Decomposition
Consider a convex polygon having vertices, . We arbitrarily decompose the polygon into triangles having all the vertices among the vertices of the polygon,such that no two of the triangles have interior points in common. We paint in black the triangles that have two sides that are also sides of the polygon, in red if only one side of the triangle is also a side of the polygon and in white those triangles that have no sides that are sides of the polygon. Find the difference between the number of white and black triangles.
The answer is 2.
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1 solution
Let there be b black triangles, r red triangles and w white triangles.
Note that for any decomposition of an n-gon, the sum of angles in all the triangles is equal to the sum of angles in the n-gon. ( b + w + r ) ∗ 1 8 0 ∘ = ( n − 2 ) ∗ 1 8 0 ∘ b + w + r = n − 2
Considering sides of the polygon, 2 b + r = n
Subtracting the two equations, b − w = 2