Kirby-Paris hydra

The Kirby-Paris hydra game is a one-player game that can last for an extraordinary number of turns, leading to very large numbers. The game is played as follows:

We start from an arbitrary unlabeled rooted tree T. At step n, Hercules selects an arbitrary leaf vertex of the tree (call it \(\alpha\) and call its parent vertex \(\beta\)), and the following happens:

1. The edge \(\beta\alpha\) is deleted from T.

2. If \(\beta\) is the root of the tree, nothing else happens. If \(\beta\) is not the root, let \(\gamma\) be the parent vertex of \(\beta\), and let \(T_{\beta}\) be the subtree of T with \(\beta\) as the root. T then grows an additional n copies of \(T_{\beta}\) starting from \(\gamma\).

Although it appears that the size of the hydra initially grows quite rapidly, it turns out that, regardless of what strategy Hercules uses, he will always eventually defeat the hydra. Kirby and Paris proved that, while this theorem is true, it is unprovable in Peano Arithmetic.

Let us define Hydra(n) as the number of turns it takes Hercules to defeat a hydra consisting of a path of length n, assuming he always cuts the rightmost edge each step. Then

Hydra(0) = 0 Hydra(1) = 1 Hydra(2) = 3 Hydra(3) = 37 Hydra(4) \(> f_{\omega*2 +4}(5)\) > Graham's number Hydra(5) \(> f_{\omega^{omega*2 + 4}}(5)\)

and so on.