Opinion – Marcelo Viana: Four mathematical puzzles

1. Lost in a dark labyrinth, Malu comes across three imposing stone doors. On the left door is engraved “Exit is here”. On the middle door, “Exit is one of these doors”. On the right door, “Exit is at the middle door”. A deep voice echoes in the labyrinth: “Only one of the doors tells the truth. You will have only one chance: if you fail, you will stay here for all eternity…”. What must the unfortunate Malu do?

2. Abel travels with a dog, two chickens and a bag of poultry feed. Arriving at a river, he discovers that the only way to cross is using a rowing boat, with three seats. Both he and the animals occupy one seat each, but the bag is big, it takes two seats.

Abel, the only one who rows, could make several trips to take everyone. But if he leaves any of the hens with the bag unattended, she will eat the feed. The dog is also hungry, and if he is left alone with any of the chickens, he will eat them. But the animal is also fearful: if it is left alone with the two chickens, it will run away and never come back.

How can Abel get everyone safely to the other side?

3. Marcos always picks up his daughter Ana from school at 5 pm. Methodical, he drives with constant speed. But today Ana was not feeling well, she left early and came walking towards home. She was walking for 1h when her father found her on the road. They made a U-turn and arrived home 40 minutes earlier than usual. How much school time did Ana miss?

4. John and Mary love to have fun with math. This time, Maria came up with the following game. Consider a two-digit number, for example 45. Square each digit and add the results: in this case it gives 4^two+5^twowhich is 41. Then, repeat this operation with the result obtained, successively: 4^two+1^two=17, then 1^two+7^two=50 etc.

Maria states that the sequence always ends up being periodic, that is, it ends up repeating itself in a cycle. Will it be true? John points out that in some cases the sequence simply becomes constant equal to 1. When does this happen? Excited, Maria states that in all other cases the sequence ends in the same cycle, which is repeated indefinitely. What cycle is this?

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