August 4, 2024

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# Puzzle 158 | Science and Mathematics

Problem proposed by Bruno Holland and Samuel Vitosa

The warden showed three prisoners in the same cell five cardboard circles: three white and two black. Then he drew a white circle on the back of each inmate so that no one could see the color of the paper they had. Every minute the jailer would ask if anyone knew the color of the circle itself and announced that the first to guess it would be set free. After a while, one of the detainees announced that he had a white circle and was released. How did he come to this conclusion?

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157.

At first, notice that everyone sees their friends’ hats but doesn’t see their own hat, so everyone can see 4 of the 5 hats. Suppose Arnold is telling the truth, that is, he is wearing a black hat. Thus, we have a group of 4 black hats (4 people telling the truth) and 1 white hat (1 person lying). This is a contradiction because we must have Bernaldo and Sernaldo lying, as they said there was more than one white hat. Arnold is lying so he wears a white hat. Let’s say Bernaldo is telling the truth, so he wears a black hat and everyone else is wearing white hats. This means that each fellow sees 1 black hat and 3 white hats, so Sernaldo must tell the truth, which leads to another contradiction. Bernardo also lies and wears a white hat. Now, if Cernaldo is telling the truth, he is wearing a black hat, and from Dernaldo’s speech, he should wear a white hat. The survival of the black hat of Ronaldo.

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If Ciraldo is lying, he is wearing a white hat. So, Dernaldo is also lying and should wear a white hat. To determine the color of Ronaldo’s hat, we must analyze the letters of our colleagues. If Ronaldo wears a white hat, we will have 5 white hats and that means that Bernaldo is telling the truth, which is not possible. So Ronaldo is wearing a black hat and that means that Arnaldo is telling the truth, and that is also not possible. Therefore, this option cannot occur. It follows that Ciraldo is telling the truth. So we conclude that:

Arnold lied. Soon, Arnold wears a white hat;
Bernaldo lied. Soon, Bernaldo wears a white cap;
– Sernaldo said the truth. Soon, Cernaldo wears a black cap, Dernaldo wears a white cap and Ronaldo wears a black cap.

Bruno Holland Professor of Federal University of Goiás and Samuel Vitosa Professor at the Federal University of Bahia.