## FANDOM

10,843 Pages

This is an attempt at rewriting Wythagoras' dollar function as a hydra.

The dollar function $$\(H)$$ takes one argument, a hydra (tree) where the root node and all the leaf nodes are labeled with nonnegative integers.

## Basic rules

At each step we mutate the hydra until it is reduced to only a root node. The output of the hydra is the label of the root node at this step.

Let $$n$$ be the label of the root node, and let $$X$$ be the leftmost node of $$T$$, or the first leaf. Let $$P$$ represent its parent.

1. If $$P$$ is the root node and $$X$$ has label $$a$$, chop off $$X$$ and relabel the root node to $$n + a$$.
2. If $$P$$ is not the root node:
1. If $$X$$ has label 0:
1. Chop off $$X$$.
2. If $$P$$ is a leaf node now, label $$P$$ with $$n$$.
2. If $$X$$ has label $$a + 1$$ for nonnegative $$a$$:
1. Relabel $$X$$ with $$a$$.
2. Add $$n - 1$$ copies of $$P$$ to the right of $$P$$.

We note that the dollar function up to this point is known to be at the $$\varepsilon_0$$ level, as is the suspiciously similar Kirby-Paris hydra.