There is a small town in the midwest with exactly 2 barbershops, one on each side of town. The barbershop on the west side of town is pristine. Its floors are spotless, the windows are always perfectly clear, and the air always smells fresh. The barber has a friendly smile, shined shoes, a well-groomed head of hair, and a fancy shirt. The barbershop on the east side of town is a mess. Its floors and windows are dirty, and the air smells of garbage. The barber always has a grimace on his face. His skin is oily, his hair is short and ragged, and he has food on his clothes all the time.
A man travelling through the town realizes he needs a haircut. Knowing the stories of the two barbers, the man decides to go to the dirty barbershop on the east side of town.
Why does he do this?
Because there are only two barbers in the town, the barbers must cut each-other's hair. The barber on the west side of town has a nice haircut, so the east-side barber must be a good barber. On the other hand, the barber on the east side of town has ragged hair, meaning the west-side barber must not be very good. So the man goes to the east-side barber to get a better haircut.
There are 1 million closed school lockers in a row, labeled 1 through 1,000,000.
You first go through and flip every locker open.
Then you go through and flip every other locker (locker 2, 4, 6, etc...). When you're done, all the even-numbered lockers are closed.
You then go through and flip every third locker (3, 6, 9, etc...). "Flipping" mean you open it if it's closed, and close it if it's open. For example, as you go through this time, you close locker 3 (because it was still open after the previous run through), but you open locker 6, since you had closed it in the previous run through.
Then you go through and flip every fourth locker (4, 8, 12, etc...), then every fifth locker (5, 10, 15, etc...), then every sixth locker (6, 12, 18, etc...) and so on. At the end, you're going through and flipping every 999,998th locker (which is just locker 999,998), then every 999,999th locker (which is just locker 999,999), and finally, every 1,000,000th locker (which is just locker 1,000,000).
At the end of this, is locker 1,000,000 open or closed?
Locker 1,000,000 will be open.
If you think about it, the number of times that each locker is flipped is equal to the number of factors it has. For example, locker 12 has factors 1, 2, 3, 4, 6, and 12, and will thus be flipped 6 times (it will end be flipped when you flip every one, every 2nd, every 3rd, every 4th, every 6th, and every 12th locker). It will end up closed, since flipping an even number of times will return it to its starting position. You can see that if a locker number has an even number of factors, it will end up closed. If it has an odd number of factors, it will end up open.
As it turns out, the only types of numbers that have an odd number of factors are squares. This is because factors come in pairs, and for squares, one of those pairs is the square root, which is duplicated and thus doesn't count twice as a factor. For example, 12's factors are 1 x 12, 2 x 6, and 3 x 4 (6 total factors). On the other hand, 16's factors are 1 x 16, 2 x 8, and 4 x 4 (5 total factors).
So lockers 1, 4, 9, 16, 25, etc... will all be open. Since 1,000,000 is a square number (1000 x 1000), it will be open as well.
With pointed fangs I sit and wait; with piercing force I crunch out fate; grabbing victims, proclaiming might; physically joining with a single bite. What am I?
Two men working at a construction site were up for a challenge, and they were pretty mad at each other.
Finally, at lunch break, they confronted one another.
One man, obviously stronger, said "See that wheelbarrow? I'm willin' to bet $100 (that's all I have in my wallet here) that you can't wheel something to that cone and back that I can't do twice as far. Do you have a bet?"
The other man, too dignified to decline, shook his hand, but he had a plan formulating.
He looked at the objects lying around: a pile of 400 bricks, a steel beam, the 10 men that had gathered around to watch, his pickup truck, a stack of ten bags of concrete mix, and then he finalized his plan.
"All right," he said, and revealed his object.
That night, the strong man went home thoroughly teased and $100 poorer.
What did the other man choose?
He looked the man right in the eye and said "get in."
You overhear a man talking to a clerk in a hardware store. The clerk says "One will cost you 12 cents, ten will cost your 24 cents, and one hundred will cost you 36 cents."
What is the man buying?
The man is buying physical numbers to nail to the front of his house. Each number costs 12 cents, and so "1" will cost 12 cents, "10" will cost 24 cents, and "100" will cost 36 cents.
