Solution:

I had 10 candies in the beginning.

Explanation:
Let us take and solve this problem in the reverse direction.

At the end, I was left with 1 candy. Let me take the extra one candy back from my younger son. Now since I gave him half of what I had, we must be having the same number of candies. I have 2, which means he also have 2. This implies that I was left with 4 candies after I gave some to the elder son.

Let me take back the extra candy from my elder son now. Now I have 5 candies. Since I gave him half of what I had, he must be having 5 candies as well. Thus, I had 10 candies in the beginning.

Solution:

It is a lateral thinking puzzle and thus it's not about anything logical.

The man who walked in the movie theater was once a father of twin brothers. Despite of the fact that they were twins, their nature was starkly opposite. While David was quite peaceful and generous, Robin was very rude and violent.

On a certain day, David came running to the man and told him that Robin had killed a man. The man rushed to the crime scene with his son and saw that the man was dead with Robin near the body. He started fighting with Robin and accidentally killed him. The police arrested the man and he was sent to prison for ten years. While in his first month at the prison, he received a letter from his son which said that he was not interested in meeting him ever and that he can't forgive his father for what he did to his brother.

After ten years, he was released and came out of the prison only to know that his son was now a small time movie star and his latest movie was running in a nearby theater. While watching the movie, he smiled when he saw his son David on screen. But the next moment, the camera shifted on his arm where he saw a birthmark that belonged to Robin. He realized that on that day the one who came to him was not David but Robin and he actually killed David.

Filled with a grief and shock he decided to take his own life. He would have never forgiven himself for what he did to his son David.

Solution:

When fifteen minutes have passed, the hour hand has already moved exactly one fourth of an hour. Thus the angle between the hour hand and minute hand will be of 7.5 degrees.

Solution:

Yes it is possible only under the following condition or situation.

When the twins were born, their mother was on a ship or boat. The elder one was born on 1st of March. But just after his birth before the younger one could be born, the boat crossed the international time zone and thus, the younger one was born on 28th of February.

Now whenever, a leap year falls, the younger brother will celebrate his birthday two days before the elder one.

Solution:

12 minutes

Explanation:
Yes, surely they can all make it out of the cave.

First, father and mother will go. Time taken = 2 minutes.
Father comes back. Time taken = 2 + 1 = 3 minutes.
Both children go. Time taken = 3 + 5 = 8 minutes.
Mother comes back. Time taken = 8 + 2 = 10 minutes.
Mother and Father go. Time taken = 10 + 2 = 12 minutes.

Thus in 12 minutes, all of them will be out of the cave through the tunnel.

Solution:

If we place a decimal between the numbers it will become 5.9 which is what we needed.

Solution:

5832/17496
This fraction will give the desired result i.e. 1/3.

Solution:

Just form an equilateral triangle, planting trees at the vertices. Now find the center of the triangle and plant the fourth three on that point. In this manner you will form a shape of tetrahedron and all the trees will have equal distance between each other.

Solution:

If you look into the picture, you can see the cut lines. If the four people cut along with the lines, each one of them will have 1/5th of the cake.

Solution:

512

Explanation:
The cube that remains will be 8 x 8 x 8. This will make a total of 512. This is because, when a layer is taken, it will be taken from all sides of the cube which will reduce the dimension be two and not one.