I came across below problem in this pdf:
Given a TM M, whether M ever writes a non blank symbol when started on the empty tape.
Solution given is as follows:
Let the machine only writes blank symbol. Then its number of configurations in the com computation on w is q × 2, where q is the number of states of M; the factor 2 is for the choices re. the direction of heads movement; there is no factor for the written symbol because that is always blank. So the problem is decidable, decided by the following machine: input (M,w), run M on w for q × 2 steps; if it M ever writes a non blank symbol, stop with yes answer; if M never writes a non blank symbol, stop with no answer
Q1. How be sure all q x 2 configurations will happen while running q x 2 steps on w? Some configuration may get repeated in q x 2 steps.
Q2. Question says "when started on the empty tape", but the answer tried to simulate TM on non empty string w. How does it makes sense?