Why Mathematicians Sometimes Eat Their Mugs and Drink out of Their Bagels

Topology (from the Greek τόπος, "place", and λόγος, "study") is the mathematical study of shapes and spaces. It is a major area of mathematics concerned with the most basic properties of space, such as connectedness, continuity and boundary. It is the study of properties that are preserved under continuous deformations including stretching and bending, but not tearing or gluing.

There is a common joke among mathematicians that a topologist might accidentally mix up his mug and doughnut because two spaces are homeomorphic if one can be deformed into the other without cutting or gluing, as is the case.

See http://en.wikipedia.org/wiki/Topology

and http://www.georgehart.com/bagel/bagel.html.

For more about topology, try reading http://www.math.wayne.edu/~rrb/topology.html.

Eggs in a Basket

There are six eggs in a basket. Six people each take one egg. How can it be that one egg stays in the basket?

Multiple Choice Paradox

If you choose an answer to this problem at random, what is the chance that you will get it right?
A. 25%
B. 50%
C. 60%
D. 25%

Islanders again...

On a certain island, the culture dictated that wives stayed home, and men did "business". The society  also dictated that if a man found out his wife was cheating, she was humiliated and kicked out by the next morning. Once a week, there was a market, and the men gossiped. It was taboo to tell a man about his wife, but other than that everyone found out about everything no later than a week after it happened.
Once, a merchant came to the island and walked around at the market. Because this was an important merchant, the islanders asked him to give a speech at the end of his stay. They were all flattered by his compliments ("What a wonderful town," etc.), but surprised to hear that he knew that there were unfaithful wives on the island.
Seven weeks later, all the unfaithful wives had been kicked out. On the seventh market day after this, there was a big announcement. "There are no more unfaithful wives on the island."
There was no polygamy on the island.
How many unfaithful wives were there originally?

The Maker

x=2

24

Create the number 24 using (all of) 1, 3, 4, and 6.
You may add, subtract, multiply, and divide. Parentheses are free. You can (and must) use each digit only once. Note that you may not glue digits together. (14 - 6) * 3 is not a solution. 1^3 * 4 * 6 is not a solution either (powers not allowed).  

Nonsense

Charles Lutwidge Dodgson was a mathematician in the nineteenth century. He was also a writer, photographer, and deacon. His pen name was Lewis Carroll.

His talent was playing with logic. Here are a few extracts from his book, Alice in Wonderland. 

     "Take some more tea," the March Hare said to Alice, very earnestly.
     "I've had nothing yet," Alice replied in an offended tone: "so I can't take more."
     "You mean you can't take less," said the Hatter. "It's very easy to take more than nothing." (Chapter VII).

Do you agree with Alice or the Hatter?

Here's some more:

About the ordered execution of the Cheshire cat, who only had a head at this point in the book:


"The executioner's argument was, that you couldn't cut off a head unless there was a body to cut it off from: that he had never had to do such a thing before, and he wasn't going to begin at his time of life.
The King's argument was that anything that had a head could be beheaded, and you weren't to talk nonsense. "

Again, who do you agree with?

Wonky Scale...

A farmer badly in need of money has decided to sell walnuts. He has a brick that weighs exactly 1 kilogram and a wonky lever balance: the distances are different. A customer comes and asks for 2 kg of walnuts.

The farmer thinks that if he puts the the weight in the left pan and measures out walnuts to balance it out, then switches the walnuts and the weight, and adds the two amounts of walnuts, he'll have the two kilograms.
Is he right?

If not, then how is it possible to measure the two kilograms?

Happy Birthday

"The day before yesterday I was 10 years old, and next year I'll turn 13."

Could this be the truth?
If yes, then how?

Pythagoras

(Skip down to the Legends and Anecdotes part if you already know the facts).

Facts
Pythagoras, one of the most famous and controversial Greek philosophers, lived from around 570 to around 490 BC. He spent his early years on the island of Samos, off the coast of modern Turkey. At the age of forty he emigrated to the city of Croton in southern Italy and most of his philosophical activity occurred there. Pythagoras wrote nothing, nor were there any detailed accounts of his thought written by contemporaries.
The popular modern image of Pythagoras is that of a master mathematician and scientist. The early evidence shows, however, that, while Pythagoras was famous in his own day and even 150 years later in the time of Plato and Aristotle, it was not mathematics or science upon which his fame rested. Pythagoras was famous as an expert on the fate of the soul after death (he thought the soul was immortal and went through a series of reincarnations), as an expert on religious ritual, as a wonder-worker, and as the founder of a strict way of life that emphasized dietary restrictions, religious rituals, and rigorous self-discipline.

The Pythagorean School
A group of 300 people formed around him, male and female. This was the school of Pythagoras.
The school of Pythagoras was every bit as much a religion as a school of mathematics. A rule of secrecy bound the members to the school, and oral communication was the rule. The Pythagoreans had numerous rules for everyday living. Here are a few of them:
-To abstain from beans.
-Not to pick up what has fallen.
-Not to touch a white cock.
-Not to stir the fire with iron.
...
-Do not look in a mirror beside a light.

