Hydra Game

In mathematics, a hydra game is a single player iterative game played on a mathematical tree called a hydra, in which the player's goal is to kill the hydra by removing its nodes, called heads, one by one, while the hydra simultaneously grows new heads according to a fixed rule, resembling the battle between Hercules and the Lernaean Hydra from which it takes its name. The rules of the game guarantee that the player can always eventually win, but the number of steps required grows extremely rapidly as the size of the starting tree increases, so the game can be used to generate very large numbers or infinite ordinals, or to demonstrate the strength of certain mathematical theories. Unlike combinatorial counterparts such as TREE and SCG, no search is needed to compute how the game plays out: a player need only keep applying the fixed transformation rule to the tree until the game ends.