MYSTERIES OF THE EQUILATERAL TRIANGLE - HIKARI Ltd

110 Mathematical Recreations
Figure 4.12: Two-Color Map [123]
Recreation 12 (Two-Color Map [123]). How can a two-color planar map
be drawn so that no matter how an equilateral triangle with unit side is placed
on it, all three vertices never lie on points of the same color [123]?
A simple solution is shown in Figure 4.12 where the vertical stripes are
closed on the left and open on the right [123]. It is an open problem as to how
many colors are required so that no two points, a unit distance apart, lie on
the same color. However, it is known that four colors are necessary and seven
colors are sufficient.
(a) (b)
Figure 4.13: Sphere Coloring: (a) The Problem. (b) Five Colors Are Sufficient.
[131]