What is both possible and impossible at the same time?

What can run but never walk, have a mouth that never speaks, have a head that never weeps, and have a bed but never sleeps?

In The 1st month of this year(the year 2013), I noticed that the date 13-1-13 (13 January 2013) is special.

This date is special as:

Date * Month = Year.

Can you figure out which year of this century has a maximum number of such dates?

Below the four parts have been reorganised. The four partitions are exactly the same in both arrangements. Why is there a hole?

Can you find out what is missing?

1 3 5
2 4 ?

Hint: It's not six. Think out of the box.

What has hands, but can’t clap?

Luis Garavito killed Density x Volume.

Who is Luis Garavito ?

John’s parents have three sons: Snap, Crackle, and what’s the name of the third son?

The host of a game show offers the guest a choice of three doors. Behind one is an expensive car, but behind the other two are goats.
After you have chosen one door, he reveals one of the other two doors behind which is a goat (he wouldn't reveal a car).

Now he gives you the chance to switch to the other unrevealed door or stay at your initial choice. You will then get what is behind that door.

You cannot hear the goats from behind the doors, or in any way know which door has the prize.

Should you stay, or switch, or doesn't it matter?

A wealthy man lives alone in a small cottage. Being partially handicapped he had everything delivered to his cottage. The mailman was delivering a letter one Thursday when he noticed that the front door was ajar. Through the opening he could see the man's body lying in a pool of dried blood. When a police officer arrived he surveyed the scene. On the porch were two bottles of warm milk, Monday's newspaper, a catalog, flyers, and unopened mail. The police officer suspects it was foul play. Who does he suspect and why?