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“The Two Sheriffs” puzzle was already discussed here. This puzzle has only one difference: we have six suspects instead of eight.

Two sheriffs in neighboring towns are on the track of a killer, in a case involving six 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?

Is this even possible? If so, how to do it? Is it possible for five (or even four) suspects? (It is not possible with three suspects.)

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    $\begingroup$ I'm assuming you're disallowing joke answers, such as (pulling from "answers" to the original Puzzling SE question), they discuss a place to meet in person, one reads their pair of names verbatim to the other, they use RSA encryption and read the encoded string over the phone, etc. $\endgroup$
    – bobble
    Commented Sep 18, 2023 at 4:01
  • $\begingroup$ You say "both end up knowing" but for an arrest, only one has to know $\endgroup$
    – BigMistake
    Commented Sep 18, 2023 at 4:54
  • $\begingroup$ @BigMistake Puzzles don’t have to accurately describe reality. $\endgroup$
    – mathlander
    Commented Sep 18, 2023 at 5:48

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This was WAY easier than I expected.

Sheriff 1 can say something like "My list is one of {a,b}, {c,d}, and {e,f}."

Sheriff 2 can say "My list is one of {a,b} or {d,e}."

Case 1: Neither of them has list {a,b}.

In this case, Sheriff 1 tells the name of the criminal directly to Sheriff 2, such as in "The criminal is e." They hang up and go arrest the criminal.

Case 2: Both of them have list {a,b}.

In this case, Sheriff 1 chooses a decoy randomly from {d,e} and states that he is the criminal. They hang up and cry because both of them have the same list.

It can't be the case that one of them has list {a,b} and the other doesn't because then their lists wouldn't intersect.

The sheriffs know which case they are in, but the lynch mob doesn't. Therefore, the lynch mob doesn't know whether the criminal is a, b, or the decoy, but the sheriffs know whether it is the decoy or in the set {a,b}.

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