A square tile, a regular hexagon tile, an equilateral triangle tile, and a regular n-sided polygon tile share a common vertex. There are no gaps or overlaps between the tiles. What is n?

The angle of the n-sided polygon is \(360-90-120-60=90\)degrees. Since a square is the only regular n-sided polygon with angles of 90 degrees, \(\boxed{n=4}\)