Consider the following region: It is bounded by a regular hexagon whose sides are of length 1 unit. Show that if any 7 points are chosen in this region (hexagon), then 2 of them must be no further apart than 1 unit.
If I draw a line segment from the centre to each vertex how to prove that 2 points MUST BE NO FURTHER APART THAN 1 UNIT length..
From the center of the hexagon, draw a line to each vertex. This will partition the hexagon into 6 equilateral triangles, each with side of length 1. If 7 points are chosen, then there must be 2 points being in a same triangle according to pigeonhole principle and the distance between these two points should not be greater than 1. Please see the figure below.