When an object penetrates into granular matter near a boundary, it experiences a horizontal repulsion due to the intrudergrain-wall interaction. Here we show experimentally that a square cuboid intruder, released from rest with no initial velocity, near a vertical wall to its left, goes through three distinct kinds of motion: it first tilts clockwise, then "slides" away from the wall, and finally tilts counterclockwise. This dynamic highly favors both repulsion and penetration of the cuboid intruder as compared to that observed from its release farther away from the wall.