2, 3, 5, 9, 17, _ What is the next number in the sequence?

You are on your way to visit your Friend, who lives at the end of the hill. It is his birthday, and you want to give him the cakes you have made. Between your house and his house, you have to cross 5 bridges, and as it goes in the land of make believe, there is a troll under every bridge! Each troll, quite rightly, insists that you pay a troll toll. Before you can cross their bridge, you have to give them half of the cakes you are carrying, but as they are kind trolls, they each give you back a single cake. How many cakes do you have to leave home to make sure that you arrive at a friend's house with exactly two

Replace the letters of words with a number so that the below equation holds true

BASE +
BALL
---------
GAMES
----------

A man is shown a portrait painting. He looks closely, then tells, “Brothers and sisters have I none, but that man’s father is my father’s son!” Who is the man in the portrait?

Can you figure out the logic I used to decide the order of the following words:
gun, shoe, spree, door, hive, kicks, heaven, gate, line, den

P is the father of Q and S is the son of R. T is the brother of P and has a daughter U. If R is the sister of P, then what is the relation between U, Q and S?

In the picture given below, can you find out which digit will take place of the question mark?

A convention is held where all the big logicians are summoned. The master places a band on everyone's forehead. Now all of them can see others bands but can't see his own. Then they are told that there are different colours of bands. All the logicians sit in circle and they are further explained that a bell will ring at regular intervals. The moment when a logician knew the colour of band on his forehead, he will leave at the next bell. If anyone leaves at the wrong bell, he will be disqualified.

The master assures the logicians that the puzzle will not be impossible for anyone of them. How will the logicians manage ?

You are given four results as below:
1111 = F
2222 = E
3333 = T
Then, can you find out how to code 4444?

I have Rs.50 and I spend it in the way as below.