, 2003

Combinatorics Difficulty 4.6 AIME Find the answer Italy

Problem:

In the country of $\mathrm{Oz}$ , besides normal people (who may lie or tell the truth), there live knights (who always tell the truth) and knaves (who always lie). A strange law requires that in every marriage the spouses are either both normal, or one a knight and the other a knave. Arriving at the semi-detached house of the Allegri and Bianchi couples, you ask whether Mr. Bianchi, who is absent, is a knight, and you get an affirmative answer both from Mrs. Bianchi and from both members of the Allegri couple. For how many spouses can you determine the type?

Pick one

Solution

Solution:

The answer is $\mathbf{( E )}$ . If Mr. Bianchi were not normal, he would be a knave or a knight. In the first case, his wife would be a knight and would have lied by stating that her husband is a knight, and this is absurd. In the second case (Mr. Bianchi a knight) his wife would be a knave and would have told the truth, and this is again absurd. Therefore Mr. Bianchi is normal, and so is his wife as well. We must then also consider the Allegri couple. Since they state the same thing, they can only be normal. In conclusion, all the people are normal.

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.