User blog comment:DWither/Doubt about busy beaver/@comment-5529393-20180904212819/@comment-35606647-20180904221729

By state I didn't mean location of anything, but just the state of the head, input taken and the outputs.

Let's see that example of a Turing machine that keeps going to the right forever in my algorithm.

It can be done with only one state, which would be

If 0: write 1, move to the right, go to state 1

If 1: (doesn't matter because it won't find a 1)

We have our infinite tape with the head represented above it.

v

...0|0|0|0|0...    s=0  According to Step 2.1, we do one step so it turns into

v

...0|0|1|0|0...    s=1   It didn't halt and b=False so we check for the sequences of steps to compare, but s is 1 so n can only go up to 0.5, not even 1.

Go back to the beggining of step 2.1: do one step so it turns into

v

...0|0|1|1|0...    s=2   Now back to comparing. We take the sequences from s-(n-1) to s and compare them to their respective s-(2n-1) to s-n for all n up to s/2, which is only 1 in this case.

the only pair of sequences is 1 to 1 and 2 to 2, which is just comparing step 1 to step 2.

first sequence:

first (and last) step: state 1, input 0, outputs 1, move right and switch to state 1

second sequence:

first (and last) step: state 1, input 0, outputs 1, move right and switch to state 1

They are exactly the same in everything, so now b=True, currs=2 and diff=1

Now that b=True we keep doing steps until s=10^currs, so until until s=100

When we are at step 100, we check all the sequences currs-(diff-1) to currs, currs+1 to currs+diff etc until we reach 100 to 100. Then, we compare them all and find that they are all equal, so it stops trying to do more steps in that turing machine and it goes to whatever is the next combination in whatever BB(n) we are in (BB(1) in this case).