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In a regular hexagon, if all 6 corners are red, then we have an equilateral triangle.

If 1 is blue and 5 are red, we still have such a triangle.

To get 4 red and 2 blue we can add one more blue one step clockwise away from the first blue, but then we have one blue pair but also some red pairs, and either of those pairs will make an equilateral triangle with the center point, which must be either red or blue.

If we instead add one more blue two steps clockwise away from the first one, then we are back to the same kind of triangle that we had before.

If we instead add one more blue three steps clockwise away from the first one, then we get 2 opposite blue points and 2 red pairs on both sides of the line connecting the blue points. Now, let the center point of the hexagon be mirrored in the line connecting a red pair. That new point makes a triangle with those two red points, but also with the two blue points, and since it must be either red or blue we again have a triangle.

Trying to instead add one more blue at any other place is just a mirror image of what we have already tried.

So all that is remaining now is 3 red and 3 blue (because 2 red is equivalent to 4 red, etc.). With 3 of each we have two possibilities. Either we put every second red and every second blue, which clearly gives us triangles.

Or, if we don't put them like that, we will get pairs of red, and pairs of blue, either of which will make a triangle with the center point.