What is the word or phrase?

Three people enter a room and have a green or blue hat placed on their heads. They cannot see their own hat but can see the other hats.
The colour of each hat is purely random. They could all be green, blue, or any combination of green and blue.
They need to guess their own hat colour by writing it on a piece of paper, or they can write 'pass'.
They cannot communicate with each other in any way once the game starts. But they can have a strategy meeting before the game.
If at least one of them guesses correctly they win $10,000 each, but if anyone guesses incorrectly they all get nothing.
What is the best strategy?

I am under you when I am whole.
I shift above you if you remove the first letter.
I come all around you if you remove the second as well.

Can you tell me who I am?

There are three boxes on a table. One of the box contains Gold and the other two are empty. A printed message contains in each box. One of the message is true and the other two are lies.

The first box says "The Gold is not here".

The Second box says "The Gold is not here".

The Third box says "The Gold is in the Second box".

Which box has the Gold?

I am taken from a mine and shut up in a wooden case, from which I am never released, and yet I am used by almost every person. What am I?

Decode below rebus
D
N
A
T
S , D

How can you drop a raw egg from a height onto a concrete floor without cracking it?

The more it dries, the wetter it becomes. Can you tell what it is?

There is a brick of gold and a brick of iron in a boat (both 10-inch blocks), if they are both dropped into the water which will make the water level higher?

I am Huge on Sunday and Saturday.
I am small from Tuesday to Thursday.
I do not exist on Monday and Friday.
Who am I?