Hydra game

In mathematics, specifically in graph theory and number theory, a hydra game is a single-player iterative mathematical game played on a mathematical tree called a "hydra" where the player's goal is to "kill" the hydra by removing its nodes ("heads") one-by-one while the hydra simultaneously expands itself (this resembles a battle between Hercules and the Lernaean Hydra, hence the name). The rules of the game allow the player to eventually win, but the number of steps required to reach the goal grows very rapidly as the size of the initial tree increases, so the game can be used to generate large numbers or infinite ordinals or prove the strength of certain mathematical theories.

Unlike their combinatorial counterparts like TREE and SCG, no search is required to compute these fast-growing function values—one must simply keep applying the transformation rule to the tree until the game says to stop.