Continuous Coloring

Geometry Level 5

Let TT be an equilateral triangle of side length 11, whose vertices are located at (0,0),(1,0)(0,0), (1,0) and (12,32)\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right). Let (fi),i=1,2,3,(f_i), i=1,2,3, \ldots be continuous functions such that, fi:[0,1]Tf_i : [0,1] \to T i.e., each point in [0,1][0,1] is mapped to some point in the triangle TT by each function fif_i.

Think of each fif_i as a pen with color ii. So that each fif_i colors its range (which is some portion of the triangle TT ) with the color ii. E.g., if f1(1/3)=(12,12)f_1(1/3)=(\frac{1}{2},\frac{1}{2}), this means that the point (12,12)(\frac{1}{2},\frac{1}{2}) is colored '1' by the function f1f_1.

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

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