roza / hydra

The combinatorial Hydra game

The hydra game

The Hydra is a rooted tree. The object of the game is to chop off all its heads (the blue disks). At step n, when you cut a head, the Hydra will grow n new copies of the tree growing from the neck at which the head was cut.

A theorem by Paris and Kirby [1] states that you always win, no matter how you chop the heads, but it takes a rather long time to chop down a hydra. Paris and Kirby also showed that Peano arithmetic does not prove that hydra always loses.

The Hydra game is implemented in Java. I have released the source code into the public domain. Please send me any enhancements you implement on top of my code.

You may run the program in several ways:

1. Run the game as an applet in your browser (you may have to convince the brower that this is not a secuirty risk).

2. Download the JAR file hydra.jar and run it on your computer by double-clicking on it or from the command line

   java -jar hydra.jar
   

3. Run HydraWindow from your IDE if you know what that means

4. Change to the repository directory, compile the program

    javac -d bin -sourcepath src src/hydra/HydraWindow.java
   

   and run it with

    CLASSPATH=bin java hydra.HydraWindow
   

References

[1] Laurie Kirby and Jeff Paris: Accessible Independence Results for Peano Arithmetic. Bull. London Math. Soc. 1982; 14: 285-293.