I know that the halting problem is undecidable in general but there are some Turing machines that obviously halt and some that obviously don't. Out of all possible turing machines what is the smallest one where nobody has a proof whether it halts or not?

here is an interesting/deeper perspective/angle in addition to those worthwhile/more standard/surface answers so far. there is known to be a strong correspondence between proofs and programs/algorithms. this was formalized decades ago in the Curry-Howard correspondence. a proof in many ways is ...

the two fields of CS and mathematics are becoming increasingly intertwined especially on the theoretical side and once "more sharp" boundaries are getting blurred by various active/ongoing research programs and developments, and one would expect this trend to continue and heighten gradually over ...