TITLE: Can We Decide Whether Small Computer Programs Halt?
QUESTION [26 upvotes]: The undecidability of the halting problem states that there is no general procedure for deciding whether an arbitrary sufficiently complex computer program will halt or not. 
Are there some large $n$ and interesting computer languages (say C++, Scheme) for which we have a constructive halting algorithms for programs of length up to $n$? Note that for any fixed bound, there is some finite lists of zero's and one's that stores whether all the programs less than that bound halt or not, so non-constructively for any $n$ there is always a decision procedure for the bounded problem. But I would like to know whether we have some $n$ for which we can constructively write a decision procedure for programs of length less than that bound. By stipulating large $n$, I'd like to avoid $n$ and languages which are trivial in some sense, for instance, decision procedure for languages where the first number of programs all halt and we can easily see that they halt. 
Note that the usual proof of the undecidability of the halting program prima facia does not apply, since the decision procedure for programs bounded by $n$ might have length much larger than $n$, so our decision procedure need not correctly apply to our diagonlization program.
What kind of work has been done on this? Are there any good references out there? Thank you!

REPLY [2 votes]: as noted in other answers this question/problem is similar/nearly equivalent to the Busy beaver problem. the short answer is that even for "small" Turing Machines even with "not that many states", there are some known to be provably universal and therefore the halting problem is uncomputable on them.[1] in fact for "small" TMs it has been shown recently that the halting problem leads to questions about number-theoretic problems similar to the unresolved Collatz conjecture (as noted in another answer).[3] a few useful refs along these lines
[1] Small TMs and the generalized busy beaver competition, Michel (note: could only find .ps fmt for this paper)
[2] Small weakly universal TMs Woods/Neary
[3] Problems in number theory from the busy beaver competition Michel