Five points in spheres Do there exist five points in the euclidean space ${\mathbb R}^3$ such that
every four of these points are in a spherical ball of radius 1, but that the five points are not in a ball of radius 1?
Do there exist five points in the euclidean space ${\mathbb R}^3$ such that
every four of these points are on a sphere of radius 1, but that the five points are not on a sphere radius 1?
 A: If you mean "any four lie in some ball of radius 1", then the same holds for all five points due to Helly's theorem (the unit balls centered in these points must have a common point). 
A: Clearly we can't have three collinear points on a sphere. Look at the "on a sphere" case and assume there is a configuration where they do not all lie on a single sphere.
Any three points define a plane. The locus of points equidistant from these three is a line perpendicular to the plane (through the circumcentre). There are at most two points on such a line which are unit distance from the original three, $P$ and $Q$ say. These are the centres of two unit circles, and one of the remaining two points must lie on each sphere.
Note that $P$ and $Q$ are related by a reflection in the original plane. There are five sets of four points in the original configuration. Each set defines a unit sphere, and if two spheres are the same, then all are. So there are five spheres and the centres are related by reflections in the planes defined by triangles.
[From here is a bathtime intuition which, per comments doesn't work. However note that each pair of centres is related by a reflection, which may not map other centres to centres, and therefore creates an infinite group. I thought I could see the group acting on the centres, but I can't make it work - thanks for comments to put me right]
There is just one group of order 5 - cyclic - and this would imply that the centres of the five spheres formed a regular pentagon. Since this does not provide a suitable configuration, none exists. [Would need three points in each of five planes meeting in a single line, no three collinear]
