# Math Brain Teasers: 5 Famous Paradoxes that Will Make you Think

Who doesn’t love a good mathematical riddle? Challenge yourself to one of these brain teasers, and whip them out at your next social gathering.

Q: Say you have 100 pounds of potatoes, which are 99% water by weight. You let them dehydrate until they’re 98% water. How much do they weigh now? The surprising answer is 50 pounds!

A: For 100 pounds of potatoes with 99% water weight, you have 99 pounds of water and 1 pound of solids. This is a 1:99 ratio. If the water decreases to 98%, then you have 2% of solids. This is a 2:98 ratio, which reduces to 1:49. The weight of the solids never changed, so you have 1 pound of solids and 49 pounds of water, so the new total weight is 50 pounds.

### The Birthday Problem

Q: Imagine you have n randomly-chosen people in a room. What is the probability that some pair of them share a birthday?

A: This problem has an interesting and unexpected solution. It follows from the solution that there’s a 50.7% chance that in a room with n=20 people, a pair will share a birthday. And there’s a 99.9999% chance that if you have n=200 people in a room, there will be a pair that shares a birthday.  So, if you’re ever in a room with 19 other strangers, there’s a greater chance that you share a birthday with one of them than if you were to flip a coin and get tails.

### The Monty-Hall Problem

via clipartkid.com

Q: Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

A: Yes, you should switch! Counterintuitively, you would have a ⅔ chance of winning the car if you switch your choice, and a ⅓ chance if you stick with your choice.

### Hilbert’s Paradox of the Grand Hotel

Q: Consider a hypothetical hotel with a countably infinite number of rooms, all of which are occupied. Can the hotel accept any more guests?

A: Yes, in fact, it can accept infinitely many more new guests. You can think about it this way: in order to make room for the new guests, move the guest occupying room 1 to room 2, the guest occupying room 2 to room 4, and in general, the guest occupying room n to room 2n. This would make all the odd-numbered rooms free.