Why 1729 is a magic number? Answer Logically.

In the given picture, you can find two letters missing. When two particular letters are placed in the missing spots, you get an eight-letter word while reading in the anti-clockwise direction. Can you find out the missing letters and the missing word eventually?

Look at me. I can bring a smile to your face, A tear to your eye, Or even a thought to your mind. But, I can't be seen. What am I?

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?

No matter how little or how much you use me, you change me every month. What am I?

Take 9 from 6, 10 from 9, 50 from 40, and leave 6.

How Come ??

You have a roll of cloth 1km long. You have a machine which cuts this roll into pieces of 10-meter-long cloth.

How long would it take for the machine to cut the roll if each cut took 4 secs?

Rectify the following equality 101 - 102 = 1 by moving just one digit.

Andrew’s doctor gives him three pills and tells him to take one every half hour. How much time will have passed by the time Andrew’s taken all three pills?

Can you complete the sequence?

192, 021, 222, 324, 252, 627, 2__, 9__?