What is both possible and impossible at the same time?

There is a box. The area of its top is 240 square units, the area of the front is 300 square units and the area of the end is 180 square units.

Can you calculate the dimensions of this box with the given data?

I shave every day, but my beard stays the same. What am I?

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 ?

Take away my first letter, then take away my second letter. Then take away the rest of my letters, yet I remain the same. What am I?

Spelt forwards I’m what you do every day, spelled backwards I’m something you hate. What am I?

I throw two dice simultaneously.

What is the probability of getting a sum as 9 of the two numbers shown?

As we know that white starts the game of chess. Can you find the scenario shown in the picture below is possible when all the white pieces are at the original place while the black pawn is not as in the below picture?

Can you find the ages of a son, father and grandfather based on the following facts?
The sum of the ages of grandfather, father and son is 140 years.
The age of the son in months is the same as the grandfather in years.
The age of the son in days is the same as the father in weeks.

In Greek mythology, the Sphinx sat outside of Thebes and asked this riddle of all travellers who passed by. If the traveller failed to solve the riddle, then the Sphinx killed him/her. And if the traveller answered the riddle correctly, then the Sphinx would destroy herself. The riddle:

What goes on four legs in the morning, on two legs at noon, and on three legs in the evening?

Oedipus solved the riddle, and the Sphinx destroyed herself.