A regular 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?

Hey there Caleb!

To start of, the sentence was clear on that it said . . . 

[tex]\boxed{A \ regular \ hexagon \ rotates \ counterclockwise \ about \ its \ center }[/tex]

The statement listed above is the (key) point. We need to keep this in mind.

So, now that we know this.

We would use the mathematic (Ф)

So,sense we know that a (hexagon) had it's (6) shapes, we would do . . . 

Ф = 360° : 6 = 60°

Your correct answer to this question would be . . . 

[tex]\boxed{\boxed{60}}[/tex]

Hope this helps.
~Jurgen