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There are 9 cities in a country. Some of them are connected by the mail flights. The number of stamps on an envelope should be the same as the number of flights needed to deliver the letter from one city to another (the itinerary used should require the least number of flights). It is known that even if two cities don’t have direct flight connections, it is always possible to send a letter from one city to another. 

On New Year's Eve the Mayors of all these cities have sent each other Greeting Cards.

What’s the biggest number of stamps all of them might need?

 

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    The least efficient configuration is if $A$ is only connected to $B$ and $B$ is connected also to $C,$ (an nowhere else) and all the other connections are in one line. – 2017-02-21
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    The gist of the proof for why that is the least efficient follows from the fact that the graph formed by the cities and connections is connected and so there exists a [spanning tree](https://en.wikipedia.org/wiki/Spanning_tree). The number of stamps needed can only increase by removing some edges, so the number of stamps needed for the spanning tree is greater than or equal the number of stamps actually needed, but the stamps needed for a spanning tree is less than or equal the number of stamps needed for the graph Doug describes (*requires a bit of explanation as to why*). – 2017-02-21

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