Recreational Math Dump
Some fun questions I've been trying to solve on and off this month
- The Fibonacci numbers count the number of ways a natural number can be expressed as the sum of 1s and 2s while considering the order. What is the average number of 1s and 2s needed to create a sum equal to any natural number? Ref: an average number of 1s and 2s by Joshua Bowman.
- Choose random numbers between 0 and 1, uniformly and independently, until the sum of the numbers exceeds 1. What is the expected number of terms?
- Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a continuous function such that for all real numbers $x$ and $y,$ \(f(x+y)-f(x)-f(y)=xy+x^2y+xy^2\) and $f(6)=69.$ Find $f(2\sqrt{6})$.