What is the fewest number of coins that would be required in order to make sure each and every coin touches exactly three other coins?

Solve the mathematical equation in the picture below.

An apple seller is hosting a competition. He offers 1000 apples and 10 boxes to the people who pass by. The challenge is to put those 1000 apples in the 10 boxes in such a manner that if he asks for any amount of apples, the person can directly give him the boxes or a combination of boxes. If the person can do it, he promises to give a thousand apples for free.

If you happen to pass by the apple seller, will you be able to win a thousand apples?

A Japanese ship is sailing in the Indian Ocean. The captain of the ship feels tired and decides to take shower to be rejuvenated. He goes to take the shower after keeping his diamond-studded Rolex watch and gold chain on the table.
When he comes back after 15 minutes, he finds them missing. There are 4 crew members on the ship and he calls them one by one to ask them what they were doing in that duration.

Following are the answers:
1. Crew 'A' who is the cook replies that he was cooking mutton for the crew members.
2. Crew 'B' who is a housekeeper tells that he was sleeping in his room.
3. Crew 'C' who is the engineer says that he was checking on the generator.
4. Crew 'D' who is another housekeeper says that he was correcting the hoisted flag that was put upside down.

It took only a couple of seconds for the captain to find the thief. Can you find the thief too?

John, a 5-year-old boy, was really fond of the chocolates. He asked his Mother to give him some money to buy his favourite chocolates. His Mother gave him $45. He went to the shopkeeper and asked, "How much is one chocolate for?". The shopkeeper said $3 for one chocolate. Also, if you give me the wrappers of three chocolates, I will give you one for the exchange.
In total, how much chocolate could John eat?

Can you complete the sequence puzzle?

AR TA GE CA LE VI LI _

The below mentioned letters have been placed in a order using a logic.

DEAD, BALL, FEAR, SHOP

Can you find the logic?

An American nightclub called 'The Coconut Grove' had a terrible fire in which over 400 people died. A simple design flaw in the building led to a high death toll. Subsequently, regulations were changed to ensure that all public buildings throughout the country eliminated this one detail, which proved deadly. What was it?

There's a day, that comes around once a year.
To celebrate the man, you hold dear.
He gave you life, love and will pick you up if you fall.
When it comes to guys like him, He's the best of all.
Who is he?

The Federal bank of London is abducted by the robbers. The head of the robbers asked the cashier to empty their money vault to them and when suddenly cashier got a call from her father. To avoid any suspicion, the robber asked the cashier to pick the call and reply her father in the shortest manner possible.

The cashier told her father "Is there an emergency father, Call me when you are free and I will help you in your furnishing" and then the cashier hung up the phone.

After 10 minutes, police arrived at the crime scene.

How did the police know about the robbery?