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 one can ask 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?}$

• $\text{Let $[n]$ denote $\{1, 2, \dots, n\}$. Consider a collection of distinct subsets of $[n]$ given by $C = \{C_1, C_2, \dots, C_m\}$}$
  $\text{such that $C_i \cap C_j$ contains an even number of elements for every $C_i, C_j \in C$.}$
  $\text{What is the maximum possible value of $m$ for which such a collection $C$ could exist? }$

  I discovered this problem through a friend. For a hint, try modeling the problem with a vector space over a finite field.

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