2003 Indonesia MO Problems/Problem 6

Problem

A hall of a palace is in a shape of regular hexagon, where the sidelength is $6 \text{ m}$ . The floor of the hall is covered with an equilateral triangle-shaped tile with sidelength $50 \text{ cm}$ . Every tile is divided into $3$ congruent triangles (refer to the figure). Every triangle-region is colored with a certain color so that each tile has $3$ different colors. The King wants to ensure that no two tiles have the same color pattern. At least, how many colors are needed?

$[asy] pair C=(25,14.434); draw((0,0)--(50,0)--(25,43.301)--(0,0)); draw(C--(0,0)); draw(C--(50,0)); draw(C--(25,43.301)); [/asy]$

Solution

First, count the number of tiles in the hall. The hall can be made up of six equilateral triangles with side length 600 cm. Since $600/50 = 12$ equilateral triangles with side length 50 cm can make up a side of the larger equilateral triangle, there are $12 + 2(11 + 10 + 9 + \cdots 1) = 144$ smaller triangles in each large triangle. In total, there are $144 \cdot 6 = 864$ tiles needed.

Let $x$ be the number of colors used. Given three colors, there are only two tiles using the color scheme that do not have the same color pattern. With this information, we can write an equation. $\begin{align*} 2 \binom{x}{3} &\ge 864 \\ 2 \cdot \frac{x(x-1)(x-2)}{6} &\ge 864 \\ x(x-1)(x-2) &\ge 2592 \end{align*}$ With some trial and error, we find that the King needs $\boxed{15}$ different colors.

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

2003 Indonesia MO (Problems)
Preceded by Problem 5	1 • 2 • 3 • 4 • 5 • 6 • 7 • 8	Followed by Problem 7
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2003 Indonesia MO Problems/Problem 6

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