Consider the following integral:
INTEGRAL (1/x) dx
Perform integration by parts: let

u = 1/x , dv = dx
du = -1/x2 dx , v = x

Then obtain:
INTEGRAL (1/x) dx = (1/x)*x - INTEGRAL x (-1/x2) dx
= 1 + INTEGRAL (1/x) dx

which implies that 0 = 1.
What's wrong with this calculation?

The digits of Pi are fascinating. As the ratio of the circumference of a circle to its diameter, Pi has such a fundamental definition, and yet this ratio is irrational and so its decimal expansion never repeats. It is easy to be mesmerized by the digits of the decimal expansion:
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510...

and many people have tried to memorize digits of pi for fun.

Did you know that whenever you shuffle a deck of cards, it is quite likely that you are making history?

A deck of 52 cards can be ordered in 52! = 52 x 51 x 50 x...x 2 x 1 ways. This is because there are 52 ways to choose the first card, 51 ways to choose the 2nd, 50 ways to choose the 3rd, etc. But 52! is a very large number: larger than

Here's a challenging problem with a surprisingly easy answer: can you show that for any 5 points placed on a sphere, some hemisphere must contain 4 of the points?

How about an easier question: can you show that if you place 5 points in a square of sidelength 1, some pair of them must be within distance 3/4 of each other?

Suppose that you are worried that you might have a rare disease. You decide to get tested, and suppose that the testing methods for this disease are accurate 99 percent of the time (regardless of whether the results come back positive or negative). Suppose this disease is actually quite rare, occurring randomly in the general population in only one of every 10,000 people.

If your test results come back positive, what are your chances that you actually have the disease?

Do you think it is approximately: (a) .99, (b) .90, (c) .10, or (d) .01?

When you flip a coin, what are the chances that it comes up heads? If the coin is "fair" then we expect to see heads 50 percent of the time. But is this really the case?

In an interesting 2007 paper, Diaconis, Holmes, and Montgomery show that coins are not fair--- in fact, they tend to come up the way they started about 51 percent of the time!

How many people do you need in a group to ensure at least a 50 percent probability that 2 people in the group share a birthday?

Let's take a show of hands. How many people think 30 people is enough? 60? 90? 180? 360?

How many shuffles does it take to randomize a deck of cards?

The answer, of course, depends on what kind of shuffle you consider. Two popular kinds of shuffles are the random riffle shuffle and the overhand shuffle. The random riffle shuffle is modeled by cutting the deck binomially and dropping cards one-by-one from either half of the deck with probability proportional to the current sizes of the deck halves.

Here's a nice mathematical magic trick based on properties of the Fibonacci sequence.

Give your friend a card with ten blank lines, numbered 1 through 10. Have your friend think of two numbers between 1 and 20 and write them down on the first 2 lines of the card. Now in each of the successive lines, have your friend write the sum of the previous two lines. For instance, in line 3, write the sum of lines 1 and 2. In line 4, write the sum of lines 2 and 3, etc. until finally in line 10, your friend has written the sum of lines 8 and 9.

The Fibonacci numbers are generated by setting F0=0, F1=1, and then using the recursive formula

Fn = Fn-1 + Fn-2

to get the rest. Thus the sequence begins: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, ... This sequence of Fibonacci numbers arises all over mathematics and also in nature.