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Everyone who lives on the Island of Careful Thought is either sane (believing all true statements, and no false statements) or insane (believing all false statements, and no true statements). The inhabitants of the island never intentionally lie, but they are either straightforward (able to say statement S if and only if they believe S) or introspective (able to say statement S if and only if they believe that they believe S).

When I arrived at the island, some of its inhabitants came out to greet me. I took the opportunity to ask them some questions.

  • First, I asked each of them, "Are you straightforward?" and all but three of the islanders said that they were.
  • Second, I asked each of them, "Do you believe that you are introspective?" and all but two of the islanders said that they did.
  • Finally, I asked each of them, "Are you the only islander of your exact type present here to greet me?" and all but one of the islanders said that they were.

How many islanders came out to greet me when I arrived?

This is a reflavoring of an original puzzle I wrote for a summer program. As usual, logic puzzles of this type can be solved with brute force (though in this case, the brute force is not so straightforward) but there is an intended solve path that is cleaner than others. To clarify a possible point of confusion, the "exact type" of an islander is the pair of adjectives (sane or insane) + (straightforward or introspective) describing them; there are four exact types.

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All statements made by introspective people are true. Only sane straightforward people claim to be straightforward; only sane introspective people claim to believe they are introspective. There cannot be 4 or more islanders present, as then 2 would be sane introspective, and neither could then claim to be the only one of their exact type. So there must be 3 islanders present. (One insane straightforward, one sane introspective, and one insane introspective.)

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  • $\begingroup$ given the definition of "exact types" I would have been tempted to insert a reference to the well-known Myers-Briggs classification types :-) but it's your puzzle not mine. $\endgroup$ Commented Dec 22, 2023 at 21:49
  • $\begingroup$ I don't follow your third sentence. Could there not be two of some other type? $\endgroup$ Commented Dec 23, 2023 at 1:13
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    $\begingroup$ rot13("bayl fnar vagebfcrpgvir crbcyr pynvz gb or vagebfcrpgvir", naq nyy ohg gjb vfynaqref znqr guvf pynvz.) $\endgroup$ Commented Dec 23, 2023 at 2:06
  • $\begingroup$ Well done! Concise and correct. I will only mention one observation which was not needed in this solution, but which I think is a fun observation to make - rot13(Vs bayl bar crefba pynvzf abg gb or gur bayl bar cerfrag bs gurve rknpg glcr, gung crefba'f pynvz vf snyfr!) $\endgroup$ Commented Dec 23, 2023 at 16:48

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