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?

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When a hexagon maps onto itself, their vertices must map to vertices and sides to sides. Please look at the picture attached: the nearest vertex for the A1 is A2. There are 6 angles between neighbour vertices, they all are equal (because a hexagon is regular) and their sum is 360°. Thus each angle has a measure of 360°/6=60°.

Each subsequent rotation by 60° also maps a hexagon onto itself. There are 5 such rotations: by 60°, 120°, 180°, 240° and 300° (the next is 360° which isn't allowed by the conditions). So the answer is 5.

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