Why is this TM problem decidable

Question

Please explain to me (like I'm an idiot) how the problem Does M on input Λ ever write a nonblank symbol on the tape? given TM M as an input is decidable.

To clarify, how do I show that the question is decidable

Answer 1

Start simulating the TM. Let $N$ be the number of its states.

If the simulation reaches halt or actually does write a nonblank, we are done with the obvious result.

If the simulation ever erases a previously non-blank field we are done because we may assume by induction that we have decidability for all inputs with less nonblank fields.

If we ever stay to the right of the last (or to the left of the first) nonblank tape position for more than $N^2+N+1$ consecutive steps, we are done: The machine can move at most $N$ steps to the right before repeating its state (note: it sees blank input all the time). If the second occurrence is closer to "home" than the first, it will keep moving closer with a net gain of at least one position per $N$ steps, thus will be back after less than $N^2+N+1$ steps. Thus being "far out" for $N^2+N+1$ steps mean that the repeated state happens at the same or a further ut position; hence the machine will loop indefinitely, either essentially in-place, or drifting away into infinity.

Therefore, the whole simulation need only work with a finite tape, namely the input area, extended to the left and right by $N^2+N$ blank fields. But with everything finite, it is clear that the simulation is eventully periodic.