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Each inhabitant of an island are either a truth teller who always speaks the truth, or a liar, who always speaks falsely.

Three boys P, Q and R who live in that island, are each either a truth teller, or a liar. It is known that P, Q and R together have 6 marbles. A passing visitor asked them how many marbles each had, to which P responded that he had only 1 marble, and Q responded that P had 2 marbles, while R said that P had more than 3 marbles. However all the three boys agreed that Q had precisely two marbles.

If each of the three boys had at least 1 marble, determine the respective number of marbles possessed by each of the three boys.

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Solution:

P=3, Q=1, R=2.

Let's start with a case-by-case analysis:

Case1: person Q has exactly 2 marbles. This case implies that all person in the island are truth-teller, but then the claim of P=2 and P>3 are contradictory. So is this case.

As for the second case:

we assume here that the person Q does not have exactly 2 marbles. This means, that all three person on the island are liars. So in particular, the claims of P=1, P=2 and P>3 are all false, implying P!=1, P!=2 and P<=3 so all in all P=3 must hold. If Q>=3 then R does not have any marbles, so it remains that Q=1, and R=2.

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