n people know each a different piece of gossip.

They can phone each other and exchange all they know so that after the call the parties know anything that either of them knew before the call.

What is the smallest number of calls needed so that everyone knows everything and how is this number achieved?

(In reply to re(2): Solution by Hugo)

Hugo,
I applaud you on finding the solution, yet in your post, "Solution", you infer that four persons are to designated A, B, C, D and any additional persons assigned numbers. This was unclear, and your example lacked explanation in making it understood clearly. I assume your example for 5 people was meant to infer the following:

Let p{g} be the gossip known by person p such that {g} is the set of gossip elements known by him. Let {*} represent the complete set of all the gossip elements.

A{A}, B{B}, C{C}, D{D}, 1{1}
1. A calls 1:
   A{A,1}, B{B}, C{C}, D{D}, 1{A,1}
2. A calls B:
   A{A,B,1}, B{A,B,1}, C{C}, D{D}, 1{A,1}
3. C calls D:
   A{A,B,1}, B{A,B,1}, C{C,D}, D{C,D}, 1{A,1}
4. A calls C:
   A{*}, B{A,B,1}, C{*}, D{C,D}, 1{A,1}
5. B calls D:
   A{*}, B{*}, C{*}, D{*}, 1{A,1}
6. A calls 1:
   A{*}, B{*}, C{*}, D{*}, 1{*}

Posted by Dej Mar on 2012-07-29 09:50:23