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A puzzle was asked in a mental ability test.

Arstil and Byzinte are two neighbouring islands in a far-off corner of the world. The natives of Arstil (the Arstillians) and the natives of Byzinte (the Byzintians) sometimes visit each other's sland. The strange thing is that whenever the islanders are on their home island, they tell the truth , but whenever they are on the other island, they lie.

One day an explorer landed on one of the islands, but he didn't know which one. He then met two islanders, but didn't know whether they were from the same island or not.

One of them said: "We are from different islands."
Then the explorer asked: " What island are we now on?"
The other replied: "Arstil".
The smart explorer then knew which island he was on.

The question: Which island is it?


Another visitor came to one of the two islands. She met two islanders names Abs and Bags. She asked about them. They made the following statements:

  1. Abs: At least one of us is Arstilian.
  2. Bags: Abs is Byzintian.

The questions are:

  • On which island did she, the wanderer, land?
  • What about the nativity of Abs and Bags?
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First problem

  • Case 1

If the first tells the truth, then "We are from different islands" is true, so the second one lies and we are not in Arstil but in Bizynte.

  • Case 2

If the first lies, that means they are from the same island and the second one lies too. As he says we are in Arstil, that means we are in Bizynte.

So

We are in Bizynte.

Second problem

  • Case 1

If Abs tells the truth, so one of them or both of them are Arstilian. Plus, we are on Abs's island. If Bags tells the truth and Abs is Bizyntian, that means we are in Byzinte and Bags is Arstilian (because at least one of them is Arstilian). So, Bags lied => paradox.

If we consider Bags is lying, that means Abs is Arstilian and we are on Arstil, and Bags is Byzintian. It works.

  • Case 2

Abs lies (so we are in Arstil), and both are Byzintian, so Bags lies too. But Bags is saying Abs is Byzintian, which is true => paradox.

So

We are on Arstil, Abs is Arstilian, Bags is Byzintian.

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  • $\begingroup$ Got the same solution for 2nd, just 1 minute to late it seems :D $\endgroup$ – print x div 0 May 30 '16 at 13:40
  • $\begingroup$ @printxdiv0 That means next time I will find one minute later than you ;) $\endgroup$ – Mayo May 30 '16 at 13:42
  • 2
    $\begingroup$ Perfect explanation. +1 and accepted :) $\endgroup$ – ABcDexter May 30 '16 at 16:42
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They are on Byzinte

because

If they are on Arstil, the second tell the truth
---- If the first tell the truth, he is from Byzinte, so they are both from the same island, so he must be lying => absurd
---- If the first lied, he is from Arstil, so he should tell the truth => absurd
Let's see if there is an absudity if they are from Byzinte
If they are on Byzinte, the second lie, so he is from Arstil
---- If the first tell the truth, he is from Byzinte, and indeed, the other is from Arstil
---- If the first lied, he is from Byzinte, both are from Byzinte

Second problem

Same kind of logic, Abs is Arstilian saying the truth, Bags is Byzintian lying, they are on Arstilian .

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Second Probleme:
1.

if abs says a lie, then there are 2 Byzintians and they are on Byzinte as abs isn't Byzintian. We should check that Bags is lying: Abs isn't Byzintian... Oh wait, yes he is from first statement, that's must be incorrect then.

2.

If abs says the truth there is at leats 1 Arstillian. 2 possibilities:
- Bags is right so Abs is Byzintian and then Bags is Arstillian: they can't be both saying the truth. It's impossible.
- Bags is false so Abs is Arstillian and as they come from different countries Bags is Byzintian. That is perfectly possible.

so the answer is:

she land on Arstil, Abs is Arstillian and Bags is Byzintian.

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Analysing these 2 islanders order of speaking:

So it could be the native and then the foreigner
Or it could be 2 foreigners
Because 2 natives would not say that they are from different islands
And if foreigner would speak first, he'd say that they are from the same island
Anyway, the "other" islander is a foreigner, so he lied.

Therefore, an explorer landed on

Byzinte


Assumption 1:

If Abs would lie, then they both are Byzintians. Then, Bags would say the truth that Abs is Byzintian. But if they are both Byzintians, they should both either speak the truth or a lie.

Assumption 2:

So Abs said the truth. If Bags would also say the truth, then Bags is Arstilian and Abs is Byzintian and they both said the truth on the same island.

Assumption 3:

That leaves us with one scenario left: Abs said the truth and Bags lied. That means, that Abs is Arstilian and Bags is Byzintian. And because Abs said the truth, she landed on Arstil.

Answer:

Abs is Artsilian
Bags is Byzintian
They are on Artsil

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