A drawing in the wall

Logic Level 2

Grandma Vera has 5 granddaughters: Emilia, Luisa, Marilia, Rafaela and Vitoria. One of the granddaughters made a drawing in grandma's Vera wall and now she wants to find the culprit. She asked to her granddaughters and they replied:

Emilia: It wasn't me;

Luisa: The culprit is either Marilia or Rafaela;

Marilia: The culprit is neither Rafaela nor Vitoria;

Rafaela : It wasn't Luisa;

Vitoria: Luisa is lying.

If only one of the 5 granddaughters is lying, who made the drawing in the wall?

This problem is from OBMEP (Brazilian Math Olympiad) 2018.

Luisa Rafaela Emilia Vitoria Marilia

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Victor Paes Plinio by Victor Paes Plinio , 21, Brazil

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2 solutions

Let's analyze what Vitoria said. If she is telling the truth, then Luisa is lying . It means that the others are telling the truth.

It means that the culprit is neither Marilia nor Rafaela. Also, Marilia is telling the truth, which means that the culprit is neither Rafaela nor Vitoria. Emilia will be telling the truth meaning that she is innocent. We conclude that Emilia , Marilia , Rafaela and Vitoria are innocents. Then, the culprit must be Luisa , BUT Rafaela is telling the truth, and she said that Luisa is innocent . This is a contradiction .

We can conclude that Vitoria must be lying!

Then, it means that Marilia is telling the truth and the culprit is neither Rafaela nor Vitoria. Rafaela is telling the truth which means that the culprit isn't Luisa. Emilia is telling the truth, then she is innocent. Hence, if Rafaela , Emilia , Luisa and Vitoria are innocents, the culprit must be Marilia \boxed{\text{Marilia}}

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I answered this by guess and check or the trial and error method.

If Victoria is lying,

  1. Emilia is telling the truth, so Emilia is not the culprit.

  2. Luisia is telling the truth, so Marilia or Rafaela can be the culprit.

  3. Marilia is telling the truth, so Rafaela is not the culprit and Victoria is not the culprit. From here the possible culprits are Marilia and Luisa.

  4. Rafaela is telling the truth, so Luisa is not the culprit.

So Marilia is the culprit.

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