The weight of a pendulum is released very close to a wall. how many return swings will strike the wall?

The pendulum will not strike the wall. This is because of the principle of conservation of energy.

The pendulum is released close to a wall with no initial velocity. At the point of release, the pendulum has no kinetic energy, but has a potential energy  E. This potential energy is converted into kinetic energy, when it reaches the mean position. The pendulum then swings to the other side and it stops momentarily at the other extreme position, when its kinetic energy is converted into potential energy. It then begins its return swing.

If no energy is lost during the oscillation of the pendulum, when it reaches the position where it was released, its kinetic energy would be entirely converted into potential energy. Therefore, when it reaches the point close to the wall, where it was released,it would have zero velocity. After this point, the pendulum turns back due to the restoring force acting on it, which is directed towards the mean position.

To travel further and strike the wall, it needs more kinetic energy, which is not provided to it. Hence it would never strike the wall on its return swings.