Two men ride their horses to the town blacksmith to ask for his daughter's hand in marriage. To help decide who will get to marry her, the blacksmith proposes a very strange race:
"You will race your horses down the mile-long road from here to the center of town, and the man whose horse passes through city hall's gates LAST will get to marry my daughter."
The men have no idea how to proceed, but after a few minutes of thinking, they come up with a great idea to abide by the blacksmith's rules. 30 minutes later, one of the men is gloating, having won the daughter's hand in marriage.
What was the idea the men had?
Each man rides the other man's horse. They race as they normally would. The blacksmith said the man whose horse crosses last would win, so the man who wins the race would have his horse finish last.
You are visiting NYC when a man approaches you.
"Not counting bald people, I bet a hundred bucks that there are two people living in New York City with the same number of hairs on their heads," he tells you.
"I'll take that bet!" you say. You talk to the man for a minute, after which you realize you have lost the bet.
What did the man say to prove his case?
This is a classic example of the pigeonhole principle. The argument goes as follows: assume that every non-bald person in New York City has a different number of hairs on their head. Since there are about 9 million people living in NYC, let's say 8 million of them aren't bald.
So 8 million people need to have different numbers of hairs on their head. But on average, people only have about 100,000 hairs. So even if there was someone with 1 hair, someone with 2 hairs, someone with 3 hairs, and so on, all the way up to someone with 100,000 hairs, there are still 7,900,000 other people who all need different numbers of hairs on their heads, and furthermore, who all need MORE than 100,000 hairs on their head.
You can see that additionally, at least one person would need to have at least 8,000,000 hairs on their head, because there's no way to have 8,000,000 people all have different numbers of hairs between 1 and 7,999,999. But someone having 8,000,000 is an essential impossibility (as is even having 1,000,000 hairs), So there's no way this situation could be the case, where everyone has a different number of hairs. Which means that at least two people have the same number of hairs.
You're standing in front of a room with one lightbulb inside of it. You cannot see if it is on or off. Outside the room, there are 3 switches in the off positions. You may turn the switches any way you want to. You stop turning the switches, enter the room and know which switch controls the lightbulb. How?
You turn 2 switches "on" and leave 1 switch "off" and wait about a minute. Then enter the room, but just before you enter, turn one switch from "on" to "off". Once in the room, feel the lightbulb - if it is warm, but off, it has to be the last switch you turned off. If it is on, it has to be the switch left on. If it is cold and is off, it has to be the switch you left in the off position.