This is a compilation of curious exercises and problems, designed to think a while before finding the correct solution.
Some of them are taken from the saga of Professor Layton, which inspired me so much when I was little, and many others are simply interesting problems that I find browsing the Internet; I hope you enjoy them and they make you think a lot until you find the solution ;p
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The 5 Hats
We have a box with five hats, of which three are white and two are black.
Three people enter the room (A, B and C), each of whom picks up a hat at random, without seeing the color. These three people sit in such a way that A can see the hats of B and C, but not his own; B can only see C's hat, and C cannot see any hat.
They are asked to try find out the color of the hat they are wearing over the head.
The first to speak is A, after looking at his classmates he says that he can't figure out what color his hat is. Next he takes the B word, who admits that he cannot deduce the color of his hat either. Finally, C intervenes and says: “I do know what color my hat is.”
What color is C's hat?
The Camera and Case
A camera with a case costs €310. The seller tells you that the camera costs €300 more than the case, and that the case costs what he asks for minus what the camera costs. If you buy only the cover and pay with a €100 bill, how much change do you have to return?
Cats and Mice
Five cats can catch five mice in five minutes. How many cats does it take to catch 100 mice in 100 minutes?
The Island of Blue Eyes
On an island there are 100 inhabitants. They all have either blue eyes or brown eyes. Everyone sees the color of others, but not their own color. They can't talk about it and there are no mirrors. Of course: a law establishes that if someone discovers that they have blue eyes, they must leave the island at 8 in the following morning. All islanders have the same ability to reason and are all capable of using impeccable logic.
One day, a person arrives to visit the island and, while looking at everyone, he says, without pointing to anyone in particular: “How good it is to see at least one person with blue eyes after being on the high seas for so long.” !”
What consequences did this comment have for the inhabitants of the island?
The ferris wheel
A ferris wheel with 10 two-seater cabins rotates so that every minute, one of the cabins reaches the departure platform.
If it started operating at 10:00 and closed 30 minutes later, how many people can it have accommodated at most?
Chairs for the pavilion
The Secret Password
A group of police officers investigate the headquarters of a group of criminals. They want to infiltrate, but they need the password, so they keep watch to try to find it out. A guy approaches the door. From inside they say: “18”. Answer: “9”. The door opens and they let him in. Another one arrives. They tell him: “8” and he answers: “4”. They also let him in. A third arrives. To the number “14” he answers “7” and they open the door.
The police believe they have found the key: you just have to divide the number they say by two. So, they decide to send an undercover agent. When you get to the door they say “0”. Answer: “0”. Not only does the door not open, but they shoot and kill him. They try it again with another agent. From inside you hear: “6”. Answer: “3”. They kill him again.
What is the mistake that the police made?
The train and the bridge
A 100 m long train takes 30 seconds to cross a 400 m bridge. If its speed is constant, at what speed (in km/h) is this train moving?
The virus
A virus fills a jar in one hour, dividing in two once a minute. How many minutes would it take for the jar to fill if you started with two viruses?
Too many mice
You buy a little mouse that can give birth to 12 babies a month, and in turn, its offspring can have their own babies within 2 months of being born.
How many mice will you have in 10 months?
Apples and Oranges
You have a fruit store and they have given you three boxes: one has only apples; another, only oranges; the third, apples and oranges. Each box has a label: “apples,” “oranges,” and “apples and oranges.” None of the boxes have the corresponding label. How can you tell what fruit each of the boxes contains by taking a single piece out of just one of them?
The two doors
They give you the opportunity to get out of prison, but there are two doors: one leads to your freedom, the other leads to your death.
Next to each door you have a guard, and you know that there is one who always tells the truth and another who always lies. They know which door is good, and they are just there to answer the only question you are allowed to ask before choosing a door. You, on the other hand, don't even know which guard is; You just know that one is always going to lie, and the other is always going to tell the truth.
What is the question you have to ask to escape with a 100% security?
Warden and Prisoner
The warden of a prison reports that he will let a random person out of prison to celebrate his 25 years of work.
They choose a man and tell him that he will be free if he takes a white ball out of a box. Inside this box there are a total of 9 black balls and 1 white one.
The prisoner finds out through a tip-off that the warden will turn all the balls black. The next day, the game is played and the prisoner is released.
How did he manage to get out of jail if all the balls were black?
A waiter and coins that disappear
Three friends eat at a restaurant and at the end they are given the €30 bill. Each friend puts in a €10 bill but in the end the waiter tells them that the dessert (which cost €5 and they had already paid for on the menu) was on the house.
The waiter gives them those leftover €5 in €1 coins.
The friends, not being able to share those €5, decide to keep €1 each and tip the waiter €2, so that in the end each one has paid €9, and the waiter has kept €2, a total 9×3+2 = €29. If at the beginning they put €30... Where is the missing coin?
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