What strategy can you use to color using only 6 colors the lines in an infinite hexagonal tiling such that no two sides of the same hexagon have the same color?
All the sides of the tiling are aligned in one of three directions. Simply assign two colors two each direction, and alternate the colors in every infinite row of sides that are "side by side". This guarantees that no hexagon will have two sides of the same color.
For example, having one row alternate between black and red, one between green and blue, and one between gold and violet: