Brain-Teasers

You have a stack of 23 cards. Each card has an image of a cross on one side, and an image of a circle on the other side. You know that 14 cards in the stack are laying with the cross up, and therefore 9 cards are with the circle up. But you don't know in which order the cards are. Moreover, you are in a completely dark room, so you can't see anything.

There is a story that Archimedes, the Greek mathematician, was asked to find out if the new golden crown of the king was made of pure gold, while keeping the crown intact. Sitting in a public bath and thinking about it, Archimedes noticed the displacement of the water caused by sinking his body lower into the water. He suddenly realized that he had found the solution: if the crown was made of pure gold, it should displace the same volume of water as a bar of pure gold with an equal weight. Excited, he jumped out of the bath and ran home shouting "Eureka!" ("I've found it!"), forgetting that he was still naked...

We do not know if the story is true. But we do know that Archimedes discovered the first law of hydrostatics: when a body is immersed in a fluid, it experiences an upward buoyant force which is equal to the weight of the fluid displaced by the immersed part of the body.

Can you solve the following questions, and have your "Eureka!" moments, using this famous law?

In an aquarium filled with water, a block of ice floats. We mark the current water level.

Here you see a pool with six happy frogs. You must exchange the positions of the brown and green frogs, using the following rules:

• a frog can only jump to an empty water-lily;
• a frog can only jump over at most one other frog, that has another color;
• a frog can only jump forward (so the green frogs jump right and the brown frogs jump left).

Four men want to cross a bridge. They all begin on the same side. It is night, and they have only one flashlight with them. At most two men can cross the bridge at a time, and any party who crosses, either one or two people, must have the flashlight with them.
The flashlight must be walked back and forth: it cannot be thrown, etc. Each man walks at a different speed. A pair must walk together at the speed of the slower man. Man 1 needs 1 minute to cross the bridge, man 2 needs 2 minutes, man 3 needs 5 minutes, and man 4 needs 10 minutes. For example, if man 1 and man 3 walk across together, they need 5 minutes.

Molly has a set of four alphabet blocks. Each side of these blocks is printed with a different letter, making 24 in total. Molly notices that by rearranging the blocks, she can spell each of the following words:

BOXY, BUCK, CHAW, DIGS, EXAM, FLIT,
GIRL, JUMP, OGRE, OKAY, PAWN, ZEST

You are sitting with one opponent at an empty, round table. Taking turns, you should place one euro on the table, in such a way that it touches none of the coins that are already on the table. The first player that is not able to place a euro on the table, has lost. By tossing a coin it has been decided that you may start.

Two whole numbers, m and n, have been chosen. Both are unequal to 1 and the sum of them is less than 100. The product, m × n, is given to mathematician X. The sum, m + n, is given to mathematician Y. Then both mathematicians have the following conversation:

X: "I have no idea what your sum is, Y."
Y: "That's no news to me, X. I already knew you didn't know that."
X: "Ahah! Now I know what your sum must be, Y!"
Y: "And now I also know what your product is, X!"

You are a participant in a quiz. The quizmaster shows you three closed doors. He tells you that behind one of these doors there is a prize, and behind the other two doors there's nothing. You select one of the doors, but before you open it the quizmaster deliberately picks out a remaining empty door and shows that there is nothing behind it. The quizmaster offers you a chance to switch doors with the remaining closed door.

Julia is as old as John will be when Julia is twice as old as John was when Julia's age was half the sum of their present ages.

John is as old as Julia was when John was half the age he will be 10 years from now.

The five pieces shown below must be put together to a square.

We have 12 coins and a balance. 11 coins are of the same weight, but one coin differs in weight (note that you do not know whether the coin with different weight is heavier or lighter!). You may perform three weighings to find out which coin has a different weight, and whether this coin is heavier or lighter.

Thanks to Hassan Issa from Lebanon, we can present you the following puzzle:

You have the chance to take your room in the "Masters Plaza", a hotel in which 5 masters (5 of the most intelligent people who ever lived) are present. The hotel consists of 5 rooms and a small restaurant that contains 5 tables. Each master has a rank which shows his level of thinking with respect to the whole group. The master with the first rank is said to be the head master, and he is not you. Rooms, as well as tables, are successively numbered from 1 to 5 in a way that each master lives in a room and eats on a table different in number from his rank. To avoid confrontation, masters with successive ranks are neither allowed to live in rooms next to each other nor to eat on tables next to each other. The four present masters are: Albert Einstein, Galileo Galilei, Hassan Issa and Archimedes. To have your room in the Plaza, you just have to know your rank, table number and room number knowing that:

• Archimedes doesn't eat on the fifth table.
• Einstein is not the head master.
• Archimedes has exactly the middle rank between Hassan and You.
• Einstein is more intelligent than Archimedes.
• Galileo eats on a table next to that of Einstein.
• Hassan does not eat on a table with the same number as his room number.

You have an unlimited number of coins with a diameter d and you stack them. The goal is to let the topmost coin stick out as far as possible.

In a small village in the middle of nowhere, three innocent prisoners are sitting in a jail. One day, the cruel jailer takes them out and places them in a line on three chairs, in such a way that man C can see both man A and man B, man B can see only man A, and man A can see none of the other men. The jailer shows them 5 hats, 2 of which are black and 3 of which are white. After this, he blindfolds the men, places one hat on each of their heads, and removes the blindfolds again. The jailer tells his three prisoners that if one of them is able to determine the color of his hat within one minute, all of them are released. Otherwise, they will all be executed. None of the prisoners can see his own hat, and all are intelligent. After 59 seconds, man A shouts out the (correct) color of his hat!

