Logic Riddles

Question: You put me on a table and cut me, but you don't eat me.

What am I?

Question: There was once three trees. The first tree did whatever the third tree did, and the third tree did what ever the second one does. What happens if tree number two gets cut down?

Question: A woman is in the hospital for a routine procedure. The doctor leaves the room locking the door before he goes, the woman is alone. The doctor comes back to find the woman has died and there is another person is in the room. The door was still locked.
How is this possible?

Question: A man drops an egg onto the cement. The egg doesn't break after falling 3 feet.

How could this be?

Question: Which triangle will have a larger perimeter : 3,4,5 or 3,4,7 ?