# You arewere might still be a sea captain.

On the 14th of December, 1214, your sailors rebelled. 6 sailors and your first mate get in a boat to row you to what they believe is an abandoned island. Before leaving to return to the ship, however, your first mate decide to give you one last chance. You are given the following information:

1. The sailors are divided into 3 categories:
• Liars (Sailors who always lie)
• Truth tellers (Sailors who always tell the truth)
• Unfortunate in-betweens (Sailors who tell the truth or lie at their will or randomly)
2. 2 of the sailors are liars
3. 2 of the sailors are in-betweens
4. 2 of the sailors are truth tellers
5. The sailors are then divided into 2 groups:
• One liar, one truth teller, and one in-between
• One liar (this one led the mutiny), one truth teller, and one in-between
6. The sailors know who led the mutiny and who fits in what category (liars, truth-tellers, in-betweeners)

The first mate then tells you that you are allowed to ask all three of the sailors in the first group one yes/no question each only about sailors in their group. You may then select one of the three and ask him one yes/no question about the sailors in the second group.

You may then ask one of the sailors in the second group one yes/no question about any of the sailors in his own group, after which you must determine who led the mutiny.

If you guess correctly, they will restore you as captain of the ship. If you guess incorrectly, they will leave you to die on this abandoned island.

## How can you determine who started the mutiny?

N.B.: Yes, it is solvable - I've gone through the solution 4 times to make sure it works. :)

• Does the second group, by definition, include the leader of the mutiny, or is it possible for the leader to be in the first group? I can't tell if the bullets under item 5 are supposed to be interpreted as ordered or unordered. Mar 8, 2017 at 18:34
• @jwodder, the former. Both groups include 1 liar, 1 truth teller, and 1 in-between, but the leader of the mutiny is included in the second group. Mar 8, 2017 at 18:35
• Do the sailors know who are liars and truth-tellers? Mar 8, 2017 at 20:52
• @GreenstoneWalker, I didn't say, did I? :) Yes, they do. Mar 8, 2017 at 20:54

The first question I believe can be something like the question stack reader used to start his answer:

You ask all 3 members of the first group: Is there a liar in your group?
If only 1 say no, he is the liar. Choose him and reverse his next answer.
If only 1 say yes, he is the truth teller, use him to ask the next question.

Then we find the culprit with the last two questions, by cleverly introducing a new kind of information about the sailors:

Ask the chosen member if the members in the second group are standing in either TJL, LJT or LTJ order. (you can numerate them out loud to define the order)

If he says yes, ask the number 1. if there is a liar in his group. If he says yes, we have TJL and the number 3. is the liar and culprit. If he says no, he is the culprit.

If he says no, you either have a TLJ, JTL or JLT order. Ask the 2. member if there is a liar in his group. If he says yes, we have JTL and number 3. is the culprit again. If he says no, he is the culprit.

• You beat me to it :) Mar 8, 2017 at 17:54
• That works! Pretty close to the solution I thought out. Mar 8, 2017 at 18:04
• @anonymous2 I'm eager to hear the intended solution then! :) Mar 8, 2017 at 18:06
• @DushDushDush, posted below. :) Mar 8, 2017 at 18:09

The following trick lets you get a truthful answer out of any concrete sailor, that is, a sailor who either always tells the truth or always lies. To find out whether statement P is true, ask

"Is exactly one of the statements P and "You are a liar" true?"

Both truth tellers and liars will say yes exactly when P is true. Though I won't mention it, I will apply this construction to all of my questions, allowing me to treat liars as honest.

1. There are three sailors in group 1, Left, Middle and Right. Start by asking Left if Middle is an in-betweener. If Left says "yes", then either Left or Middle is an in-betweener, so we know Right is concrete. If Left says no, we similarly know that Middle is concrete. Either way, we found a concrete sailor.

2. Ask the concrete sailor in the first group the following: "Is the arrangement of the second group either LIT, LTI, or TLI?" In other words, "is the liar to the left of the in-betweener?" If this is true, then you now know the leftmost sailor in the second group is concrete, and you know the liar is in the middle or left. If false, then the possibilities are ILT, ITL, or TIL, so you instead know the right-most sailor is concrete and that the liar is either in the right or middle.

3. Finally, ask the concrete sailor you found in the second group if the middle sailor is a liar. If so, the liar is in the middle, if not, the liar is on the right or left, according to how the second question was answered.

We accomplished this task using only three questions instead of the given five.

• Excellent, this works too! Mar 8, 2017 at 18:07

You ask all 3 members of the first group. Is there a liar in your group.
If only 1 says no, he is the liar. Choose him and reverse his next answer.
If only 1 says yes, he is the truth teller, use him to ask the next question.
Ask the chosen if member 1 of group 2 is an in between.
If he is then use someone else for you next question, if not, use that one.
Ask the a non in between member of group 2 if there is a liar in the group. If he says yes, then the other non in between is the liar and culprit. If he says no then he is the culprit.

