I am an odd number. Take away a letter and I become even. What number am I?

A man is walking down a road with a basket of eggs. As he
is walking he meets someone who buys one-half of his eggs
plus one-half of an egg.
He walks a little further and meets another person who buys
one-half of his eggs plus one-half of an egg.
After proceeding further he meets another person who buys
one-half of his eggs plus one half an egg. At this point, he
has sold all of his eggs, and he never broke an egg.
How many eggs did the man have to start with?

I possess a head

I flaunt a tail

But I don't have mouth or legs

Can you find out who I am?

Can you solve this amazing hard number series problem?
7, 8, 10, 12, 16, 18 ?

A man is trapped in a room. The room has only two possible exits doors. Through the first door there is a room constructed from magnifying glass. The blazing hot sun instantly fries anything or anyone that enters. Through the second door there is a fire-breathing dragon. How does the man escape?

What has one letter and starts with an E?

I am lying and I always lie. Tell me whether I am lying or telling the truth.

A man was found murdered on Sunday morning. His wife immediately called the police. The police questioned the wife and staff and got these alibis: The Wife said she was sleeping. The Cook was cooking breakfast. The Gardener was picking vegetables. The Maid was getting the mail. The Butler was cleaning the closet. The police instantly arrested the murderer. Who did it and how did they know?

Who has married many women but was never married?

What is both possible and impossible at the same time?