If two's company, and three's a crowd, then whats four and five ?

I am the one who remains awake when you sleep. But I need to rest when you awake. Who Am I?

What time is it when an elephant sits on a fence?

Things are just as they seem, right? Well, check again.
Can you count the number of tigers in this picture ?

Count the number of triangles in the below picture?

Two friends decide to get together; so they start riding bikes towards each other. They plan to meet halfway. Each is riding at 6 MPH. They live 36 miles apart. One of them has a pet carrier pigeon and it starts flying the instant the friends start traveling. The pigeon flies back and forth at 18 MPH between the 2 friends until the friends meet.

How many miles does the pigeon travel?

If you had an infinite supply of water and 5-quart and 3-quart pails, how would you measure exactly 4 quarts? What is the least number of steps you need?

You’re out on the water and see a boat filled with people. You look away for a second and look back again, but this time you don’t see a single person on the boat. Why? Hint: The boat did not sink.

There was a blind beggar living on the footpath of a street. Suddenly one day, the beggar's brother died. What was the relation of the blind beggar with the person who died?

PS: Brother is not the answer.

There are people and strange monkeys on this island, and you can not tell who is who (Edit: until you understand what they said - see below). They speak either only the truth or only lies.
Who are the following two guys?
A: B is a lying monkey. I am human.
B: A is telling the truth.