Legends and Anecdotes
It is said that Pythagoras had a fear of irrational numbers so pronounced, that when one of his students proved their existence, he decided that the gods were punishing him for some sin and sacrificed one hundred bulls and (it is said) the student who proved their existence.
Another legend states that as some guards were chasing him for a political mistake (they'd set his house on fire and sent him running), he came upon a field of beans where he stopped, and declared that he would rather die than enter the field. The guards slit his throat.

http://www.youtube.com/watch?v=X1E7I7_r3Cw

The above is a good video about Pythagoras...

Thank you, WIkipedia, Youtube, etc. :)

Lift

A person walks into an elevator on the first floor, goes up eleven floors, gets out, and walks up four flights of stairs. This person does this every day. Why?

The lift goes up to the top of the building (say 20 floors), and the person isn't crazy, or badly in need of exercise. 

The answer will be at answersdiscarded.blogspot.com someday.

Skipping over the busy-work

Many people know this math problem, but I'm going to post it here anyway, because I've always liked the solution.

Add numbers 1 through 100.

Of course, you could just go through and add...

1+2=3

3+3=6

6+4=10

10+5=15*

... but that would be a lot of busy-work, and mathematicians don't approve of busy-work. 

*One small cool thing about the sequence you get from going through and adding everything (1, 3, 6, 10, 15, 21, 28...): these are the triangular numbers. 

So how can you do this in a much faster, easier, and more effective way?

To find the answer, go to answersdiscarded.blogspot.com. As usual, this is where I beg you to try to figure it out by yourselves before checking the answer.

If you've already done that problem, do this one:
Add all numbers divisible by three from 100 to 200.

Leonardo Fibonacci and the Fibonacci Sequence

Leonardo Fibonacci, or Leonardo of Pisa, wrote a book called Liber Abaci, in which he used a sequence of numbers he learned during his earlier travels as an example. He was a mathematician and was born in 1170. He died in 1250.  [more]

Math problem:

Fibonacci's Problem:

"Start with a pair of rabbits, (one male and one female) born on January 1. Assume that all months are of equal length and that :

1. rabbits begin to produce young two months after their own birth;

2. after reaching the age of two months, each pair produces a mixed pair, (one male, one female), and then another mixed pair each month thereafter; and

3. no rabbit dies.

How many pairs of rabbits will there be after one year?"

Solution: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...

So 144 pairs of rabbits.

The Fibonacci Sequence.

Why is the Fibonacci sequence beautiful?

(quick summary.. way too quick).
Fibonacci numbers really do appear in nature. For example, some plants branch in such a way that they always have a Fibonacci number of growing points. Flowers often have a Fibonacci number of petals, too: daisies can have 34, 55, or even 89 petals.
The next time you look at a sunflower, look closely at the arrangement of the seeds. They appear to be spiraling outwards both to the left and the right. There are a Fibonacci number of spirals!

Another amazing thing about the sequence is the fact that, the higher up in the sequence, the more closely the ratio of two consecutive Fibonacci numbers approaches the golden ratio (approximately 1:1.618 or 0.618:1).

Divisibility:
Every 3rd number of the sequence is even and, more generally, every kth number of the sequence is a multiple of Fk. Thus the Fibonacci sequence is an example of a divisibility sequence.

Comment if you don't understand something.

Counterfeit Coins

You have nine identical coins. One of them is counterfeit. The counterfeit one is a tiny bit lighter than the normal ones. 

In two weightings (see picture), how can you figure out which one is counterfeit?

The answer will soon be at answersdiscarded.blogspot.com.

http://www.smithsonianmag.com/science-nature/the-natural-beauty-of-math-174842751.html

How to slice a cake

This is one of the easier math problems that requires either one good look or several minutes' worth of squinting at it and trying to force your mind to think out of the box.  To many mathematicians, the answer seems obvious as soon as they hear the problem, but for some people, it takes awhile.

How to cut a generic birthday cake into eight pieces in no more than three straight cuts? No special tools or anything.

The answer is at answersdiscarded.blogspot.com.

Please, please, please try to think of the answer before looking it up.

Wise People

On a distant planet, far across the galaxies, a mighty ruler decided to test her wise people to find out which one of them was the wisest. She invited them to her planet,  and offered them wine and food.  The wise people, whose talents were much unappreciated,  had not seen so much food in all of their lives, and were soon sound asleep.  When the wise people woke up, they all saw that everyone's faces were colored green, but they could not see their own faces. They all immediately started laughing, and so did the ruler, quietly laughing behind her secret spot under a table loaded with dessert.
But then one stopped, and then another.
They all gradually stopped laughing, realizing that they, too must have paint on their faces.
The ruler named the first one to stop her prime minister.

Question: how did the wise people find out that they, too had been colored green?
Hint: try deciding on a number of wise people.

