Once upon a time, in a very far land, there was a king who made a will  when he had three children. In his will, he wrote that the land should be divided into connected regions for each of his children with the condition that each pair of children should be able to visit each other without crossing the land of the other, in other words, every pair of regions should share a piece of boundary, not just a point. Can the land be divided in this way? Later in life, he had another son and did not change the will. Can the land still  be divided in this way? 

What if this  king’s mythical land was like the surface of a donut? or of a pretzel? What is the maximum of connected regions in which a given surface can be divided with the condition that every pair of regions share a piece of boundary?