Kill the hydra!

Click on a red leaf node to chop it off. The hydra will regrow according to the Kirby-Paris rules.

Rather surprisingly, the hydra is always defeated in a finite number of turns! Even though the hydra seems to regrow nodes exponentially, the depth of the tree never actually increases. Eventually only the root node remains and the battle ends.

Rules The game starts at turn $t = 1$.
Each turn, until the hydra is defeated:

  1. Choose a leaf node $V$.
  2. Delete $V$ from tree.
  3. Then, attach $t$ copies of the subtree rooted at $V$'s parent to $V$'s grandparent*.
  4. Increment $t$ by $1$.
*If the grandparent does not exist, do nothing. (In this simulation: if reduced growth is on, only one copy is attached.)

These rules guarantee termination. However, this cannot be proven in Peano Arithmetic as the proof requires transfinite induction on large ordinals.


References