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Knights and rogues

Knights and rogues

An island is inhabited by two kinds of people: the gentlemen who always tell the truth and the rogues who always lie. The three inhabitants of the island: Adrián, Benjamín and Conrad are talking:

Adrian says:
- All of us are rogues.

But Benjamin rectifies him and says:
- Exactly one of us is a gentleman.

Could you tell us how many gentlemen and how many rogues are in the group?

Solution

Adrian cannot be a gentleman because if he were a gentleman his statement would be false, something contradictory to the definition of gentleman. Therefore Adrian is a rogue and his claim is consequently false. This means that they cannot all be rogues and that at least one must be a gentleman.

If Benjamin is a gentleman he would be telling the truth and therefore there would be only one gentleman on the island who would be himself and Conrad would be another rogue.

If Benjamin lies, it would be a rogue and his claim Exactly one of us is a gentleman it would be false this would mean that there would be no gentleman or that there would be more than one. If there were no gentlemen, Adrian's phrase would be true, which is impossible since we have concluded that Adrian is a rogue. If there were more than one gentleman and since we know that Adrian is rogue, Benjamin and Conrad would be gentlemen, which contradicts our second hypothesis that Benjamin would be lying and would be a rogue.

So the only possible solution is that On the island there is a gentleman who is Benjamin and two rogues who are Adrian and Conrad.