Consider 6 people $p\_i$, $i=1,\dots 6$, standing on a sphere $S^2$. We label the positions of these people by $p\_i$ again. Suppose no pair of these points $p\_i$ are antipodal. At each point $p\_i$ each person divides the sphere in two 4 identical regions by two orthogonal planes passing through the center of the sphere. The four regions $R\_i^j$, $j=1,2,3,4$ can be uniquely ordered in counterclockwise order. Assume the person $p\_i$ sees everyone standing in regions $R\_i^{1}$ and $R\_i^{3}$ green and everyone standing in regions $R\_i^{2}$ and $R\_i^{4}$ red or vice versa. Indeed the odd number and even number regions are seen the same color. Also assume that no one is standing exactly on the plane that any other person has used to divide the sphere into 4 regions.


Can we show that there exists two people $p\_i, p\_j$ such that they both see each other the same color?



Six people standing on earth
It is possible for all pairs of these people to see each other in different colors.


Consider six people all on the arctic circle, with longitudes 0°, 61°, 122°, 183°, 244° and 305° east of the prime meridian. Suppose they all color the directions northwest and southeast of them red, and they all color the directions northeast and southwest of them blue.


The arcs connecting these people are geodesics which go somewhat north of the arctic circle, and not to the north pole. So when two of these people look at each other along these arcs, one sees the other to the east in blue, and one sees the other to the west in red.


