At a dinner party, many of the guests exchange greetings by shaking hands with each other while they wait for the host to finish cooking.
After all this handshaking, the host, who didn't take part in or see any of the handshaking, gets everybody's attention and says: "I know for a fact that at least two people at this party shook the same number of other people's hands."
How could the host know this? Note that nobody shakes his or her own hand.

Assume there are N people at the party.
Note that the least number of people that someone could shake hands with is 0, and the most someone could shake hands with is N-1 (which would mean that they shook hands with every other person).
Now, if everyone at the party really were to have shaken hands with a different number of people, then that means somone must have shaken hands with 0 people, someone must have shaken hands with 1 person, and so on, all the way up to someone who must have shaken hands with N-1 people. This is the only possible scenario, since there are N people at the party and N different numbers of possible people to shake hands with (all the numbers between 0 and N-1 inclusive).
But this situation isn't possible, because there can't be both a person who shook hands with 0 people (call him Person 0) and a person who shook hands with N-1 people (call him Person N-1). This is because Person 0 shook hands with nobody (and thus didn't shake hands with Person N-1), but Person N-1 shook hands with everybody (and thus did shake hands with Person 0). This is clearly a contradiction, and thus two of the people at the party must have shaken hands with the same number of people.
Pretend there were only 2 guests at the party. Then try 3, and 4, and so on. This should help you think about the problem.
Search: Pigeonhole principle

Why are manhole covers round? Do manhole covers really need to be circular?

Manhole covers are round so that they won't fall through the hole into the sewer below them. No matter how you turn the cover, you won't be able to push the cover through the hole.
However, if you were to have square manhole covers, you would be able to rotate the cover such that one of the edges of the square cover is lined up with the diagonal line of the square hole, which would allow the cover to fall through, causing countless problems that the general public would rather avoid.

An old man wanted to leave all of his money to one of his three sons, but he didn't know which one he should give it to. He gave each of them a few coins and told them to buy something that would be able to fill their living room. The first man bought straw, but there was not enough to fill the room. The second bought some sticks, but they still did not fill the room. The third man bought two things that filled the room, so he obtained his father's fortune. What were the two things that the man bought?

The wise son bought a candle and a box of matches. After lighting the candle, the light filled the entire room.