User blog:Edwin Shade/The Challenge Of Minimum Rings

Suppose you have a rod, and upon it is a series of rings. They come in a multitude of hues, and your job is to remove all the rings from the rod. You must however, obey these rules:

1.) You may only remove one ring at a time, from the top.

2.) Every time you remove a ring of a given color it is to be replaced with a number of rings of a lesser color equal to the score, so for instance if your score is 37 and you take off a green ring, you have to replace it with 37 blue rings, since blue is one color lesser than green.

3.) A red ring gives you 1 point and won't be replaced with any other rings.

4.) The colors in ascending order are red, blue, green, yellow, and so on as far as you like.

5.) Your starting score is 0.

If you have observed by now this is just the fast-growing hierarchy phrased differently, then congratulations ! That isn't all though, because there is now a question I ask, namely, what is the minimum number of rings needed to demonstrate step by step the process of the rod emptying, if the rod has a yellow ring with three red rings above it ?

Now don't be fooled, the question is more challenging than it looks. We know the final score will be $$f_3(3)$$, but that doesn't mean we need $$f_3(3)$$ rings total, as we can just reuse the rings we've already used in the beginning in further steps.

(As a footnote, I want to demonstrate to a friend the rules of the fast-growing hierarchy without the math, so far I've made 46 paper rings of various colors so I can show how a number like $$f_{\omega+3}(3)$$ would be calculated.)