What Comes Next: 1, 2, 4, 8, 16, … hint it’s not 32

A Kick in the Discovery

Interesting Puzzles

  1. How many ways are there to get a royal flush in poker?
  2. How many ways are there to flip a coin and get exactly 10 heads in 20 flips of the coin?
  3. How many groups can you make with three women and two men out of a class of 20 women and 15 men?
  4. Can you modify the proof given here to show that \lim_{x\rightarrow c}T(c) = 0 for all c?
  5. Can you use induction to prove that 1^2 + 2^2 + 3^2 + \cdots n^2 = \frac{1}{6} (1+n)(1+2n)?
  6. Prove that the ratio of consecutive Fibonacci elements converges to something finite and then find that something!
  7. Use the Euclidean Algorithm to first find the greatest common divisor of 304 and 399. Then use your work to find the integer solutions to 304x+399y = 95. (Note there may be no solution!)
  8. Prove there is no integer solution to x^2 + y^2 = N with N = 3,\;43,\; 103, \;,207,\;248 without guessing and checking!
  9. Can you prove using the definition of a limit that \frac{n}{n+1}\rightarrow 1?
  10. What’s the supremum of the set S = \{ \frac{n}{n+1} \,:\, n \in \mathbb{N}\}?
  11. Prove that the ratio of two Fibonacci numbers converges to the golden ratio. (Hint: The Fibonacci sequence is defined inductively by F_{n} = F_{n-1}+F_{n-2} . With F_1 = 0 and F_2 = 1 . The golden ratio is defined to be \phi = \frac{1 + \sqrt{5}}{2}. . Also, consider using the monotone convergence theorem!)

Here is also a great source of tough problems!

https://www.erdosproblems.com/

Hello,

Welcome to A Kick in the Discovery! If you’re curious where this phrase came from, that’s good! It came from the great physicist Richard Feynman. He was curious about anything and everything and he loved to explain science to others. I want to communicate the same enthusiasm about things that I find amazing! More than that, I want to practice teaching and explaining more and more complex ideas. Communication is very important and very challenging, and I hope this blog helps me learn this skill. With all this said, please bear with me while I practice my writing skills! It will be a process.

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