What is both possible and impossible at the same time?

I am sure no one can have two things for dinner. what are they?

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

Here is what you have to do. You have to throw a ball as hard as you can but it must return back to you even if it does not bounce at anything. Also, you have nothing attached to the ball. There is no one on the other end to catch that ball and throw it back at you.

How will you do it?

Can you find the largest Three-digit number whose sum is 14 and whose square is less than 5,00,000?

There were five men at church, and it started raining while they were outside. The four that ran still got wet, but the one that was still stayed completely dry. Why did he stay dry?

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?

There was once a troop of 5 elves. The 5 elves were very dedicated on finding the magical treasure of 1000 coins. However, being elves, they were super geniuses, very greedy and they did not hesitate in taking lives of other elves. The 5 elves were named Aye, Bee, Cee, Dee and Ee, ranked from high to low respectively, from Aye to Ee. One fine day, their efforts brought results and they found 1000 coins. Now they had to split it in between them as per their ranks. The lowest ranked elf has to make the proposal. If the proposal is accepted by majority, it is agreed, or the suggesting elf is killed.

What proposal should elf Ee make ?

If we tell you that there is a relation between the numbers and letters in the given figure, can you analyze it and find the missing letter in the last box?

Two girls were born to the same mother, on the same day, at the same time, in the same month and year and yet they're not twins. How can this be ?