11 posts on each side.

We reached 121 square posts with this one!

Think of every post published right now as being in a grid. It’ll probably be a grid with size 11 by 11, which is a perfect square! We did it! Perfect square post!

Fun fact: if you sum up all the digits in 121, you get 4, which is also a perfect square.

This post isn’t about 121 or 11 or even 4, though. It’s one of those math posts again, and it will be good, I hope.

Alright… Fish number 4.

b r u h

Cast your mind back to Fish number 3. We’re going to be introducing a new map function! Yay… The s'(1)f map, which maps a function f to a function g. This map, according to the Googology wiki, is “a busy beaver function for an oracle machine having an oracle which calculates function f. That is, the maximum possible numbers of ones that can be written with an n-state, two-colour oracle Turing machine is s'(1)f(n).”

A mouthful? Yes.

Okay, let me explain. Cast your mind back to the Beaver function post, where I talked about the Turing machine. The maximum number of ones (think of it as lights turned on) that can be written in an n-state (for the beaver: scared, normal, daylight saving, etc.), two colour (black and white, on and off, 1 and 0…) is s'(1)f(n).

Simple? Yes… no.

Define the s'(n)f map to be s'(n-1)^xf(x), where s'(1)^xf is the order-n busy beaver function.

Okay, now define ss'(1)f to be s'(x)f(x). Now, new rule!

ss'(n)f = [ss'(n-1)^x]f(x) for n > 1. Huh?

Basically, it means (I think) iterate the ss'(n) map before the chosen n x times for f(x). Make more sense? I hope so. Now, it’s going to get confusing.

F₄(x) = ss'(2)⁶³f; f(x) = x + 1

In short, this line defines Fish function 4, which takes the ss'(2) map for a function f(x), which, in this case, is x + 1. So, for example, F₄(1) would be ss'(2)⁶³f(1), which is… uhh… I have no idea. Don’t look at me…

Fish number 4 is F₄⁶³(3). This means: iterate the F function 63 times, with the starter input being 3. Simple, right?

Right…

Fun fact: Fish number 4 is the smallest Fish number that is defined using an uncomputable function. This means… Fish function 4 is uncomputable?

I’d think so.