



Eureka!
 
 

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. However, 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.
 

The Question:
When the ice has molten completely, will the water level be higher, lower, or still the same?
 

The Answer:
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Another Question:
In an aquarium filled with water, a block of ice block floats, with a bar of gold frozen in it.
We mark the current water level.
When the ice has molten completely and the bar of gold has sunk to the bottom of the aquarium,
will the water level be higher, lower, or still the same?
 

A Hint :
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Another Answer:
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Yet Another Question:
In an aquarium filled with water, a block of wood floats.
On top of the block of wood, a brick has been glued.
We mark the current water level.
If the block of wood is turned around (so that the brick hangs under it), will the water level rise, fall, or stay the same?
 

Yet Another Answer:
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The Fourth Question:
We have a pair of scales, with a block of lead on the left scale, and a block of wood on the right scale.
Both blocks have the same weight, so the scales are in balance.
We take the scales with the blocks and immerse them in an aquarium filled with water.
Will the scales stay in balance, will they turn left, or will they turn right?
 

The Fourth Answer:
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The Fifth Question:
In an aquarium filled with water, a block of wood floats.
On top of the block of wood, there lies a bar of gold.
We mark the current water level.
If the bar of gold falls into the water and sinks to the bottom of the aquarium, will the water level rise, fall, or stay the same?
 

The Fifth Answer:
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The Sixth Question:
In an aquarium filled with water, a sponge floats.
While the sponge slowly absorbs water (but stays floating), will the water level rise, fall, or stay the same?
 

The Sixth Answer:
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Alphabet Blocks
 
 

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
 

The Question:
Which letters are on each block?
 

The Answer:
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To Know or not To Know
 
 

Two whole numbers, m and n, have been chosen.
Both are greater than 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: "Aha! Now I know what your sum must be, Y!"
Y: "And now I also know what your product is, X!"
 

The Question:
What are the numbers m and n?
 

The Answer:
Click here!...
 

Another Question:
Thanks to Yiheng Wang, we can present you the following puzzle:
There is a professor with three of her equally highly intelligent students (Amy, Brad, and Charles) and they are playing a puzzle game.
The professor puts a piece of paper on each student's forehead, and on each piece of paper, there is a positive integer number.
Each student can see the numbers on the other two students' foreheads, but not the one on him/herself.
The professor tells the students: out of these three positive integer numbers, one number equals to the sum of the other two.
The students cannot speak until the professor starts to ask question and the three students answer in order.
Professor: "Do you know the number on your forehead (for sure, no guessing)?"
Amy: "I don't know."
Brad: "I don't know."
Charles: "I don't know."
Then the professor starts the second round of questioning.
Professor: "Do you know the number on your forehead (for sure, no guessing)?"
Amy: "I don't know."
Brad: "I don't know."
Charles: "Yes. It's 144."
Question to you, the reader: what are the three numbers?
 

Another Answer:
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The Masters Plaza
 

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 five masters (five of the most intelligent people who ever lived) are present.
The hotel consists of five rooms and a small restaurant that contains five 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 allowed neither 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 does not 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 is.
 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.
 

The Question:
What are the ranks, room numbers, and table numbers of the five masters?
 

The Answer:
Click here!...
Copyright © 2005 by Hassan Issa, Lebanon. Published on this website with permission.
 

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3 Heads & 5 Hats
 
 

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 hat, and all are intelligent.
After 59 seconds, man A shouts out the (correct) color of his hat!
 

The Question:
What is the color of man A's hat, and how does he know?
 

The Answer:
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Colorful Dwarfs
 
 

In a distant, dark forest, lives a population of 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.
Striking however, is that none of the dwarfs knows what the color of his or her hat is.
This is because it is forbidden to speak about the color of the hats, and there no mirrors in the forest.
Nevertheless, the dwarfs do know that there is at least one dwarf with a red hat.
Once every century, 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 not to return to the party the next day, as soon as they know that they are wearing a red hat.
 

The Question:
How many days did it take before there were no more dwarfs with a red hat left at the last party,
if you know that there were then 250 dwarfs with a red hat and 150 dwarfs with a blue hat?
 

The Answer:
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Another Question:
The dwarfs prefer a faster way to separate the red hats from the blue hats, how can they accomplish this?
 

A Hint :
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Another Answer:
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Coconut Chaos
 
 

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 the coconuts.
At one o'clock in the morning, the first sailor wakes up.
He realizes that he cannot 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 have 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.
 

The Question:
What is the smallest amount of coconuts that there could have been in the original pile?
 

The Answer:
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Numbers and Dots
 
 

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'.
 

The Question:
How close can you get to the number 82?
 

The Answer:
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