1. **Problem Statement:** An artist designs a logo using a regular hexagon with identical star icons at each vertex. We need to find how many times the logo appears unchanged when rotated clockwise around its center before completing a full 360° rotation, and list all such rotation angles between 0° and 360°. 2. **Formula and Concept:** A regular hexagon has 6 vertices equally spaced around a circle. Rotating the hexagon by the angle between adjacent vertices maps the hexagon onto itself. The angle between adjacent vertices is given by: $$\text{Angle} = \frac{360^\circ}{6} = 60^\circ$$ 3. **Important Rule:** The logo appears unchanged at rotations that are multiples of this angle, excluding the full 360° rotation (which is the starting position). 4. **Calculations:** The rotations where the logo appears unchanged are: $$60^\circ, 2 \times 60^\circ = 120^\circ, 3 \times 60^\circ = 180^\circ, 4 \times 60^\circ = 240^\circ, 5 \times 60^\circ = 300^\circ$$ 5. **Number of times unchanged:** There are 5 such rotations before completing a full 360° rotation. **Final answer:** The logo appears unchanged 5 times at the angles $60^\circ, 120^\circ, 180^\circ, 240^\circ,$ and $300^\circ$.