• You're solution almost works, but what if the one you chose in the second group is the truth teller? You don't know which of the other two is the liar. Mar 8, 2017 at 15:40
• @n_palum, He's still missing a step. Suppose member 1 is the truth teller. Chosen from group one will inform you that he is not an in between. If you then ask person 1 if there is a liar in the group, he will reply yes. You still don't know, however, if the liar is person 2 or person 3. Mar 8, 2017 at 16:32
• If 1 is the truth teller, you could just ask if member 2 could tell you the truth. If they say yes then that's the joker and 3 is the liar. If no then 2 is the liar. Mar 8, 2017 at 16:38
• @n_palum, but you need an extra question then to figure out if he is indeed the truth teller. Mar 8, 2017 at 16:40
• I was just stating that based off your presumption that 1 was the truth teller... I see where the bit of flaw is though in the original. Mar 8, 2017 at 16:43

Since the riddle has been answered correctly by two users, I'll post the answer I thought of here:

So you ask each of the sailors in the first group,

"Would you consistently say that you are a liar?"

• The truth teller will reply no.
• The liar will reply yes.*
• The in-between will reply yes / no.

If the in-between replies "yes," you will have two "yes's", so select the person who said "No," and move on to the next group.

If the in-between replies "no," you will have two "no's", so select the person who said "yes," and move on to the next group.

If you have the truth teller (the one who said "No"), ask him if sailor 1 in the second group would consistently say that sailor 2 led the mutiny.

If he replies "yes," either sailor 1 is the liar, or sailor 1 is the truth teller and sailor 2 is the liar. Proceeding is relatively simple: simply ask sailor 1 if sailor 3 led the mutiny.

• If sailor 1 is the liar, he will say "yes" - sailor 1 led the mutiny.
• If sailor 1 is the truth-teller, he will say "no," and you know that sailor 2 is the liar - sailor 2 led the mutiny.

On the other hand, if he replies "no," you can deduce that sailor 1 is not the liar. If he was, he would always say that everyone else led the mutiny. Therefore, either:

• Sailor 1 is the in-between, and you can determine nothing about sailor 2, or
• Sailor 1 is the truth teller and sailor 2 is the in-between.

From that point it is simple: you ask sailor 3 if sailor 1 led the mutiny.

• If sailor 1 is the truth-teller, sailor 3 is a liar and will say "yes."
• If sailor 1 is the in-between and sailor 3 is the liar, he will still say "yes."
• If sailer 1 is the in-between and sailor 2 is the liar, sailor 3 is a truth teller, and he will say "no."

If you have the liar (the one who said "Yes"), simply do the inverse. Ask him if sailor 1 would consistently say that sailor 2 led the mutiny.

If he replies "no," you know that sailor 1 actually would consistently say that sailor 2 led the mutiny. Therefore, either sailor 1 is the liar, or sailor 1 is the truth teller and sailor 2 is the liar. Proceeding is relatively simple: simply ask sailor 1 if sailor 3 led the mutiny.

• If sailor 1 is the liar, he will say "yes" - sailor 1 led the mutiny.
• If sailor 1 is the truth-teller, he will say "no," and you know that sailor 2 is the liar - sailor 2 led the mutiny.

On the other hand, if he replies "yes," you can deduce that sailor 1 is not the liar. If he was, he would always say that everyone else led the mutiny. Therefore, either:

• Sailor 1 is the in-between, and you can determine nothing about sailor 2, or
• Sailor 1 is the truth teller and sailor 2 is the in-between.

From that point it is simple: you ask sailor 3 if sailor 1 led the mutiny.

• If sailor 1 is the truth-teller, sailor 3 is a liar and will say "yes."
• If sailor 1 is the in-between and sailor 3 is the liar, he will still say "yes."
• Otherwise, sailor 2 is the liar: sailor 3 is a truth teller, and he will say "no."

* This is counterintuitive, but think it through. The liar would consistently say that he is not a liar: thus he would not consistently say that he is a liar. Since he is a liar, he will say that he would.

Well, the first part is the same as always, there are multiple questions that lead to the same result - you get truth teller or liar and know which is it. Or even use that form in another answer. For the second part, alternative solution (Assuming you have truth telling sailor in group 1, for lies process is similar):

Is sailor 1 in group liar or 2 in-between?

There is another form that works, but 3rd combination doesn't.

On NO, you know that 2nd person cannot be in-between and 1st cannot be liar. Ask 2nd if the first one stole money. Yes = he stole it. No = 3rd stole it. On YES, you know that the 1st person cannot be in-between and 2nd cannot be liar. Ask 1st if 2nd stole. Yes = he stole it. No = 3rd stole it.

Now to show impossibility about a single person question that doesn't involve others. Suppose we asked person 1 what he is:

T? Y= ask if 2nd stole money. N= L can be any out of 3. No way to figure out with 1 question. L? Y= you already did it. N= he is T/IB. Not going to work as any question is going to come out arbitrary due to IB. IB? Y= ask any other if 1st stole money. N = L can be any out of 3. 1 question is insufficient.