Aregular hexagon rotates counterclockwise about its center. it turns through angles greater than 0° and less than or equal to 360°. at how many different angles will the hexagon map onto itself?

You can draw lines from the center of the hexagon to the peak. And you can find that this hexagon is divided to 6 parts. Every 60° can the hexagon map onto itself. So at 60,120,180,240,300 and 360 total 6 times.