I am looking for an undecidable problem that I could give as an easy example in a presentation to the general public. I mean easy in the sense that the mathematics behind it can be described, well, without mathematics, that is with analogies and intuition, avoiding technicalities.
"Are these two real numbers (or functions, or grammars, or mathematical statements) equivalent?"
(See also word problem)

"Does this statement follow from these axioms?"
(Hilbert's Entscheidungsproblem)

"Does this computer program ever stop?"
"Does this computer program have any security vulnerabilities?"
"Does this computer program do <any non-trivial statement>?"
(The halting-problem, from which all semantic properties can be reduced)

"Can this set of domino-like tiles tile the plane?"
(See Tiling Problem)

"Does this Diophantine equation have an integer solution?"
(See Hilbert's Tenth Problem)

"Given two lists of strings, is there a list of indices such that the concatenations from both lists are equal?"
(See Post correspondence problem)



There is also a large list on wikipedia.
