add a dot to the new loop/line (no dot can have more than three lines attached to it.)
5. The game is finished when you will have to cross a loop/line to add a new loop/line

Step 1
Played with 3 dots
Step 2
Step 3
First Player
Step 5
Second Player
Step 6
Step 4
First Player
Second Player
First Player

Step 7
Played with 3 dots
Step 8
Second Player
First Player wins because there is no move for Player Two.

Step 1
Played with 1 dot
Step 3
Step 2
First Player
Second Player wins because there is no move for Player
One.

• Every game that starts with n dots;
– has a maximum of 3 n -1 moves
• Proof
– Each move uses 2 of the openings; one at the end and one at the beginning of the curve
– At the same time it adds another dot and therefore adds one more opening
– Each time a player goes it decreases the number of openings by one, and a move requires filling two openings
– So the game ends when there is only one opening left

• This means that if a game starts with 4 dots then it must end in at most 11 moves.
• This means that;
– Player One will always win games involving
3, 4 or 5 dots
– Player Two will always win games involving
1, 2, or 6 dots

Try a game with a partner using more than 6 dots and see who wins.