How many people must be gathered together in a room, before you can be certain that there is a greater than 50/50 chance that at least two of them have the same birthday?

A time long back, there lived a king who ruled the great kingdom of Trojan House. As a part of the renovation of the kingdom to meet future security needs, he asked his chief architect to lay down a new play in a manner that all of his 10 castles are connected through five straight walls and each wall must connect four castles together. He also asked the architect that at least one of his castles should be protected with walls. The architect could not come up with any solution that served all of King's choices, but he suggested the best plan that you can see in the picture below. Can you find a better solution to serve the king's demand?

What goes up but never comes down?

How can you take 2 from 5 and leave 4?

My location can be determined by the below rebus.
So where am i ?

Can you find out the number that should replace the question mark with a valid reason?

Can you count the number of lines in the below picture?

A girl was standing near the window thinking something. All of a sudden she decides something and throws something out of the window. She dies very soon after throwing it. She was perfectly healthy and had no disease or allergy. No one killed her and she did not commit suicide.

Can you think of any possible explanation that is logical as well for what happened there?

I throw two dice simultaneously.

What is the probability of getting a sum as 9 of the two numbers shown?

I came first on earth but second on the heaven.
I also came twice in a week but found just once in a year.
I stay away from months but you can find me in February.

What am I?