There are n lines in a plane ( may or may not be concurrent ) . Find out the maximum and the least number of regions that can be formed ?

What is the maximum number of regions can be there by connecting six points on a circle?

Example

A cuboid of dimension 7 cm * y cm * 15 cm is given. Find the minimum value of y so that at least two cones of maximum volume having some dimensions and a height of 15 cm, can be cut from this solid.

A unit sphere (radius = 1) is out on a flat plane in the rain. Find the side length of the largest cube that can hide underneath it and not get wet.