# The Expedition Into The Ancient Cave

You are a archeologist and have taken 8 of your students on a expedition trip to a ancient cave to try and find ancient treasure. After you and your team entered the cave, you travel as a group exploring the entire cave. There is only one tunnel left in the cave to explore. As you reach the end of the tunnel one of your students accidentally leans on a panel that opens up a secret entrance before you.

As you enter you see ancient writing on the wall and a giant gem sitting on a pedestal. After sometime you and your students deciphered the ancient writing, it says "Beware of the curse! The darts will control whether you lie or tell the truth and the green gas will turn you into a hideous beast. The only way to return back to normal is to escape alive". Some of your students laugh and call it a bluff designed to scare and prevent theft of the treasure hidden in the cave.

You decided to take a chance and take the gem off of the pedestal. As you do, two darts fly out of nowhere, barely missing you, and hit two of your students. You jumped away from the pedestal just in time as green gas sprays all 8 of the students. After the gas clears, everyone but you, look exactly the same. You can't tell them apart. One of the students points behind you and says "What's that?!". You turn around to see purple gas starting to fill the room, naturally you and your students start running back to the entrance.

After escaping the tunnel you came from, you have to make a choice between 4 other tunnels. From what you recall it took around 4 minutes to travel from one end of the tunnel to the other. The gas will reach where you are in about 10 minutes. You decide to split up into groups, check each tunnel, and report back here to see which tunnel is the correct way out. Which means there is only have 2 minutes to discuss and make a decision before the gas reaches the groups location.

Remember, two of the students can't control if they lie or not, and they all look the same so you can't tell them apart. How many groups should there be when you all split up? How many people per group? and how can you tell which is the correct tunnel to escape through?

• Sorry about the tags if I have chosen the wrong ones, I don't know what to choose – DemonicBirdFlu Mar 11 '16 at 13:26
• I may have misread it... but the darts only control "TELLING" truths/lies, right? But they can still act as instructed of their will? If so, what prevents us to split evenly in 2/2/2/3 with the instruction "If you find the exit don't come back, if you don't come back" ? Then everyone that comes back just goes for the path no one came back from? – Diego Martinoia Mar 11 '16 at 13:56
• @DiegoMartinoia but what if there are booby-traps and more than one person doesn't come back due to unforeseen circumstances? – Daedric Mar 11 '16 at 14:05
• @DemonicBirdFlu I assume "two of the students can't control if they lie or not" also prevents the usual trick questions like "If I asked you ... would you say ... ?". – Sleafar Mar 11 '16 at 14:54
• OK - think I figured it out now so long as we (the teacher) get to explore one of the tunnels. – Trenin Mar 11 '16 at 15:53

EDIT: Handled the unpredictable nature of the affected students this time.

Thinking quickly, you decide to send

3 students down one tunnel, 3 down another, 2 down a third, and you check the forth.

In 8 minutes all return back and you make your decision.

You know for certain the status of your own tunnel, so if it is the exit gather them up and leave.

Otherwise, that was not the exit, so it must be in the other tunnels. Listen to what the students have to say. The possibilities are:

* Both 3 person groups are unanimous in their report
* One 3 person group is in disagreement and the 2 person group is unanimous
* One 3 person group is in disagreement and the 2 person group is in disagreement
* Both 3 person groups are in disagreement

But these are all easily resolvable:

* For the first case, both 3 person groups have a truth teller and thus are not lying. If neither indicate the exit, the 2 person tunnel will be the exit, regardless of how they answered.
* For the second case, we don't know if there are two liars in this group, or if there are two truth tellers. Regardless, we know all the other groups are telling the truth since they have truth tellers and are in agreement. Thus, infer the truth value of this group based on the other two groups. For example, if the other two groups didn't find the exit, then we know this group did find the exit regardless of how they answer.
* For the third case, we know that there are 2 truth tellers in the 3 person group and only one liar since the other liar must be in the 2 person group. So use the majority decision of both 3 person groups to infer the status of the two person tunnel.
* For the last case, if both 3 person groups are in disagreement, then they both have a single affected person. But that means the other two people in the group are unaffected, so simply use the majority decision of all three groups.

Thus, you can easily infer the status of the tunnels and can safely make your escape!

• I just came to the same conclusion. This is the only possible solution, because you need at least 5 students for any combination of 2 tunnels. – Sleafar Mar 11 '16 at 15:58
• Yeah I did think something along those lines too. Though I still prefer my option: it's more practical and it's the more realistic to think of in a high-pressure situation (bc you'd be pressed to save at least someone, even if you have an unforeseen flaw in your strategy) :D – Diego Martinoia Mar 11 '16 at 16:59

Bonus points: I think this solution would work even if you, the professor, had also been affected by the green gas or if EVERYONE has been darted.

My answer is based on the assumption that the darted students are still in control of their actions, they can't just decide whether to lie or tell the truth. It also assumes that the tunnels aren't able to prevent involuntary non-return (such as being booby-trapped), as pointed out in the comments to the question post by Daedric.

The solution is simple:

Split the group in 2 / 2 / 2 / 3 (one with you, doesn't matter which) and send a party down each tunnel. Instruct the parties to leave through the exit if they find it within 4 minutes of walking, and to come back to the center if it's a dead-end or it goes deeper in the mountain.

What will happen is that:

Only one party won't return to the center, and those have found the exit. Have everyone else walk down the tunnel people did not come back from. Everybody lives

• So when you ask remaining people what tunnel they used, how do you know they are telling the truth? You can't tell them apart and it is possible that you are the last ones to return to the cave. – Trenin Mar 11 '16 at 17:41
• If they are allowed to act without the dart affecting them, then instead of having them answer verbally, get them to nod. I think the spirit of the puzzle is that those affected may try to foil you. In which case you'd return to the cave and have an argument as to which tunnel no one returned from. – Trenin Mar 11 '16 at 17:45
• You have a point regarding telling them apart, but if the affected ones just won't abide to the strategy you tell them to do, all solutions would crumble (what if they don't come back? what if they murder the third of their party? etc...) – Diego Martinoia Mar 12 '16 at 20:19
• @Trenin That can be solved by you not going down a path at all - instead, you stay back and see for yourself which path each group comes back from. – Brilliand Mar 10 '17 at 19:46

Assuming that the darts would control who lies, so the two students hit by darts will lie. Split the students and us into three groups of three. Each group goes down one tunnel, which will cover three of the four possibilities.

Assuming the students hit with darts don't know that they're lying, and we can only ask them one question after returning, then when everyone returns we can ask everyone from every group if they found the exit. If any group has all three people say "yes", then that's the exit, otherwise there are two possibilities;

1: two groups have mixed answers, so both groups have a liar. If either group had two "yes" then that's the exit. If both had two "no", neither of them have the exit, and the exit is through the last, unchecked tunnel.

2: one group has mixed answers, so it has both liars, so if it has two "no", then those are the two lairs and that's the exit. If it has two "yes" then those are the two liars, and it's not the exit, and we go through the last, unchecked tunnel.

• The second case doesn't work. One of the affected students could tell the truth (they are unpredictable), then two "no" would mean no exit. – Sleafar Mar 11 '16 at 14:35
• @Sleafar I didn't realize that they "could" lie, i thought they "would" lie. I'll think about it, thanks :) – Piotr Pytlik Mar 11 '16 at 14:39