There is a puzzle:
Find a finite set of points with the following property:
1. Points are placed in space (not on a single plane)
2. If you take any pair of points A and B there are different points C and D such that AB segment is parallel to CD and doesn't lay on the same line.
Known solution to this problem:
Take a regular hexagon. Obviously it satisfies rule #2. Now take another hexagon, which has common center O with the first one. This will be the required set of points.
Indeed, if you take A and B on the same hexagon then there are C and D on it; if you take A and B on different hexagons then take C symmetric to A relatively to O and D symmetric to B relatively to O, obviously ABCDO are in the same plane and AB || CD.
I asked myself a question: what if we want to minimise number of points in the set? What is the minimal number to build a set with the described properties? The given solution allows you to get a set with
6+6-2 = 10 points. Just make 2 points of the hexagons coincide.
my question is: Can you find a smaller number points or to prove that this is not possible?