Spreading gossip
Each of elderly dons knows a piece of gossip not known to any of the others. They communicate by telephone, and in each call the two dons concerned reveal to each other all the information they know so far.
What is the smallest number of calls that can be made so that, in the end, all the dons know all the gossip if
The answer is 1996.
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1 solution
In general, the smallest number of calls for $n>3$ dons is $2n-4$. Given $n=4$ call them A, B, C, D. Using the calls AB, CD, AC, DB they know all the gossips. For $n>4$ select 4 dons A, B, C, D, and make all the $n-4$ call A. Among the selected $4$, we can share all the information in $4$ steps and then A can distribute these informations to the remaining $n-4$ dons.
For the proof of the optimality, one can look at the following link, which has several papers published around the same time, with different proofs
https://www.math.uni-bielefeld.de/~sillke/PUZZLES/gossips.pdf