TITLE: An example of an easy to understand undecidable problem
QUESTION [37 upvotes]: 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.

REPLY [6 votes]: I think the Post correspondence problem is a very good example of a simple undecideable problem that is also relatively unknown.
Given a finite set of string tuples
(a , bba) X
(ab,  aa) Y
(bba, bb) Z

the problem is to determine if there is a finite sequence of these tuples , allowing for repetition, such that the concatenation of the first half is equal to the concatenation of second half
(bba, bb) Z
(ab,  aa) Y
(bba, bb) Z
(a,  bba) X
------------ gives
(bbaabbbaa, bbaabbbaa)

The only big issue I have with this problem is that the only undecideability proof I know of falls back on simulating a Turing machine - it would be nice to find a more elementary alternate version.