Interesting Problems

Here are some interesting math problems I've come across during my time at IISER-K. It may help to first consider a special case, say $n = 1$, and then extend the idea to solve the general case. I'll put up solutions when I find time. Happy solving :)

• $\text{Can you find a continuous function $f \colon \mathbb{Q} \to \mathbb{Q} $ such that $ f(f(q)) = -q $ for all $ q \in \mathbb{Q}$ ?}$

  This problem was originally posed to me by a person in the #math channel on Libera. A related question is whether there exists an odd homeomorphism from $\mathbb{Q}$ to $\mathbb{Q} \times \mathbb{Q}$. I am unaware if such a homeomorphism exists - ensuring continuity at the seam $(0, \mathbb{Q})$ doesn't seem to be easy.

• $\text{A needle of length $ n $ where $ n \in \mathbb{N} $ is dropped onto a square lattice.}$
  $\text{What is the probability that the needle touches exactly $ n $ unit-squares of the lattice?}$

  To be precise, you may assume that the position of the tip of the needle is uniformly distributed over $[0, 1) \times [0, 1)$, and the angle is uniformly distributed over $[0, 2\pi)$.

• $\text{Let $[n]$ denote $\{1, 2, \dots, n\}$. What is the largest integer $m$ for which there exists a}$
  $\text{collection $C = \{C_1, C_2, \dots, C_m\}$ of distinct subsets of $[n]$ such that $C_i \cap C_j$ contains}$
  $\text{an even number of elements for every $C_i, C_j \in C_m$?}$

  This problem is called the eventown problem. For a hint, try modeling the problem with a vector space over a finite field . For a further hint, look at the space $\mathbb{F}_2^n$ with the standard inner product.

• $\text{Let $J_n$ be the unit $n$-ball centered at the origin, i.e., $J_n = \{p \in \mathbb{R}^n : |p| \le 1 \}$. You form}$
  $\text{a (possibly finite) sequence of points $(p_i)$ by randomly picking uniformly distributed}$
  $\text{points from $J_n$. You stop this process as soon as you pick a point $p_N$ such that}$
  $\text{$|p_N| \gt |p_{N-1}|$, and when this happens you obtain a sequence of length $N$. What is the}$
  $\text{expected length of the sequence you obtain?}$

  You may assume that the volume of $J_n$ is equal to $\dfrac{\pi^\frac{n}{2}}{\Gamma(\frac{n}{2} + 1)}$ where $\Gamma$ denotes the gamma function.