# Need Help NIMO-2007 Q19

1. Feb 21, 2012

### Jjjxy

I really don't get this, question says ''on an island, there are only two kinds of people: those who are always honest and those who always lie. Three of the inhabitants are talking; Andrea says: 'Barbara is honest', Barbara says 'Andrea and Ciro are both honest' and Ciro says 'Andrea is a liar'. We know that:
A. All three are honest
B. Andrea and Barbara are honest
C. Andrea is honest, Ciro and Barbara are liars
D. Andrea and Barbara are liars, Ciro is honest
E. All three are liars''

2. Feb 21, 2012

### sg001

To take a guess i would say E.
The reason for this is that three inhabitants produce conflicting statements/
ie Andrea's statement conflicts with Ciro's statement.
Although the biggest give away is Barbaras statement.
focus on this statement.
Assume she is not "honest", but taking her word anyway we beleive Ciro's statement, which leaves Andrea as a lier. Therefore if Andrea is lying about her statement, then barbaras statement is not true either, meaning Ciro is'nt honest either.

Therefore all three are liars.
all though i find it hard to "prove" any definite staements.
However, with "E" you cant "prove" it wrong, where as the other choiuces can be proved wrong

Therefore E

3. Feb 21, 2012

### Staff: Mentor

Why not D? If Ciro is honest then Andrea is a liar. If Andrea is a liar then Barbara is a liar too. So look to Barbara's statement to prove whether it's D or not.

4. Feb 21, 2012

### sg001

I dont agree with D.

The reason is because it can be disproved with the given information.
ie, If Barbara is lying then her statement is incorrect.

Barbara says " Andrea and Ciro are both honest".
If Barbara was a lier, then it would be fare to say that
Andrea is a lier and Ciro is honest
or
Ciro is lier and Andrea is honest
or
Ciro is a lier and Andrea is a lier.

These are the only possible outcomes from barbaras original statement if she was lying.

While there is plause that if she was giving a false statement it could lead to Ciro being honest.

You cant prove that statement, because you cant predict what she hasnt already said.

With E from the given information you can say that it is a viable option, without having to predict more information then what was given.

do you follow?

5. Feb 21, 2012

### Norwegian

jedish has got it, sg001 has not

6. Feb 21, 2012

### sg001

So if option D is correct you believe

D. Andrea and Barbara are liars, Ciro is honest

which means Andrea says: 'Barbara is honest' is false

Therefore Barbara must be a lier.

which means, Barbara says 'Andrea and Ciro are both honest' is false.

Coming back to the earlier problem i mentioned about the viable options of barbara's satement to make it true or "honest"

There are three possible outcomes to make this statement true or "honest"

Andrea is a lier and Ciro is honest
or
Ciro is lier and Andrea is honest
or
Ciro is a lier and Andrea is a lier.

This is building of the premise of D (Ciro is honest, and barbara & Andrea are liers).

Therefore you have 3 options available all of which are equally viable.
However, with the information given you cannot say D, or else you would be taking one of the assumptions from above.

So the answer is E, Assume they are all liers.
Statistically speaking this is the most probable outcome with the given information.
As you do not have to make Assumptions within other Assumptions.
ie. that Ciro is honest & the second assumption from barbaras plausible "honest" statement outcomes.

7. Feb 21, 2012

### sg001

The way I see it there is no decisive answer one can make without making assumptions.
From the list D & E are the only viable options.
Although as you can see E is the most probable outcome.

8. Feb 21, 2012

### MrAnchovy

I have seen this question a number of times, unfortunately for something that is presumably supposed to test the ability of the answerer to make precise logical deductions the question and answer choices contain not one but two imprecisions.

Firstly it is necessary to add that each of the islanders has perfect knowledge of the status of the other islanders, otherwise very little can be inferred from the statements.

Secondly it is necessary to explicitly resolve the ambiguous statement "Andrea and Ciro are honest". The 'correct' solution requires this statement to have the meaning "Andrea and Ciro are both honest". The alternative, liguistically equally valid, interpretation "Both Andrea is honest and Ciro is honest" leads to inconsistency.

Armed with these corrections, the question is easily answered:

If Andrea is honest then from her statement Barbera is honest, so from Barbera's statement both Andrea and Ciro are honest. But then from Ciro's statement Andrea must be a liar so we have reached inconsistency.

So Andrea must be a liar and from her statement so must Barbera be. The statement "both Andrea and Ciro are honest" must therefore be false. If we interpret this statement to mean that "both Andrea is honest and Ciro is honest", as Andrea always lies she must be lieing about the status of both Andrea and Ciro. Ciro is therefore a liar, so from Ciro's statement, Andrea must be honest and again we have inconsistency.

Instead therefore we must interpret "Andrea and Ciro are honest" to mean "Andrea and Ciro are both honest". The negation of this statement is that "at least one of Andrea and Ciro is a liar". We have already assumed that Andrea is a liar so this statement gives us no knowledge of Ciro's status. If Ciro is a liar, his statement must be false and so Andrea is honest: inconsistancy. Ciro must therefore be honest and we can see that his statement is true.

The only consistent resolution of these statements is therefore that Andrea and Barbera are liars and Ciro is honest, D.

What is the learning point from this? That in solving a problem it is OK to make arbitrary additional assumptions if they lead you to an answer that is within your expected solution space?

9. Feb 21, 2012

### sg001

If we are able to make arbitrary assumptions to lead to a possible solution then you can not say there is "one" correct answer. The correct statement would be such that under certain assumptions Ciro is honest. And under other conditions or assumptions we can also say that they are all liers.

This is almost starting to sound like wave/particle duality.
Intentional?

10. Feb 21, 2012

### HallsofIvy

check each possibility.

This is not possible because then Ciro could not say "Andrea is a liar".

This is not possible because then Barbara could not say "Andrea and Ciro are both honest".

This is not possible because then Barbara could not say "Andrea and Ciro are both honest".

I see no problem with this- though that alone does not prove it is correct.

This is not possible because then Ciro could not say "Andrea is a liar".

Since these are the only possibilities, the correct one must be D.

11. Feb 21, 2012

### MrAnchovy

Oops, I jumped in with both feet there - I see the question as posed here included the statement "Andrea and Ciro are both honest" rather than the ambiguous "Andrea and Ciro are honest".

As you were.

12. Feb 21, 2012

### sg001

This is not possible because then Ciro could not say "Andrea is a liar".

Since these are the only possibilities, the correct one must be D.

That makes sense.
My mistake D is correct.

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