Turing machines moving left at least once Is the following language decidable?
  $$ L = \{ \langle M,w \rangle \mid \text{$M$ moves its head left at least once when run on $w$}\}. $$


I feel like this is a decidable language. But I don't know the algorithm to prove that it is decidable. Any help would be appreciated.
Let's assume for simplicity that at each step, the head moves either left or right (but doesn't stay in place).

Suppose that $M$ never moves its head left when run on $w$.
After the machine traverses the length of $w$, it reaches a blank cell, and from this point on, will always fall on a blank cell. Therefore the only information that changes from step to step is the current state of the machine. After at most $|Q|$ steps (where $Q$ is the set of states), the machine will either halt or repeat a state, in which case it enters an infinite loop.

This gives a simple algorithm for deciding your language, that I'll let you spell out.

If the head is also allowed to stay put, the argument becomes a bit more complicated, but the conclusion is identical.