Many Caves
Let a concave angle be an interior angle in a concave polygon that measures more than As shown below, a quadrilateral can have at most 1 concave angle, a pentagon can have at most 2 concave angles, and a hexagon can have at most 3 concave angles.
What is the most number of concave angles a dodecagon (12 sides) can have?
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1 solution
Consider concave angles which are just slightly more than 1 8 0 ∘ . Putting those angles adjacent to each other, you end up with a (nearly) straight segment. Since you need at least 3 segments to make a polygon (and a triangle is always convex), you need at least 3 angles < 1 8 0 ∘ to make a concave polygon, regardless of how many sides it has.
Therefore, if a concave polygon has n sides, then the most number of concave angles it can have is n − 3 . In the case of a 12-sided polygon, the most number of concave angles it can have is 9 .