# Continuous Coloring

Geometry Level 5

Let $$T$$ be an equilateral triangle of side length $$1$$, whose vertices are located at $$(0,0), (1,0)$$ and $$\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$$. Let $$(f_i), i=1,2,3, \ldots$$ be continuous functions such that, $f_i : [0,1] \to T$ i.e., each point in $$[0,1]$$ is mapped to some point in the triangle $$T$$ by each function $$f_i$$.

Think of each $$f_i$$ as a pen with color $$i$$. So that each $$f_i$$ colors its range (which is some portion of the triangle $$T$$ ) with the color $$i$$. E.g., if $$f_1(1/3)=(\frac{1}{2},\frac{1}{2})$$, this means that the point $$(\frac{1}{2},\frac{1}{2})$$ is colored '1' by the function $$f_1$$.

Find the minimum number of continuous functions $$f_i$$ needed to color the entire triangle.

×