# Is it decidable whether a given Turing machine moves its head more than 481 cells away from the left-end marker, on input ε?

So, while reading some problems on decidability, I came across the following resource: https://www.isical.ac.in/~ansuman/flat2018/tm-more-undecidable.pdf

Here, on page no 12, it is written that the problem is decidable and with the following argument:

"Yes, Simulate M on for upto m^481 · 482 · k steps. If M visits the 482nd cell, accept, else reject."

I am quite confused with the step count. Can anyone please explain what does this mean, or maybe point to some resources where I can find a proper explanation!!!! Image of the slide

In particular, if a configuration $$c$$ yields a runs that gets back to $$c$$, at some point, then the run is stuck in a loop (note that this is not a necessary condition, a run might not halt while never repeating a configuration!).
Now, consider all the possible configurations that use up to 481 cells. There is a finite amount of those, namely $$m^{481} \cdot 482 \cdot k$$ (where $$m$$ is the size of the alphabet, and $$k$$ is the number of states).