What are examples of paradoxes?

The time-traveling grandma-killer.

Suppose you build a time machine, go back in time to find your grandmother when she was three years old, and then drown her in the bathtub.

So she never grows up, and never has any kids.

Therefore, one of your parents was never born.

Therefore, YOU were never born.

Therefore, you never traveled back in time and killed your infant grandma.

So she DID grow up and have kids.

Therefore, your parent WAS born.

Therefore, YOU were born.

Therefore, you traveled back in time and killed your infant grandma.

Ad infinitum.

[Paradoxes are really hard to understand]

Jiwon Park, 5th grader
"I know that I know nothing"

"Drugs--the higher you get, the further they drag you down."
The information paradox has to do with black holes and were the sucked in material goes. Search Steven Hawking information paradox to learn more

A Good Example of a paradox

You walk up to the bridge protector and ask for passage. He says for asking you will die. Then after a moment he says, "if you say a true statement i will choke you to death, if you say a false statement then i will drown you to death. Your choice. The young adventurer calmly replies, "You will drown me to death.

The man doesn't know what to because if he drowns the man then its a true statement and he should have choked him. If he where to choke the man then the statement would have been false and he should have drowned the man.
you can not win for loosing or maybe the silence is so loud.
An example of a paradox might include "Nobody goes to that restaurant because it is too crowded." It is a paradox because both no one and everyone are at the restaurant. A jumbo shrimp and bittersweet are also examples of paradoxes.
There are many kinds and definitions of paradoxes, and many are variations of others. The most famous is Russel's Paradox, which revolutionised logic and set theory.

Russel's: Does the set of sets that don't contain themselves contain itself?

Barber: The barber shaves men who do not shave themselves. Who shaves him?

Liar's: "My fellow citizens, I am now lying to you"

Coastline: The smaller the ruler, the longer an island's coast is.

Arrow information: You need to understand the info you're buying before you buy it.

...and many more. There are economic, political, topological, and even astronomical paradoxes that are actually quite entertaining.
go back in time and kill your mother, then you would not exist anymore, but then your mom wouldn't be dead, then you would exist and kill your mother, ect.

Also, Banach - Tarski Paradox states that you can have a 3-D sphere and you divide it up into 5 pieces. You rearrange them and you will get 2 balls identical with the original.

Unlike the first one, this one is not contradictory. It's a paradox because it betrays basic intuition and logic (or "common sense") on volume, but who said this sphere have volume, or dimension for that matter?
There is the well-known French paradox:

French diet consists of fattening foods (cheese, butter), and yet they never seem to gain weight.

Paradox of Choice:

The more options you have, the less happy you will be with the choices you make.