30/12/2020

In Folha, five mathematical challenges to close out the year.

Foto: Freepik

Reproduction of Marcelo Viana's column in Folha de S.Paulo.

This is the 200th edition of this column, and the last one of 2020. The magic of these numbers made me think that, after this surreal year, we all deserve a little fun. Since New Year's Eve parties are out of the question, I decided to propose some mathematical challenges for readers to solve as a family. I hope you have fun and I will be happy to receive the solutions by email at vianafolhasp@gmail.com.

1. In a game show, there are three doors, and behind one of them is a car. If the dear reader guesses which one it is, they win the prized possession. Each door has a clue: door 1 says "it's here," door 2 says "it's not here," and door 3 says "it's not behind door 1." The problem is that only one of the clues is true… Where is the car?

2. The king of Far Away loves his daughters so much that he decided it would be better for the kingdom if there were more women than men. Therefore, he decrees that all couples must continue having children until a girl is born. However, concerned about overpopulation, he determines that every couple must stop procreating as soon as they have a daughter. The probabilities of birth of both sexes are equal. Will the king achieve his goal? What will be the proportion of women in Far Away as a result of these laws?

3. The lucky reader inherits five chains from his great-aunt Joana, each made of four solid gold links. He immediately thinks that, put together, they would make a beautiful necklace to give as a gift to his wife. The goldsmith charges R$100 for each link he has to break to make the join. How much will the work cost?

4. Little Joãozinho, from Rio de Janeiro, is the darling of his two grandmothers. The problem is that they live in different states. To resolve the dispute, he proposes the following: every weekend, he will go to the bus station at a random time and take the first bus. If it's going to Minas Gerais, he'll visit Grandma Maria; if it's going to São Paulo, Grandma Vera. The buses leave punctually every 20 minutes in both directions. But after a while, Grandma Maria starts complaining that she only sees her grandson one weekend out of every five. What's going on?

5. Madalena prepares a game for her children Amanda, Bernardo, and Carolina. She sticks a positive number on each of their foreheads, explaining that the three numbers are distinct and that one of them is the sum of the others. She asks Amanda what number she has. Amanda notices that Bernardo has 20 and Carolina has 30 and replies "I don't know." Bernardo thinks and replies the same, and the same happens with Carolina. Madalena teases: "I beat you all!" But Amanda exclaims, "Wait, now I know my number!" What is it, and how did she find out?

To read the full text, visit the newspaper's website.