Math Is Fun Forum

Theorem: All horses have infinitely many legs.
Proof (i): Everyone would agree that all horses have an even number of legs. It is also well-known that horses have forelegs in front and two legs at the back. 4 + 2 = 6 legs, which is certainly an odd number of legs for a horse to have! So, since we have shown the number of legs on a horse to be both even and odd, there must be infinitely many of them. QED.

The great logician Bertrand Russell once claimed that he could prove anything if given that 1+1=1. So one day, an undergraduate demanded: "Prove that you're the Pope." Russell thought for a while and proclaimed, "I am one. The Pope is one. Therefore, the Pope and I are one."

Q: Why do you rarely find mathematicians spending time at the beach?
A: Because they have sine and cosine to get a tan and don't need the sun!

It is only two weeks into the term that, in a calculus class, a student raises his hand and asks: "Will we ever need this stuff in real life?"
The professor gently smiles at him and says: "Of course not - if your real life will consist of flipping hamburgers at MacDonald's!"

A physicist, a mathematician and a computer scientist discuss what is better: a wife or a girlfriend.
The physicist: "A girlfriend. You still have freedom to experiment."
The mathematician: "A wife. You have security."
The computer scientist: "Both. When I'm not with my wife, she thinks I'm with my girlfriend. With my girlfriend it's vice versa. And I can be with my computer without anyone disturbing me..."

Solution to problem # k + 9

The side of the square is 2.
Let a length 'c' be cut from the two ends of a side.
We have right angled triangles of sides 'c'.
The hypotenuse would be h² = 2c²
or h = (√2)c
Since it is a regular octagon.
2 - 2c =  (√2)c
c = 2/(2+√2)
h is the side of the octagon,
h = 2 (√2)/(2+√2)
or h = 2/(1+√2)

Because, the resultant is always 3² or 33² or 333² or 3333² etc.
Follow every step of the proof carefully, you can understand the reasoning

Old mathematicians never die; they just lose some of their functions. 

The highest moments in the life of a mathematician are the first few moments after one has proved the result, but before one finds the mistake. 

A cat has nine tails.
Proof:
No cat has eight tails. A cat has one tail more than no cat. Therefore, a cat has nine tails. 

An insane mathematician gets on a bus and starts threatening everybody: "I'll integrate you! I'll differentiate you!!!" Everybody gets scared and runs away. Only one lady stays. The guy comes up to her and says: "Aren't you scared, I'll integrate you, I'll differentiate you!!!" The lady calmly answers: "No, I am not scared, I am e^x ."

The following problem can be solved either the easy way or the hard way.

Two trains 200 miles apart are moving toward each other; each one is going at a speed of 50 miles per hour. A fly starting on the front of one of them flies back and forth between them at a rate of 75 miles per hour. It does this until the trains collide and crush the fly to death. What is the total distance the fly has flown?
The fly actually hits each train an infinite number of times before it gets crushed, and one could solve the problem the hard way with pencil and paper by summing an infinite series of distances. The easy way is as follows: Since the trains are 200 miles apart and each train is going 50 miles an hour, it takes 2 hours for the trains to collide. Therefore the fly was flying for two hours. Since the fly was flying at a rate of 75 miles per hour, the fly must have flown 150 miles. That's all there is to it.
When this problem was posed to John von Neumann, he immediately replied, "150 miles."
"It is very strange," said the poser, "but nearly everyone tries to sum the infinite series."
"What do you mean, strange?" asked Von Neumann. "That's how I did it!" 

Q: Why didn't Newton discover group theory?
A: Because he wasn't Abel. 

Q:What is a dilemma?
A: A lemma that proves two results.

I don't know what made you think of this idea. In base-n system, numbers greater than or equal to n are 'banned'.
Anyway, the illustration given by you was interesting.

A math professor is talking to her little brother who just started his first year of graduate school in mathematics.
"What's your favorite thing about mathematics?" the brother wants to know.
"Knot theory."
"Yeah, me neither." 

"Divide fourteen sugar cubes into three cups of coffee so that each cup has an odd number of sugar cubes in it."
"That's easy: one, one, and twelve."
"But twelve isn't odd!"
"It's an odd number of cubes to put in a cup of coffee..."

An engineer, a physicist, and a mathematician are trying to set up a fenced-in area for some sheep, but they have a limited amount of building material. The engineer gets up first and makes a square fence with the material, reasoning that it's a pretty good working solution. "No no," says the physicist, "there's a better way." He takes the fence and makes a circular pen, showing how it encompasses the maximum possible space with the given material.
Then the mathematician speaks up: "No, no, there's an even better way." To the others' amusement he proceeds to construct a little tiny fence around himself, then declares:
"I define myself to be on the outside."

Two Secrets for Success
1. Never tell anyone everything you know.

There are 10 kinds of people in the world:
Those who understand Binary and those who don't.

Problem # k + 9

A square, whose side is 2 meters, has its corners cut away so as
to form an octagon with all sides equal. Then, what is the length of each side of the octagon, in meters?

The solution to problem # k + 7 given by kylekatarn is correct.
The sum of the roots is 7 and their product is 6.

Problem # k+8

In a number system the product of 44 and 11 is 2124. The
number 3111 of this system, when converted to the decimal number system, becomes____________.

a^b/a^c = a^(b-c)

Problem # k+7

Mike and Jim attempted to solve a quadratic equation. Mike
made a mistake in writing down the constant term. He ended up with the
roots (4, 3). Jim made a mistake in writing down the coefficient of x. He
got the roots as (3, 2). What are the roots of the original quadratic equation?

Problem # k+6

A woman and her grandson have the same birthday. For six consecutive birthdays, she is an integral multiple of his age. How old is the grandmother at the sixth of these birthdays?

(This was true for me and my maternal grandmother, although we didn't have the same birthday. For most of 6 consecutive years, her age was an integral multiple of mine)

Thats what I thought of immediately on seeing the problem!
I am not saying you're correct, I'm not saying you're wrong....
I leave the problem open for some days