Math problems that are impossible to solve I recently read about the impossibility of trisecting an angle using compass and straight edge and its fascinating to see such a deceptively easy problem that is impossible to solve. I was wondering if there are more such problems like these which have been proven to be impossible to solve.

Constructing an algorithm to solve any Diophantine equation has been proven to be equivalent to solving the halting problem, as is computing the Kolmogorov complexity (optimal compression size) of any given input, for any given universal description language.
In general I think what you're looking for is either problems that are proven to be equivalent to the halting problem, or problems that are formally independent from, say, ZFC, such as the Continuum Hypothesis or more generally Gödel's incompleteness theorems.
