Colored nonagon problem

Let the vertices of a regular 9-gon be colored black or white.

(A)Show that there are two adjacent vertices of same color

(B)Show that there are three vertices of same color forming an isoceles triangle.

4 years, 1 month ago

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For (A), if two consecutive vertices are not found having the same colour, the worst case scenario is when $$8$$ vertices are alternating between black and white. In this case, the $$9^{th}$$ vertex will share a colour with one of its two neighboring vertices.

For (B), let a point $$A$$ be the reference vertex. From $$A$$, no point on its left must be equidistant to it as a point from its right, otherwise the two points and $$A$$ will form an isosceles triangle.Thus, let us assume that the vertex adjacent to $$A$$, to the left, is the same colour as $$A$$. Thus, a point to the right of $$A$$ can be at least $$2$$ vertices away. This pattern continues until the $$4^{th}$$ and $$5^{th}$$ vertices from $$A$$. By the pattern, they share the same colour as $$A$$, and thus, form an isosceles triangle. Also, in the case of $$3$$ or more consecutively coloured points, there is always an isosceles triangle of the same coloured vertices.

- 4 years, 1 month ago

Nice job!!

- 4 years, 1 month ago

Thank you!

- 4 years, 1 month ago