If you drop me I’m sure to crack, but give me a smile and I’ll always smile back. What am I?

what does the below math rebus puzzle mean?

I am a seven-letter word but if you remove four of my letters I will have one left. What word am I?

10 people came into a hotel with 9 rooms and each guest wanted his own room. The bellboy solved this problem.
He asked the tenth guest to wait for a little with the first guest in room number 1. So in the first room, there were two people. The bellboy took the third guest to room number 2, the fourth to number 3, ..., and the ninth guest to room number 8. Then he returned to room number 1 and took the tenth guest to room number 9, still vacant.
How can everybody have his own room?

Birbal was jester, counsellor, and fool to the great Moghul emperor, Akbar.
The villagers loved to talk of Birbal's wisdom and cleverness,
and the emperor loved to try to outsmart him.
One day Akbar (emperor) drew a line across the floor.
"Birbal," he ordered, "you must make this line shorter, but you cannot erase any bit of it."
Everyone present thought the emperor had finally outsmarted Birbal.
It was clearly an impossible task.
Yet within moments the emperor and everyone else present had to agree that Birbal had made the line shorter without erasing any of it.
How could this be?

What is the next number in the series?

15 29 56 108 208 ?

Count the number of squares in the picture below.

Look at the picture and find out which square is not linked with any of the other squares.

12345 a famous South Indian actor
567 is a path in jumbled form
8 & 10 are same letters
9 is one among the first six alphabets.
WHO AM I?

Once while in his court, King Akbar asked Birbal to write something on a wall that makes one sad when read in good times and makes one happy when read in sad times.

He took only a few moment and wrote something that fit the requirements. What did he write?