> The four color theorem states (informally) 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.” (from wikipedia). The mathematics statement is that any planar graph can be colored with four colors. Thus, the first part of the “warm up” has a solution; in fact the world map can be colored with four colors. The four color theorem is deceivingly simple- it can be explored by a kindergartner, but it turns out to have a lengthy proof. In fact, the proof of the theorem requires extensive case checking by computer. Not every map can be colored with three colors (for an example illustrating why see here). It is therefore natural to ask for a characterization of which maps can be 3-colored. Of course any map can be tested for 3-colorability by trying all possibilities, but a “characterization” would involve criteria that could be tested by an algorithm that is polynomial in the number of countries. The 3-colorability of planar graphs is NP-complete.
This is the background and context for the problem for kindergarten. It would be good if educators and writers could provide a version of this for young children and their teachers.