# The Island of Careful Thought

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.