Maths challenge 4

[Gustavus]

I posted the following puzzle here once before and no one seemed interested. Perhaps some of you may find it interesting this time.

There is a genre of puzzles known as "island problems". Typically, in these problems, there are two tribes of natives, one of which always speaks the truth and one of which always lies and in various puzzle settings the problem is to devise a set of questions that will result in an answer you can trust. Several years ago I devised what I modestly called "the ultimate island problem" -- although that title must now go to an island problem created by a colleague who has done me one better. I don't have his permission to give his problem before he publishes it so here is my "penultimate island problem".

On an island there are three tribes. The members of one tribe speak only the
truth. The members of another tribe always lie. The members of the third
tribe, like most of us, tell the truth sometimes and lie sometimes. The
social custom on the island is that in any gathering of three or more
individuals each tribe must be represented.

An explorer lost on the island, but familar with the strange customs of the
natives, comes on three individuals in a clearing in the jungle; call them A,
B and C for convenience. While he is trying to figure out what questions to
ask, and of whom, to get directions back to his camp that he can trust, the
natives speak to him.

A says, "Ask C, he always tells the truth".
B says, "Oh no, you can't believe anything C says".
C says, "That advice ought to confuse you".

The explorer now knows who to ask for directions. Do you?

 

[Adul]

Bangs head on wall...

where is my grid..

 

[Adul]

Lets see now

two are beng truthful.

My answer would be c is the truthful one
__L_S_T
a_x_o_x
b_o_x_x
c_x_x_o

 

[Moonbeam]

Without thinking I choose C, now please eliminate one wrong choice so I can switch.

 

[Adul]

I determined b was the liar.

1. C statent about confusion is true. It was confusing.
2. a States that c always says the truth.
3. B states nothing can be believed of C, where in fact see is teelling the truth. Which makes b the liar. leave a and b for the truth teller. Well I assume A is telling the truth because we determined b is a liar. So that make a the middleman. and C the truthful one.

ok now I am confused

 

[prodigy]

<< A says, &quot;Ask C, he always tells the truth&quot;. >>



&quot;A&quot; can't be the full-time liar. If he was, this statement would mean that C always lies, and of course you can't have 2 who lie all the time. &quot;A&quot; is the one who both lies and tells the truth, in this case he's telling the truth.




<< B says, &quot;Oh no, you can't believe anything C says&quot;. >>



&quot;B&quot; is the full-time liar, therefore his statement means that you can believe what &quot;C&quot; says.




<< C says, &quot;That advice ought to confuse you&quot;. >>



&quot;C&quot; is the one who tells the truth.


Am I right?

 

[Xede]

The truthful one could be either B or C, depending on whether you consider &quot;That advice ought to confuse you&quot; to be true or false. If you answer &quot;Yes, this is a confusing problem&quot;, then C is the truthful one. If you aren't confused by the problem, B is the truthful one. So I guess it all depends on how smart the explorer is.

 

[Moonbeam]

Maybe some might be right, but one has to be right. I gotta walk my dog.

 

[somethingwitty]

I'm stumped...here's what I did so far, but then the last two situations both seem to work. argh, too late in evening to continue thinking about this one.

A B C
1 T F B X
2 T B F X
3 F T B X
4 F B T X
5 B T F
6 B F T

Now, here's what I'm thinking. If A always tells the truth, then C would have to tell the truth full time-but that doesnt fit, cuz they'd both be from the same tribe. So that knocks out situations 1 and 2. Next, we know that B claims C is a lying scumbage. If B always tells the truth, then that means C MUST be F. So situation #3 is out. Next: A says C tells the truth-if A always lies, then C most certainly is NOT a truthteller. #4 is out.

Now, down to the last two situations, knowing I should either ask B or C (b/c A will either lie or tell the truth, and I dont know). Let's say A was lying when he says C is always right-that would mean that C always lies, and in turn, B tells the truth. C knows that A is lying this time and that B is telling the truth-so in fact, if I, the explorer knew, I'd know that this situation seems to make perfect sense. A is F-ing with me, B tries to help me out, then C goes ahead and lies even though I shouldnt be confused at all now. If, on the other hand, A is telling the truth this time, then B is the one who lies. So A says C always tells the truth, B follows his role in lying, and C is kind enough to point out that I'd be confused right about now...argh!

I know my logic is flawed on the last two cases. nevermind, I can find my way to my bed well enough. I'll check back later.

 

[tom3]

&quot;A says, &quot;Ask C, he always tells the truth&quot;.
B says, &quot;Oh no, you can't believe anything C says&quot;.
C says, &quot;That advice ought to confuse you&quot;.

The explorer now knows who to ask for directions. Do you?&quot;

The problem has 2 types of components, Truth teller and Liars, and True statements and False statements.

Since there is 1 Full time truth teller, 1 full time liar, and 1 half/half, there must be 1 true statement and 2 false statements, OR 2 false statements and 1 true statement.

1. Suppose A is a full time truth teller, then his statement must be true, which would make C a full time truth teller. INCORRECT

2. Suppose A is a full time liar, then all we know is that C does not always tell the truth. Then B has to be the full time truth teller, that leaves C as the half/half, since A is already the full time liar. Yet B's statement would not be correct, since he says &quot;you cannot believe ANYTHING C says&quot; while C is a half/half, which means he SOMETIMES tell the truth. INCORRECT

3. That means A HAS to be the half/half. The tricky part follows:

4. (CASE 1) IF A happened to be telling the truth at the time, then that makes C a full time truth teller, and B the full time liar. (CASE 2) IF A happened to be telling a lie, then that means C is NOT the full time truth teller, which makes him the full time liar, and B the full time truth teller.