The answer is at answersdiscarded.blogspot.com.

PLEASE THINK ABOUT IT BEFORE YOU LOOK IT UP.

Grad students

The following joke/riddle is a mathematician's way of disgusting potential future mathematicians, to see what the impact will be...

N grad students were sitting in a train compartment. They were discussing deep intellectual subjects (free food, math, free food, professors, politics, and the best places for free food), all the while eyeing a platter previously full of (free) meatballs, now containing six meatballs.  Each grad student had a fork and a knife, and was very aware of the fact that taking a meatball would be impolite, however much he/she wanted to. At that moment the train went into a tunnel and everything went dark for a couple of seconds. There was a loud yelp, and the light came back, revealing a platter containing a cut-off hand with a fork stuck in it.  How many grad students were there?


The answer might be at answersdiscarded.blogspot.com. Answer number 5.

How to boil water

This is what I just did while doing a math problem:

Problem number 1.   How do you boil water given a teapot, a turned off stove, and water?
Solution: Fill the teapot with water, turn on the stove, put teapot on stove, wait to boil.

Problem number 2.  How do you boil water given a teapot full of water, a turned on stove, and water?
Solution. Empty the teapot, turn off the stove, and repeat solution to Problem 1.


I feel ridiculous. 

Knights and Knaves (starting easy)

     I think that this may be my favourite type of math/logic riddle.  In a world where there are only knights and knaves...

Knights can only tell the truth.  Knaves can only lie.

     You are on an island with two towns, walking  along a road. One town is where all the knaves live, and the other is where all the knights live. You have to post a letter to your friend, but there is only one post office, and it's in the knights' town.
     You have come to a fork: one road leads to the knaves' town and the other to the knights' town. You see a man sitting at the fork, but you do not know whether he is a knight or a knave.  On this peculiar island, you are always allowed to ask one question to another person.  You have to ask him a question which determines which town is the knights' town.  Remember: you do not know whether he is a knight or a knave.


This is one of the easier questions of this type. Again, the answer is at discardedlogic.blogspot.com, answer number 5.  PLEASE TRY TO FIND THE ANSWER before looking it up. (!)

Are the horizontal lines parallel?

I like this illusion because it actually fools me... even though I always know the answer.

 

And this one's pretty cool too, as long as you're not looking at it.

Choosing a computer

In a very peculiar shop, there are three computers, one each of three types:
-American computers, which always answer any questions truthfully,
-Chinese computers, which always lie,
-Russian computers, which sometimes lie, and sometimes tell the truth.

You want a computer that can answer yes/no questions, and you don't care if it's an American computer or a Chinese one. You do not, however want a Russian one.
The biggest problem with these computers is that they are exactly alike on the outside. The shopkeeper tells you that you are allowed to ask ONE question to ONE computer.

In this one question, you have to find out whether or not this computer is Russian.

What is the question?

To find the answer, go to discardedlogic.blogspot.com, answer number 4, where there may be the answer to the question of the question. (It depends on when you look and when I put the answer up).

How Many Legs?

Princess' Groom

      In a faraway kingdom, the beautiful princess, daughter of the mighty (and unjust) ruler, subject of the courting of many princes and even kings from even farther away kingdoms, suddenly fell in love.  This was a completely new feeling for her, and she really didn't know what to do.  After many hours of thinking and rethinking, forced to use parts of her lovely little head which were normally left idle, the poor creature decided to tell her maid, an elderly woman with much experience. She immediately told the girl to confess to the fortunate man. Little did the maid know that the subject of the princess' love was a swineherd.
      The swineherd, a very sharp young man who hoped to do great things and had already taught larger part of the royal pig population the times two table, reacted very strangely indeed.  Instead of dressing up in furs and gold, bringing gifts to the king, and sucking up to the princess (as did all of her previous suitors), he responded by kissing her. On the whole, thought the princess, happily (and muddily) returning home, it was a most productive evening.
      After about half a year's worth of such evenings, the king found out.  He immediately ordered the poor swineherd's execution and the princess' marriage to the baldest king on the planet. No pleading or tears from his daughter could change his mind. However, his chief counselor advised him that maybe it would be a good idea to appear merciful to his people. He suggested that the condemned man should be publicly given a choice of two papers which he would blindly draw from a bag, and which would determine whether he should be executed or allowed to marry the princess. To make sure that he would die, both papers would have "guilty" written on them.
      The princess, hearing only the first part of the plan, immediately ran to find her beloved in the dungeons.  He, being an optimistic young man, rejoiced and said that it was better than no chance at all.  However, the princess soon found out about the trickery. In tears, she told her swineherd all about it.  Instead of being discouraged, he told her that the fates had turned their way, and he was to be soon united with her for the rest of their lives.
      How?
      Write your answers in comments, but try not to read other people's comments before thinking yourselves.
      I have put the answer on answersdiscarded.blogspot.com. Answer #3.

Please, please, please think about it before looking at the answer.  (!)