The Three-Color Theorem

"In mathematics, the four color theorem, or the four color map theorem states that, given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. Two regions are called adjacent only if they share a border segment, not just a point." (Wikipedia) However, if all the shapes are rectangles, and each rectangle is either wholly contained within another rectangle or shares all of its vertices with adjacent rectangles, then three colors are enough, as demonstrated in this woven-patterned tie from Bachrach.