Why do people go to bed?

I am a five-letter word under you,
Remove the first letter and I am above you,
Remove the second as well and I am around you.

Who am I?

There is one four-digit whole number n, such that the last four digits of n2 are the original number n

The Puzzle: Here is a famous prize problem that Sam Loyd issued in 1882, offering $1000 as a prize for the best answer showing how to arrange the seven figures and the eight 'dots' .4.5.6.7.8.9.0. which would add up to 82

In the Wild West, you are challenged into a death match by two bounty hunters nicknamed Golden Revolver (GR) and Killer Boots (KB). You accept the challenge. None of you want to waste any of the bullet and so a certain rules are laid down:

1) All of you will shoot in a given order till the last man standing.
2) Each of you shoots only once upon his turn.
3) If any one of you is injured, the other two will finish him off with an iron rod.
4) The worst shooter of all (which is you) shoots first and the best one shoots at the last.

Now, how will you plan things if you know that you hit every third shot of yours, KB hits every second shot and GR hits every shot ?

Distances from you to certain cities are written below.

BERLIN 200 miles

PARIS 300 miles

ROME 400 milesAMSTERDAM 300 miles

CARDIFF ??? miles

How far should it be to Cardiff ?

French Police inspects a room where there are no windows, no doors, no tables and is almost empty expect there is just a puddle of water. They found a dead man who obviously hung herself from the ceiling, but they are not able to figure out how this had happened.

John and Jacob entered the crime scene and easily able to solve the case.

How did the man die?

The barber of Town shaves all men living in the town. No man living in the town is allowed to shave himself. The barber lives in that town. Who then shaves the barber of the town?

If in a car race, the man who came two places in front of the last man finished one ahead of the man who came fifth, how many contestants were there?

We know that the number 7 is the prime followed by a cube. Which next number is also a prime followed by a cube?