Hard math questions to challenge friends?

Hey guys! My friends and I love to challenge each other with difficult math problems. Do any of you have any hard math questions we could try to tackle for fun? Just trying to sharpen our skills and have a good time. Thanks!

Hey there! I'm glad you and your friends have a shared interest in math challenges. Here are a few problems that should get your brains working and keep you entertained:

1. The Monty Hall Problem: 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. They then ask you, "Do you want to switch your choice to door No. 2?" Is switching a good strategy or should you stick with your original choice?

2. The Four 4's Problem: Using exactly four 4's and any mathematical operations you can think of, represent each of the numbers from 1 to 100. For example, to represent the number 3, you could use the following expression: (4 + 4 + 4) / 4 = 3.

3. Sum of Consecutive Odd Integers: Prove that the sum of the first n odd positive integers is equal to n^2. For example, 1 + 3 + 5 + 7 + 9 = 25, which is 5^2.

4. The St. Petersburg Paradox: A fair coin is tossed until it lands heads up. Your winnings depend on how many times the coin was flipped before the head appeared. If it was flipped n times, you win 2^n dollars. If the coin shows heads in the first try, you win 2 dollars; in the second try, you win 4 dollars; in the third try, you win 8 dollars, and so on. What is the expected value of your winnings considering the potentially unlimited number of flips?

5. The Birthday Paradox: In a group of people, how many individuals do you need so that there is a more than 50% chance that at least two of them share the same birthday? Assume there are 365 days in a year and disregard leap years.

Feel free to discuss and collaborate with your friends on these problems, and most importantly, have fun! Mathematics offers countless creative ways to challenge yourselves, so remember, the sky is the limit when it comes to finding new problems to tackle. Good luck and happy problem-solving!