Saturday, 29 November 2014

logical deduction - 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.


Ancient Cave


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?



Answer



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!


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