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One day, while flying your single-engine seaplane out over the Pacific, the electrical system shorts out, all of your electrical systems (including the radio) fail, and the engine dies. You are in a spot of trouble. Fortunately, as you are slowly gliding over the ocean, losing altitude at an alarming pace, you spot a small, low-lying, scrub-covered cay below you. Even better, it appears that someone has built a large radio transmission tower on the island. You quickly decide to land the plane on a sheltered bay on the south side of the cay, then make your way to the radio tower to in the hopes of calling for help.

After bringing the plane in, you pull out your hand guidebook of the Pacific to try to get a sense of where you have landed. According to the guidebook, this part of the Pacific is populated by several groups of people:

  • European Colonizers. Over the last couple of centuries, a large number of Europeans have ended up settling in this part of the Pacific. They are always happy to chat with outsiders, and will always tell you the truth.
  • HMS Abundance Mutineers. Or, more accurately, the descendants of the mutineers. At the end of the 18th Century, the crew of the HMS Abundance mutinied and, in order to avoid facing justice in England, settled in the nearby islands. While this happened more than 200 years ago, the descendants of the mutineers are still quite wary of outsiders, and will alway lie to you.
  • Native Islanders. The natives of this part of the Pacific have been dealing with both colonizers and mutineers for centuries, and are pretty fed up with both of them. They don't really want to deal with outsiders, but, when forced, will do what they can just to mess with outsiders. Their favorite game to play it "Two Truths and Lie (and Another Lie)". When talking to outsiders, they will always alternate back and forth between telling the truth and lying to you.

According to the guidebook, there has been quite a lot of intermarriage between these groups. They all look and dress alike, and they all speak the same language (which happens to be fluent English, though each of the groups has some unique idiosyncratic linguistic features). The only way to distinguish members of a group is to talk to him or her and see how they answer your questions.

Hopefully, you won't have to deal with any of these folk.

You stow the guidebook, drag the plane onto the beach, and secure it to the one tree you can see with a long rope. You then start making your way to the radio tower along a level path which runs away from the the beach.

After walking for a few minutes, you come across a group of five islanders, who are standing at a fork in the road, and arguing about something (seemingly, the pronunciation of the word "ghoti"). Clearly, these guys aren't all members of the same group.

Drat. It seems that you are probably going to have to deal with some of the locals, after all.

Question: What is the least number of questions that you can pose to the members of this group of arguing islanders in order to determine the correct path to the radio tower?

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  • 1
    $\begingroup$ Finally got a Puzzling account, eh? +1 Nice question ;) $\endgroup$ – Mr Pie Aug 16 at 16:32
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    $\begingroup$ @Feeds Yeah. A colleague was teaching Knights and Knaves problems as part of a lower division "Learn How to Math" class this summer, and I trolled her with this one. She threatened to beat me when I gave her the answer, then gave an evil chuckle and went to give the problem to her students, as well. While I don't claim to be the originator of this problem, I don't know where I learned it, and didn't see it here. So I figured I would post it. $\endgroup$ – Xander Henderson Aug 16 at 16:41
  • $\begingroup$ Hahah. I think I've seen a similar Knights and Knaves question on an mathematics competition test question, but simpler. I think it's a famous concept for truth-teller problems like this :P $\endgroup$ – Mr Pie Aug 16 at 16:44
  • $\begingroup$ Could the path directions be as simple as North, East, etc.? $\endgroup$ – Abbas Aug 16 at 17:08
  • $\begingroup$ @Abbas Certainly. "Left" and "right" would also work. Or "This Way" and "That Way." $\endgroup$ – Xander Henderson Aug 16 at 17:11
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I suspect you need

0 questions, as the island is flat and scrub-covered, thus the large radio tower is likely visible from the fork.

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  • 2
    $\begingroup$ Yes, this was my intended answer. $\endgroup$ – Xander Henderson Aug 16 at 17:23
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    $\begingroup$ This is terrible. I love it. :) $\endgroup$ – Rubio Aug 17 at 20:55
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You can do it in

1 question

as follows:

Ask a single person the question: "If, instead of asking you this question, I asked you whether I should take the left path, would you say "yes"?"

If they are a truth-teller, or a switcher currently in the truth-teller state, then they would say 'yes' if the left path is correct and 'no' if the right path is correct, and so they will answer 'yes' for left and 'no' for right.

If they are a liar, or a switcher currently in the liar state, then they would say 'no' if the left path is correct and 'yes' if the right path is correct, and so they will lie and answer 'yes' for left and 'no' for right.

(This is a fairly trivial modification of the regular solution to this problem: the only difference is that you have to specify "instead of this question" so the switchers have the right parity.)

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  • $\begingroup$ Guvf nccrnef gb or n pbeerpg fbyhgvba, ohg vg vf cbffvoyr gb qb orggre. V'z arj urer, fb V znl abg unir sbhaq nyy bs gur nccebcevngr gntf, ohg abgr gur "yngreny guvaxvat" gnt. $\endgroup$ – Xander Henderson Aug 16 at 17:23

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