There are 5 adjacent houses. Each house has a unique color, and each owner has a different nationality. Each owner keeps a different pet, drinks a different type of beverage, and has a different occupation. The Brit lives in the red house, the Swede keeps a dog, and the Dane drinks tea. The green house is on the immediate left of the white house. In the green house they drink coffee. The postman has birds. The fireman lives in the yellow house. In the middle house they drink milk. The Norwegian lives in the leftmost house. The baker lives in the house next to the house with the cats. The fireman lives in the house next to the house with the horse. The bus driver drinks beer. The German is plumber. The Norwegian lives next to the blue house. They drink water in the house that lies next to the house where the baker lives. One of the owners keeps a zebra.

On the planet Gnirica grows the peculiar cube plant (cubus vulgaris gniricae), see the image on the right. On the perfect cubical flower of this plant, the cube creature (ambulator cubi gniricae) lives. This creature is born at the bottom of the flower (point A), and his whole life it walks along the sides of the cubical flower. The cube creature walks one whole side in one year, and if it arrives in a vertex, it chooses a new direction (it can also decide to walk back along the same side it came from). As soon as the cube creature arrives at the top of the cubical flower (point B), it dies. Consequently, cube creatures live at least three years.

In a distant, dark forest, lives a population of 400 highly intelligent dwarfs. The dwarfs all look exactly alike, but only differ in the fact that they are wearing either a red or a blue hat. There are 250 dwarfs with a red hat and 150 dwarfs with a blue hat. Striking however, is that the dwarfs don't know these numbers themselves and that none of them knows what the colour of his hat is (there are for example no mirrors in this forest). But the dwarfs do know that there is at least one dwarf with a red hat.

During a certain period of their year, there is a big party in this village, to which initially all dwarfs will go. However, this party is only intended for dwarfs wearing a blue hat. Dwarfs with a red hat are supposed never to return to the party again, as soon as they know that they are wearing a red hat.

A pirate ship captures a treasure of 1000 golden coins. The treasure has to be split among the 5 pirates: 1, 2, 3, 4, and 5 in order of rank. The pirates have the following important characteristics:

• Infinitely smart.
• Bloodthirsty.
• Greedy.

Starting with pirate 5 they can make a proposal how to split up the treasure. This proposal can either be accepted or the pirate is thrown overboard. A proposal is accepted if and only if a majority of the pirates agrees on it.

Someone shows you two boxes and he tells you that one of these boxes contains two times as much as the other one, but he does not tell you which one this is. He lets you choose one of these boxes, and opens it. It turns out to be filled with $10. Now he gives you the opportunity to choose the other box instead of the current one (and skip the $10 of the first box), because the second box could contain twice as much (i.e. $20).

On an early morning, three rivals get together on an open spot in a dark wood to compose a quarrel by means of guns. A kind of duel, but with three persons: A, B and C. The rules of the game are:

• They draw lots who may fire first, second and third.
• Next, they will continue firing at each other in this order until only a single person is alive.
• Every person decides himself at which person he fires.
• Everyone knows that A hits (kills) in 100% of all shots, B hits (kills) in 80% of all shots and C hits (kills) in 50% of all shots.
• Each person chooses his ideal strategy.
• No one is killed by a stray bullet.

Below you see a square with fifteen numbered, movable tiles and an empty space in the lower right corner. The tiles are placed in the correct order, except for the tiles numbered "14" and "15", which have been swapped.

Five sailors survive a shipwreck and swim to a tiny island where there is nothing but a coconut tree and a monkey. The sailors gather all the coconuts and put them in a big pile under the tree. Exhausted, they agree to go to wait until the next morning to divide up the coconuts.

At one o'clock in the morning, the first sailor wakes up. He realizes that he can't trust the others, and decides to take his share now. He divides the coconuts into five equal piles, but there is one coconut left over. He gives that coconut to the monkey, hides his coconuts (one of the five piles), and puts the rest of the coconuts (the other four piles) back under the tree.

At two o'clock, the second sailor wakes up. Not realizing that the first sailor has already taken his share, he too divides the coconuts up into five piles, leaving one coconut over which he gives to the monkey. He then hides his share (one of the five piles), and puts the remainder (the other four piles) back under the tree.

At three, four, and five o'clock in the morning, the third, fourth, and fifth sailors each wake up and carry out the same actions.

In the morning, all the sailors wake up, and try to look innocent. No one makes a remark about the diminished pile of coconuts, and no one decides to be honest and admit that they've already taken their share. Instead, they divide the pile up into five piles, for the sixth time, and find that there is yet again one coconut left over, which they give to the monkey.

This is a famous problem from 1882, to which a prize of $1000 was awarded for the best solution. The task is to arrange the seven numbers 4, 5, 6, 7, 8, 9, and 0, and eight dots in such a way that an addition approximates the number 82 as close as possible. Each of the numbers can be used only once. The dots can be used in two ways: as decimal point and as symbol for a recurring decimal. For example, the fraction ¹/₃ can be written as

The dot on top of the three denotes that this number is repeated infinitely. If a group of numbers needs to be repeated, two dots are used: one to denote the beginning of the recurring part and one to denote the end of it. For example, the fraction ¹/₇ can be written as

Note that '0.5' is written as '.5'.