Imagine looping each (interior) point's three hexagons in counterclockwise fashion, and counting the transitions of colors (R to B, B to R and so on). the main observation is that any edge between two hexagons (that are not both on the boundary) is counted twice, in opposite orientations. moreover, if the three hexagons meeting at each point are either all the same color (then we have no color transitions) or of two colors (then we have two opposite transtions) then this would imply that on the boundary the number of counterclockwise transitions (say) R to B is the same as B to R. this is not the case as one can count (3 is not 1).