Can you solve below mathematical equation?

2^1234 - 2^1233

What can go up a chimney down, but can’t go down a chimney up?

John and Jacob are twin brothers. One of them is a liar whereas the other one is a truth-teller.

I called their home number and one of them picked up.
I asked. 'Does John lie?' The person replied with a yes.

Whom did I talk with, John or Jacob?

Can you replace the question mark with the correct number in the below table sequence?

He digs out tiny caves and stores gold and silver in them. He also built bridges of silver and made crowns of gold. They are the smallest you could imagine. Sooner or later everybody needs his help, yet many people are afraid to let him help them. Who is he?

Peter wakes up daily to pick up his cycle and crosses the border between Spain and France daily with a bag on his shoulder. He is investigated daily by the officials but they don't find anything suspicious.

If we tell you that he is smuggling something what would it be?

Buy me, smell me or deliver me, I do not change.

Who am I?

I look at you, you look at me I raise my right, you raise your left. What am I?

You have a three-gallon and a five-gallon measuring device. You wish to measure out four gallons.

What does below Rebus mean ?

esgg sgeg gegs gsge