The Three-Color Triangle Problem (Essays in Memory of the Late Professor Yukio Ito)

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  • <Article> The Three-Color Triangle Problem (Essays in Memory of the Late Professor Yukio Ito)
  • The Three-Color Triangle Problem

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Abstract

Place elements in the form of an inverted triangle, and color them from top to bottom by row according to some rule. When doing so, is it possible to predict the color of the element that becomes the triangle's botttom vertex? It turns out that when you begin with a triangle of 4,10,28, ... elements in its top row, knowing the colors of that row's leftmost and rightmost elements in sufficient to predict the color of that bottom vertex, and this article presents an elegant proof of this surprising result. Coloring triangles is simple even for children, but this problem provides a springboard to learning about some advanced mathematics, including Abelian groups, the superposition principle, and fractal structures.

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