5. Now here's the part I am not sure about. Depending on whether the statement C made was the truth or a lie, both cases 1 &amp; 2 from above can be correct, if you think about their statements.. Therefore, I've taken a step further and determined that C was not telling the truth, because he said &quot;that statement ought to confuse you&quot;, yet the explorer was not confused. In fact in the original problem, he knew who to ask for a truthful answer after the brief conversation between the 3.

6. Therefore, A is the half/half, B is the full time truth teller, and C is the full time liar.

(credit goes to mahpoh as we thought through this together)

now please tell us if we are right!!!

edit fixed typos

 

[tom3]

come on Gustavus..

 

[Gustavus]

tom3 (and other interested parties)
Sorry I haven't been back before this. I will explain the logic to solving my version of the island problem, but more importantly, my colleague, Prof. David-Olivier Jacquet-Chiffelle of Switzerland, has given me persmission to share his much better version of the island problem with you. More about that later.

In my island problem, when A says to ask C since C always tells the truth, that rules out A being the truth teller, T, since T couldn't say that about anyone else. It also says that A can't be the liar, L, and C the truthteller, since the statement would then be true and L never speaks the truth. When B says, you can't believe anything C says, that says B cannot be T and C the native who sometimes does tell the truth, S. A is therefore the sometime truth teller S. somethingwitty got this far. That leaves two possibilities: either B is the lier and C the truth teller or vice versa. The first two statements are not enough for the explorer to identify T the truth teller, and therefore he is still confused (not certain of which native, B or C, to ask directions). Since C is either the lier or the truth teller, the true statement that the explorer must be confused by the advice he has received could have only been said by T, so C is the truthteller.

This is a neat puzzle (I think) but not too surprising in it's resolution by anyone who has ever thought about what are called weighing problems in which there is a bad coin (whose weight is different) in a bunch of good ones and the problem is to find the bad coin in a minimum number of weighings using a two pan balance. As you know if you have ever solved any such problems, you try to divide the classes in two on each successive weighing so that n weighings can frequently be devised to find the one exceptional object out of 2^n objects in all. Since we have 3!=6 assignments of the natives to tribes and three statements which could conceivably have identified a unique assignment out of even eight the result doesn't appear to be paradoxical. Prof. David-Olivier Jacquet-Chiffelle version does appear to be a paradox though. In his problem there are four tribes and four natives so there are 4!=24 possible assignments of individual natives to tribes and still only three statements -- but it is still possible to uniquely identify the assignment. The paradox is that three statements would normally only serve to identify one out of eight elements, but in his puzzle you can uniquely identify one out of twenty four. This time I will not return to give you the solution -- which I found without even putting pencil to paper.

On an island there are four tribes. The members of one tribe tell only the
truth. The members of another tribe always lie. The members of the third
tribe, like most of us, tell the truth sometimes and lie sometimes. The
member of the fourth tribe always lie but never answer a question?
The social custom on the island is that in any gathering of four or more
individuals each tribe must be represented.

An explorer lost on the island, but familar with the strange customs of the
natives, come on four individuals in a clearing in the jungle; call them A,
B, C and D for convenience. While he is trying to figure out what questions
to ask, and to whom, to get directions to get back to his camp that he can
trust, the natives speak to him.

A says, &quot;Ask D, he always tells the truth&quot;.
B says, &quot;Don't ask me anything, I never answer a question&quot;.
C says, &quot;That's true! B never answers a question&quot;.

The explorer now knows who to ask for directions. Do you?

Have fun.

 

[urbantechie]

Ahhh!! MATH!!! Wheres the door!!!!???

 

[prodigy]

*ahem*

I think I was the first to answer correctly, was I not?

 

[RGN]

Nobody for the second one?

On an island there are four tribes. The members of one tribe tell only the
truth. The members of another tribe always lie. The members of the third
tribe, like most of us, tell the truth sometimes and lie sometimes. The
member of the fourth tribe always lie but never answer a question?
The social custom on the island is that in any gathering of four or more
individuals each tribe must be represented.

An explorer lost on the island, but familar with the strange customs of the
natives, come on four individuals in a clearing in the jungle; call them A,
B, C and D for convenience. While he is trying to figure out what questions
to ask, and to whom, to get directions to get back to his camp that he can
trust, the natives speak to him.

A says, &quot;Ask D, he always tells the truth&quot;.
B says, &quot;Don't ask me anything, I never answer a question&quot;.
C says, &quot;That's true! B never answers a question&quot;.

The explorer now knows who to ask for directions. Do you?


I think the always truth teller is &quot;D&quot;

&quot;B&quot; is lying, but is not the 'no questions guy' This is because the 'no questions guy' is a liar. An he would not say that he doesn't answer questions.

That means &quot;C&quot; is also lying. Because its not &quot;B&quot; that doesn't answer questions. And...

If &quot;A&quot; was the truth teller, it would mean that &quot;D&quot; is the half truth guy. Which would mean that &quot;A&quot; lied.

So, that means that &quot;D&quot; must be the one to ask for directions...


Any thoughts?

 

[jeremy806]

D is the truth teller.

Similar reasoning.

Jeremy806