Submitted by yonatan zilpa on

## Four bottles - Equidistant mouth problems

How to place four identical bottles in the plane such that the distance between each pair of mouths will be the same?## Solution

Denote the height of each of the (identical) bottles by $h$. Draw a square with side length $h$.
Put each bottle on each of the vertices of the square, where each pair of bottles standing on the same segment stand in such a way that one bottle is standing on the base and the other is standing on the mouth. This way the bottles form a cube, the distances of the mouths are the

**diagonal**of the 6 facets of the cube (see picture). Since the facets are congruent squares the mouths of the bottles are equidistant to each others.**Notice:**this solution was made possible by the fact that there are ${{4}\choose {2}}=6$ possible combinations between the distances, this allow us to select the diagonals of each of the 6 facets of the cube. Note that the solution for 5 bottles require us to consider ${{5}\choose{2}}=10$ combinations of distances. ■