A pencil, a sheet of paper, two people. In 1967, John Conway and Michael Paterson at the University of Cambridge invented it.
That year is written in a 1991 report from Carnegie Mellon University. The report is called Computer Analysis of Sprouts. Ben Orlin includes the game in Math Games with Bad Drawings.
You already know
Put some dots on paper and join them with lines. You could do that when you were little.
The odd part
Every game comes out looking different, all bends and loops. It looks as if the winner is pure luck.
You will ask
Can anyone really calculate a mess like this?
Learn to play. Three rules.
1
Start by drawing some dots. However many you draw, that is the game you are playing.
2
On your turn, draw a line between two dots, or from a dot back to itself. The line must not touch another line. Then put a new dot on the line. That new dot is a sprout.
3
Each dot can grow at most three lines. A line from a dot back to itself uses two. If it is your turn and there is nowhere to draw, you lose.
What the mathematicians say
Sprouts looks like a scribble, and a pattern is hiding in it. Find the pattern, and you can tell the ending before you play.
Why do you say that?Follow one game, from the start to the end.
OPENING
Count the lives on each dot, and you know the most moves this game can last
Major premise
Each move, the whole board loses exactly one life. The two ends spend two lives, and the new sprout brings one back.
Minor premise
n dots start with 3n lives. The sprout from the last move still has one life, so at least one life is left when the game stops.
Conclusion
n dots last at most 3n−1 moves. One dot is 2 moves. Two dots are 5 moves. The game always ends.
MIDDLE
The bend of a line does not matter. What matters is that a loop splits the paper into inside and outside
Mathematicians look at which dots are joined. Think of the paper as a rubber sheet. Stretch it or squash it. If the joins stay the same, it is still the same game.
Once a line closes a loop, the paper splits into inside and outside. A line cannot cross a line, so a dot inside can never reach a dot outside.
Each loop gives the remaining dots their own rooms. A strong player counts the lives left in each room.
ENDING
Fix the number of dots, and the winner is fixed, in a cycle of six
Play the one-dot game yourself. It ends in two moves, and the second player wins.
Line the numbers up. 1 and 2 go to the second player. 3, 4, and 5 go to the first. 6 goes to the second again. Then the same cycle repeats.
Computers have checked every game from 1 dot through 44, and the cycle never broke. Nobody has proved it keeps going for every larger number.
Opening Each dot has three lives, and each move spends one
Tap me
The game starts with 1 dot. It can grow three lines, so it has 3 lives.
Lives on the board 3
The number by a dot is how many lives it has left. A gray dot has none left.
Middle One loop splits the paper into inside and outside
Think of the paper as a rubber sheet. You can pull the loop into a square, a flat oval, or a wobble. Inside stays inside. Outside stays outside.
Mathematicians call this way of looking topology. They care about what is joined, not about the shape. To them, a doughnut and a mug with a handle are the same thing, because each has one hole.
Try me
Ending Fix the number of dots, and the winner is fixed
Drag me
Lives at the start6 lives
Moves at most5 moves
When neither player makes a mistake, the second player wins
The move limit comes from the lives above. Draw a game and count.
Here are 1 through 18, six in a row. What do you see?
1
2
3
4Tap me
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Orange first player wins.Blue second player wins.
Numbers in the same column leave the same remainder when divided by 6. Remainder 3, 4, or 5: the first player wins. Remainder 0, 1, or 2: the second player wins. 1 divided by 6 leaves 1, so the second player wins, the same as the game you just walked through. From 2 dots through 44, Julien Lemoine and Simon Viennot checked every one by computer in 2011, and the colors match. The 6-dot game was worked out by hand by Denis Mollison, in 47 pages, and written up in Winning Ways. Nobody has proved the pattern for every number of dots.
The lines can tangle,and the pattern stays.
The moment you find it,you see the beauty of math.