“The Two Sheriffs” puzzle was already discussed here Two Sheriffs and Eavesdroppers. This puzzle has only one difference: we have seven suspects instead of eight.
Two sheriffs in neighboring towns are on the track of a killer, in a case involving seven suspects. By virtue of independent, reliable detective work, each has narrowed his list to only two. Now they are engaged in a telephone call; their object is to compare information, and if their pairs overlap in just one suspect, to identify the killer.
The difficulty is that their telephone line has been tapped by the local lynch mob, who know the original list of suspects but not which pairs the sheriffs have arrived at. If they are able to identify the killer with certainty as a result of the phone call, he will be lynched before he can be arrested.
Can the sheriffs, who have never met, conduct their conversation in such a way that they both end up knowing who the killer is (when possible), yet the lynch mob is still left in the dark?
Original problem is takken from: Mathematical Puzzles, a Connoisseur's Collection, Peter Winkler. The solution for seven suspects belongs to Yoav Kallus.